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
17
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
2
3
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).
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