MATH- TEST PAPERS

Monday, 21 June 2021

REVISION QUESTIONS- XII. (2021/ NOVEMBER)

Revision (Differentiation)

1) 3x⁵+ 7x⁴ - 2x² - x +6.

2) 1+ x + x²/2! +x³/3! + x⁴/4!

3) √x + 2√x² + 3√x³+ 4√x⁴ +5√x⁵.

4) logₐx + log xᵃ+ e ˡᵒᵍ ˣ + log eˣ + e ¹⁺ˣ.

5) x² log x

6) 10ˣ. x¹⁰

7) (x²+7)(x³+10).

8) √x eˣ. sec x

9) x sec x log(x eˣ).

10) (1+sin x)/(1- sin x).

11) (sinx+ cos x)/√(1+sin 2x).

12) (cos x- cos 2x)/(1- cos x)

13) (x³-2+ 1/x³)/(x-2+ 1/x)

14) (eˣ+ e³ˣ)/(eˣ+1/eˣ)

15) (tanx)/x log (eˣ/xˣ).

16) √(2x) - √(2/x) + (x+4)/(4-x) at x= 2.

17) {¢(x)}ⁿ.

18) √(log x)

19) tan⁵x.

20) (tan⁻¹x)².

21) ₑax²+ bx + c

22) ₑ√(x+1) - ₑ√(x-1)

23) ₇x²+2x

24) Log(sin x).

25) log(ax² + bx+c)

26) log(sec x+ tan x).

27) logₛᵢₙ ₓ x.

28) logₑ{x + √(x²± a²)}

29) sin ¢(x)

30) Cos(ax + b)

31) sin x°

32) sinx sin 2x sin 3x.

33) Sin⁻¹(x/a).

34) cot⁻¹(cosec x + cot x)

35) log cos x²

36) 2 tan⁻¹√{x -a)/(b-x)} then show that (dy/dx)² + 1/{(x-a)(x-b)}

37) cot⁻¹{√(1+x²) - x}

38) cos⁻¹{(1-x²)/(1+x²)}

39) sin⁻¹{2x/(1+x²)}

40) tan⁻¹{2x/(1- x²)}

41) tan⁻¹{1/√(x² - 1)}

42) tan⁻¹{(3x-x³)/(1- 3x²) at x= 1

43) tan⁻¹{cos x/(1 - sinx)}

44) cos(sin⁻¹x) + tan(cot⁻¹x)

45) sin(cos⁻¹x) + 1/2 sin⁻¹{2x/(1+ x²)}

46) Find f(log x) where f(x)= log x

47) tan⁻¹{cosx/(1+ sinx)}+ sin(eˣ)

48) logₓ(tan x)

49) Log √{(1- cosx)/(1+ cosx)} + aˣ

50) sinx/(1+ cosx)

51) tan⁻¹√{(1-x)/(1+x)}

52) cos[2 sin⁻¹(cosx)]

53) x √(x²+a²)+ a² log{x+ √(x²+a²)} at x= 0

54) eᵃˣ sin bx.

55) log₁₀ √{1- cosx)/(1+ cos x)}

56) If cos y= x cos(a+y), prove that dy/dx= {cos²(a+y)}/sin a

57) tan⁻¹√{(1+ cosx)/(1 - cos x)}

58) If sin y= x sin(a+y) then show that dy/dx= {sin²(a+y)}/sin a= sin a/(1- 2x cos a + x²)

59) ₑ√cot x

60) ₃√(1+x+x²).

61) log sec(ax+b)³

62) log[2x+4+ √(4x²+16x-12)]

63) Tan log sin(ₑx²)

64) xˣ

65) ₓ1+x+x²

66) (tan x)ˢᶦⁿ ˣ at x=π/4

67) xᶜᵒˢ ˣ + sin(logₑx)

68) sin⁻¹{2x/(1+x²)} w.r.t cos⁻¹{(1-x²)/(1+x²)}

69) ₓsin⁻¹x w.r.t sin⁻¹x.

70) tan⁻¹x/(1+ tan⁻¹x) w r.t. tan⁻¹x

71) cos⁻¹{1-x²)/(1+x²)} w r.t tan⁻¹{2x/(1-x²)}

72) cos⁻¹{(1-x²)/(1+x²)} w.r.t sin⁻¹{2x/(1+x²)}.

73) tan⁻¹{2x/(1-x²)} w.r.t. sin⁻¹{2x/(1+x²)}

74) Log[eˣ{(x-1)/(x+1)}³⁾²].

75) {x³ √(x²-12)}/³√(20-3x) at x=4.

76) [x/{1+ √(1- x²)}]ⁿ.

77) cos⁻¹(8x - 8x²+1).

78) If xʸ = eˣ ⁻ ʸ then show dy/dx= log x/(1+ log x)² = log x/(log eˣ)²

79) 1/{√(x+a) + √(x+b)}

80) log{(x²+x+1)/(x²- x+1)}

81) tan⁻¹{(cosx - sinx)/(cosx+ sinx)}

82) sin⁻¹{2ax √(1- a²x²)}.

83) cos⁻¹(3+ 5 cosx)/(5+ 3 cos x)

84) If √(1-x²) + √(1- y²)= a(x - y) then show dy/dx= √{(1-y²)/(1-x²)}

85) log{(a cosx - b sinx)/(a cosx + b sinx)}.

86) sin[2tan⁻¹√{(1-x)/(1+x)}]

87) tan⁻¹[√{(a-b)/(a+b)} tan(x/2)]

88) x + 1/[x + 1/{x+ 1/x}]

89) 3x⁴ - x²y + 2y³= 0

90) x³ + y³ = 3axy.

91)91) eˣʸ - 4xy = 2.

92) eˣʸ - 4xy = 4

93) ₓ cos ⁻¹x.

94) xy= cos(xy) when x=π/2, y= 0.

95) xʸ. yˣ = 1.

96) (x+2)/{(x-1)(x+5)}

97) ₓ cos²x.

98) y logₑ(xy).

99) xʸ + yˣ = 1.

100) ₐx² (a> 0).

101) (sinx)ᶜᵒˢ ˣ + e³ˣ.

102) (sin x)ᶜᵒˢˣ + (cos x)ˢᶦⁿ ˣ.

103) log(xy)= eˣ⁺ʸ + 2,

104) (cos x)ʸ= (sin y)ˣ.

105) xᵖ yᑫ = (x+y) ᵖ⁺ ᑫ.

106) ax² + by² + 2hxy + 2gx + 2fy +c= 0

107) If x= at², y= 2at.

108) When x= a cos t, y= b sin t.

109) When x= sin²t, y= tan t.

110) when x= a(2t + sin 2t), y= (1- cos 2t).

111) when tan y= 2t/(1-t²), sin x= 2t/(1+t²).

112) tan⁻¹{t/√(1-t²)} and x= sec⁻¹ {1/(2t² -1)}.

113) y= sin⁻¹(3t - 4t³), x= sec⁻¹{1/(1- 2t²).

114) y= ₑsin⁻¹t , x= ₑ- cos⁻¹t.

115) y= a sin³t, x= a cos³t.

116) x= a(cos t + t sin t) and y= a(sin t - t cos t) at t= 3π/4.

117) tan y={2t/(1- t²) and cos x= {(1-t²)/(1+ t²)}.

118) x= 3at/(1+t²), y= 3at²/(1+t³)

119) If f(x)= {(a+x)/(b+x)}ᵃ⁺ᵇ⁺²ˣ. Then show that f'(0)= {2 log(a/b) + (b²-a²))ab }{a/b)}ᵃ⁺ᵇ

120) If f(x)= {(a+x)/(1+x)}ᵅ⁺¹⁺²ˣ, find the value of f'(0).

121) y= 1 + a₁/(x-a₁) + a₂x/{(x- a₁)(x - a₂)} + a₃x²/{(x-a₁)(x-a₂)(x- a₃)} then show that dy/dx= y/x{a₁/(a₁ -x)(a₂/(a₂ - x)(a₃/(a₃ - x)}

122) If x= (t - ₑ-t²)/2t², show that t dx/dt + 2x = 1/2t + ₑ-t²

123) ((tan x)ᵗᵃⁿ ˣ) ᵗᵃⁿˣ.at x=π/4

124) ₑ x sin x³ + (tan x)ˣ.

125) tan⁻¹[{√(1+t²)+ √(1-t²)}/{√(1+t²)- √(1-t²)}.

126) sin(π eˣʸ /6) at x= 0.

127) If y= x²/2 + x/2 √(x²+1) + log√{x+√(x²+1)} show that 2y= x dy/dx + log(dy/dx).

128) If y= 2 sin⁻¹ {(x-2)/√6 - √(2+ 4x-x²) then show dy/dx at x= 2 is 2/√6

129) tan ⁻¹[{√(1+x²) -1}/x] w.r.t. tan ⁻¹[{2x√(1-x²)}/(1- 2x²)] at x= 0.

130) cot ⁻¹{2√(1+x²) - 5√(1-x²)}/{5 √{(1+ x²+2) √(1-x²)} w.r.t. cos⁻¹√(1- x⁴) is -1/2.

131) If x= a sin 2t(1+ cos 2t) and y= a cos 2t(1- cos 2t), then show that 1 + (dy/dx)²= sec²t

Revision (Maximum & Minimum)

1) The condition for maxima of a function y= f(x) at a point where d²y/dx² ≠ 0

A) when f"(c)< 0 B) when f"(c) > 0

C) when f"(c)≤ 0 D) when f"(c)≥0

2) The condition for minima of a function y= f(x) at a point where d²y/dx² ≠ 0

A) when f"(c)< 0 B) when f"(c) > 0

C) when f"(c)≤ 0 D) when f"(c)≥0

3) The maximum value of x³ - 9x² + 24x - 12 is

A) 2 B) 4 C) 6 D) 8

4) minimum value of x³ - 9x² + 24x - 12 is

A) 2 B) 4 C) 6 D) 8

5) the maximum value of (x²-7x+6)/(x-10) is

A) 1 B) 2 C) 12 D) 25

6) minimum value of (x²-7x+6)/(x-10) is

A) 1 B) 2 C) 12 D) 25

7) For what value of x will (x-1)(3-x) have its maximum

A) 1 B) 2 C) 12 D) 25

8) maximum value of the function x+ 1/x is ____ than its minimum value.

A) less B) more C) equal D) none

9) The maximum value of f(x)= x¹⁾ˣ is

A) e B) 1/e C) e¹⁾ᵉ D) eᵉ

10) The maximum values of x³+ 1/x³ is..

A) -1 B) -2 C) 2 D) 1

11) The minimum values of x³+ 1/x³ is..

A) -1 B) -2 C) 2 D) 1

12) find the point of x for which the function x⁴ - 8x³+ 22x² - 24x +5 is maximum.

A) 1 B) 2 C) -1 D) -3

13) find the value of x for which the function x⁴ - 8x³+ 22x² - 24x +5 is maximum.

A) 1 B) 2 C) -1 D) -3

14) find the point of x for which the function x⁴ - 8x³+ 22x² - 24x +5 is minimum.

A) 1,3 B) 2,3 C) -1,-3 D) 1, -3

15) find the value of x for which the function x⁴ - 8x³+ 22x² - 24x +5 is minimum.

A) 1 B) 2 C) -4 D) 4

16) find x for which the function (x²+x+1)/(x²-x+1) is the maximum.

A) 1 B) -1 C) 3 D) 1/3

17) find the values of x for which the function (x²+x+1)/(x²-x+1) is the maximum.

A) 1 B) -1 C) 3 D) 1/3

18) find x for which the function (x²+x+1)/(x²-x+1)is the minimum.

A) 1 B) -1 C) 3 D) 1/3

19) find the values of x for which the function (x²+x+1)/(x²-x+1)is the minimum.

A) 1 B) -1 C) 3 D) 1/3

20) The function f(x)= x³- 3x²+ 9x-1 has

A) maximum B) minimum

C) neither a maximum and minimum

D) either maximum or minimum

21) Given x/2 + y/3= 1 the maximum value of xy is

A) 2 B) 3 C) 2/3 D) 3/2

22) Given x/2 + y/3= 1 then minimum value of x²+ y² is

A) 13 B) 36 C) 36/13 D) 13/36

23) The greatest rectangle inscribed in a circle is a

A) triangle

B) right angled triangle

C) square

D) circle

24) All rectangles of a given area, the ____ has the least perimeter.

A) triangle

B) right angled triangle

C) square

D) circle

25) maximum value of cos²x + cosx +3 in the interval 0≤x≤π/2 is

A) 0 B) 5 C) π/2 D) no maximum value

26) minimum value of cos²x + cosx +3 in the interval 0≤x≤π/2 is

A) 0 B) 5 C) π/2 D) No minimum value

27) The perimeter of a triangle is 8 cm. if one of the sides is 3 cm., find the length of the other sides so that area of the triangle maybe a maximum.

A) 1/2, 1/2 B) 3/2, 1/2 C) 5/2, 5/2 D) 1/2, 5/2

28) A policeman chases a thief as the letter running with uniform velocity u overtakes him. If the police starts from rest and moves with uniform acceleration f, then the distance between the thief and the policeman is maximum after u/f measured from the instant the pursuit starts and that, this maximum distance is.

A) u/f B) u²/f C) u/2f D) u²/2f

29) The perimeter of a rectangle is 100cm. if the area of the rectangle is maximum find the length of its sides.

A) 20,25 B) 25,25 C) 25, 50 D) 50, 100

30) what will be the radius of the base of a solid cylinder of volume 16 π for which the total surface area will be the smallest.

A) 2 B) 4 C) 6 D) 8

31) All rectangle of given perimeter, the square has the ___ area

A) smallest B) greatest C) equal D) none

32) Given xy= 4, find the maximum value of 4x+ 9y

A) 24 B) -24 C) 3 D) -3

33) Given xy= 4, find the maximum value of 4x+ 9y

A) 24 B) -24 C) 3 D) -3

34) divided 10 into two parts such that the product is a maximum.

A) 4,6 B) 5,5 C) 7,3 D) 2,8

35) find the maximum value of the product of two numbers if their sum is 12.

A) 6 B) 24 C) 36 D) 48

36) the space described in time by a particle moving in a straight line is given by s= t⁵ - 40t³ + 30t² + 8ot - 250.

find the minimum value of acceleration.

A) 220 B) 480 C) 260 D) -260

37) find the coordinates of a point on the parabola y= x²+7x+2 which is nearest to the straight line y= 3x -3.

A) (1,7) B) (-1,-7) D) (2,8) D) -2,-8)

38) On the parabola y= x², find the point of least distance from the straight line y= 2x-4.

A) (1,1) B) (1,-1) C)(-1,1) D) (-1,-1)

39) A wire of length l is to be cut into two pieces, one being to form a square and the other to form a circle. how should the wire be cut if the sum of the areas enclosed by two pieces to be a

A) minimum B) maximum C) inflation D) minimum as well maximum

40) Find the point and straight line 2x+ 3y = 6 which is closest to the origin.

A) (12/18, 12/13)

B) (12/13, 18/13)

C) (13/18, 18/14). D) (1,1)

41) For what value of x, the function y= 2 sinx + cos 2x (0≤ x ≤ 2π) attains maximum and minimum values.

A) π/6,5π/6 and π/2, 3π/2

B) -π/6,5π/6 and -π/2,3π/2

C) -π,π and π, -π D) none

++++++((((((+++++++++)))))

Test paper--3

1) If x is real, then the minimum value of x² - 8x +17= 0 is

A) -1 B) 0 C) 1 D) 2

2) If y= 3x - (cosx)/2, then d²x/dy² is..

A) -2 cosx/(6+ sinx)²

B) -4 cosx/(6+ sinx)²

C) -4 cosx/(6+ sinx)³

D) -4 sinx/(6+ sinx)³

3) The value of the derivative of | x-1| + |x-3| at x -2 is

A) -2 B) 0 C) 2 D) 1/2

4) Two function f and g are continuous at x= a. The function f+ g, f - g, f/g [g(a) ≠ 0] and f x g are also continuous at x= a. If a function F is defined as:

F(x)= (eˣ + e⁻ˣ-2)/(x sinx) on {-π/2, π/2}, then which one of the following is correct?

A) F(x) is continuous in [-π/2, π/2]

B) F(x) is not continuous in [-π/2, π/2]/{0}

C) F(x) is continuous in [-π/2, π/2]/{0}

D) F(x)= is continuous in (-π/2, π/2)

5) The value of:

Lim ₓ→ₐ {log (x-a)/(log (eˣ - eᵃ) is

A) 1 B) eᵃ C) e⁻ᵃ D) -1

6) If f(x)= 2 + 1/x, then which of the following is correct?

A) as x increases, f(x) also increases.

B) As x becomes larger and larger, f(x) assumes values nearer to 2.

C) As x increases, f(x) takes values nearer and nearer to 3 for sufficiently large values of x

D) As x becomes larger and larger, it is not possible to find the value of f(x).

7) Let y²= 4ax, a≠ 0. Now consider the following statements:

I. y= 2√(ax) express y as a function of x.

II. y= - 2√(ax) express y as a function of x.

III. y= ±2√(ax) express y as a function of x.

Which of these is/are correct?

A) I and II

B) I and III

C) II and III

D) III only

8) If f(x)= log{(1+x)/(1-x)}, then f{2x/(1+x²)} is equal to

A) f(x)= {(1-x)/(1+x)}

B) f(x²) C) 1 D) 2f(x)

9) Let f: R--> R be defined as f(x)= x |x|. Which one of the following is correct ?

A) f is only onto

B) f is only one-one

C) f is neither onto nor one-one.

D) f is one-one and onto.

10) What is the value of:

lim ₓ→∞ {(x+3)/(x-1)ˣ⁺³ ?

A) e² B) e³ C) e⁴ D) e⁻⁴

11) Let f(x)=[x], where [x] denotes the greatest integer contained in x. Consider the following statements.

I. f(x) is not onto

II. f(x) is continuous at x= 0.

III. f(x) is discontinuous for all positive integral values of x.

Which of the statements given above are correct?

A) I and II. B) I and III

C) II and III D) I, II and III

12) If (x+y)ᵐ⁺ⁿ = xᵐ yⁿ then what is the value of dy/dx.

A) x/y B) xy C) y/x D) 1

13) What is the dy/dx of logₓ² x.

A) 1/x B) 0 C) 1/(2x) D) 1/2

14) For what value of p so as the function:

px²+1 if x ≤ 1

f(x)= x+ p if x > 1 is derivable at x= 1 ?

A) 1/2 B) 2 C) -1/2 D) -2

15) A function f: R-- R satisfies f(x+y)= f(x) + f(y) for all x, y belongs to R and f(x)≠ 0 for all x belongs to R. If f(x) is differentiable at x= 0 and f'(0)= 2 then f'(x) is equal to which one of the following?

A) f(x) B) -f(x) C) 2f(x) D) f(x)/2

16) Let f(x) be a differentiable even function. Consider the following statements.

I. f'(x) is an even function.

II. f'(x) is an odd function.

III. f'(x) may be even or odd.

A) I only B) II only C) I and III D) II and III

Revision Continuity

_____________

1) A function f(x) is defined as:

= x² when x< 1

f(x)= 2.5 when x= 1

= x²+2 when x> 1

Is f(x) continuous at x= 1?

2) A function f(x) is defined as:

= 3+ 2x when -3/2≤x< 0

f(x)= 3- 2x when 0≤ x< 3/2

= -3x -2x when x ≥ 3/2

Show that f(x) continuous at x= 0 and discontinuous at x=3/2.

3) A function f(x) is defined as:

=(Sin 3x)/2x when x≠ 0

f(x)= 2/3 when x= 0

Is f(x) continuous at x= 0 ?

4) Find the points of discontinuity of the function (x²+2x+5)/(x²-7x+12).

5) A function f(x) is defined as:

f(x)=(x⁴+4x³+2x)/sin x when x≠0

And f(0)=0

Show that f(x) continuous at x= 0

6) A function f(x) is defined as:

f(x)= x sin(1/x) when x≠ 0

= 0 when x = 0

Show that f(x) continuous at x= 0

7) The function f(x) is defined as:

=x²-2x+3 when x< 1

f(x)= 2 when x= 1

=2x²-5x+5 when x> 1

Is f(x) continuous at x= 1 ?

8) f(x)= (x²-1)/(x³-1) is undefined at x= 1. What should be the value of f(x) at x= 1 such that f(x) may be continuous at x= 1 ?

9) For what value of f(4), the function f(x) = (x²-16)/(x-4) is continuous at x= 4 ?

10) Examine the continuity of the function f(x) at x= 1.

f(x)= |x-1|/(x-1) when x≠1

= 0 when x = 1

Is f(x) continuous at x= 1?

11) Test whether the following function is continuous at x= 0 :

f(x)= |x|/x when x≠ 0

= 1 when x = 0

12) A function f(x) is defined as:

= x+1 when x≤ 1

f(x)= 3- ax² when x > 1

Find the value of a for which f(x) will be continuous at x= 1 ?

13) A function f(x) is defined as:

f(x)=(x³+x²-16x+20)/(x-2)² at x≠2

= K at x= 2

If f(x) continuous for all values of x then find the value of K.

14) Determine the values of a so that the following function is continuous at x= 1

f(x)= ax+3, when x ≥ 1

= x²+a² when x < 1

15) Two function f(x) and g(x) are defined as:

f(x)= x²+4 when x ≤ 2

= x+6 when x > 2

And g(x)= 2x when x ≤ 2

= 4 when x > 2

Show that f(x). g(x) is continuous at x = 2.

16) The function f(x)={log(1+ax) - log(1- bx)}/x is not defined at x= 0. Find the value of f(0), so that f(x) is continuous at x= 0.

_________________________________

Revision (inverse trigonometry)

1) state whether the inverse circular function sec⁻¹(√2) is
A) Defined B) not defined
C) infinite D) none

2) Statement is correct sin⁻¹2 + cos⁻¹2=π/2.
A) yes B) no C) not defined D) none

3) Statement is correct sin⁻¹1/2 + cos⁻¹1/2=π/2
A) Yes B) no C) not defined D) n

4) Statement is correct sin cos⁻¹ tan 45°.
A) yes B) No C) not defined D) n

5) Statement is correct sin⁻¹cos tan⁻¹√3.
A) yes B) no C) not defined D) n

6) Value of sin⁻¹tan(3π/4).
A) π/2 B) -π/2 C) π/4 D) -π/4

7) If sin⁻¹x= A then cosec⁻¹{1/√(1-x²)} is
A) π+A B)π -A C) π/2 + A D) π/2-A

8) If cos⁻¹1/5 = A then cosec⁻¹√5
A) π+A B) π+A C) π/2+A D) π/2 - A

9) Value of sin⁻¹3/5+ cosec⁻¹5/4
A) π/2 B) -π/2 C) π D) -π

10) Value of Cos(sin⁻¹1/2 + sec⁻¹2).
A) π B) π/2 C) π/4 D) 0

11) Value of Sin(sin⁻¹1/3 + sec⁻¹3) + cos(tan⁻¹1/2 + tan⁻¹2).
A) 0 B) 1 C) -1 D) none

12) Value of Tan[1/2 sin⁻¹{2x/(1+x²)} + 1/2 cos⁻¹{+1- y²)/(1+y²)}].
A) (x+y)/(1-xy) B) (x-y)/(1-xy)
C) (x+y)/(1+xy) D) (x+y)/(1+xy)

13) Value of Cos (2 cos⁻¹x + sin⁻¹x), at x= 1/5, where 0 ≤ cos⁻¹x ≤ π; -π/2≤ sin⁻¹x≤π/2.
A) 2√6/5 B) -2√6/5 C) -2√6/7 D) 2√6/5

14) If tan⁻¹y = 4 tan⁻¹x then the algebraic relation between x and y is
A) Y={4x(1-x²)}/(1-6x²+ x⁴)
B) Y={4x(1+x²)}/(1-6x²+ x⁴)
C) Y={4x(1-x²)}/(1 +6x²+ x⁴)
D) Y={4x(1+x²)}/(1+6x²+ x⁴)

15) Two angles of a triangle ABC are tan⁻¹2 and tan⁻¹3. Then the third angle is
A) π/2 B) π/8 C) π/4 D) π/6

16) value of Sin⁻¹12/13 + cos⁻¹4/5 + tan⁻¹63/16.
A) π B) -π C) π/2 D)-π/2

17) value of cos⁻¹x + cos⁻¹[x/2 + √(3- 3x²)/2] (1/2≤ x ≤ 1).
A) π/3 B) π/4 C) π/6 D) π/2

18) If sin⁻¹cos sin⁻¹x =π/3 then the value of x is
A) 0 B) 1 C) 1/2 D) 1/3

19) If sec⁻¹x= cosec⁻¹y then the value of cos⁻¹1/x + sin⁻¹1/y is
A) π/2 B) π C) -π/2 D) -π

20) If | x|≤ 1 then the value of Sin cosec⁻¹cot tan⁻¹x is
A) 0 B) 1 C) -1 D) x

21) If x≠ 1 then value of tan[1/2 sin⁻¹{2x/(1+x²)} + 1/2 cos⁻¹ {(1-x²)/(1+x²)}] is
A) 2x/(1- x²) B) - 2x/(1- x²)
C) 2x/(1+ x²) D) -2x/(1+ x²)

22) If A= tan⁻¹{1/√cos 2x} - tan⁻¹√cos 2x then value of tan²A is
A) sin⁴x/cos 2x B) sin⁴x/cos x
C) sin²x/cos 2x D) sin⁴x/cos² 2x

23) If A= tan⁻¹{1/√cos 2x} - tan⁻¹√cos 2x then value of
secA is
A) ± 1/2 (√sec 2x + √cos 2x)
B) ± 1/2 (√sec 2x - √cos 2x)
C) ± 1/2 (sec 2x + cos 2x)
D) ± 1/2 (sec 2x - cos 2x).

24) If cos⁻¹x/a + cos⁻¹y/b = A then the value of x²/a² - 2xy/ab cosA + y²/b² is
A) sinA B) sin²A C) cos A D) cos²A

25) If sin⁻¹x/a + sin⁻¹y/b = sin⁻¹c²/ab then the value of b²x² + 2xy √(a²b² - c⁴) + a²y² is
A) c B) c² C) c³ D) c⁴

26) If cos⁻¹x + cos⁻¹y + cos⁻¹z=π then x²+ y²+ z²+ 2xyz is
A) 0 B) 1 C) -1 D) none

27) If cos⁻¹x + cos⁻¹y + cos⁻¹z= π then the value of x+ y+ z is
A) 1 B) 1/2 C) 3/2 D) none

28) If cos⁻¹x + cos⁻¹y + cos⁻¹z= π then the relation between x, y ,z is
A) x= y = z B) 2x= y = z C) x= 2y = z D) x= y = 2z

29) value of tan⁻¹1/2 + tan⁻¹1/3 is
A) π B) π/2 C) π/3 D) π/4

30) value of tan⁻¹1/2 + tan⁻¹2/11
A) 0 B) 1 C) cos⁻¹4/5 D)sin⁻¹4/5

31) Value of sin⁻¹cos sin⁻¹x+ cos⁻¹sin cos⁻¹x
A) π B) π/2 C) π/3 D) π/4

32) value of sin⁻¹{2x/(1+x²)} is
A) tan⁻¹x B) 2tan⁻¹x C) tan⁻¹2x D) 2tan⁻¹2x

33) value of tan⁻¹{2x/(1-x²)} is
A) tan⁻¹x B) 2tan⁻¹x C) tan⁻¹2x D) 2tan⁻¹2x

34) value of cos⁻¹{(1-x²)/(1+x²)}
A) tan⁻¹x B) 2tan⁻¹x C) tan⁻¹2x D) 2tan⁻¹2x

35) value of 2 tan⁻¹1/3 + tan⁻¹1/7
A) π B) π/2 C) π/3 D) π/4

36) Value of tan[2 tan⁻¹√{(1+cos 2x)/(1- cos⁻¹x)}]+ tan x
A) 0 B) 1 C) -1 D) ±1

37) Value of 1/2 cos{(5cosx+3)/(5+ 3 cosx)} is
A) sin⁻¹(1/2 tan x/2)
B) tan⁻¹(1/2 tan x/2)
C) cos⁻¹(1/2 tan x/2)
D) cot⁻¹(1/2 tan x/2)

38) Value of cos⁻¹{(cosx + cosy)/(1+ cosx cosy)}
A) tan⁻¹(tan x/2 tan y/2)
B) 2 tan⁻¹(tan x/2 tan y/2)
C) tan⁻¹(tan x tan y)
D) 2 tan⁻¹(tan x tan y)

39) Value of 2 tan⁻¹[√{(m-n)/(m+n). Tan x/2]
A) cos⁻¹{(n cosx + m)/(n + m cosx)}
B) cos⁻¹{(m cosx - n)/(m + n cosx)}
C) cos⁻¹{(m cosx + n)/(m + n cosx)}
D) cos⁻¹{(m cosx - n)/(m - n cosx)}.

40) Value of sin⁻¹ 4/5 + sin⁻¹5/13 + sin⁻¹16/65
A) π B) π/2 C) π/3 D) π/4

41) Value of cot⁻¹1/2 - 1/2 cot⁻¹4/3
A) π B) π/2 C) π/3 D) π/4

42) Value of 4(tan⁻¹1/3 + cos⁻¹2/√5)
A) π B) π/2 C) π/3 D) π/4

43) Value of tan⁻¹1/x - tan⁻¹1/(x+ y)
A) tan⁻¹{y/(x² + xy+1)}
B) cot⁻¹{y/(x² + xy+1)}
C) sin⁻¹{y/(x² + xy+1)}
D) cos⁻¹{y/(x² + xy+1)}

44) Value of tan⁻¹1/2 + tan⁻¹2/11
A) tan⁻¹1/3 B) 2 tan⁻¹1/3
C) - tan⁻¹1/3 D) 2 tan⁻¹1/3

45) value of cos⁻¹2/√5 + sin⁻¹1/√10
A) π B) π/2 C) π/3 D) π/4

46) value of cos⁻¹15/17 + cos⁻¹3/5
A) 0 B) 1 C) cos⁻¹13/85 D) none

47) value of 2 sin⁻¹3/5 + sin⁻¹7/25
A) π B) π/2 C) π/3 D) π/4

48) Value of tan⁻¹1/3 + tan⁻¹1/7 + tan⁻¹1/5 + tan⁻¹1/8
A) π B) π/2 C) π/3 D) π/4

49) Value of 4 tan⁻¹1/5 - tan⁻¹1/239
A) π B) π/2 C) π/3 D) π/4

50) Value of tan⁻¹1 + tan⁻¹2 + tan⁻¹3
A) π B) π/2 C) π/3 D) π/4

51) value of 2(tan⁻¹1/2 + tan⁻¹1/3 + tan⁻¹1)
A) π B) π/2 C) π/3 D) π/4

52) Value of tan⁻¹(1/2 tan 2A) + tan⁻¹(cot A) + tan⁻¹(cot³A)
A) 0 B) π C) 1 D) π/2

53) Value of tan⁻¹{x cosA/(1 - x sinA)} - cot⁻¹{cos A/(x - sinA) is
A) 0 B) 1 C) x D) A

54) value of tan(π/4 + 1/2 cos⁻¹a/b) + tan (π/4 - 1/2 cos⁻¹a/b)
A) b/a B) 2/a B) 2b/a C) 2a/b

55) Value of tan⁻¹(cotx) + cot⁻¹(tanx)
A) 2π - x B) 2π +x C) π - 2x D) 2x

56) Value of sec²(cot⁻¹2) + cosec²(tan⁻¹3)
A) 85/36 B) 8/3 C) 25/36 D) none

57) Value of cot⁻¹(tan 2x) + cot⁻¹(- tan 3x)
A) 0 B) 1 C) x D) none

58) Value of sin cos⁻¹tan sec⁻¹x
A) √(2-x²) B) (2-x²) C) √(x²-2) D) √(2+ x²)

59) Value of cos tan⁻¹sin cot⁻¹x
A) √{(x²+1)/(x²+2)}
B) √{(x²-1)/(x²+2)}
C) √{(x²+1)/(x²-2)}
D) {(x²+1)/(x²+2)}

60) Value of 2 tan⁻¹[ tan A/2 tan(π/4 -B/2)] is
A) tan⁻¹{(sinA )/(cosA+ sinB)]
B) tan⁻¹{(cosB)/(cosA+ sinB)]
C) tan⁻¹{(sinA cosB)/(cosA+ sinB)]
D) none

61) value of √(a²+ b²) cos(A - tan⁻¹b/a)
A) b cosA + a sinA
B) b cosA - a sinA
C) cosA + sinA
D) a cosA + b sinA

62) find x: 2 sin⁻¹x + sin⁻¹(1-x) =π/2.
A) 0,1/2 B) 1,1/2 C) 1/2, 1/3 D) n

63) value of x: tan⁻¹(1/2 sec x) + cot⁻¹(2 cos x) =π/3.
A) 2nπ ± π/6 where n= 0, ±1, ±2,..
B) nπ ± π/6 where n= 0, ±1, ±2,....
C) 1 D) none

64) solve x: tan⁻¹{(1-x)/(1+x)} = 1/2 sin⁻¹{x/√(1+x²)}.

A) 1 B) √3 C) 1/√3 D) -1/√3

65) solve x: sin⁻¹5/x + sin⁻¹12/x =π/2.

A) 1 B) 3 C) 13 D) 15

66) Solve x: cot⁻¹x + cot⁻¹(a² - x+1)= cot⁻¹(a -1).

A) a B) a² - a+1 C) -a D) a²+a+1

67) Solve x: tan⁻¹(x+1) - tan⁻¹(x -1)= tan⁻¹2.

A) ±1 B) 0 C) ±2 D) none

68) Solve x: tan⁻¹1/3 + tan⁻¹1/5 + tan⁻¹1/8 + tan⁻¹1/x = π/4.

A) 7 B) 5 C) 3 D) 1

69) solve x: sin⁻¹x + sin⁻¹2x =π/3.

A) 1/2 B) √(3/7) C) 1/2√(3/7) D)n

70) solve x: 3 sin⁻¹{2x/(1+x²)} - 4 cos⁻¹{(1- x²)/(1+x²)} + 2 tan⁻¹{2x/(1-x²)}=π/3.

A) 1 B) √3 C) 1/√3 D) none

71) Tan⁻¹(cotx) + cot⁻¹(tanx)= π/4.

A) π/8 B) 3π/8 C) 5π/8 D) π

72) Solve x: Sin⁻¹x + sin⁻¹(1-x)= cos⁻¹x.

A) 0 B) 1/2 C) 0, 1/2 D) none

73) solve x: 3 cot⁻¹{1/2-√3)} + cot⁻¹x =π/2.

A) 0 B) 1 C) -1 D) none

74) solve x: Tan⁻¹(x-1) + tan⁻¹x + tan⁻¹(x+1) = tan⁻¹3x.

A) 0, B) ±1/2 C) 0, ±1/2 D) none

75) Silve x: Sin⁻¹x - sin⁻¹y , cos⁻¹x + cos⁻¹y = 2π/3.

A) √3/2 B) 0 C) √3/2, 0 D) none

_________________________________

Revision Test (increasing and decreasing)

Test

1) On which of the following intervals is the function f(x)= x¹⁰⁰ + sinx -1 increasing ?

A) (0,π/2) B) (π/2,π) C) (0,1) D) (-1,1)

2) Which of the following functions are decreasing on (0,π/2) ?

A) cosx B) cos 2x C) tanx D) cos 3x

3) The function f(x)= log(x³ +√x⁶+1)) is of the following types:

A) even and increasing

B) odd and increasing

C) even and decreasing

D) odd and decreasing

4) Let f(x)= x³ + ax² + bx + 5 sin²x be an increasing function on the set R. Then, a and b satisfy

A) a²- 3b-15> 0

B) a²- 3b+15> 0

C) a²- 3b+15< 0

D) a> 0 and b>0

5) The function f(x)= x⁹+ 3x⁷+64 is increasing on

A) R B) (-∞,0) C) (0, ∞) D) R₀

6) The function f(x)= - x/2 + sinx defined on [-π/3, π/3] is

A) increasing B) decreasing

C) constant D) none

7) function f(x)= x³ - 27x +5 is monotonically increasing when

A) x < -3 B) |x|> 3

C) x ≤ -3 D) |x|≥ 3

8) Every invertible function is

A) monotonic function

B) constant function

C) Identity function

D) not necessarily monotonic function

9) If the function f(x)= x³ - 9kx² + 27x + 30 is increasing on R, then

A) -1≤ k<1 B) k< -1 or k>1

C) 0 <k < 1. D) -1< k < 0

-----------+

Revision (increasing and decreasing)

1) Find intervals in which the function f(x)= 3x⁴ - 4x³ - 12x²+5 is

A) strictly increasing. (-1,0),(2,∞)

B) strictly decreasing. (-∞,-1), (0,2)

2) Find intervals in which the function f(x)= sin x+ cosx, 0 ≤ x ≤ 2π is

A) strictly increasing. (0,π/4),(5π/4, 2π)

B) strictly decreasing. (π/4,5π/4)

3) Find intervals in which the function f(x)= x/2 + 2/x, x≠ 0 is

A) increasing. (-∞,-2)U(2,∞)

B) decreasing. (-2,0)U(0,2)

4) Determine the values of x for which f(x)= xˣ, x> 0 is

A) increasing. (1/e, ∞)

B) decreasing. (0, 1/e)

5) Determine the values of x for which f(x)= 2 log(x-2) - x²+ 4x +1

A) increasing. (2,3)

B) decreasing. (3, ∞)

6) If a, b, c are real numbers, then find the intervals in which

x+ a² ab ac

f(x)= ab x+ b² bc

ac bc x+ c² is

A) Increasing. (-∞, -2/3 (a²+b²+c²))U(0, ∞)

B) decreasing. (-2/3 (a²+b²+c²),0)

7) Show that f(x)= tan⁻¹(sinx + cosx) is an increasing function on the interval (0, π/4)

8) Find the least value of a such that the function x² + ax +1 is increasing on [1,2]. -2

9) Find the values of a for which f(x)= (a+2)x³ - 3ax² + 9ax -1 decreases for all real values of x. (-∞, -3)

10) Find the values of k for which f(x)= kx³ - 9kx² + 9x +3 is increasing on R. (0,1/3)

Revision Test (mean theorem)

1) If the function f(x)= log(x²+2) - log 3 on [-1,1] satisfy the Rolle's theorem. Find a point in the interval where the tangent to the curve is parallel to the x-axis.

A) 2/3 B) 3/2 C) log(2/3) D) log(3/2)

2) If the function f(x)= x(1- log x) on [1,2] satisfy the Lagrange's mean value theorem then the value of C is

A) e B) 4 C) 4/e D) e/4

3) If from Lagrange's mean value theorem, we have f'(x₁)= {f'(b)- f(a)}/(b-a), then

A) a<x₁≤b B) a≤x₁<b C) a<x₁<b D) a≤x₁≤b

4) Rolle's theorem is applicable in case of €(x)= aˢᶦⁿ ˣ, a> a in

A) any interval

B) the interval [0,π]

C) the interval (0, π/2) D) none

5) When the tangent to the curve y= x log x is parallel to the chord joining the points (1,0) and (e,e), the value of x is

A) e¹⁾⁽¹⁻ᵉ⁾ B) e⁽ᵉ⁻¹⁾⁽²ᵉ ⁻¹⁾

C) e⁽²ᵉ⁻¹⁾/⁽ᵉ⁻¹⁾ D) (e-1)/e

6) The value of c in Rolle's theorem for the function f(x)= x³ - 3x in the interval [0, √3] is

A) 1 B) -1 C) 3/2 D) 1/3

7) If f(x)= eˣ sin x in [0,π], then c in Rolle's theorem is

A) π/6 B) π/4 C) π/2 D) 3π/4

8) If f(x)= Ax² + Bx + C is such that f(a)= f(b), then the value of c in Rolle's theorem is

A) a B) b C) a+ b D) (a+b)/2

_______________________________

Revision (MEAN THEOREM)

1) Discuss the applicability of Rolle's theorem on the function f(x)= x³ - 6x² + 11x -6 on the interval [1,3]. 2± 1/√3

2) Verify Rolle's theorem for the function f(x)= √(4-x²) on [-2,2]. 0

3) Verify Lagrange's mean value theorem for the function on the indicated interval f(x)= 2sin x + sin 2x on [0,π]. π/3

4) Using Lagrange's mean value theorem, find a point on the curve y=√(x-2) defined on the interval [2,3], where the tangent is parallel to the chord joining the end points of the curve. (9/4,1/2)

5) Verify Rolle's theorem for the function f(x)= x² - 5x+6 on the interval [2,3]. 2.5

6) Verify Rolle's theorem for the function f(x)= log[(x²+ab)/{x(a+b)}] on (a,b), where 0< a< b. √(ab)

7) Verify f(x)= sinx - sin 2x on [0,π]. 32°32', 126°23'

8) Verify Lagrange's mean value theorem for the function f(x)= (x-3)(x-6)(x-9) on the interval [3,5]. 4.8

Revision Test (Tangent and Normal)

1) The straight line lx + my= 1 is a normal to the hyperbola x²/a² - y²/b² = 1, Then

A) a²/l² - b²/m²= (a²+b²)².

B) a²/l² - b²/m²= (a²+b²)

C) a²/l² - b²/m²= (a+b)².

D) a/l - b/m= (a²+b²)

2) The straight line lx+ my+ n= 0, touches the parabola y²= 4ax, then

A) am²= nl B) a²m= nl C) am= n²l D) am= nl²

3) If normal to the hyperbola y²= 4ax at the point (ak², 2ak) meets the parabola again at the point (am², 2am), then the value of m- k is

A) -2/k B) 2/k C) 1/k D) -1/k

4) For what value of c, 3x+ y= c will be a normal to the parabola y² = 4(x+1).

A) 2 B) 3 C) 4 D) none

5) The gradient of normal at t= 2 of the curve x= t²-3, y= 2t +1 is

A) -1/2 B) 1/2 C) 2 D) -2

6) The equation of the tangent to the parabola y²= 4x at (1,2) is

A) 2x+3y= 4 B) 2x-3y= 4 C) 2x+3y= 5 D) none

7) The slope of the tangent to the circle x²+ y² = a² at the point (m,n) is

A) n/m B) - n/m C) - m/n D) m/n E) none

8) Determine the points on the curve y= x + 1/x, where the tangent is parallel to x-axis.

A) (1,-2) B) (-1,2) C) (1,2) D)(-1,-2)

9) The equation of the tangent to the parabola y²= 4x+ 5, which is parallel to the straight line y= 2x+7 is

A) y-2x-3=0 B) y+2x-3=0 C) y+2x+3=0 D) y-2x+3=0

10) The equation of the normal to the curve y= x³ - 3x at the point (2,2) is

A) x+9y-20=0 B) x-9y-20=0 C) x+9y+20=0 D) 2x+9y-20=0

TANGENT AND NORMAL

--------------------------------

1) At what point on the curve y= x² does the tangent make an angle of 45° with the x-axis? (1/2,1/4)

2) show that the line x/a + y/b = 1 touches the curve y= b eˣ⁾ᵃ at the point when the curve in intersects the axis of y.

3) find the point/s on the curve x²/9 + y²/4= 1, where the tangent is parallel to y-axis. (±3,0)

4) Find the ratio of the tangent and normal to the curves x²/a² - y²/b³ = 1 at the point (√2 a, b).

5) find the equation of the tangent to the curve y² = x² - 2x+7 which is

A) parallel to the line 2x-y+9= 0.

B) perpendicular to the line 5y - 15x -13 = 0. 2x-y+3= 0, 5y- 15x = 13 or 36y +12x -227= 0.

6) Find the equation of the tangent to the curve 4x² +9y²= 36 at the point (3 cos s, 2 sin a). 2x cos a + 3y sin a - 6= 0

7) Find the slope of the tangent to the curve y= 3c² - 4x at the point whose x-cordinate is 2. 8

8) find the points on the curve y= x³ - 11x + 5 at which the equation of the tangent is y= x -11. (2,-9) and (-2,19)

9) find the equation of the tangent to the curve x² + 3y= 3, which is parallel to the line y - 4x +5= 0. 4x -y+13= 0.

10) find the equations of the normal to the curve y= x³ + 2x +6, which is parallel to the line, x+ 14y +4= 0. x+ 14y = 254 and x+ 14y = -86.

11) find the equations of the tangent and normal to the curve x= 1 - cos k, y= k - sin k at k=π/4. 4 √2 x +(8- 4 √2) y=π(2 - √2).

12) Show that the equation of the normal at any point t on the curve x= 3 cos t - cos³t and y= 3 sin t - sin³t is 4(y cos³t - x sin³t)= 3 sin 4t.

13) The equation of the tangent at (2,3) on the curve y² = ax³ + b is y= 4x -5. Find the values of a and b. 2, -7

14) Find the points on the curve y= x³ at which the slope of the tangent is equal to the x-cordinate of the point. (0,0) and (3,27)

15) Show that the area of the triangle formed by the tangent and the normal at the point (a,a) on the curve y²(2a -x)= x³ and the line x = 2a, is 5a²/4 sq.units.

16) Find the equation of the tangent and the normal to the curve y² = 4ax at the point (at², 2at). tx + y= 2at + at³

17) At what points on the curve x² + y² - 2x -4y+1=0 the tangent are parallel to the y-axis? (2,2) and (-1,2)

18) Find the equation of the tangent line to the curve y= √(5x-3) -5, which is parallel to the line 4x -2y+5= 0. 80x- 40y-223= 0.

19) Find the equation of tangent to the curve y= cos(x+y), -π ≤ x≤ 2π that are parallel to the line x + 2y= 0. 4y + 2x + 3π= 0

20) Find the equation of the normal to the curve ay² = x³ at the point whose x-cordinate is am². 2x+ 3my - am²(3m²+2)= 0

21) Find the value of p for which the curves x² = 9p(9 - y) and x² = p(y+1) cut each other at right angles. 4

For November

TEST

___________________________*******

1) Let P(a sec m, b tan m) and Q(a sec n, b tan n) where m+ n = π/2, be two points on the hyperbola x²/a² - y²/b²= 1, if (h,k) be the point of intersection of the normal at P and Q, then the value of k is

A) (a²+b²)/a B) -(a²+b²)/a

C) (a²+b²)/b D) - (a²+b²)/b.

2) The equation of the tangent to the curve (1+x²)y = 2 - x where it crosses the x-axis is .

A) x+5y= 2. B) x-5y= 2

C) 5x- y= 2 D) 5x+y= 2

3) equations of the tangent and the normal drawn at the point (6,0) on the ellipse x²/36 +y²/9=1 respectively are

A) x=6, y= 0.

B) x+y=6, y -x= 0

C) x=0, y= 3 D) x= -6, y= 0

4) the straight line x+y=a will a tangent to the ellipse x²/9 + y²/16= 1 if the value of a is

A) 8 B) ±10 C) ±5. D) ±6

5) The equation of the tangent to the parabola y²= 8x which is perpendicular to the line x-3y-8=0, is

A) 3x+y=-2 B) 3x-y=1

C) 9x- 3y=-2 D) 9x+3y+2=0.

6) If the curves y²= 4x and xy= k cut orthogonally, then the value of k² will be

A) 16 B) 32. C) 36 D) 8

7) if the slope of the normal to the curve x³= 8a²y at P is (-2/3), then the co-ordinates of P are

A) (2a,a). B) (a,a) C) (2a,-a) D) n

8) if a> 2b > 0, then the positive value of m for which the line y= my - b √(1+m²) is a common tangent to the circle x²+ y²= b² and (x-a)² + y²= b² is..

A) 2b/√(a²-4b²).

B) √(a²-4b²)/2b

C) 2b/(a- 2b)

D) b/(a- 2b²)

9) The minimum value of f(x)= x²+ 250/x is.

A) 55 B) 25 C) 50 D) 75.

10) If f(x)= kx³ - 9x² + 9x+3 is an increasing function then --

A) k<3 B) k≤3

C) k>3. D) k is indeterminate

11) If f(x)= 1/(4x²+2x+1), then its maximum value is--

A) 2/3 B) 4/3. C) 3/4 D) 1

12) If f(x)= 1/(x+1) - log(1+x), x> 0, then f(x)=

A) a decreasing function.

B) an increasing function

C) neither increasing nor decreasing

D) increasing when x> 1

13) the function f(x)= 2x³ - 3x² -12x+4 has

A) no Maxima and minima

B) one maxima and one minima.

C) two maxima

D) two

14) the height of the cylinder of maximum volume that can be inscribed in a sphere of radius a is

A) 3a/2 B)√2 a/3 C) 2a/√3. D) a/√3

15) maximum value of (log x)/x in (0, ∞) is .

A) (log 2)/2 B) 0 C) 1/e. D) e

16) Let the function f: R--> R be defined by f(x)=2x + cos x; then f(x)=

A) has maximum value at x= 0

B) has minimum value at x=π

C) is a decreasing function

D) is an increasing function.

17) the maximum distance from the origin of a point on the curve x= a sin t - b sin(at/b), y= a cos t - b cos(at/b), both a, b> 0, is..

A) a- b B) a+ b.

C)√(a²+ b²) D) √(a²+ b²)

18) if the slope of the tangent at (x,y) to a curve passing through the point (2,1) is (x²+y²)/2xy, then the equation of the curve is..

A) 2(x²- y²)=3x.

B) 2(x²- y²)=3y

C) x(x²- y²)=6

D) x(x² + y²)= 6

19) If y= 3x²+2 and if x changes from 10 to 10.1, then the approximate change in y will be..

A) 8 B) 6. C) 5 D) 4

20) the rate of change of surface area of a sphere of radius r when the radius is increasing at the rate of 2cm/s, is proportional to..

A) 1/r² B) r² C) r. D) 1/r

21) f(x) and g(x) are two differentiable function on [0,2] such that f"(x) - g"(x)= 0, f'(1)= 2, g'(1)= 4, f(2)= 3 and g(2)= 9; then [f(x) - g(x)] at x= 3/2 is equal to..

A) 0 B) 2. C) 10 D) -5

22) If x= ₑ tan⁻¹{(y- x²)/x²} then the value of dy/dx is..

A)2x{1+ tan(log x)}+ x sec²(log x).

B) x{1+ tan(log x)}+ sec²(log x)

C) 2x{1+ tan(log x)}+x²sec²(log x)

D) 2x{1+ tan(log x)}+ sec²(log x)

23) if x= a cos⁴t, y= a sin⁴t then the value of dy/dx at t= 3π/4 is

A) 0 B) 1 C) -1 D) -2.

24) d/dx(x ˣ) is equals to ..

A) x ˣ(1- log x) B) x ˣlog x

C) x ˣ⁺¹(1- log x) D) x ˣ(1 + log x).

25) the differential coefficient of ₑ x³ w.r.t. log x is

A) ₑ x³ B) 3x³ₑ x³.

C) 3x²ₑ x³ D) 3x²ₑ x³ + 3x²

26) the second derivative of a sin³t w.r.t. a cos³t at t=π/4 is..

A) 2 B) 1/12a C) -4√2/3a. D) 0

27) the derivative of the function f(x) =|3x+2| at the point, x= -3 is..

A) -3. B) 3 C) 0 D) doesn't exist

28) let f(x) = eˣ, x belongs to [0,1]; then a number of c of Lagrange's mean value theorem is..

A) log(e+1) B) A) log(e-1)

C) log e D) none.

29) If y=√[x+ √{x +√(x +.....∞)}] then the value of dy/dx is ..

A) x/(2y-1) B) 3/(2y-1)

C) 1/(2y-1) D) x/(y-1).

30) If y= sinx + eˣ, then the value of d²x/dy² is..

A) (sinx - eˣ)/(coax+ eˣ)³

B) 1/(eˣ - sinx)

C) (sinx - eˣ)/(coax+ eˣ)³.

D) (sinx + eˣ)/(coax+ eˣ)³

31) If sin⁻¹x + sin⁻¹y = π/2, then the value of dy/dx is...

A) x/y. B) -x/y C) y/x D) -y/x

32) If dx/dy = u, and d²x/dy²= v, then the value of d²y/dx² is..

A) -v/u² B) v/u². C) -v/u³ D) v/u³

33) the value of

lim ₓ→₀ {(x²+5x+1)/(x²+x+1)}¹⁾ˣ is..

A) e B) e² C) e³. D) e⁴

34) If y= √{sin √x) then the value of dy/dx is...

A) 1/[2√{sin √x}]

B) [√{cos √x}]/2x.

C) 1/[2√{cos √x}]

D) cos√x/[4√{sin √x}]

35) If f(x)= cos⁻¹[{1+(log x)²}/{1+ (log x)²}], then the value of f'(e) is...

A) 2/e. B) 1/e C) 1 D) 1/e²

36) If y= a cos mx - b sin mx, then the value of d²y/dx² is...

A) - m²y B) m²y C) -my. D) my

37) If y= ₓ eˣ, then the value of dy/dx is...

A) y(log x + eˣ)

B) y log x(eˣ+ 1/2)

C) y eˣ(log x + 1/x)

D) y eˣ(x + log x).

38) If 2ˣ + 2ʸ = 2ˣ⁺ʸ, then the value of dy/dx at x= y=1 is..

A) 0 B) -1. C) 1 D) 2

39) f(x)= x(x-1)(x-2), 0≤ x≤ 4, then the point x= c which satisfies mean value theorem satisfies...

A) 0< c< 1. B) c> 3

C) 0< c < 1/2 D) 1< c < 3

40) Let f(x)= x| x|, then the set of points where f(x) is two differentiable is..

41) If x= sin⁻¹t, y= log(1- t²), 0≤t< 2, then the value of d²y/dx² at x= 1/3 is...

A) -9/4 B) -9/8. C) 9/4 D) 9/8

42) If y= {x+ √(1+x²)}ⁿ, then (1+x²) d²y/dx³ + x dy/dx is equal to

A) -y. B) n²y C) - n²y D) 2n²y

43) If si y + e ⁻ˣ ᶜᵒˢ ʸ = e, then the value of dy/dx at (1,π) is..

A) 0 B) 1 C) e D) -1.

44) If x= 2 cos t+ cos 2t and y= 2 sint - sin 2t, then the value of dy/dx at t=π/4 is

A) -(√2 +1) B) √2

C) √2 -1. D) 1 - √2

45) The value of

lim ₓ→∞{(n²- n+1)/(n²-n-1)} ⁿ⁽ⁿ⁻¹⁾ is equal to

A) e² B) e. C) 1/e D) 1

46) The value of the determinant

1+a 1 1

1 1+b 1

1 1 1+ c is

A) 1+ abc+ ab+ ca

B) abc(1/a + 1/b + 1/c).

C) 4abc D) abc(1/a +1/b +1/c)

47) If A= 2 -1

-1 2 and I is the unit matrix of order 2, then A² is equal to

A) 4A - 3I. B) 3A - 4I C) A-I D) A+ I

48) The multiplicative inverse of matrix 2 1

7 4 is

A) 4 -1 B) 4 -1 C) 4 -7 D)-4 -1

-7 -2 -7 2. 7 2 7 -2

49) For a real number a, let A(a) denote the matrix cos a sina

- sina cosa

Then for real numbers a₁ and a₂, the value of A(a₁) A(a₂) is..

A) A(a₁a₂) B) A(a₁ + a₂).

C) A(a₁ - a₂) D) A(a₂ - a₁)

50) A is a square matrix such that A³ = I; then A⁻¹ is equal to..

A) A². B) A C) A³ D) none

51) 1 a a² - bc

If D= 1 b b² - ca

1 c c² - ab then D is

A) 0.

B) independent of a

C) independent of b

D) independent of c

52) y x 0

If 0 y x = 0

x 0 y and x≠ 0, then which one of the following is correct ?

A) x is one of the cube roots of 1

B) y is one of the cube roots of 1

C) y/x is one of the cube roots of 1

D) y/x is one of the cube roots of (-1).

53) Assuming that the sums and products given below are defined, which of the following is not true for matrices ?

A) AB= AC doesn't imply B= C

B) A+ B= B + A

C) (AB)'= B' A'

D) AB = O implies A= O or B= O.

54) One of the root of equation

x+ a b c

b x+ c a = 0 is

c a x+ b

A) -(a+b) B) -(b+c)

C) -a D) -(a+b+ c).

55) If A= 1 0 2 and adj A= 5 a -2

-1 1 -2 1 1 0

0 2 1 -2 -2 b

Then the values of a and b are--

A) a=-4, b= 1 B) a=-4, b= -1

C) a= 4, b= 1. D) a= 4, b= - 1

56) If A= -1 0

0 2 then the value of A³ - A² is equal to..

A) I B) A C) 2A. D) 2I

57) If A= -x - y

z t, then the transpose of adj A is

A) t z B) t y C) t -z D) none

-y -x -z -x y - x.

58) Find x: x 2 -1

2 5 x = 0

-1 2 x

A) -3,1 B) 3,-1 C). 3,1 D) -3,-1

59) If A is a square matrix of 3 x 3 and K is a scalar, then adj(KA) is equal to

A) K adj A B) K² adj A

C) K³ adj A D) K⁴ adj A

60) If A= 3 5 & B= 1 17

2 0 0 -10 then |AB| is equal to

A) 80 B) 100. C) -110 D) 92

61) The inverse of the matrix

5 - 2

3 1 is

A) -2/13 5/13 B) 1 2

1/13 3/13 -3 5

C) 1/11 2/11 D) 1 3

-3/11 5/11. -2 5

62) If A is a singular matrix order n then A. (adj A) is equals to ..

A) a null matrix.

B) a row matrix

C) a column matrix D) none

63) if the determinant of the matrix

a₁ b₁ c₁

a₂ b₂ c₂

a₃ b₃ c₃ is denoted by D, then the determinant of the matrix

a₁+ 3b₁ - 4c₁ b₁ 4c₁

a₂ +3b₂- 4c₂ b₂ 4c₂

a₃ +3b₃ - 4c₃ b₃ 4c₃ will be

A) D B) 2D C) 3D D) 4D.

64) IF x-2 2x -3 3x -4

x-4 2x-9 3x-16 = 0

x- 8 2x -27 3x-64 then the value of x is ...

A) -2 B) 3 C) 4. D) 0

65) if f(x)=1 x x+1

2x x(x-1) x(x+1)

3x(x-1) x(x-1)(x-2) x(x-1)(x+1) then the value of the value of f(11) is

A) 0. B) 1 C) 11 D) -11

66) The value of the determinant

0 b³- a³ c³ - a³

a³ - b³ 0 c³ - b³

a³ - c³ b³ - c³ 0 is equals to

A) a³+ b³ + c³ B) a³- b³ - c³

C) 0. D) 2(a³+ b³ + c³)

67) If a, b ,c are all different and

a a³ a⁴ - 1

b b³ b⁴ - 1 = 0

c c³ c⁴ - 1 then the value of abc(ab+ bc+ ca) is

A) a+ b+ c. B) 0

C) a²+ b²+ c² D) a³+ b³+ c³

68) If p≠ 0, then the solutions of the equation 1 1 x

p+1 p+1 p+x= 0

3 x+1 x+2 are

A) 2,3 B) 1,p,2 C) 1,2-p D) 1,2.

69) If A and B are two square Matrices and if inverse of A and inverse of B exist, then inverse of (AB) is equals to

A) A⁻¹B⁻¹ B) AB⁻¹ C) A⁻¹B D)B⁻¹A⁻1.

70) If A= 3 -5

-4 2 then the value of A² - 5A is equal to..

A) I B)14I. C) O D) None

71) if A= 5 6 -3

-4 3 2

-4 -7 3, then the cofactor of the elements of second row are

A) 3,3,12. B) 3,-3,11

C) -29,3,-11 D) 30, -3,11

72) If A= 1 2 & B= 1 2

2 3 2 1

3 4 then

A) both AB and BA exist

B) neither AB nor BA exist

C)AB exist but BA does not exist.

D) AB does not exist but BA exist

73) If A= 2 -1 & B= 1 0

0 1 -1 -1

then (A+ B)² is not equals to

A) A²+ AB+ BA+ B²

B) A²+ AB+ BA+ B²I

C) A²I+ AB+ BA+ B²

D) A²+ 2AB+B².

74) If A be an n x n Matrix and k any scalar, than det kA is equal to..

A) k detA B) n ᵏ detA

C) kⁿ detA. D) kn detA

75) The equation of the normal to the parabola y²= 4ax at point (at², 2at) is..

A) tx + y= 2at+ at³.

B) x + ty= 2at+ at³

C) tx - y= at+ 2at³

D) x - ty= at+ 2at³

76) If the slope of the normal to the parabola 3y² + 4y+2= x at a point on it is 8, then the co-ordinates of the point are..

A) (1,-1) B) 6,-2). C)(9,1) D)(2,0)

77) If the line lx+ my+n= 0 is a tangent to the parabola y²= 4ax, then

A) an²= ml B) al²= mn

C) am²= nl. D) a²n= nl

78) The equation of the tangent to the parabola and 3x²- 5y²= 12 which are inclined at an angle 60° to the x-axis are ..

A) y=√3x ±12 B) y=√3x ±10

C) y=√3x ±15 D) y=√3x ±3.

79) The equation of tangent to the curve xy²= 4(4-x) where it meets the line y= x is ..

A) x+y+4= 0 B) x+y-4= 0.

C) x+y-2= 0 D) x- y +2= 0

80) the normal to the curve x= 3 cost - cos³t, y= 3sint - sin³t at t= π/4...

A) is at a distance of 2 units from the origin

B) is a distance of 4 units from the origin

C) passes through the origin.

D) passes through the point (2,3)

81) the point of the curve x²+2y= 10 at which the tangent to the curve is perpendicular to the line 2x - 4y -7= 0 is

A) (2,3). B) (-2,3) C) (4, -3) D) (-4,-3)

82) let x and y with two variable and x> 0, xy= 1: then the minimum value of x+y is

A) 1 B) 5/2 C) 10/3 D) 2.

83) the function f(x)= 1- x³ - x⁵ is decreasing for

A) 1≤ x≤ 5 B) all real value of x.

C) x≤ 3 D) x≥ 5

84) the function y= a(1- cos x) is maximum when x is..

A) π/2 B) -π/2 C) π. D) π/3

85) let f(x)= x³+ 6x²+ px+2; if the largest possible interval in which f(x) is a decreasing function is (-3,-1), then the value of p is..

A)3 B) 9. C) -2 D) none

86) the length of the longest interval in which the function 3sin x - 4 sin³x is increasing, is..

A) π/2 B) π C) 3π/2 D) π/3.

87) The real number x when added to its inverse gives the minimum value of the sum at x equals to

A) -2 B) 2 C) 1. D) -1

88) If the minimum value of f(x)= x² + 2bx + 2c² is greater than maximum value of g(x)= -x²- 2cx + b², then for real value of x..

A) √|c|>|b| B) |c|> √2 |b|.

C) 0< c < 2b D) none

89) Let f(x)=x³ + bx² + cx+ d, 0< b² < c, then f(x)=

A) has a local maximum

B) has a local minimum

C) is strictly decreasing

D) is strictly increasing.

90) If v= 4πr³/3, then the rate (in cubic unit) at wage v is increasing when r= 10 and dr/dt= 0.01, is

A) 4π B) π C) 40π D) 4π/3

91) if the time rate of change of the radius of the sphere is 1/2π, then the rate of change of its surface area (in square cm), when the radius is 5 cm, is

A) 20. B) 10 C) 4 D) 5

92) two perpendicular tangents to y²= 4ax always intersect on the

line..

A) x= a B) x= -a.

C) x= -2a D) x= 2a

93) if the gradient of the tangent at any point (x ,y) of a curve which passes through the point (1,π/4) is y/x - sin²(y/x), then the equation of the curve is

A) y= cot⁻¹(log x)

B) y= cot⁻¹(log x/e)

C) y= x cot⁻¹(log xe).

D) y= cot⁻¹(log e/x)

94) the number of tangents that can be drawn from the point (6,2) on the hyperbola x²/9 + y²/4 = 1 is .

A) 0. B) 1 C) 2 D) 4

95) The equation of the tangent to the curve x²⁾³ + y²⁾³ = a²⁾³ at the point (a cos³t, a sin³t) is .

A) x cost + y sint= a sint cost

B) x cost - y sint= a sin 2t

C) x sin t - y cos t= a sin 2t

D) x sin t + y cos t= a sint cost.

96) The equation of the tangent to the curve y= be⁻ˣ⁾ᵅ at the point where it crosses the y-axis is..

A) bx+ay = ab . B) ax+by = 1

C) bx - ay = ab D) ax - by = 1

97) the equation of the two common tangents to the circle x²+ y²= 2a² and the parabola y²= 8ax are

A) x= ±(y+2a) B) y= ±(x+2a).

C) x= ±(y+a) D) y= ±(x+a)

98) The equation of the normal to the ellipse x²/a² + y²/b² = 1 at the point (a cos t, b sin t) on it is ..

A) ax sin t - by cos t= a² - b²

B) ax sin t + by cos t= a² - b²

C) ax cos t - by sin t= (a² - b²) sin t cos t

D) ax sin t - by cos t= (a² + b²) sin t cos t.

99) the point on the curve √x+√y= √a, the normal at which parallel to the x-axis is .

A) (0,0) B) (a,0)

C) (0,a). D) (a)4,a/4)

100) the slope of the tangent to the curve at x= 3t² + 1, y= t³ - 1 at x= 1 is...

A) 1/2 B) 0 C) -2 D) undefined.

101) if the line x+ y= a is tangent to the parabola y² - y+ x= 0, then the point of contact is...

A) (0,1). B) (a,0)

C) (1,1) D) (-1,0)

102) the angle between the curve y= sinx and y= cos x is..

A) tan⁻¹(5√2) B) tan⁻¹(3√3)

C) tan⁻¹(3√2) D) tan⁻¹(2√2)

103) The function f(x)= cos x - 2ax is a monotonically decreasing when

A) a< 1/2 B) a>1/2. C) a<0 D) a>0

104) if PQ and PR are the two sides of a triangle, then angle between them which gives maximum area of the triangle is ..

A) π/4 B) π/3 C) π/2. D) 2π/3

105) the function f(x)= x³+ 3x² + 4x +7 is increasing for...

A) all real values of x. B) x< 0

C) x > D) x= 0

106) if x+ y= 60, x, y> 0, then the maximum value of xy³ is...

A) 30 B) 60 C) 45(15)³ D) 15(45)³.

107) If the function f(x)= 2x³ - 9ax² + 12a²x +2, where a>0 attains its maximum and minimum at x= p and x= q respectively, such that p²= q, then the value of a is..

A) 1/2 B) 3 C) 1 D) 2

108) A land in the form of a circular sector has been faced by wire of 40 metres length, The area of the land will be maximum when the radius of the circular sector (in metre) is

A) 25 B) 20 C) 10. D) 15

109) the maximum value of the function f(x)= 3 cos x - 4 sinx is..

A) 5. B) 4 C) 3 D) 2

110) The function f(x)=(k sinx + 6 cosx)/(2sinx + 3 cosx) is monotonic increasing when....

A) k> 1 B) k> 4. C) k<1 D) k < 4

111) The values of x, at which the first derivative of the function (x + 1/x) w.r.t.x is 3/4, are...

A) ±1/2 B) ±2/√3 C) ±√3/2 D) ±2

112) If f(x)= sin 3x cos 4x, then the value of f"(x) is..

A) 24 B) 25 C) -25 D) -24

113) let f(x) be a differentiable even function; consider the following statements:

i) f'(x) is an even function

ii) f'(x) is an odd function

iii) f'(x) may be even or odd

which of the above statement is/are correct ?

A) (i) only B) (i) and (iii)

C) (ii) only. D) (i) and (ii)

114) find which function does not obey mean value theorem in [0,1]..

A) f(x)= 1/2 - x when x< 1/2

(1/2 - x)² when x≥ 1/2.

B) f(x)= |x|

C) f(x)=x|x|

D) f(x)= (sinx/x) when x≠ 0

1 when x= 0

115) the derivative of sec⁻¹{1/(2x² - 1) w.r.t √(1-x²) at x= 1/2 is

A) 2 B) 4. C) 1 D) -2

116) the derivative of the function f(x)= log₅(log₇x) (x> 7) is

A) 1/(x logx) B) 1/xlog5 (log7)

C) 1/xlog5 log 7(log₇x). D) none

117) If 2y= (x-a) √(2ax - x²) + a²sin⁻¹{(x-a)/a}, then the value of dy/dx is equal to

A) √(ax - x²) B) √(x² - ax)

C) √(x² - 2ax) D) √(2ax- x²).

118) For a function f(x) if f(1)= 3 and f'(1)= 9, then the value of

k the value of lim ₓ→₀ [f(1+x)/3]¹⁾ˣ is equal to...

A) e² B) e³ C) 1/e². D) 1/e³

119) If y= x³ then the value of

d²y/dx³/[1+ (dy/dx)²]³⁾² at the point (1,1) is ..

A) 3/5√10. B) 5/3√10

C) 4/3√10 D) 3/4√10

120) If x= 1/z, y= f(x) and d²y/dx² = kz³ dy/dz + z⁴ d²y/dz², then the value of k is...

A) -1 B) 1 C) 2. D) -2

121) If xy= ax² + b/x , then the value of x² d²y/dx² + 2x dy/dx is

A) y B) -y C) -2y D) 2y.

122) The value of c in Rolle's theorem when f(x)= 2x³ - 5x² - 4x +3, x belongs to [1/2,3] is..

A) -1/3 B) 2/3 C) 2. D) -2

123) the value

lim ₓ→₀ {(1-tanx)/(1-sinx)}ᶜᵒˢᵉᶜˣ is equal to...

A) 0 B) 1. C) e D) 1/e

124) If A= 3 5 & B= 1 17

2 0 0 -1 then |AB|=

A) 80 B) 100 C) -110 D) 92

125) The value of the determinant of

b²c² bc b+c

c²a² ca c+a

a²b² ab a+b is

A) abc(a²+b²+c²)

B) 0.

C) abc(bc+ca+ab)

D) (a+b+c)(a²+b²+c²) (ab+bc+ca)

126) If x+1 1 1

2 x+2 2 = 0

3 3 x+3 then the value of x is..

A) 0,6 B) 0,6. C) 0,2 D) 2,-1

127) If A= 1 2

3 -5 find inverse of A

A) -5 -2 B) -5/11 -2/11

-3 1 -3 1

C) 5/11 2/11 D) 5 2

3/11 -1/11 3 -1

128) If A= -1 2 & B= 5

2 -1 7 and AX= B, X is equal to...

A) 19 17 B) 19/3

17/3.

C) 19/3 17/3 D) 19

129) If A≠ O and B≠ O are two 2 x 2 metrices such that AB= O, then which of the following is correct?

A) det A= 0 or det B= 0

B) det A= 0 and det B= 0.

C) det A= det B≠ 0 D) none

130) If A= log₃512 log₄3

log₃8 log₄9

B= log₂3 log₈4

log₃4 log₃4 find AB

A) 7 B) 17 C) 13 D) 10

131) dy/dx of (sec x+ tanx)/(sec x- tanx) is

A) 2sec x((sec x+ tanx)

B) 2sec² x((sec x+ tanx)²

C) 2sec x((sec x+ tanx)².

D) sec x((sec x+ tanx)

132) If y= sin x° and z= logx, then the value of dy/dz is

A) (x° cos x°)/log e.

B) x cos x°/log 10

C) x cos x°/log e

D) x° cos x°/log 10

133) d/dx[ tan ⁻¹{√x(3-x)/(1-3x)}] is

A) 3/{2(1-x)√x} B) 3/{2(1+x)√x}.

C) 2/{(1+x)√x} C) 3/{(1+ x)√x}

134) If x= sin t and y= cos pt, then which of the following is true?

A)(1-x²)d²y/dx² +x dy/dx+p²y= 0

B)(1-x²)d²y/dx² +x dy/dx-p²y= 0

C)(1+x²)d²y/dx² -x dy/dx+p²y= 0

D)(1-x²)d²y/dx² -x dy/dx+p²y= 0.

135) In the mean value theorem f(b) - f(a) = (b -a) f'(c)(a<c<b), if a=4,b=9 and f(x)= √x, then the value of c is

A) 8 B) 5.25 C) 4 D) 6.25.

136) If the function f(x)= 4x³ + ax² + bx -1 satisfies all the conditions of Rolle's theorem in -1/4 ≤ x≤ 1 and if f'(1/2)= 0, then the values of a and b are..

A) a=2, b= -3 B) a=2, b= -4.

C) a=-1, b= 4 D) B) a=-4, b= -1

137) lim ₓ→∞{(n-3)/(n+2)ⁿ is equal

A) 1/e⁵. B) 1/e⁴ C) 1/e² D) 1/e

138) If f"(0)= 4, then the value of lim ₓ_0 {f(x)- 3f(2x)+ f(4x)}/x² is..

A) 6 B) -6 C) 12 D) -12

139) If y= (√x)^∞ and x dy/dx= f(y)/(2 - y log x), then the value of f(y)

A) y log y B) log y C) 2y D) y².

140) If x= eᵗ Sin t and y= eᵗ cos t, then the value of (x+y)² d²y/dx² - 2x dy/dx is..

A) 2y B) -2y. C) 4y D) -4y

141) If y = f(x²) and f'(x)=√(3x²+1) then the value of dy/dx at x=2 is

A) 4√13 B) 2√13 C) 28. D) 14

142) If x= sec t - cos t, y= secⁿt- cosⁿt and (x²+4)(dy/dx)²= k(y²+ 4) then the value of k is..

A) n². B) 2n C) - n² D) -2n

143) If f(x) is a differentiable function for all x> 0 and lim ᵧ→ₓ {y²f(x)- x²f(y)}/(y-x) = 2, then the value f'(x) is..

A) 1/x²[x f(x) -1]

B) 2/x²[x f(x) - 2]

C) 1/x²[x f(x) - 2]

D) 2/x²[x f(x) - 1].

144) If y² = 4ax, then the value of d²y/dx² . d²x/dy² is..

A) 2a/y³ B) - 2a/y³.

C) -a/y³ D) a/y³

145) the point on the curve y²= x, the tangent makes an angle 45° with the x-axis is..

A) (0,0) B) (1/4,1/2)

C) (1/2,1/4) D) (2,4)

146) The equation of the normal to the hyperbola x= a sec t, y= b tan t at the point (a sec t, b tan t) is .

A) ax cos t + by cot t= a²+ b²

B) ax cos t + by tan t= a²+ b²

C) ax sin t - by cot t= a²- b²

D) ax cos t - by tan t= a²- b²

147) If the straight line lx + my= 1 is a normal to the parabola y²= 4ax, then..

A) al²+ 2lm= m²

B) al³ - 2alm= m²

C) al³ + 2alm= m²

D) al² + 2alm= m²

148) A cone of height h is inscribed in a sphere of radius R; If the volume of the inscribed cone is maximum, then the value of h: R will be ...

A) 3/4 B) 4/3. C) 2/3 D) 3/2

149) the minimum value of f(x)= 2x² + x -1 is..

A) -1/4 B) 3/4 C) 9/4 D) -9/8.

150) The point on the curve xy²= 1 that is nearest to the origin is..

A) (1,1) B)(4,1/2)

C)(1/³√2,⁶√2). D)(1/4, 2)

151) The number of values of x for which f(x)= cos x+ cos √2 x attains its maximum value, is

A) 1. B) 0 C) 2 D) infinite

152) the function f(x)= 2x³ - 3x²+ 90x + 174 is increasing in the interval..

A) 1/2< x < 1

B) 1/2 < x < 2

C) 3 < x < 59/4

D) - ∞ < c <∞.

153) The coordinates of the point for minimum value of z= 7x - 8y subject to the conditions x+ y ≤ 20, y≥ 5, x ≥ 0 are

A) (20,0) B)(0,20). C)(15,5) D)(0,5)

154) If M and m are the maximum and minimum values respectively of the function f(x)= x + 1/x, then the value of M - m is

A) 0 B) 2 C) 4 D) -4

155) The interval in which the function f(x)= 2x² - log|x| (x≠0) is increasing, is...

A) 0<x < 1/2 B) x < -1/2

C) -1/2 < x < 0 D) none

156) the sides of an equilateral triangle are increasing at the rate of 2cm/s.; then the rate at which the area (in cm²/s) increases when the side is 10 cm, is

A) √3 B) 10 √3. C) 10 D) 10√2

157) Air is being pumped into a spherical balloon at the rate of 30 cm³/s. Then the rate (in cm/s) at which the the radius increases when it reaches the value 15 cm, is...

A) 1/30π B)1/15π C)1/20 D) 1/25

1/8/21

** Discuss the applicability of Rolle's theorem..

1) f(x)= sin(1/x) for -1≤x≤1. No

2) f(x)= 2x²-5x+3 on [1,3]. No

3) f(x)= -4x+5, 0≤x≤1

2x-3, 1<x≤2. No

** Veryfy Rolle's Theorem::

4) x(x-4)² on [0,4]

5) x²+5x+6 on [-3,-2].

6) x² - 4x+3 on [1,3].

7) (x- 1)(x-2)² on [1,2].

8) (x² -1 )(x-2) on [-1,3].

9) cos 2(x-π/4) on (0,π).

10) eˣ sinx on [0,π]

11) sin 3x on [0,π]

12) ₑ(1-x²) on [-1, 1]

13) log(x²+2) - log 3 on [-1, 1]

14) 6x/π - 4 sin²x on [0,π/6]

15) 4ˢᶦⁿ ˣ on [0,π].

16) Using Rolle's Theorem, find points on the curve y= 16 - x², x belongs to [-1, 1], where tangent is parallel to x-axis. (0,16)

17) Using Rolle's Theorem, find points on the curve y= ₑ1-x², x belongs to [-1, 1], where tangent is parallel to x-axis. (0,e)

18) Using Rolle's Theorem, find points on the curve y= 12(x+1)(x - 2), x belongs to [-1, 2], where tangent is parallel to x-axis. (1/2,-27).

** Verify Lagrange's mean value theorem:::

19) x³- 2x²-x+3 on (0,1)

20) x²- 3x +3 on (-1,2)

21) (x-1)(x-2)(x-3) on 0,4)

22) x+ 1/x on (1,3)

23) x²+x-1 on (0,4)

24) x(x+4)² on (0,4)

25) sinx - sin2x - x on (0,π)

26) Show that the Lagrange's mean value theorem is not applicable to the function f(x)=1/x on (-1,1).

27) find a point on the curve y= x³+2 where the tangent is parallel to the chord joining (1,2),(3,28).

28)If f(x)=Ax²+Bx + C is such that f(a)=f(b), then write the value of c in a Rolle's theorem. (a+b/2

29) n is a positive integer. if the value of c prescribed in Rolle's theorem for the function f(x)=2x(x-3)ⁿ on the interval (0, 2√3) is 3/4, write the value of n. 3

30) find the value of C prescribed by Lagrange's mean value theorem for the function f(x)=√(x²-4) defined on (2,3). √5

31) If from lagrange's mean value theorem we have f'(x₁)={f'(b)- f(a)}/(b-a), then

A) a<x₁≤b. B)a≤x₁<b C) a<x₁<b D) a≤x₁≤b

32) Rolle's theorem is applicable in case of φ(x)= aˢᶦⁿˣ, a> a in

A) any interval B) the interval [0,π]

C) the interval (,π/2) D) none.

33) the value of c in Rolle's theorem when f(x)= 2x³- 5x²- 4x+3, on (1/3,3)

A) 2, B) -1/3 C) -2. D) 2/3

34) When the tangent to the curve y= x logx is parallel to the chord joining the points (1,0) and (e,e), the value of x is .

A) e¹/⁽¹⁻ᵉ⁾ B) e⁽ᵉ⁻¹⁾⁽²ᵉ⁻¹⁾ C) e⁽²ᵉ⁻¹⁾/⁽ᵉ⁻¹⁾ D) (e-1)/e

35) the value of c in rolle's theorem for the function f(x)= {x(x+1)}/eˣ defined on (-1,0) is..

A) 0.5 B) (1+√5)/2 C) (1-√5)/2 D) - 0.5

36) The value of c in Lagrange's mean value theorem for the function f(x)= x(x-2) on (1,2) is..

A) 1. B) 1/2 C) 2/3. D) 3/2

37) the value of c in Rolle's theorem for the function f(x)= x³-3x in the interval (0,√3)

A) 1 B) -1 C) 3/2. D) 1/3

38) If f(x)= eˣ sinx in (0,π), then c in Rolle's theorem is..

A) π/6 B) π/4 C) π/2 D) 3π/4

39) Use lagrange's mean value theorem to determine a point P on the Curve y=√(x²-4) defined in the interval [2,4] where the tangent is parallel to the chord joining the end points on the curve. (√6,√2)

40) verify rolle's theorem for the function f(x)= e²ˣ (sin 2x - cos2x) defined in the interval (π/8,5π/8)

41) given f(x)= (x-3) logx prove that there is at least one value of x in the interval (1,3) which satisfies the equation x logx= 3 - x.

42) Examine the validity and conclusion of rolle's theorem for the function. f(x)= eˣ sinx, on(0,a).

43) Examine the validity and conclusion of Lagrange's mean value theorem for the function f(x)= x(x-1)(x-2) for every x belongs to (0,1/2).

44) verify rolle's theorem for the function f(x)= log[(x²+ab)/{x(a+b)}], x belongs to (a,b) and 0 is not belong to (a,b).

45) verify Rolle's theorem for the function f(x)= sinx+ cosx -1 in (0,a/2) and find the point where the derivativation vanishes. π/4

46) verify Lagrange's mean value theorem for the following function. f(x)= 2x - x², 0≤ x ≤ 1.

47) check if Lagrange's mean value theorem is applicable to. f(x)= 4 - ³√(6-x)² in (5,7).

48) Find c of the Lagrange's mean value theorem for the function f(x)= x(x-2) in the interval (1,2).

25/7/21

1) If tan⁻¹{√(1+x²) - √(1-x²)}/{√(1+x²)+ √(1+x²)}= a then a² is... Sin2a

Find the value of:

2) Tan(cos⁻¹1/5√2 - sin⁻¹4/√17). 3/29

3) 2tan⁻¹{cosec(tan⁻¹x)- tan (cot⁻¹x). cot⁻¹1/x

4) If cos⁻¹x/a + cos⁻¹y/b= k then x²/a² - (2xy cosk)/ab + y²/b² is.. sin²k

5) sin[cot⁻¹{tan(cos⁻¹x)}]. x

6) If x<0, y<0 such that xy= 1, then tan⁻¹x + tan⁻¹y equals. -π/2

7) If u= cot⁻¹√(tanx) - tan⁻¹√(tanx) then, tan(π/4 - u/2)=? √tanx

8) If cos⁻¹x/3 + cos⁻¹y/2 = k/2, then 4x² - 12xy cos(k/2) + 9y²= ? 18 - 18 cosk

9) If a= tan⁻¹{√3 x/(2y-x)}, b= tan⁻¹{(2x-y)/√3y}, then a- b is.. π/6

10) Let f(x)= ₑ cos⁻¹{sin(x+π/3)} , then, f(8π/9) is .. ₑ13π/18

11) tan⁻¹1/11 + tan⁻¹2/11 is.. 1

12) If cos⁻¹x/2 + cos⁻¹y/3= k, then 9x² - 12xy cosk + 4y² is.. 36sin²k

13) sin⁻¹(cos 33π/5). -π/10

14) cos⁻¹(cos 5π/3) + sin⁻¹(sin 5π/3) is . 0

15) sin{2 cos⁻¹(-3/5)} is... -24/25

16) If k= sin⁻¹[sin(-600°)], then one of the possible values of k is. π/3

17) If 3 sin⁻¹{2x/(1+x²)} - 4 cos⁻¹ {(1- x²)/(1+x²)}+ 2tan⁻¹{2x/(1-x²)}= π/3 is equal to.. 1/√3

18) sin(1/4 sin⁻¹(√63)/8) is... 1/2√2

19) cot(π/4 - 2 cot⁻¹3) is... 7

SOLVE:

20) tan⁻¹x + cos⁻¹{y/√(1+y²)} = sin⁻¹3/√10. 1,2

21) Sin⁻¹x - cos⁻¹x = π/6. √3/2

22) tan⁻¹3 + tan⁻¹x= tan⁻¹8,. 1/5

23) If 4 cos⁻¹x + sin⁻¹x=π. √3/2

24) If tan⁻¹{(x+1)/(x-1)} + tan⁻¹{(x-1)/x}= tan⁻¹(-7). 2

25) tan⁻¹(cotx)= 2x,. ±π/6

26) The number of solutions of the

a) tan⁻¹2x + tan⁻¹3x =π/4

A) 2. B) 3 C) 1 D) none

b) √(1+ cos2x)= √2 sin⁻¹(sinx), -π≤ x ≤ π is

A) 0 B) 1 C) 2. D) infinite

27) If a= tan⁻¹(tan 5π/4) and b= tan⁻¹(-tan 2π/3), then

A) 4a= 3b. B) 3a= 4b C) a-b=7π/12 D) n

Solve:

28) tan⁻¹{(x-1)/(x-2)} + tan⁻¹{(x+1)/(x+2)}= π/4. ±1/√2

29) tan⁻¹(x+1)+ tan⁻¹(x-1) = tan⁻¹(6/17), for x> 0. 1/3

Prove:

30) tan⁻¹1/2+ tan⁻¹1/3=π/4

31) tan⁻¹1/2+tan⁻¹1/4=tan⁻¹6/7

32) tan⁻¹3/5+ tan⁻¹1/4=π/4

33) tan⁻¹1+ tan⁻¹2+ tan⁻¹3 =π = 2(tan⁻¹1/2+ tan⁻¹1/3+tan⁻¹1).

34) (tan⁻¹1+ cot⁻¹1/2+ tan⁻¹3) =π = 2(tan⁻¹1+ tan⁻¹1/2+ cot⁻¹3).

35) tan⁻¹x+ cot⁻¹(x+1)= tan⁻¹(1+x+x²)

36) tan⁻¹x+ cot⁻¹y= tan⁻¹{(xy+1)/(y-x)}

37) tan⁻¹{(2a-b)/b√3} + tan⁻¹2{(2b-a)/a√3}= π/3

38) tan⁻¹{1/(p+q)}+ tan⁻¹{q/(p²+pq+1)} = tan⁻¹1/p

39) tan⁻¹a - tan⁻¹c = tan⁻¹{(a-b)/(1+ab)} +tan⁻¹{(b-c)/(1+bc)}

40) tan⁻¹{(a-b)/(1+ab)}+ tan⁻¹{(b-c)/(1+bc)}+ tan⁻¹{(c-a)/(1+ca)} = tan⁻¹{(a²-b²)/(1+a²b²)} + tan⁻¹{(b²-c²)/(1+b²c²)} +tan⁻¹{(c² -a²)/(1+c²a²)}.

41) sin(cos⁻¹x)= cos(sin⁻¹x)

42) sin⁻¹cos(sin⁻¹x)+ cos⁻¹ sin(cos⁻¹x) = π/2, |x|≤1

43) sin cosec⁻¹cot tan⁻¹x= x. For |x|<1

44) sin⁻¹77/85 - sin⁻¹3/5=cos⁻¹ 15/17.

45) cot⁻¹43/32 - tan⁻¹1/4 = cos⁻¹12/13

46) tan⁻¹1/2 + tan⁻¹2/11 = cos⁻¹4/5

47) sin⁻¹4/5 + tan⁻¹3/4=π/2

48) sin⁻¹4/5 +sin⁻¹5/13+ sin⁻¹16/65 = π/2

49) 2 tan⁻¹1/3 + tan⁻¹1/7=π/4

50) sin cot⁻¹tan cos⁻¹x= x.

51) Cos⁻¹{(b+ a cosx)/(a+b cos)}= 2tan⁻¹[√{(a-b)/(a+b)} tan(x/2).

52) If cos⁻¹x + cos⁻¹y+ cos⁻¹z=π, show that x²+y²+z²+2xyz=1.

53) sin⁻¹√{(x-q)/(p-q)}= cos⁻¹√{(p-x)/(p-q)} = cot⁻¹√{(p-x)/(x-q)}

54) sin(sin⁻¹1/2 - cos⁻¹1/4) = (1- 2√6)/6

55)1/2 cos⁻¹{5 cosx+3)/(5+ 3cosx)} = tan⁻¹(1/2 tan x/2).

56) 2tan⁻¹x + 2 tan⁻¹y=sin⁻¹[{2(x+y)(1-xy)}/{(1+x²)(1+y²)}

57) 1/2 cos⁻¹{(1+2 cosx)/(2+ cosx)}= tan⁻¹{1/√3 tan (x/2)}

58) cot⁻¹{(tan 2x)}+cot⁻¹{-tan 3x}= x.

59) sec²(tan⁻¹2)+ cosec²(cot⁻¹3) =15.

60) If tan⁻¹x+tan⁻¹y+tan⁻¹z = π, show that x+y+z=xyz

61) If tan⁻¹x +tan⁻¹y+tan⁻¹z=π/2, show that xy+ yz + zx=1.

62)If Sin⁻¹x+ sin⁻¹y+ sin⁻¹=π, prove that x√(1-x²)+ y√(1- y²)+ z√(1-z²)= 2xyz.

Solve:

63) 2 tan⁻¹{2x/(1-x²)}=π/3. 2-√3

64) 2 cot⁻¹{(1+x)/(1-x)= tan⁻¹x. ±1/√3

65) tan⁻¹{2x/(1-x²)} = sin⁻¹{2a/(1+a²)} - cos⁻¹{(1-b²)/(1+b²). (a-b)/(1+ab).

66) 2 sin⁻¹x+ sin⁻¹(1-x)=π/2. 0,1/2

Find the value of

67) Tan(1/2)(tan⁻¹x + cot⁻¹x).

68) sin(sin⁻¹1/4 + sec⁻¹3)+ cos(tan⁻¹1/2 + tan⁻¹2).

69) Show tan[2 tan⁻¹√{(1+cos2x)/ (1-cos2x)} + tanx]= 0

70) If a= cot⁻¹√cos2x - tan⁻¹√cos2x, prove that sin²a= tan²x.

71) If tan⁻¹y= 4 tan⁻¹x, find an algebraic relation between x and y. y= 4x(1-x²)(1-6x²+x⁴)

72) Solve: tan⁻¹(1/2 secx)+ cot⁻¹(2cosc) =π/3. √(3/2),-1/2√3

73) If tan⁻¹x+ tan⁻¹y+ tan⁻¹z= π/2 and x+y+z=√3, then prove that x= y= z

74) Which of the following two statements, is correct?

a) sin cos⁻¹ tan45° defines an angle

b) sin⁻¹ cos tan⁻¹√3 defines an angle.

Prove:

75) cos⁻¹{cosx + cosy)/(1+ cosx cosy)}= 2 tan⁻¹( tan x/2 tan y/2).

76) 2 tan⁻¹[√{(m-n)/(m+n)} tan x/2] = cos⁻¹{(m cosx + n)/(m+ n cosx).

77) If a= tan⁻¹{1/√cos 2x} - tan⁻¹√cos 2x, then show tan²a= sin⁴x/cos2x and seca= ±1/2(√sec2x + √cis2x)

78) cot⁻¹1/2 - 1/2 co⁻¹4/3=π/4.

79) 4(tan⁻¹1/3 + cos⁻¹2/√5)=π

80) tan⁻¹1/x + tan⁻¹1/(x+y) = tan⁻¹(y/(x²+ xy+1).

81) 4 tan⁻¹1/5 - tan⁻¹1/239=π/4

82) tan⁻¹(1/2 tan2A) +tan⁻¹(cotA) +tan⁻¹(cot³A)= 0.

83) Show that value of tan⁻¹{(x cosa)/(1- x sina) - cot⁻¹{(cosa)/(x - sina)} independent of x and find it. a

84) tan (π/4+ 1/2 cos⁻¹a/b) + tan (π/4 - 1/2 cos⁻¹a/b)= 2b/a.

85) tan⁻¹(cotx)+ cot⁻¹(tanx)= π-2x.

86) sin cos⁻¹ tan sec⁻¹x =√2-x²).

87) Evaluate: cos(2 cos⁻¹x+ sin⁻¹x), at x= 1/5, where 0≤ cos⁻¹x≤π; -π/2 ≤ sin⁻¹x ≤π/2. -2√6/5

88)If sin⁻¹x/a + sin⁻¹y/b = sin⁻¹c²/ab then show b²x²+ 2xy √(a²b²- c⁴) + a²y²= c⁴.

89) value of cos⁻¹x+ cos⁻¹[x/2 + √(3- 3x²)/2] (1/2≤x≤1). π/3

90) Prove 2 tan⁻¹[tan a/2 tan(a/2 - b/2)]= tan⁻¹x{(sina cosb)/(cosa+ sinb).

91) If cos⁻¹x+cos⁻¹y+cos⁻¹z=π, then Prove x²+ y²+ z²+2xyz =1.

92) Prove: a cosx + b sinx = √(a²+b²) cos (x - tan⁻¹b/a).

93) If sin(π cosy)= cos(π siny) then show y= ± 1/2 sin⁻¹3/4.

94) If cos⁻¹x= 2 sin⁻¹x then show that x= 1/2.

95) If sin⁻¹cos sin⁻¹ x=π/3 then find the value of x. 1/2

** Solve:

96) tan⁻¹(1/2 secx) + cot⁻¹(2cosx) = π/3. x= 2nπ± π/6

97) tan⁻¹{(1-x)/(1+x)}= 1/2 sin⁻¹{x/ √(1+x²)}. 1/√3

98) cot⁻¹x+cot⁻¹(a²- x+1)= cos⁻¹(a-1). a, a²-a+1

99) tan⁻¹1/3 +tan⁻¹1/5 +tan⁻¹1/8 + tan⁻¹ 1/x= π/4. 7

100) if tan⁻¹{(x-1)/(x-2)} + tan⁻¹ {(x+1)/(x+2)}= π/4. Then x is. ±1/√2

101) 3 sin⁻¹{2x/(1+x²)} - 4 cos⁻¹{(1-x²)/(1+x²)} + 2 tan⁻¹{2x/(1-x²)=π/3. 1/√3

102) tan⁻¹(cotx) +cot⁻¹(tanx)=π/4. 3π/8

103) 3 cot⁻¹1/(2-√3) +cot⁻¹x=π/2. 1

104) tan⁻¹(x-1) +tan⁻¹x+tan⁻¹(x+1)= tan⁻¹3x. 0, ±1/2

105) sin⁻¹x - sin⁻¹y= π/3, cos⁻¹x+cos⁻¹y= 2π/3. x=√3/2, y=0

106) If tan⁻¹2x+ tan⁻¹3x=π/4. Then the value of x is. 1/6, -1

Prove:

107) 4tan⁻¹1/5 + tan⁻¹1/70 + tan⁻¹1/99= π/4.

108) sin⁻¹[cot{sin⁻¹√{(2-√3)/4} + cos⁻¹√12/4+ sec⁻¹√2)]= 0

Solve)

109) Sin(2 cos⁻¹(cot(2tan⁻¹x)))= 0. ±1, ±1±√2.

110) (tan⁻¹x)²+ (cot⁻¹x)²= 5π²/8. -1

111) Sin⁻¹6x + sin⁻¹6√3 x= -π/2. ±1/12

112) 2 tan⁻¹(cosx)= tan⁻¹(2cosecx). nπ+π/4.

113) x= tan⁻¹(2tan²x)- 1/2 si{(3sin2x)/ (5+4 cos2x). nπ+ tan⁻¹(-2)n

114) If sin⁻¹x+ sin⁻¹y+ sin⁻¹z=π, prove x⁴+y⁴+z⁴+4x²y²z²=2(x²y²+y²z²+z²x²).

115) If sin⁻¹x/a + sin⁻¹y/b= sin⁻¹c²/ab, show b²x²+ 2xy√(a²b²- c²)+ a²y²= c⁴

116) If sin⁻¹x + sin⁻¹y= π/2, show 2(x² - x²y²+ y²)= 1+ x⁴+ y⁴.

* Find the value of::

117)Sec²(tan⁻¹2)+cosec²(cot⁻¹3)+2. 17

118) tan[sin⁻¹(3/5)+ cot⁻¹(3/2)]. 17/6

13/7/21

1) Points on the curve y²=4a(x+ a sin(x/a)) at which the tangent to the curve are parallel to the x-axis lie on a/an --

A) circle B) parabola.

C) ellipse D) hyperbola

2) the equation of the tangent to the curve x= 2cos³t and y= 3 sin³t at the point t=π/4 is...

A) 2x+3y= 3√2. B) 2x-3y= 3√2

C) 3x-2y= 3√2. D) 3x+2y= 3√2

3) On the ellipse 4x²+9y²=1, the points at which the tangent are parallel to the line 8x+=9y are--

A) (2/5,1/5),(-2/5,1/5)

B) (-3/5,-2/5),(3/5,2/5)

C) (-2/5,1/5),(2/5,-1/5).

D) (-3/5,2/5),(3/5,-2/5)

4) if a tangent to the curve y= 6x - x² is parallel to the line 4x-2y=1, then the point of tangency on the curve is...

A) (2,8). B) (8,2) C)(4,8) D) (8,4)

5) if the line 3x + y +c=0 is a normal to the circle x²+y²-4x-6y +3=0 then the value of c.

A) 8 B) -8 C) 9 D) -9.

6) the equation of the normal to the parabola y²= 4(x-1) at point (5,4) on it is...

A) 2x - y +=6 B) x -2 y +3=0

C) 2x + y= 14. D) x + 2y =13

7) if x is real, then the maximum value Of (x²-x+1)/(x²+x+1) is.

A) 1 B) 2 C) 4 D) 3.

8) the difference between the maximum and the minimum values of the function f(x)= x³)3 - 2x² + 3x +1 is .

A) 4 B) 2 C) 1 D) 4/3.

9) If y= sec(tan⁻¹x), then dy/dx is

A) - x/√(1+x²) B) x/√(1-x²)

C) x/√(1+x²). D) 2x/√(1+x²)

10) If y=(logx)/x, then [d²y/dx²] at x=1 is....

A) -3. B) -2 C) -1 D) 3

11) dy/dx of logₓ2x is..

A) -(log 2)/(x logx)

B) (log 2)/x(logx)²

C) (log 2)/(x logx)

D) -(log 2)/x(logx)².

12) If y= tan⁻¹(secx - tanx), then dy/dx is...

A) 1/2 B) -1/2. C) 2 D) -2

13) If f(x)= logx, then the differential coefficient of f(sinx) w.r.t.x is..

A) tanx B) cotx. C) f(cosx) D) 1/x

14) If y=√(x+1) - √(x-1), then the value of (x²-1) d²y/dx² + x dy/dx is....

A) 2y B) -2y C) y/4. D) y/2

15) In the mean value theorem f(b) - f(a) = (b-a)f'(c), (a< c< b), if f(x)= x³ - 3x -1, a= -11/7, b= 13/7 then the value of c is...

A) 0 B) 1 C) -1 D) ±1.

16) lim ₓ→ₐ (aˣ - xᵃ)/(xˣ -aᵃ)= -1, then the value of a is...

A) -2 B) 2 C) 1. D) -1

17) If xʸ = eˣ ⁻ʸ, then the value of dy/dx at x= e

A) 1/4. B) 1/2 C) 1 D) -1/2

18) If x⁴ +5xy+ y=2, then the value of d²y/dx² at the point (0,1) is...

A) -3/5 B) 2/5 C) 3/5 D) -2/5.

8/7/21

1) Prove:

A) 2 tan⁻¹1/3 + tan⁻¹1/7=π/4

B) sin⁻¹12/13+ cos⁻¹4/5+ cos⁻¹63/16= π

C) 2sin⁻¹3/5 + sin⁻¹7/25=π/2

D) tan (2 tan⁻¹p)= 2 tan(tan⁻¹p + tan⁻¹ p³).

2) a) Solve: sin⁻¹5/x + sin⁻¹12/x = π/2. 13

b) sin⁻¹x + sin⁻¹(2x)= π/3. 1/2 √(3/7)

3) If sec⁻¹x = cosec⁻¹y then Show that 1/x² + 1/y² = 1

4) If sec⁻¹x = cosec⁻¹y then Show that cos⁻¹1/x + cos⁻¹1/y = π/2.

7/7/21

1) If y= (x-1)eˣ, then dy/dx at x= 1

A) e B) 2e C) 1 D) 0

2) If y= 1+ cos2x, then d²y/dx² +4y = -------

3) solve by Cramer's rule: x+y-3= 0; x+ 2y- 5= 0;

4) If x²+ y²= t+ 1/t, x⁴+ y⁴= t²+ 1/t², then Show that dy/dx= -1/x³y

5) If y= tan⁻¹[1+tanx)/(1-tanx)], then find the value of d²y/dx²

6) show that the maximum value of the function f(x)= 4x - x²-1 is 3.

7) Show a+b+2c a b

c b+c+2a b

c a c+a+2b

= 2(a+b+c)²

6/7/21

1) d/dx (1/2 sin⁻¹{2x/(1+x²)}.

2) If A= 3 2 & B= 4 1

1 -1 2 3 with the relation yXA = B. Find matrix X

3) Given A= 2 2

4 4 Prove A. A⁻¹= I

4) (sinx)ˣ w.r.t.x

5) (eˣ +1).y= eˣ -1. Find dy/dx

6) Given y= sinx - cosx. Prove d²y/dx² + y= 0.

7) The cofactor of b in the determinant a b

c d is....

5/7/21

1) If y= √(sin√x), dy/dx is.

2) f(x)= cos⁻¹[{1-(logx)²}/{1+(logx)²}], then f'(e) is

3) If y= a cos mx - b sin nx, then d²y/dx² is

4) If y= ₓeˣ, then dy/dx is

5) If 2ˣ + 2ʸ= 2 ˣ⁺ʸ, then find the value of dy/dx at x= y= 1 is

6) If f(x)= x(x -1)(x-2), 0≤x ≤ 4, then the point x= c which satisfies mean value theorem satisfies--

A) 0<c<1 B) c> 3 D) 0<c<1/2 D)1< c< 3

7) If x= sin⁻¹t, y= log(1-t²), 0≤ t< 1, then the value of d²y/dx² at t= 1/3 is.

8) if y= (x +√(1+x²))ⁿ Then (1+x²)y₂ + x y₁ is

9) If siny + e⁻ˣ ᶜᵒˢ ʸ = e, then the value of dy/dx at (1,π) is

10) If x= 2 cost + cos 2t and y= 2 sint - sin 2t, then the value of dy/dx at t=π/4 is...

10) lim ₓ→∞ {(n²- n+1)/(n²-n-2)} ⁿ⁽ⁿ⁻¹⁾ is equal to.

4/7/21

1) prove: sec²(tan⁻¹2)+ cosec²(cot⁻¹3) = 15.

2) prove tan⁻¹(yz/xr) + tan⁻¹(zx/yr) + tan⁻¹(xy/zr) = π/2, where x²+ y² + z² = r².

3) Show that f(x)= 5x -4; 0< x ≤1

4x³-3x; 1< x< 2 is continuous at x= 1

4) d/dx log{(a + b sinx)/(a- b sinx)}.

5) If y= √(sinx+ y) find dy/dx. Cosx/(2y -1)

3/7/21

1) 1, if x ≤ 3

If f(x)= ax+ b, if 3< x < 5

7, if 5≤ x. Determine the values of a and b so that f(x) is continuous. 3, -8

2) Prove: 2tan⁻¹(1)5) + sec⁻¹(5√2/7) + 2tan⁻¹(1/8)= π/4

3) x= a{cost + 1/2 log tan²t/2} and y= asin t. Find dy/dx. tan t

4) If siny = x cos(a+y). Find dy/dx.

5) If y² = ax²+ bx + c, then find the value of y³ d²y/dx² .

2/7/21

1) If A= 0 1 1

1 2 0

3 -1 4 Prove that A.(adj. A)= det. A.I

2) Solve by Cramer's Rule: 1/x+ 1/y +1/z = 1; 2/x +5/y +3/z= 0; 1/x+2/y + 4/z = 3. 1,-1,1

3) cot⁻¹{(1+x)/(1-x)} w.r.t.x. -1/(1+x²)

4) find dy/dx when xˣ + x¹⁾ˣ. xˣ (1+ logx)+ x¹⁾ˣ{(1- logx)/x².

5) If y= eᵃˣ. Sin bx. Show that y₂ - 2ay₁ +(a² + b²)y = 0.

1/7/21

1) If y= x log{x/(a+bx)}, Prove x³ d²y/dx² = {x dy/dx - y)².

2) If x = a sect, y= b tan t, prove that d²y/dx²= - b⁴/a²y³.

3) If f(x)= mx +1, x ≤π/2

Sinx + n, x > π/2 is continuous at x=π/2, then find the value of m, n. n= mπ/2

4) ₓxˣ w.r.t.x. ₓxˣ ₓx{(1+logx). logx +1/x}

5) If A= 1 2 3 4 & B= 1

4 find AB

6) Discuss the continuity of

f(x)= 2x -1; if x< 2

3x/2; if x ≥ 2. Yes

30/6/21

1) Find f(0), so that f(x)= x/{1-√(1-x)} becomes continuous at x=0. 2

2) Solve the equation :. 0,-1

tan⁻¹{√(x²+x)} + sin⁻¹√(x²+x+1)=π/2

3) if f(x)= 5x -4 ; 0<x ≤1

4x²+3bx; 1 <x<2 is continuous at x=1, find b. -1

4) value of cos⁻¹1/2+3sin⁻¹ 1/2. 2π/3

5) Determine whether the function continuous at x= 0 or not

f(x)= (sin x²)/x; x≠ 0

0 ;. x= 0. Yes

6) (1+cosx)ˣ w.r.t.x.

7) If y= tanx + secx, prove d²y/dx² = cosx/(1- sinx)²

29/6/21

1) Find the inverse of 0 1 2

1 2 3

3 1 1

2) tan⁻¹√{1-x)/(1+x)} w.r.t.x. -1/{2√(1-x²)

3) prove 1 1 1

1 1+x 1 = xy

1 1 1+y

4) prove: b+ c a b

c + a c a

a+ b b c = (a+b+c)(a- c)²

5) If y= Aᵏˣ + B⁻ᵏˣ, Prove d²y/dx² = k²y.

6) If y= (sin⁻¹x)/√(1-x²), show (1-x²) dy/dx - xy = 1.

28/6/21

1) log(x + ₑ√x) w.r.t.x. (2√x + ₑ√x)/{2√x(x + ₑ√x)}

2) If A=1 - 1 and B= -1 2

2 -1. -1 2 , find AB

3) Solve by Cramer's rule: 2x+y + z= 1; x-y +2z= -1; 3x+2y -z=4. 1,0,-1,

4) If A=3 -1 & B= 3 & C= 1

1 2 1 -2 . Find X such that AX = 3B + 2C. 3

-2

5) sin⁻¹{1-x²)/(1+x²)} w.r.t.x. -2/(1+x²).

6) find dy/dx: log xy= x² + y².

7) If y= 2x + 4/x, Prove x² d²y/dx² + x dy/dx - y= 0.

27/6/21

1) sin(cos⁻¹x) w.r.t.x. -x/√(1-x²)

2) Show that (1-x²)d²y/dx² - x dy/dx + 4y = 0, when y = sin(2sin⁻¹x).

3) ₑx² w.r.t.x. 2xₑx²

4) sin(x² eˣ). (x² eˣ)(x+2)Cos(x² eˣ)

5) If A= -4 -3 -3

1 0 1

4 4 3 show that adj.A = A.

6) If A= 1 2 1 & B= 1 4 0

1 -1 1 -1 2 2

2 3 -1 0 0 2 find the value of AB - 2B. -3 0 6

4 -2 -4

-1 14 0

7) 1) Using Cramer's rule Solve the equation: 2x - z= 1; 3x+ 4y -z= 1;

x - 8 y - 3z= -2. 1,0,1

26/6/21

1) Using Cramer's rule Solve the equation: x+y+z= 1; ax+ by +cz= k;

a²x + b² y + c²z= k²

2) tan⁻¹{(1- cosx)/(1+cosx)}.w.r.t.x. 1/2

3) x= eˣ⁾ʸ show xlogx dy/dx= x - y.

4) sin⁻¹(3x- 4x³) w.r.t.x. 3/√(1-x²)

5) If A= 1 4 & X= a and B= -7

5 3 b -1 , the relation: AX= B , find a,b. 17/14,33/14

6) Prove: a² 2ab b²

b² a² 2ab

2ab b² a² is a perfect square.

7) If A= 5 & B= a & C= 1

-1 b 8

2 c 12 and the relation: 2A+ 3B= C, then find a,b,c. -3, 10/3, 8/3

25/6/21

1) Prove: x y z

x² y² z²

x³ y³ z³

= xyz(x - y)(y - z)(z - x)

2) If y= x/2 √(a² - x²) + a²/2 sin⁻¹(x/a), prove dy/dx= √(a² - x²)

3) If y= xʸ , prove x dy/dx= y²/(1-y logx)

4) If x= a cos³t , y= a sin³t. Prove d²y/dx²= 1/3a sec⁴t cosect

5) If y= x/√(1+x²), show that d²y/dx²= -3x/√(1+x²)⁵

6) Given A= 4 1

1 3 Evaluate (A -2I)(A -3I), Where I is the unit matrix of order 2. 3 2

2 1

24/6)21

1) xˣ sin⁻¹√x w.r.t.x. xˣ/{√x(x-1)) + xˣ(1+ logx) sin⁻¹√x.

2) For the matrix 2 1 -1

1 0 3 show that (A')'= A.

3) Solve the following equation using Cramer's rule:

2x+3y-z= 9

x+ y +z= 9

3x - y - z= -1. 2,3,4

4) Determine the metrices A and B if 3A + 2B= 11 11

-3 6

and A - B= 7 2

-1 -3. 5 3 -2 1

-1 0 & 0 3

5) Find dy/dx

A) log(tan x)ˢᶦⁿˣ.

B) - cot²(x/2)- 2 log sin(x/2). Cot³(x/2)

6) If P= 9 1 & B= 1 5

4 3 7 12 and 5P + 3Q + 2R be a null matrix, find the matrix R. -24 -10

-41/2 -51/2

7) tan⁻¹[x/{1+√(1-x²)}] w.r.t.x. 1/{2√(1-x²)}

23/6/21

1) Show that

a a+b a+b+c

2a 3a+2b 4a+3b+2c = a³

3a 6a+3b 10a+6b+3c

2) y= log{eˣ(eˣ +1)}. Show that dy/dx= 1/(1+ eˣ)

3) If A= -1 -2

2 -2 prove that A²+3A+4I= 0

4) If R(t) = cos t sin t

- sin t cos t prove that R(s). R(t) = R(s+t).

5) Solve the following set of equations using cramer's rule: x+ y + z= 6 ; 2x- y + 3z = 9 ; x+ 3y - 2z= 1

6) y= xᶜᵒˢˣ find dy/dx

7) ₑ tan⁻¹x

22/6/21

1) x= ₑ tan⁻¹{(y-x²)/x²}, then dy/dx

A) 2x[1+tan(logx)+ xsec²(logx)]

B) x[1+tan(logx)+ xsec²(logx)]

C) 2x[1+tan(logx)+ x²sec²(logx)]

D) 2x[1+tan(logx)+ sec²(logx)

2) If x= a cos⁴t, y= a sin⁴t then find dy/dx at t= 3π/4 is--

A) 0 B) 1 C) -1 D) -2

3) d/dx (xˣ) is

A) (xˣ)(1- logx). B) (xˣ) logx

C) xˣ⁺¹(1- logx). D) xˣ(1+ logx)

4) ₑx³ w.r.t. log x is..

A) ₑx³ B) 3x³ ₑx³

C) 3x² ₑx³. D) 3x²ₑx³+ 3x²

5) a sin³t w.r.t. a cos³t find d²y/dx² at t=π/4.

A) 2 B) 1/12a C) 4√2/3a D) 0

6) If f(x)=3|x+2| find f'(x) at x=-3 is

A) -3 B) 3 C) 0 D) doesn't exist

7) If y= √[x+√{x+√(x +.....)}] Then dy/dx is..

A) x/(2y-1). B) 2/(2y-1)

C) 1/(2y-1). D) x/(y-1)

8) If y= sinx + eˣ, then d²y/dx²

A) (sinx - eˣ)/(cosx+ eˣ)

B) 1/( eˣ- sinx)

C) (sinx - eˣ)/(cosx+ eˣ)²

D) (sinx + eˣ)/(cosx+ eˣ)³

9) If sin⁻¹x + sin⁻¹y = π/2, then dy/dx is..

A) x/y B) -x/y C) y/x D) - y/x

10) If dx/dy= u and d²x/dy²= v, then value of d²y/dx² is .

A) -v/u² B) v/u² C) -v/u³ D) v/u³

21/6/21

1) If a > 0 and discriminant of ax² + 2bx + c= 0 is negative, then the value of

a b ax+ b

b c bx +c

ax+b bx+c 0 is

A) positive

B) (ac-b²)(ax²+ bx+c)

C) negative D) 0

2) value of 1+a 1 1

1 1+b 1

1 1 1+c is

A) 1+abc+ab+bc+ca

B) abc(1+1/a +1/b + 1/c)

C) 4abc D) abc(1/a +1/b +1/c)

3) If A= 2 -1

-1 2 and I is the unit matrix of order 2, then A² is equal to..

A) 4A - 3I B) 3A - 4I

C) A - I D) A + I

4) The roots of the Equation

x 3 7

2 x -2 = 0 are....

7 8 x

A) -2, -7,5 B) -2,-5,7

C) 2,5,-7 D) 2, 5,7

5) If f(x)= sinx cosx tanx

x³ x² x

2x 1 1 then the value of lim ₓ→₀ f(x)/x² is..

A) -3 B) 3 C) -1 D) 1

6) Inverse of 2 1

7 4 is...

A) 4 -2 B) 4 -1 C) 4 -7 D) -4 -1

-7 -2 -7 2 7 2 7 -2

7) If the system of equations x+2y+3z= 0, 2x+ky+5z= 1, 3x+4y+7z= 1 has no solutions, then-

A) k= -1 B) k=1 C) k=3 D) k=2

8) A is square matrix such that A³= I; then Inverse of A is equal to...

A) A² B) A C) A³ D) none

9) If D= 1 a a²-bc

1 b b²-ca

1 c c²-ab, then D is..

A) 0 B) independent of a

C) independent of b

D) independent of c

10) y x 0

If 0 y x = 0 and x≠0, then

x 0 y Which one of the following is correct?

A) x is one of the cube roots of 1.

B) y is one of the cube roots of 1.

C) y/x is one of the cube roots of 1.

D) y/x is one of the cube roots of (-1).

Thursday, 13 May 2021

REVISION Maths(X) ICSE

Revision (Similarity)

1) A vertical stick 12m long casts a shadow 8m long on the ground. At the time a tower casts the shadow 40m long on the ground. Determine the height of the tower. 60m

2) The perimeter of two similar triangles are 30 and 20cm respectively. If one side of the first triangle is 12cm, determine the corresponding side of the second triangle. 8cm

3) The perimeter of two similar triangles ABC and PQR are respectively 36cm and 24cm. If PQ= 10cm, find AB. 15cm.

4) Two triangles BAC and BDC, right angled at A and D respectively, are drawn on the same base BC and on the same side of BC. If AC and DB intersect at P, prove AP x PC= DP x PB.

5) P and Q are points on sides AB and AC respectively of ∆ ABC. If AP= 3cm, PB=6 cm, AQ= 5cm and QC = 10cm, show BC = 3PQ

6) Two poles of height a metres and b metres are p metres apart. Prove that the height of the point of interaction of the lines joining the top of each pole to the foot of the opposite pole is given by ab/(a+b) metres.

7) In trapezium ABCD AB//DC and DC= 2AB. EF drawn parallel to AB cuts AD in F and BC in E such that BE/EC = 3/4, Diagonal DB intersects EF at G. Prove that 7 FE = 10 AB.

8) A vertical stick 10cm long casts a shadow 8cm long. At the same time a tower casts a shadow 30m long. Determine the height of the tower. 37.5m

9) The perimeters of two similar triangles are 25cm and 15cm respectively. If one side of first triangles is 9cm, what is the corresponding side of the other triangle ? 5.4cm

10) in ∆ ABC and DEF, it is being given AB= 5, BC= 4cm and CA= 4.2cm, DE= 10cm, EF= 8cm and FD= 8.4cm. If AL perpendicular to BC and DM perpendicular to EF, find AL : DM. 1:2

11) D and E are the points on the side AB and AC respectively of ∆ ABC such that AD= 8cm, DB= 12cm, AE= 6cm and CE= 9cm. prove that BC=5/2 DE.

12) DE is the midpoint of the side BC of ∆ABC. AD is bisected at the point E and BE produced cuts AC at the point X. prove that BE: EX= 3:1.

Revision Test (November)

SECTION A (Marks 40)

(Attempt all questions)

1)a) Ravi bought a radio for ₹840(GST at 7%) and a watch for ₹825(GST at 10%). Calculate the total amount. Ravi had to pay to settle the bill.

b) Find the eighteenth term of 9,5,1,......

2)a)

3a) solve the inequation: -2 ≤7- 3x < 1; x belongs to R. Represent the solution set on a number line.

b) ABC is a triangle XY|| to BC which is same side of BC, area of∆ AXY= 49cm², area of trapezium BCYX= 24cm², BC= 10cm, find XY.

4a) A girl of height 90cm is walking away from the base of a lamp-post at a speed of 1.2m/s. If the lamp is 3.6m above the ground, find the length of her shadow after 4 seconds.

b) Given A= 2 0

1 3 evaluate A²+2A.

SECTION B (Marks: 40)

6a) Given x belongs to I, solve the inequation 3≥ (x-4)/2 +x/3 ≥2. Graph the solution set a number line.

b) How many terms of the geometric series 1+4+16+64+ ... will make the sum 5461 ?

7)a) A train travels a distance of 300 km constant speed. If the speed of the train is increased by 5 km an hour, the journey would take 2 hours less. find the speed of the train.

b) Ravi opened a Recurring Deposit Account in a bank and deposited ₹500 per month for 3 years. The bank paid him ₹20220 on maturity. Find the rate of interest paid by the bank.

8a) In an AP, the first term is 2 and the sum of first five terms is one-fourth of the next five terms. Show that its 20th term is -112.

b) i)If A= -1 3 & B= a & C= 10

4 k 5 20

And expression AB= C, find the values of a and k.

ii) If P= -3 1 & Q= -3 2

2 0 1 4 , calculate PQ= Q²

9a)Solve for x: 3x²- 2x- 1=0.

------------------------------------------------------

Revision Test (metrices)

1) If A= 3 2 & B= 5/3 -2/3

6 5 -2 1 then AB= ?

A) BA B) O C) I D) none

2) If A= 4 1

-1 2 then A² - 6A= ?

A) I B) 9 C) -9 D) none

3) If I= 1 0

0 1 then I³ is

A) I B) O C) 2 D) none

4) If A= 2 B= -3 C= 1

3 1 2 and the relation X+A - B= 0. Then matrix X is

A) O B) 2I C) -3I D) none

5) If each element of a matrix is zero, it is called

A) null matrix

B) Unit matrix

C) Identity matrix

D) none

6) If A= 1 2 x & B= 1 -2 y

0 1 0 0 1 0

0 0 1 0 0 1 and AB= I, then x+y is

A) 0 B) -1 C) 2 D) none

7) If A= 1 -1 & B= a 1

2 -1 b -1 and (A+ B)²= A² + B², values of a, b are

A) 4,1 B) 1,4 C) 0,4 D) 2,4

8) If A= a b

c - a is such that A² = I, then

A) 1+ a²+ bc= 0

B) 1- a²+ bc= 0

C) 1- a²- bc= 0

D) 1+ a²- bc= 0

9) 3x+7 5 = 0 y-2

y+1 2- 3x 8 4 then find the values of x, y

A) -1/3, 7 B) 7, -2/3 C) -1/3, -2/5 D) not possible to find.

10) If A= 0 2 and kA= 0 3a

3 -4 2b 24 then the values of k, a, b are

A)-6,-12,-18 B) -6,4,9 C) -6,-4,-9 D)6,12,18

Revision (Metrices)

1) a+2 b -3 = 2a -3 4b+1

-1 3+c -1. 5c-7 then find the value of a, b, c, d

2) If A= 2 x & B= y 0 & C= 8 -6

1 5 3 -1 6 14 and the 3A + B= C then find x, y

3) If A= 1 4 & B= -4 -1

2 2 -3 -2

A) find the matrix 2A+ B

B) find the matrix C such that: C+ B= O

4) If M- 2I= -3 0

12 3 and

5M + 3I= 8 -20

0 -12 Find the matrix M

5) If A= 1 3 & B= 1 2 & C= 4 3

2 4 4 3 1 2 then find

A) (AB)C

B) A(BC)

6) IF A= -2 1 & B= m & C= 5

0 -3 1 n and AB= C then find the value of m, n.

7) If A= -1 1

a b and A² = I, find a, b

8) If A= 0 1 & B= 0 1

1 -2 1 -1 then

Is A² - B²= (A+B)(A-B)

9) If A= 1 1 & B= 1 2

0 2

And the relation MA = B where M is a matrix

A) state the order of matrix M

B) Find the matrix M.

10) If A= p q & B= p

The relation AB= 25 then the value of p, q

11) If A= 1 4 & B= 2 1

1 0 3 -1 verify if IA² + IB² = A²I + B²I.

___________________________________

Write down the first five terms of the sequence, whose nth term is (-1)ⁿ⁻¹. 5ⁿ⁺¹. 25,-125,625,-3125, 15625

2) If the 3rd and 6th terms of an AP are 7 and 13 respectively, find the first term and the common difference. 3 and 2

3) find the sum of all natural numbers between 100 and 1000 which are multiple of 5. 98450

4) how many terms of the AP -6, -11/2, -5,.... are needed to give the sum -25 ? 5 or 20.

5) Determine the sum of the first 35 terms of an AP if a₂ = 2 and a₇ = 22. 2310

6) If the first term of an AP is 2 and the sum of first five terms is equal to one fourth of the sum of the next five terms, show that the 20th term is --112

7) Insert 3 arithmetic mean between 2 and 10. 4,6,8

8) The sum of three decreasing numbers in AP is 27. If -1, -1, 3 are added to them respectively, the resulting series is in GP. The numbers are

A) 5,8,13 B)15,9,3 C)13,9,5 D) 17,9,1

9) The sum of all odd numbers between 1 and 100 which are divisible by 3, is..

A) 83667 B) 90000 C) 83660 D) n

10) If 7th and 13th terms of an AP be 34 and 64 respectively, then its 18th term is.

A) 87 B) 88 C) 89 D) 90

11) If the sum of p terms of an AP is q and the sum of q terms is q, then the sum of the p + q terms will be..

A) 0 B) p-q C) p+q D) -(p +q)

12) If the sum of n terms of AP be n² - n and its common difference is 6, then its first term is..

A) 2 B) 3 C) 1 D) 4

13) Sum of all two digit numbers which when divided by 4 yield Unity as reminder is..

A) 1200 B) 1210. C)1250. D) n

14) In n AM's introduced between 3 and 17 such that the ratio of the last mean to the first mean is 3:1, then the value of n is..

A) 6 B) 8 C) 4 D) n

15) The 1st and last terms of an AP are 1 and 11. If the sum of its terms is 36, then the number of terms will be.

A) 5 B) 6 C) 7 D) 8

16) Find the sum of all odd integers from 1 to 1001. 251001

17) If the ratio between the sums of n terms of two AP is (7n+1):(4n+27) find the ratio of their 11th term. 148: 111

18) If the sum of m terms of an AP be n and the sum of n terms be m, show that the sum of m+n terms is -(m+n).

19) If the sum of n terms of an AP is (pn+ qn²), where p and q are constants, find the common difference. 2q

20) In an AP, the first term is 2 and the sum of first five terms is one-fourth of the sum of next terms. Show that the 20th term is - 112 and the sum of first 20 term is -1100.

21) If the sum of n terms of an AP is given by (3n²+ 4n), find its rth term. 6r +1

22) The digits of a three-digit numbers are in AP and their sum is 15. The number obtained by reversing the digits is 594 less than the original number. Find the number. 852

23) Between 1 and 31, m numbers have been inserted in such a way that the ratio of 7th and (m-1)th numbers is 5:9. Find the value of m. 14

24) In the arithmetic progression whose common difference is non zero, the sum of the first 3n terms is equal to the sum of next n terms. Then the ratio of the sum of the first 2n terms the next to 2n terms is

A) 1/5. B) 2/3 C) 3/4 D) none

25) If four numbers in AP are such that their sum is 50 and the greatest number is 4 times the least, then the numbers are:

A) 5,10,15,20 B) 4,10,16,22

C) 3,7,11,15 D) none

26) The first and the last term of an AP are a and l respectively. if S is the sum of all the terms of the AP. and the common difference is given by (l²-a²)/{k -(l+a)}, then k is

A) S B) 2S C) 3S D) none

27) If the sum of the first n even natural number is equal to K times the sum of the first n odd natural numbers, then k is..

A) 1/n B) (n-1)/n C)(n+1)/2n D)(n+1)/n

28) If the first, second and last term of an AP are a,b and 2a respectively, then its sum is

A) ab/{2(b-a)} B) ab/(b-a)

C) 3ab/{2(b-a)} D) none

29) If x is the sum of an arithmetic progression of n odd number of terms and y the sum of the terms of the series in odd places, then x/y is

A) 2n/(n+1) B) n/(n+1)

C) (n+1)/2n D) (n+1)/n

30) If the first term of an AP is 2 and common difference is 4, then the sum of its 40 terms is

A) 3200 B) 1600 C) 200 D) 2800

31) The number of terms of the AP 3, 7, 11, 15, ... to be so that the sum is 406 is...

A) 5 B) 10 C) 12 D) 14 E) 20

32) If a(1/b+ 1/c), b(1/c + 1/a), c(1/a + 1/b) are in AP , then

A) a, b, c are in AP

B) 1/a, 1/b, 1/c are in AP

C) a, b, c are in HP

D) 1/a, 1/b, 1/c are in GP.

33) If the sum of the three numbers in AP be 18 then what is the middle term ? 6

34) The fifth term and the 11th term of an AP are 41 and 20 respectively. Find the first term. What will be the sum of first 11 terms of the AP. ? 825/2

35) The n-th term of an AP is p. Show that sum of first (2n-1) terms is (2n-1)p.

36) The middle term, of an AP having 11th term is 12. Find the sum of the 11 terms of that progression. 132

37) There are n arithmetic means between 4 and 31. If the second mean : last mean=5: 14 then find the value of n. 8

38) If the sum of the first P terms of an AP be equal to the sum of the first Q terms then show that the sum of the first P +Q terms is zero.

39) Find the sum upto n terms of the series 1²- 2²+ 3²- 4²+ 5²- 6²+.. .. -n/2 (n+1) (n= 2r)

40) if the sum of p terms of an AP is to the sum of q terms as p²:q², show that (pth term)/(qth term)= (2p-1)/(2q-1).

40) The first term of an AP is a, the second term is b and the last term is c. Show that the sum is {(a+c)(b+c-2a)}/{2(b-a)}.

41) The sides of a right angled triangle are in AP. if the smallest side is 5cm then find the largest side. 25/3

42) find the sum of natural numbers from 1 to 200 excluding those divisible by 5. 16000

43) Show that the sum of all odd numbers between 2 and 1000 which are divisible by 3 is 83667 and of those not divisible by 3 is 166332.

44) Find the 14 A. M which can be inserted between 5 and 8 and show that their sum is 14 times the Arithmetic mean between 5 and 8.

45) Divide 25/2 into five parts in AP, such that the first and the last parts are in the ratio 2: 3. 2,9/4,5/2, 11/3, 3.

46) For what value of m, the sequence 2(4m+7), 6m + 1/2, 12m-7 forms an AP. -3/4

47) Find the 20th term of the AP 80, 75, 70,... Calculate the number of terms required to make the sum equal to zero. 33

48) Prove that if unity is added to the sum of any number of terms of the AP 3, 5,7,9...the resulting sum is a perfect square.

49) The sum of n terms of the series 25, 22, 19, 16,.. is 116. Find the number of terms and the last term. The given series is AP. 18405

50) Find the sum of all natural numbers from 100 to 300:

a) which is divisible by 4. 10200

b) excluding those which are divisible by 4. 30000

c) which are exactly divisible by 5.

d) which are exactly divisible by 4 and 5. 8200, 2200

e) which are exactly divisible by 4 or 5. 16200

1/8/21

1) Amit deposited ₹150 per month in a bank for 8 months under the recurring deposit scheme. What will be the maturity value of his deposits, if the rate of interest is 8% p.a and interest is calculated at the end of every month ? ₹1236

2) Lakshmi took a Cumulative time deposit account of ₹240 per month at 10% p.a. she received ₹3840 on maturity. Find the period for this account. 15 months

3) Manoj opened a recurring deposit account in a bank and deposited ₹500 per month for 3 years the band paid him ₹20220 on maturity. Find the rate of interest paid by the bank. 8%

4) Rajeev opens a Recurring deposit account with the bank of Rajasthan and deposited ₹600 per month for 20 months. calculate the maturity value of this account, if the bank pays interest at the rate of 10% per annum. ₹13050

5) Miss Anshu Pandey deposited ₹350 per month for 20 months under recurring deposit scheme. Find the total amount payable by the bank on maturity of the account if the rate of interest is 11 % per annum. ₹7673.75

6) Mrs. Matthew opened a recurring deposit account in a bank with ₹500 per month for 5/2 years. find the amount she will get on maturity if the interest is paid on monthly balance at 12.5% per annum. 17421.87

7) calculate the amount received on maturity of a recurring deposit of ₹150 per month for 1 year 6 months, if the rate of interest is 11% per annum. 2935.13

8) Amar deposits ₹1600 per month in a Recurring deposit for 3 years at the rate of 9% p.a. simple interest, find the amount Amar will get at the time of maturity. 65592

9) A Recurring deposit account of ₹1200 per month has a maturity value of ₹12440. If the rate of interest is 8% interest and the intrest is calculated at the end of every month, find the time (in months) of the Recurring deposit account. 10 months

10) Sujata deposited a certain sum of money, every month, for 5/2 years in cumulative time deposit account. At the time of maturity She collected ₹4965. if the rate of interest was 8% p.a. find the monthly deposit. 150

11) Sumit paid ₹300 per month in cumulative time deposit account for two years. he received ₹7875 as the maturity amount. find the rate of interest. 9%

12) Meena has cumulative time deposit account of ₹340 per month at 6% per annum. if she gets ₹7157 at the time of a maturity, find the total time for which the account was held. 20 months

13) on depositing ₹200, every month in a cumulative time deposit account, paying 9% p.a. interest, a person collected ₹2517 at maturity. find the period. 12 months.

14) Mr. Desai opens a recurring deposit of ₹2000 per month for 30 months paying simple interest of 12%p a. Calculate the amount he received at the time of maturity. 69300

15) Calculate the amount receivable on maturity of a recurring deposit of ₹800 every month for 5 years at 11 % per annum.

16)

27/7/21

1) Roots of a quadratic equation are 1/2 and -14. Find the equation. 2x²+ 27x -14= 0

Solve::

2) √(3x²+x+5)= x-3. -4, 1/2

3) 8(t²+1/t²)- 42(t- 1/t)+29=0. 15/4, 3/2

4) 5ˣ⁺¹ + 5²⁻ˣ = 126.

5) 3²ˣ - 10.3ˣ+ 9=0. 2,0

6) 2²ˣ⁻¹ - 9. 2ˣ⁻²+1 = 0. 2, -1

7) 6x² - x -14= 0.

8) x² -8x -1280 = 0

9) 1/(2y-9) = 1/(y-3) + 4/5.

10) x⁵ +242= 243/x⁵. -3,1

**Correct up to 2 decimal places.

11) x² -6x -16 = 0. -1.90, 7.90

12) 2x² +11x -10 = 0.

13) The bill of a party for a certain number of people is ₹19200. If there were 10 more persons, the bill each person had to pay would have reduced by ₹160. Find the number of people at the party. 30

14) A two digit number is such that the product of digits is 12. When 9 is added to the number the digits are interchanged. Find the number.

15) The sides of a right angled triangle are x cm, 4(x+1)cm, (4x+5) cm. Find x.

16) A man purchased some sheep for ₹4500. Three sheep were lost and the rest he sold for ₹30 more per sheep than he had paid. If his gain on the whole transaction is 8%, how many sheep did he buy?

17) the sum of the ages of a man and his son is 46 years and the product of their ages is 168 years. find the age of the Son.

18) The total surfaces area of a cylinder is 75.24cm² and its height is 3.6 cm. If its radius is x cm, find x

***Find the nature of the roots.

19) 4x² - 4x+1=0

20) If m,n are roots of x²- px+q= 0, find

a) m²+n². p²-2q

b) m³+ n². p³ -3pq

c) m-n. -√(p²-4q).

d) m⁴+ n⁴. p⁴- 4p²q+2q²

*** Solve:

21) 3x² - x- 7 = 0. 1.70, -1.37

22) 1/(x+1) + 2/(x+2)= 4/(x+4). 2± 2√3

23) 2x² + √7x -7= 0. √7/2, -√7

24) √3 x²+ 10x - 8√3= 0. 2√3/3, -4√3

25) (x+3)/(2x+3)= (x+1)/(3x+2). -3± √6

26) (x-2)/(x+2)+ (x+3)/(x -2)= 4. ±2√3

27) /(x+1)+ 2/(x+2)= 4/(x+4). 2± 2√3

28) a(x²+1)= (a²+1)x, a≠ 0. a, 1/a

29) 4x² - 4ax +(a² - b²)= 0. (a±b)/2

30) x/(x+1)- 4/(x+2)=0 3.24,-1.24

31) 5/(x -1)+ 2x/(x- 2)=0 1.61,-3.11

32) 2x - 1/x= 7. 3.64, -0.138

33) 2/(x -1)+ 3/(x+2)= 4/(x+2). 0.23, -8.77

34) (x+3)/(x-3) - (1-x)/x = 17/4. 4, -2/9

1) (x+3)/(x-3) - (1-x)/x = 17/4. 4, -2/9

35) a/(ax-1) + b/(bx-1)= a+ b, a+b≠0, ab≠0. (a+b)/ab, 2/(a+b)

Form the Quadratic equations whose roots are:

36) a) -2, 1. x²+x-2= 0

b) -3, -4. x²+7x +12=0

c) a,-b. x²-(a-b)x -ab=0

d) -2/3, 4/5. 15x²-2x -8=0

e) -3, 2/5. 5x²+13x -6=0

f) 2/5, -1/2. 10x²+x - 2=0

37) An Aeroplane travelled a distance of 480 km at an average speed of x kmhr. On the return journey, the speed was increased by 40 km/hr. Write down an expression for the time taken for

a) the onward journey

b) the return journey

If the return journey took 30 minutes less than the onward journey, write an equation in x and find the value of x. 160

38) Car A travels x km for every litres of petrol, while car B travels (x+5) km for every litres of petrol.

a) write down the number of litres of petrol used by car A and car B in covering a distance of 400 km.

b) If car A uses 4 litres of petrol more than car B in covering the 400 km, write down an equation in terms of x and solve it to determine the number of litres of petrol used by car B for the journey. 16 litres

39) In an auditorium, seats were arranged in rows and columns. The number of rows was equal to the number of seats in each row. when the number of rows was doubled and the number of seats in each row reduced by 10, The total number of seats increased by 300. Find

a) the number of rows in the original arrangement.

b) the number of seats in the auditorium after rearrangement. 30, 1200

40) A Hotel bill for a number of people for overnight stay is ₹4800. If there were four people more, the bill each person had to pay would have reduced by ₹200. find the number of people staying overnight. 8

41) A trader boy x articles for a total cost of ₹600.

a) write down the cost of one article in terms of x.

if the cost per article were ₹5 more, the number of articles that can be bought for ₹600 would be 4 less.

b) Write down the equation in x for the above situation and solve it for x. 24

42) The distance by road between two towns A and B is 216 km, and by rail it is 200 km. A car travels at a speed of x km/hr and the train travels at a speed which is 16 km/hr faster than the car. calculate:

a) the time taken by the car to reach town B from A, in terms of x

b) If the train takes two hours than the car, to reach town B, obtain an equation in terms of x, and solve it.

c) Hence, find the speed of the train. 52 km/hr

43) A train covers a distance 600 km/hr. Had the speed been (x+20) km/hr, the time taken to cover the distance would have been reduced by 5 hours. write down an equation in terms of x and solve it to evaluate x. 40

44) x/(x+1) + (x+1)/x= 34/15, x≠ 0, x≠ -1. 3/2, -5/2

45) (x+3)/(x -2) - (1- x)/x= 17/4, 4, -2/9

46) 4/x - 3 = 5/(2x+3) -2, 1

47) 2x/(x -3)+ 1/(2x+3)+ (3x+9)/{(x-3)(2x+3) =0. -1

TEST PAPER

-----------------

Section - A

___________

(Answer all the questions from this section). (40 marks)

1) (1x10)

a) Amarnath purchased a cycle for ₹1664 including GST. If the list price of the cycle is ₹1600, then the rate of sales tax is..

A) 4% B) 5%. C) 6%. D) 7%

c) the Quadratic equations whose roots are -3, -4

A) x²+7x+12=0 B) x²-7x+12=0

C) x²+7x-12=0 D) x²-7x-12=0

f) Sum of the roots of the Equation x² - 5x +6

A) 5. B) -5 C) 6 D) -6

g) Number of roots in Quadratic equations are

A) 1 B) 2 C) 3 D) 4

h) a is less than b, written as a< b, if and only if b - a is

A) positive B) negative C) both positive D) both negative.

i) A dealer from West Bengal sells goods worth ₹50000 to a dealer in Bihar at 18% GST. Then seller collect ₹9000 as IGST and the entire amount will go to the..

A) Central Government only

B) State Government only

C) both A and B

D) none

3) Solve: 3x²- x-7= 0. Correct upto 2 decimals. (6)

4) A shopkeeper buys an article whose printed price is ₹4000 from a wholesaler at a discount of 20% and sells it to a consumer at the printed price. If the sales are intrastate and the rate of GST is 12%, find

A) the price of the article inclusive of GST at which the shopkeeper bought it

B) the amount of Tax (under GST) paid by the shopkeeper to the State government.

C) the amount of Tax(under GST) received by the Central Government.

D) the amount which the consumer pays for the article. (6)

5) Given A={x: 5x-4≥6, x∈ R} and B={x:5-x > 1, x ∈ R}

Represent A and B on the number line.

Find a) A∩ B b) A' ∩ B. (6)

SECTION -- B

Attempt any 4 (40 marks)

7) a) using Quadratic formula, solve: 5 + 11x - 5x². (5)

8) a) Kriss goes to a shop to buy a leather coat costing ₹654. The rate of GST is 9%. He tells the shopkeeper to reduce the price to such an extent that he has to pay ₹654, inclusive of GST. Find the reduction needed in the price of the coat. (5)

b) a(x²+1)= (a²+1)x, a≠0. (5)

b) Find the smallest value of x which satisfies the inequation 2x + 7/2 > 5x/3 +3, x ∈I. (5)

10a) Ram paid ₹345.60 as GST on purchase of ₹3840. Find the rate of GST. (2)

c) 2x - 1/x= 7. (2)

11)a)

b) 2/(x-1) + 3/(x+1) = 4/(x+2). (5)

8/7/21

7/7/21

1) The catalogue price of a colour T. V is ₹24000. The shopkeeper gives a discount of 8% on the listed price. He gives a further off-season discount of 5% on the balance. But GST at 10% is charged on the remaining amount. Find:

A) the GST a customer has to pay.

B) the financial price he has to pay for the colour TV. 2097.60, 23073.60

3) 2x -5≤ 5x+4<11, x belongs to R.

4) find the remainder if 4x³ - 6x²+7x -2 is divided by x - 1/2. 1/2

5) X: 13 15 18 20 22 24 25

F:. 6 4 11 9 16 12 2 find a) median b) lower quartile c) upper quartile d) semi-interquartile. 21, 18, 22, 2

6) If the m th term of an AP is a and its n th term is b, Show that the sum of its m+ n terms.

6/7/21

1) By remainder theorem show that 5x³+2x²-13x+6 is divisible by x+2.

2) Given A= 30 and B= 60 verify sin(B-A)= sin B cos A - cos B sin A.

3) Given 2 tanx= 5 find (3sinx - 4 cosx)/(sinx + 4cosx). 7/13

4) The sum of the three consecutive terms of an AP is 21 and the sum of the squares of these terms is 165. Find these terms. 4,7,10

5) If the sum of n terms of an AP is 3n²+ 4n, find its rth term. 6r+1

6) Following table gives the basic salaries of person employed in an office.

Salary no. Of employees

200-300 11

300-400 10

400-500 15

500-600 8

600-700 4

A) using above information Calculate cumulative frequencies of the employees.

B) draw the ogive

C) estimate the median. 420

7) Mr. X purchased a cycle for ₹25488, which includes 10% rebate on the marked price and 18% tax on the remaining price. Find the marked price of the cycle. 24000

5/6/21

1) find the sum of all numbers between 100 to 300 which are divisible by 8. 5000

2) Find the remainder when 3x³ - 9x +4 is divided by x-1. -2

3) Find the mean, median and mode of 3,7,10, 6,9, 5, 7, 5, 16, 7. 7.5, 7, 7

4) solve: x + 1/5< 4x/3+ 8/15≤ x/5 + 5/3, x belongs to R.

5) find median

Class Frequency

00-10 7

10-20 9

20-30 13

30-40 25

40-50 16

50-60 10

6) class frequency

0-5 4

5-10 a

10-15 8

15-20 5

20-25 3 if mean is 34/3 find a.

4/6/21

1) A V. C. R is marked for sale ₹ 14280 inclusive of GST at 12%. Calculate GST in rupees. 765,765

2) Find the remainder when 2x³+ 3x² - 2x +3 is divided by x+2 with the help of factor theorem. 3

3) Solve: -8/3≤ x + 1/3< 10/3; x belongs to R. -3≤ x < 3

4) √{(1-sinA)/(1+ sinA)}= cosA/(1+ sinA)

5) class frequency

0-8 3

8-16 5

16-24 7

24-32 4

32-40 2 find mode. 19.2

6) If the 10th, 28th and last term of an AP are respectively 29, 83, 122. Find its first term, common difference and number of terms. 2,3,41

1/7/21

1) Draw the histogram and hence the mode for the following:

Class frequency

00-10 2

10-20 8

20-30 10

30-40 5

40-50 4

50-60 3. 23

2) calculate the mean, the median and mode of 6,3,6,5,4,6,8,4,5,3.

3) Student English maths

A 30 26

B 60 80

C 35 33

D 62 68

E 47 44

F 60 85

G 47 44

H 28 72

I 64 65 find median.

4) Given x belongs to I, Solve the inequation 3≥ (x-4)/2 + x/3 ≥ 2.

5) Class Frequency

00-05 2

05-10 7

10-15 18

15-20 10

20-25 8

25-30 5 find mode

6) Class Frequency

05-10 10

10-15 6

15-20 4

20-25 12

25-30 8

30-35 4

35-40 2

40-45 1

45-50 3 find median

30/6/21

1) cosA/(1- tanA) + sinA/(1- cotA) = cosA + sinA.

2) Find the value of k, given that 3x³ + 4x³ - 6x + k is divisible by x+1.

3) When ax²+ bx - 6 is divided by (x-1), (x+1), the remainder are -10, 4. Find a,b.

4) find: sin48/cos42 + (sin29 + cos61)(sin29- cos61).

5) Simplify: 2/5 ≤x - (1+2/5)<4/5, x belongs to R and graph the solution set on a real number line.

6) Find the remainder when 5x³+ 8x² -2x-9 is divided by x+2.

7) simplify: (sin63 + cos17)(sin50 - cos 40).

8) solve: -2≤7- 3x<1: x belongs to R

9) √{(1-cosx)/(1+cosx)}= cosecx- cotx

29/6/21

1) If tanx= 5/4, find the value of sin²x - cos²x. 9/41

2) Kamta purchased a cycle for₹1664 including GST. If the list price of the cycle is ₹1600, find the rate of GST. 4%

3) Find the values of m if (x-m) is a factor of x² + my -18. ±3

4) Solve: 3 ≥ (x-4)/2 + x/3 ≥2, x belongs to R. 4.8≤x ≤ 6

5) Find the remainder when 2x³ - 6x +4 is divided by x+3. 40

6) Find the mean, the median and the mode of the following data: 7,4,6,4,5,8,9,11,10,15,4,3. 7,6.5,4

8) Find the mode

Class interval frequency

0-5 2

5-10 2

10-15 18

15-20 10

20-25 8

25-30 5 16

9) Salary (₹) no.of people

400 10

600 8

800 6

1000 10

1200 10

1500 6

Find the mean salary. ₹892

28/6/21

1) Simplify: -31/3 < 5y/3+ 3 ≤ y/2+ 16/3, y belongs to R. -8<y≤2

2) prove x+1 is a factor of 3x³+5x² - 6x -8.

3) If the mean of 10 Observation is 20 and that of another 15 observation is 16, find the mean of all 15 observation. 17.6

4) Shikha bought a calculator for ₹1026, which includes 5% rebate on the marked price and then 20% GST on the remaining price. Find the marked price of the calculator. 900

5) Show that 3x -1 is a factor of 6x² + 7x -3. 1

6) Solve: 21/x² - 29/x -10= 0. -7/2,3/5

7) weight No.of students

30-35 4

35-40 16

40-45 40

45-50 22

50-55 10

55-60 8

Find median. 44

8) Prove: cotx - tanx= (2cos²x -1)/(sinx cosx).

9) Class Frequency

00-10 12

10-20 16

20-30 6

30-40 7

40-50 9

Find mean. 22

27/6/21

1) Prove: (cotx -1)/(2-sec²x)= cotx/(1+tanx)

2) Manisha has a recurring deposit account for 2 years at 10% p.a. if she receives ₹1900 as intrest, find the value of the monthly installment paid by her. 760

3) Find the mean of 25 observation, if the mean of 15 of them is 18 and the mean of remaining ones is 13. 16

4) Find the smallest value of x, which satisfies the inequation 2x+ 7/2> 5x/3 + 3, x belongs to I. -1

5) Find the remainder when 2x³+7x²-x+1 is divided by x+1. 7

6) What is the rate of GST levied on an article that was sold at a price two and half times its marked price? 150%

9) Find the median

Wages No of workers

4000-4400 8

4400-4800 12

4800-5200 20

5200-5600 25

5600-6000 17

6000-6400 10. 4800

10) calculate the mean, the median and mode of the numbers:

3,2,5,4,1,7,2,5,4,2. 3.5, 3.5, 2

11) Prove: (1-sinA)/(1+sinA)= (secA- tanA)².

26/6/21

Marks No. Of students

0-9 5

10-19 9

20-29 16

30-39 22

40-49 26

50-59 18

60-69 11

70-79 6

80-89 4

90-99 3

A) find median. 42.6, 10

B) the number of students who obtained more than 75% marks.

2) Find the mean of the following:

X: 200 300 400 500 600 700

F: 5 11 10 10 8 6 446

3) The price of a washing machine inclusive of GST at 9% is ₹10028. Find its marked price. If the sales tax is increased to 14%, How much more does the customer pay for the washing machine? 9200

4) Using remainder theorem, find the remainder when 5x² - 4x -1 is divided by (2x-1). -7/4

5) If 12 tanx= 5, then 13sinx-5 is. 0

6) Show that (x-5) is a factor of x³- x² -17x-15.

7) The mean of five numbers is 18. On excluding one number, the mean becomes 16. Find the excluded number. 26

25/6/21

1) The marks scored by 40 pupils of a class in a test were as follows:

X: 0 1 2 3 4 5

F: 2 4 5 14 11 4

Calculate mean mark. 3

3) Using the Step-deviation method, find mean:

Class frequency

50-60 9

60-70 11

60-70 10

80-90 14

90-100 8

100-110 12

110-120 11 85.8

4) Draw the histogram and hence the mode for the following:

Class frequency

00-10 2

10-20 8

20-30 10

30-40 5

40-50 4

50-60 3 23

5) Find the remainder when 7x² - 3x +8 is divided by x-4. 108

6) If x belongs to integer, find the solution set for the inequation 5<2x - 3 ≤ 14 and graph it

24/6/21

1) Mamta has a cumulative time deposit account in a bank. She deposits₹800 p.m and gets ₹15198 as maturity value. If the rate of interest be 7% p.s. find the total time for which the account was held. 18 months

2) The marks scored by 40 pupils of a class in a test were as follows:

X: 0 1 2 3 4 5

F: 2 4 5 14 11 4

Calculate mean mark. 3

3) solve: 2 ≤ 2x -3 < 5, x belongs to R and mark it on the number line. 5/2≤x< 4

5) Using the remainder theorem, find the remainder when 7x³+ 5x² - 4x - 1 is divided by x+1. 1

6) If the price of an Almirah including GST is 7884. If its marked price is₹7300, find the rate of GST. 8%

8) Using the Step-deviation method, find mean:

Class frequency

50-60 9

60-70 11

60-70 10

80-90 14

90-100 8

100-110 12

110-120 11 85.8

9) Prove: (1+cosA)/(1-cosA)= (cosecA + cotA)².

23/6/21

1) Find the remainder when 7x² - 3x +8 is divided by x-4. 108

2) If x belongs to integer, find the solution set for the inequation 5<2x - 3 ≤ 14 and graph it on a number line.

3) Find the values of p and q if g(x)= x+2 is a factor of f(x)= x³ - px + x + q and f(2)=4. 9/2, 2

4) From the following, find

A) The average wage of a worker, Give your answer, correct to the nearest paise.

B) Medain

Wages in ₹. No of workers

Below 10 15

Below 20 35

Below 30 60

Below 40 80

Below 50 96

Below 60 127

Below 70 190

Below 80 200. 44.85

6) Prove: √{(1+cost)/(1-cost)= cosect + cot t

22/6/21

1) Find the remainder when 2x³ - 3x² + 7x -8 is divided by x-1. -1

2) Find the value of the constants a and b, If (x-2) and (x-3) are both factors of x³ + ax² + bx - 12. 3,-4

3) The catalogue price of a colour TV is ₹24000. The shopkeeper gives a discount of 8% on the listed price. He gives a further off-season discount of 5% on the balance. But, GST at 10% is charged on the remaining amount. Find:

A) GST amount a customer has to pay. 2097.60

B) The final price he has to pay for the TV. 23073.60

4) x² - 10x +6= 0 Correct to decimal places. .. 9.36, 0.64

5) The distance by road between two towns A and B is 216Km, and by rail it is 208 km. A car travels at a speed of x km/hr, and the train travels at a speed which is 16km/hr faster than the car. calculate:

A) time taken by the car to reach town B from A. in terms of x.

B) time taken by the train to reach town B from A, in terms of x.

C) If the train takes 2 hours less than the car to reach Town B, obtain an equation in x and solve it. 36

D) Hence find the speed of the train. 52 km/hr

6) A trader buys x articles for a total cost of ₹600.

A) write down the cost of one article in terms of x.

If the cost for article were ₹5 more, the number of article that can be bought for ₹600 would be four less.

B) write down the equation in x for their work situation and solve it for x. 24

7) An aeroplane travelled a distance of 400km at an average speed of x km/hr. On the return journey, the speed was increased by 40km/hr. Write down expression for the time taken for:

A) The onward journey

B) The return journey if the return journey to 30 minutes less than the onward journey

If the journey took 30 minute less than the onward journey, write down an equation in x and find its value. 160

8) The work done by (x-3) men (2x-1) days and the work done by (2x+1) men in x+4 days are in the ratio of 3:10. Find the value of x. 6

Tuesday, 11 May 2021

QUICK (SIMPLIFICATION) ADD+ SUB

1) 987 - 256 . 731

2) 824 - 587. 237

3) 9217 - 858. 8359

4) 934 - 286+ 847. 697

5) 3542- 7156 - 2188 +6843. 1041

6) 53629 + 39431 +8068- 6543 + 6129. 100714

7) 7849- 2731- 4132 -1763. -777

8) 103 +607- 1146 +13846. 13410

9) 99786 -5584- 934- 88 - 9. 93171

10) 3963 - 2743 +4671- 5136. 755

11) 89561- 91436+ 9148 -3639. 3634

12) 35480+ 81563- 41341 - 51464. 24166

13) 8163+ 4769 +213 -687 -3691. 8767

14) 1971 -1636 -2148 +6136. 4323

15) 2967 - 5861 +2346 +3184. 2636

15) 86003 +3749- 41078 +3851 - 2136. 50389

16) 5671 - 2136 - 2449 -4167. -3081

17) 5136 - 3249 +6784 -2931. 5740

18) 5784- 8448 +9146- 4138. 2344

19) 48936+ 36849 -63788+ 13526 + 21673. 57196

19) 3547- 6341+ 8149 -3671. 1684

20) 8163 - 13270+ 6984+ 4986. 577

21) 65888- 13739+ 80560 -70061. 62648

22) 3780 6321 8163 3900 1680 8163 3684 4192 3164 4570 12367 31600 7250 6730 38740 3344 4433 7788 8877 9669 6338 40008 5168 8678 1529 3149 6981 12368 12538 8649 40523 3753 23490 371 1648 583 2136 5320 2147 3584 6149 8193 496 6943 8853 5329 1148 6839 9343 144 8996 3193 5143 6358 7164 2936 5943 3784 631 947 546 834 1131 136 8888 6688 7777 5937 6660 5163 3967 51 8169 7143 8232 490 372 344 466 289 183 81 7632 249 2463 3143 563 8123 426 9150 621 749 7568 213 4231 264 5457 4215 612 3649 824 7100 642 42500 6748 372 91 2648 1056 544 2149 268 6279 8290 4673 5807 7516 8975 57 97 380 22 645 435 668 843 527 763 338 947 339 746 440 835 520 611 12624 15285 16234 19724 16318 386 738 920 848 523 3268 7327 515 674 5545 9587 6372 3849 1684 4273 6748 4372 1850 9204 7373 884 7368 2846 5330 9878 3380 7295 8275 6845 5300 1782 3369 4320 7360 965 9999 2345 1230 3456 123 8250 317 964 3489 6347 5332 3856 4972 5272 443 780 992 856 738 549 700 123 8243 9963 4580 3333 10238 5792 1601 8780 3312 2298 5580 9963 16385 902 5700 6347 9552 8263 7885 438 10084 3730 923 692 IX 6565 8282 9393 319 63 3256 4163 892 1505 3940 8592 6834 3845 7263 2723 428 306 742 96.3 640 3378 9309 8247 190 7581 1636 4525 3.33 8294 647 8680 740 7263 9309 844 569 282 464 7263 993 856 2757 32060 580 17090 378 1329 3496 6300 7953 582 772 840 569 522 437 6798 7520 918 3451 692 15623 8452 19266 24075 937 3582 7660 9247 8325 1712 1584 1926 2323 4565 666 965 7649 1865 9273 1584 1975 2362 1848 6797 57 56789 12345 98765 7654 223 784 906 standard 38 638 405 716 7515 6820 9216 642 881 360 75 943 272 3643 3959 7231 248 1700 condition error 476 442 6259 759 354 752 928 336 440 672 893 5640 3753 1254 342 6123 44119 33270 6247 40 1532 6123 31407 11111 315 60 1782 33122 6134 2134 6132 6319 1237 6877 143 121 12 21212 12346 31835 10023 12 234 6124 3122 4197 6380 1290 2345 8753 213 6178 4183 2127 61882 13576 3129 88045 6163 1642 30 8943 6100 8340 2976 32134 617 178 789 895 63129 400 4216 8563 2109 632 4196 3123 3190 526 6313 4215 5739 6123 37122 14216 3127 7795 4385 9796 7863 31264 51423 12496 31240 to 12978 613 2148 3100 824 31000 824 60 1321 80 1429 312 3314 31423 6129 12121 318 213 617 212 885 642 21345 31216 230 3892 60 1243 5678 7789 8819 1012 2340 123 89101 234 5678 3892 213 612 663 1221 9768 534 12 1289 2340 8120 3876 5455 6123 1287 61853 2344 3126 123 3888 3118 6123 1294 7123 9126 6164 3132 1993 8898 1230 4143 421 1248 9631 3345 6166 2854 6432 7887 9436 63 8263 1263 40 1263 e40 1865

QUICK ADDITION

1) 342 +557 +629 +746+ 825.

A) 3099. B) 4009 C) 3909 D) 4339

2) 6965 +3246+ 1234+ 9847+ 8238.

A) 29650 B) 29530. C) 30530 D) n

3) 1598+ 5423+4627+7953 + 8675.

A) 27726 B) 27676 C)28276. D) n

4) 53421+ 67586+ 81538+ 69356.

A) 271901. B) 281901 C) 371902 D) n

5) 29388+51462+79863+ 81345.

A) 34208 B)242058. C) 142058 D) n

6) 28654+ 56120+ 38561 +93268.

A)216623. B)316623 C)416623 D) n

7) 96345+ 35671+ 68592+ 86432.

A)387040 B)287040. C)27040 D) 378040

8) 30508+ 38563+ 41634+ 86481.

A)197186. B) 9986716 C)996516 D)n

9) 9152+ 6843+ 2461+ 3008.

A)21464. B)31464 C) 21144 D) n

10) 2222 +3333+ 8888 +9999.

A)24442. B)3442 C) 34442 D) n

11) 8563+ 3947+ 5760 +5691.

A) 23961. B) 39621 C)43361 D)n

12) 2248+ 3167 +4385 +9158 +3749.

A)22707. B)33707 C)22077 D) n

13) 18916+ 38431+ 29538+ 45691.

A)132576. B)44576 C) 232576 D) n

14) 28637 +47329 +29436 +63543.

A) 168245 B)168345 C)169245. D)n

15) 2931+4367+8139+4856+6934.

A)27227. B)22277 C)27777 D) n

16) 5871 +3973 +4869+ 3654+ 8191.

A) 26558. B)25568 C)25658 D) n

17)3381+4673+8629+2736+4856.

A)24275. B) 34275 C) 23275 D) n

18)4967+8654+2167+3949+2763.

A) 32500 B)22500. C) 23500 D) n

19) 4768 + 3967 +5431+ 3670.

A)17836. B) 17936 C) 18936 D)n

20) 680+ 3056 +4109 +3008.

A)10853. B)10583 C) 18583 D) n

21)3546+2939+4867+6349+5137.

A)22838. B) 22938 C) 22338 D) n

22)2936+5138+4763+2458+8134.

A) 33329 B) 32439 C)23429. D) n

23)3157+2361+ 4761+5869+ 3148

A) 22396 B) 18296 C)19296. D) n

24)4327+6334+5886+4675+6751.

A) 28973 B)27973. C) 28983 D) n

25)5639+3946+7154+3761+9159.

A)29659. B) 20659 C) 28659 D) n

26)8765+7654+6543+5432+4321.

A) 32175 B)32715. C) 49755 D) n

27)5637+4129+3786+6149+4764.

A) 23365 B) 24465 C)24465. D) n

28)1738+8162+3954+5146+3980.

A)22980. B)22890 C) 21880 D) n

29) 642 +513+ 675+ 963.

A)2793. B) 2973 C) 2773 D) 2783

30)1325+2438+3612+4753 +6540.

A)18668. B) 19668 C) 20668 D) n

31)2342+3652+7465+6328 +8157.

A) 26945 B)27944. C) 28954 D) n

32)3425+ 564+6512 +7415 +1057.

A)18973. B) 22973 C) 21973 D) n

33) 18181818+ 181818+ 1818+18.

A) 1945472 B) 219765 C)18365472. D) 9232322

34)2184+4567+3546+2468+7531.

A) 20296. B)30296 C)75312 D) n

35) 412 +342 +564 +827+ 615.

A) 2760. B) 7620 C) 6720 D) none

36)4782+396+ 4436+ 6118+ 2631.

A)18345. B) 23445 C) 33445 D) n

37)793 +467 +307 +253 +885+ 848

A) 3353 B) 5553 C) 3553. D) 3533

38)356 +496+ 836 +779+ 589 +295

A)3351. B) 3551 C) 5531 D) 2151

39)3268+7327+8515 +2674+5545.

A) 12344 B)27324. C) 55654 D) n

40)3127+7213 +1372+2371+3217.

A) 13400 B)17300. C) 23000 D)n

41)7329 +3563+8472+9245+2967.

A)31576. B) 41516 C) 616161 D) n

42)3201+7873+5279 +4364+ 6894.

A)27611. B) 35111 C) 54211 D) n

50) 73923+ 80453+ 19685+ 23889.

A)197950. B)219750 C)564320 D)n

51) 4324 +7288 +9346 +6289.

A) 54327 B) 54337 C)27247 D) n

52)889 +2375+3742+ 1036+ 7781.

A)15823. B) 25823 C) 14823 D) n

53)354+ 631+ 248 +139+ 9765.

A)11137. B) 21117 C) 11111 D) n

54)2203 +3202+2302+2032+ 3022.

A) 12761. B) 12671 C) 12661 D) n

55) 1268+ 979+ 3244+ 1873+ 65.

A)7429. B) 4729 C) 7439 D) 7559

56) 965+ 1237 +89554+96.

A)91852. B) 99852 C) 99952 D) n

57) 374+923+884+423+179.

A) 7282 B)2783. C) 2784 D) 2785

58)4837+6255+9623+8515 +5962.

A)35192. B) 33192 C) 45192 D) n

59)8729+7184+3345+4764 +5253.

A)29275. B) 5555 C) 35455 D) n

60)6565+1113+8247+7617 +1593.

A) 35135 B) 25131 C) 51355 D) n

61)1536+8577+2345+6666+2936.

A)22060. B) 32330 D) 45600 E) n

62)6431+2194+3312+6317+123+ 2401.

A) 20778. B) 30778 C) 20878 D) n

63) 15632 +69146+ 42913 +63129

A)190820. B) 244920 C) 1900820 D)n

64) 63512+ 9123+ 62 +3146+ 331.

A) 76174. B) 56431 C) 65244 D) n

65) 3456 +4567 +5678+ 7890.

A) 21671 B) 21591. C) 76511 D) n

66)1614+2377+6412+3914+ 6152.

A)20469. B) 45679 C) 87799 D) n

67)356+453+414+ 637+ 984+ 331.

A)3175. B) 6176 C) 3177 D) 4135

68) 3456 +8852 +3312+ 6819.

A)22439. B) 88759 C) 99999 D)n

69) 352+ 457 +639+ 783 +1023.

A) 4598. B)3189 C) 4198 D) 6598

70) 3485+6123+8769+1723+ 2443.

A)22543. B) 56453 C) 67533 D) n

71) 8752 +6197+ 31872 +12964.

A)59785. B) 52315 C) 55345 D) n

72) 3652+ 9312+ 2213 +2163.

A) 17340. B) 41320 C) 18440 D) n

73) 8742+ 8957+ 6394 +3187.

A) 27280. B) 27750 C) 67580 D)n

74) 61 +342+ 366 +3842+ 33192.

A)45323 B)37803. C) 88975 D) n

75)3742+6138+9898+9654+ 3819.

A)33251. B) 54541 C) 89751 D) n

76)1987+8942+4796+9834 +3999.

A) 78648 B) 66588 C)29558. D) n

77)6719+ 8752+ 9637+ 1864.

A) 26972. B) 56772 C) 98772 D)n

78)2172+8543+9794+6142+ 3897.

A)30548. B) 67488 C) 88888 D) n

79)632+1293+ 8567+ 312+ 6178.

A) 16982. B) 78654 C) 17342 D) n

80)3122+4163+3184+2122+1296.

A)13887. B) 13557 C) 13457 D) n

81)4138+7852+4293+8562+1374.

A)26219 B) 55349 C) 87879 D) n

82)4382+6123 +1398 +1212+ 664.

A)13779. B) 14799 C) 13979 D)n

83)1623+ 3156+ 2219+ 6123.

A) 23231 B)13121. C) 32211 D) n

84)2314+2315+2416+2718+4132.

A)13895. B) 54545 C) 13555 D) n

85)123+842+631+ 928 +742+ 338.

A) 4504 B) 3504 C)3604. D) 3304

86) 6854+3219+8852+7413+6129.

A)32467. B) 56467 C) 67677 D) n

87) 19763+ 45289+ 31267+ 94374.

A)190693. B) 19993 C) 89773 D)n

Friday, 7 May 2021

REVISION (XII)

INVERSE TRIGONOMETRY

4.5.21

1) sin(2tan⁻¹1/5 - tan⁻¹5/12). 0

2) tan[ sin⁻¹1/3 +cos⁻¹1/√3]. 5/√2

3) sin[2sin⁻¹1/√26+sin⁻¹12/13]. 1

4) tan[2tan⁻¹1/5 - π/4]. -7/17

5) tan⁻¹sin cos⁻¹√(2/3). π/6

6) tan(cos⁻¹4/5 + tan⁻¹2/3). 17/6

7) tan{1/2(tan⁻¹x + tan⁻¹1/x)}. N.s

8) sin{tan⁻¹(tan7π/6)+ cos⁻¹(cos 7π/3)} is

A) 0 B) -1 C) 1 D) none

9) cos⁻¹y + cos⁻¹(x/2 + 1/2 √(3-3x²), 1/2≤x≤1. π/3

9.5.21

B) Prove:

1) 2tan⁻¹2 + tan⁻¹3 = π+ tan⁻¹1/3

2) 2cos⁻¹2/√5+ cos⁻¹3/5 =sin⁻¹24/25

3) 4tan⁻¹1/5 - tan⁻¹1/239=π/4

4) cos⁻¹8/17 + cos⁻¹3/5 + cos⁻¹36/85= π

5) cos⁻¹a - sin⁻¹b = cos⁻¹[b√(1-a²) + a √(1-b²)].

6) tan⁻¹p/q - tan⁻¹{(p-q)/(p+q)} = π/4

7) tan⁻¹x + tan⁻¹y + tan⁻¹{(1-x-y-xy)/(1+x+y- xy)}=π/4

8) tan⁻¹{(2a- b)/b√3} + tan⁻¹{2b-a)/a√3}=π/3

9) cos{1/2 cos⁻¹(-1/9)}= 2/3

10) a cosx + b sinx =√(a²+b²) cos(x - tan⁻¹b/a)= √(a²+ b²) sin(x + tan⁻¹a/b)

--------------------------------------------------------

11) 2 tan⁻¹a + 2tan⁻¹b = sin⁻¹[{2(a+b)(1-ab)}/{(1+a²)(1+b²)}

12) tan(1/2 cos⁻¹a)= √{(1-a)/(1+a)}

13) cos⁻¹√(3/5)= 1/2 cos⁻¹1/5

14) 2cos⁻¹3/√13 + cot⁻¹16/63 + 1/2 cos⁻¹7/25=π

15) cos⁻¹√(2/3) - cos⁻¹(√6+1)/2√3}= π/6.

16) 2tan⁻¹[√50 - √18 - 1/√{3- 2√2)} = π/4

17) 2tan⁻¹{tan(π/4 - a) tan b/2} = cos⁻¹{(sin 2a + cos b)/(1+ sin 2a cos b)}

18) 2tan⁻¹{tana/2 tan (π/4 - b/2)} = tan⁻¹{(sina cos b)/(cos a+ sin b)

19) tan⁻¹1+ tan⁻¹[tan²(a+b) tan²(a-b)]= tan⁻¹[1/2 (cos 2a sec 2b + cos 2b sec 2a]

20) a³/2 cosec²(1/2 tan⁻¹a/b) + b³/2 sec²(1/2 tan⁻¹b/a)= (a+b)(a²+b²).

21) tan⁻¹{(3 sin 2x)/(5+ 3cos 2x)} + tan⁻¹(1/4 tanx) = x

22) tan⁻¹{(2 sin 2x)/(1+ 2cos 2x)} - 1/2 sin⁻¹{(3 sin 2x)/(5+ 4 cos 2x)}= x

C) SOLVE:

1) tan⁻x + tan⁻¹2x + tan⁻3x= π. 1

2) cot⁻¹x + sin⁻¹1/√5=π/4. 3

3) tan⁻¹{(x-1)/(x-2)} + tan⁻¹{(x+1)/(x+2)}=π/4. ±1/√2

4) cos⁻¹8/x + cos⁻¹5/x =π/2. 16

5) sin⁻x + sin⁻¹(1-x)= sin⁻¹√(1-x²). 0, 1/2

6) tan⁻¹{(x+1)/(x-1)}+ tan⁻¹{x-1)/x}= tan⁻¹(-7), (x≠0,1). No sol.

7) sec⁻¹x/a + sec⁻¹x/b = sec⁻¹a + sec⁻¹b. ab

8) sin⁻¹x - cos⁻¹x = sin⁻¹(3x-2). 1/2,1

9) cos⁻¹{(x²-1)/(x²+1)}+ tan⁻¹{2x/(x²+1)= 2π/3. √3, 2-√3

10) sin [2cos⁻¹cot(2tan⁻¹x)]=0. ±1, 1±√2, -1±√2

11) tan⁻¹{(x-1)/(x+1)} + tan⁻¹{2x-1)/(2x+1)}= tan⁻¹7/6. 2

* If sin⁻¹x + sin⁻¹y= π/2, prove 2(x² - xy +y²)= 1+ x⁴+y⁴

* If a= tan⁻¹{x√3/(2k -x)} and b=tan⁻¹{(2x-k)/k√3}. Show that one of the values of a- b is π/6.

* If tan⁻¹yz/xr + tan⁻¹zx/yr + tan⁻¹xy/zr = π/2, prove x² + y² + z² = r²

* If sin⁻¹x + sin⁻¹y= 2π/3, find the value of cos⁻¹x + cos⁻¹y. 1

* If cos⁻¹x/2 + cos⁻¹y/3= a, prove that 9x² - 12xy cos a + 4y²= 36 sin²a

* Show that the least value of (sin⁻¹x)³ + (cos⁻¹x)³ is π³/32.

* Prove tan⁻¹√(xr/yz) + tan⁻¹√(yr/zx) + tan⁻¹√(zr/xy)=π, when x+y+z= r

* If tan⁻¹x + tan⁻¹y + tan⁻¹z=π/2 and x+y+z= √3, show that, x=y=z

* If cos⁻¹x + cos⁻¹y + cos⁻¹z=π and x+y+z= 3/2, prove that, x=y=z

* If sin⁻¹x/a + sin⁻¹y/b = sin⁻¹c²/ab, then show b²x² + 2xy√(a²b² - c⁴) + a²y² = c⁴.

* If ax + b sec(tan⁻¹x)= c and ay + b sec(tan⁻¹y)= c, show that, (x+y)/(1-xy) = 2ac/(a² - c²).

* If A= sin⁻¹(sinx + siny) - sin⁻¹(sinx - siny) and sin²y + sin²x = 1/2 (0<x<π/2), then show that, cosA= cos 2y - cos 2x.

* If tan⁻¹√{(a²-x²)/(a²+x²)} + tan⁻¹{(b² - y²)/(b²+ y²)} = k/2, prove x⁴/a⁴ - (2x²y²cos k)/a²b² + y⁴/b⁴ = sin²k

* If {m tan(a-x)}/cos²x = n tan x/cos²(a-x), then show x= 1/2[a- tan⁻¹{(n-m)/(n+m)} tana]