Short Question for XII

FUNCTION

EXERCISE -1

1) If A{1,2,3} and B ={a,b}, Write total number of functions from A to B. 8

2) If A{a,b,c} and B ={-2,-1,0,1,2}, Write total number of one-one functions from A to B. 60

3) Write total number of one-one functions from set A I={1,2,3,4} to set B ={a,b,c}. 0

4) If f: R--> R is defined by f(x)= x², Write f⁻¹(25). {-5,5}

5) If f: C--> C is defined by f(x)= x², Write f⁻¹(-4). Here C denotes the set of all complex numbers. {2i, -2i}

6) If f: R--> R is defined by f(x)= x³, Write f⁻¹(1). {1}

7) Let C denote the set of all complex numbers. A function f: C--> C is defined by f(x)= x³, Write f⁻¹(-1). {1, w,w²}

8) Let f be a function from C(set of all complex numbers) to itself defined by f(x)= x³, Write f⁻¹(-1). {-1,- w,-w²}

9) Let f: R--> R is defined by f(x)= x⁴, Write f⁻¹(1). {-1,1}

10) Let f: C--> C is defined by f(x)= x⁴, Write f⁻¹(1). {-1,-i, 1,i}

11) If f: R--> R is defined by f(x)= x², Write f⁻¹(-25). ¢

12) If f: C--> C is defined by f(x)= (x-2)³, Write f⁻¹(-1). {1,2,-w,2-w²}

13) If f: R--> R is defined by f(x)= 10x -7,, Write f⁻¹(x). (x+7)/10

14) Let f: (- π/2, π/2) --> R be a function is defined by f(x)= [cos x], Write range of (f) . {1, cos 1, cos 2}

15) If f: R--> R is defined by f(x)= 3x -4 Write f⁻¹(x). (x+4)/3

16) If f: R--> R g: R--> R are given by f(x)= (x +1)² and g(x)= x²+1, then write the value of gog(-3). 121

17) Let A= {x belongs to R: -4≤x ≤ 4 and x ≠ 0} and f: A--> R be defined by f (x)= |x|/x. Write the range of f. {-1,1}

18) Let f: (- π/2, π/2) --> A be defined by f(x)= sin x. If f is a bijection, write set A. A[-1,1]

19) If f: R--> R⁺ is defined by f(x)=a ˣ, a> 0 and a≠ 1. Write f⁻¹(x). Logₐ x

20) Let f: R--{-1}--> R--{1} is defined by f(x)= x/(x+1). Write f⁻¹(x). x/(1-x)

21) Let f: R--{-3/5}--> R--{1} be a function defined by f(x)= 2x/(5x+3). Write f⁻¹: Range of f --> R --{-3/5}. 3x/(2 - 5x)

22) Let f: R--> R, g: R--> R be two functions defined by f(x)= x²+x+1 and g(x)= 1 - x². Write fog(-2). 7

23) Let f: R---> R be defined by f(x)= (2x- 3)/4. Write fof⁻¹(1). 1

24) Let f be an invertible real function. Write (f⁻¹of)(1)+ (f⁻¹of)(2)+....(f⁻¹of)(100). 5050

25) Let A={1,2,3,4} and B={a,b} be two sets. Write total number of onto functions from A to B. 14

26) Write the domain of the real function f(x)= √{x - [x]}. R

27) Write the domain of the real function f(x)= √{[x] - x}. ¢

28) Write the domain of the real function f(x)= 1/√{|x| - x}. (- ∞ ,0)

29) Write whether f: R---> R given by f(x)= x + √(x²) is one-one, many-one, onto or into. Many one into 

30) If f(x)= x+7 and g(x) =x-7, x belongs to R, write fog(7). 7

EXERCISE -2

Short Question for XI

TRIGONOMETRY

1) 25° when measured in radians is

A) 5π/18 B) 5π/24 C) 6π/36 D) none

2) 162° when measured in radians is

A) 5π/10 B) 9π/10 C) 4π/3 D) 5π/4

3) (8π/5)ᶜ = ?

A) 272° B) 302°  C) 288°  D) 316°

4) 11ᶜ= ?

A) 315° B) 372°  C) 418°  D) 630°

5) 1ᶜ= ?

A) 56°27'22" B) 57°16'22"  C) 55°18'32"  D) 57°26'32"

6) 3°45' expressed in radians is

A) (π/36)ᶜ B) (π/54)ᶜ C) A) (π/48)ᶜ B) (5π/96)ᶜ

7) 50°37'30" =?

A) (5π/16)ᶜ B) (7π/18)ᶜ C) A) (9π/32)ᶜ B) (11π/36)ᶜ

8) In a right angled triangle, the difference between two angles is (π/15)ᶜ. The measure of the smallest angle is

A) 40° B) 45°  C) 36°  D) 39°

9) The angles of a triangle are in AP and the greatest angle is double the least. The largest angle measures?

A) 60° B) 80°  C) 75°  D) 90°

10) The angles of a triangle are in AP and the ratio of the number of degrees in the least two the number of radians in the greatest is 60:π. The smallest angle is

A) 15° B) 30°  C) 45°  D) 60°

11) In a circle, the central angle of 45° intercepts an arc of length 33 cm. The radius of the circle is 

A) 21cm B) 35cm  C) 42cm  D) 14cm

12) In a circle of radius 14cm an arc subtends an angle of 36° at the centre.. The length of the arc is 

A) 6.6cm B) 7.7cm  C) 8.8cm  D) 9.2cm

13) The minute hand of a watch is 1.4cm long. How far does its tip move in 45 minutes?

A) 6cm B) 6.3cm C) 6.6cm D) 7cm

14)  If the arcs of the same length in two circles subtend angles of 60° and 75° at their respective centres, the ratio of their radii is

A) 4:5 B) 5:4  C) 3:5 D) 5:3 

15) A wire of length 121cm is bent to form an arc of a circle of radius 180cm. The angle subtends at the centre by the arc is

A) 36°20' B) 34°49' C) 38°30' D) 39°10'

16) A horse is tied to a post by a rope. If the horse moves along a circular path, always keeping the rope tight and describes 88m when it traces 72° at the centre, then the length of the rope is

A) 35m B) 70m C) 17.5m D) 22m

17) A pendulum swings through an angle of 42° in discribing an arc of length 55cm. The length of the pendulum is

A) 56cm B) 60cm C) 75cm D) 88cm

18) The radius of a circle is 30 cm. The length of the arc of the circle whose chood is 30 cm long, is

A) 9πcm B) 10πcm C) 12πcm 13.6πcm

19) A wheel makes 180 revolutions in 1 minute. How many radians does it turn in 1 second?

A) 3π B) 4π C) 6π D) 12π

20) A railway train is moving on a circular curve of radius 1500m at a speed of 90 km/hr. Through what angle has it turned in 11 seconds?

A) 10°30' B) 11°40' C) 12° D) 16°30'

21) When a clock shows the time 7:20, what is the angle between its minute hand and the hour hand ?

A) 60° B) 80° C) 100° D) 120°

22) the angle between the hour and the minute hand of a clock at half past three is

A) 54° B) 63° C) 72° D) 75° 

23) The angle between the minute hand and the hour hand of a clock when the time is 8:25 a.m. is 

A) 107°15' B) 105° C) 102°30' D) 92°45

24) The length of a pendulum is 60cm. The angle through which its swings when its tip describes an arc of length 16.5 cm is 

A) 15°30' B) 15°45' C) 16°15' D) 16°30'

25) the angles of a quadrilateral in degrees are in AP and the greatest angle is 120°. The smallest angle is 

A) π/4 B) π/3 C) π/5 D) π/6

26) The perimeter of a sector of a circle is equal to half the circumference of the circle. The angle of the sector is

A) π/4 B) π/2 C) π-2 D) π+2

27) sin (25π/3)=

A) 1/2 B) 1/√2 C) √3/2 D) -√3/2

28) cos (41π/4)=?

A) 1/√2 B) -1/√2 C) √3/2 D) -√3/2

29) tan(-16π/3)=?

A) √3 B) -√3 C) 1/√3 D) -1/√3

30) cot(29π/4)=?

A) -1 B) 1 C) √3 D) 1/√3

31) sec(-19π/3)=?

A) -2 B) -1/2 C) 2 D) 1/2 

32) Cosec(-33π/4)=?

A) -√2 B) √2 C) 1/√2 D) -1/√2 

33)cos 15π=?

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

34) sec 6π

A) 1 B) -1 C) √2 D) -√2  

35) tan 5π/4=?

A) √3 B) 1/√3 C) 1 D) -1

36) sin(765°)=?

A) √3/2 B) -√3/2 C) 1/√2 D) -1/√2

37) cot(-600°)=?

A) -1 B) -√3  C) -1/√3 D) none

38) If sinx =-2√6/5 and x lies in quadrant III, then cot x =?

A) 1/2√6 B) -1/2√6 C) 3/2√6 D) -3/2√6

39) If cos x =- √15/4 and π/2 < x< π, then sin x= ? 

A) 3/4 B) -3/4 C) 1/4 D) -1/4

40) If sec x =- 2 and π < x< 3π/2, then sin x= ? 

A) √3/2 B) -√3/2 C) 1/2 D) -1/2

41) If cosec x =- 2/√3 and x lies in quadrant IV,  then tan x= ? 

A) 1/√3 B) -1/√3 C) √3 D) -√3

42) if cot x=√5 and x does not lie in the quadrant I, then the value of cosec x and sec x are respectively.

A) √6,(√6/5) B) -√6,√(6/5) C) -√6,-(√6/5) D) -√6,(√6/5)

43) If cos x =- 1/2 and x lies in quadrant II, and then (2sin x+ tan x) = ? 

A) 0 B) 3√3/2 C) -√3/2 D) none

44) If cos x =- 3/5 and π < x< 3π/2, then (cosec x + cot x)/(sec x - tan x) = ? 

A) 1/6 B) 2/3 C) 3/2 D) 1/3

45) If sin x = 3/5 and π/2 < x< π, then (2 sec x- 3 cot x)= ? 

A) 3/2 B) -3/2 C) 13/2 D) -13/2

46) If sec x =√2 and 3π/2 < x< 2π, then (1+ tan x + cosec x)/(1- cot x- cosec x)= ? 

A) √3/2 B) -1 C) -3/8 D) 3/4

47) If cos x =- 12/13 and π < x< 3π/2, then (cot x + cosec x) = ? 

A) 1/5 B) -1/5 C) 3/5 D) -3/5

48) If cot x =- 12/5 and π/2 < x< π, then (1+ sin x - cos x)/(1- sin x + cos x)= ? 

A) 13/4 B) -13/4 C) 15/2 D) -15/2

49) If sin x =4/5 and π/2 < x< π, then (5 cos x+ 4 cosec x + 3 tan x)/(4 cotx + 3 sec x + 5 sin x) = ? 

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

50) If sec x = 13/5 and x is acute, then (4- 3cot x)/(3+ 4 tanx) = ? 

A) 55/252 B) 44/305 C) 54/255 D) 33/215

51) cos 135 =  ? 

A) 1/√2 B) -1/√2 C) 1/2 D) -1/2

52) sec 120 =  ? 

A) √2 B) -√2 C) 2 D) -2

53) cosec 150 =  ? 

A) -2 B) 2 C) √2 D) -√2

54) sin 315 =  ? 

A) 1/√2 B) -1/√2 C) √2 D) -√2

55) cos 405 =  ? 

A) √2 B) 1/√2 C) -√2 D) -1/√2

56) tan(11π/6) =  ? 

A) -1/√3 B) 1/√3 C) √3 D) -√3

57) cot 675 =  ? 

A) -1 B)  1 C) -√3 D) √3

58) sin(31π/3) =  ? 

A) 1/2  B) -1/2 C) √3/2 D) -√3/2

59) cot (-600) =  ? 

A) -√3 B)  1/√3 C) -1/√3 D) √3

60) cosec(-1100) =  ? 

A) 2/√3 B)  -2/√3 C) 2 D) -2

61)  sec(-33π/4) =  ? 

A) -√2  B) √2 C) -√3/2 D) √3/2

62) Tan (-25π/3) =  ? 

A) -√3 B) √3 C) -1/√3 D) 1/√3

63) The value of Cot (π/3),  Cot (π/4), Cot (π/6) are in 

A) AP  B) GP C) HP D) none

64) Which is smaller, sin 64° or cos 64°?

A) sin 64° B) cos 64° C) both are equal D) can not be compared

65) Which is larger, sin 24° or cos 24°?

A) sin 24° B) cos 24° C) both are equal D) can not be compared

66) The extremum values of sin x are

A) 0 and 1 B) -1 and 0 C) -1 and 1 D) -√3/2 and 1/√2

67) The extremum values of cos x are

A) 0 and 1 B) -1 and 0 C) -1 and 1 D) -√3/2 and √2

68) The values of sec x can 

A) never be greater than 1 B) never be less than 1  C) never be equal to 1 D) never lie between -1 and 1

69) tan(150) =  ? 

A) 1/√3 B)  -1/√3 C) √3 D) -√3

70) sec(150) =  ? 

A) 2/√3 B)  -2/√3 C) 2 D) -2

71) cot(120) =  ? 

A) -1/√3 B)  1/√3 C) -√3 D) √3

72) sin 105 + cos 105 =  ? 

A) √2 B)  1/√2 C) 1/√3 D) 2/√3

73) sin 15 =  ? 

A) √3/2√2 B) (√3+1)/2√2 C) (√3-1)/2√2 D) (√2-1)/√2

74) cos 15 =  ? 

A) (√3+1)/√2 B) (√3+1)/2√2 C) (√3-1)/2√2 D) (√3-1)/√2

75) tan 15 =  ? 

A) (√3+1)/(√3-1) B) (√3-1)/(√3+1) C) (√2+1)/(√2-1)  D) (√2-1)/(√2+1)

76) sin 75 =  ? 

A) (√3+1)/2√2  B) (√3-1)/2√2  C) (√2+1)/2√2 D) (√2-1)/2√2

77) cos 75=?

A) (√3+1)/2√2  B) (√3-1)/2√2  C) (√2-1)/2√2 D) (√2+1)/2√2

78) tan(13π/12)=?

A) (2+ √3) B) (1+ √2) C) (2 - √3) D) (√2-1)

78) (sin70 cos 10 - cos 70 sin 10)=?

A) 1/√2 B) 1/2 C) √3/2  D) -√3/2

80) (sin36 cos 9 - cos 36 sin 9)=?

A) 1/√2 B) 1/2 C) √3/2  D) 1

81) (cos 80 cos 20 + sin 80 sin 20)=? 

A) 1/√2 B) 1/2 C) √3/2 D) 1

82) (cos 50 cos 10 - sin 50 sin 10)=? 

A) 1/√2 B) 1/2 C) √3/2 D) 1

83) sin(40 +x) cos(10 + x) - cos(40 +x) sin (10 + x) =? 

A) √2 B) 1/2 C) √3/2 D) none

84) sin 7π/12 cos π/4 - cos 7π/12 sin π/4 =? 

A) 1/√2 B) √2 C) √3/2 D) none

85) sin π/4  cos π/12 - cos π/4 sin π/12 =? 

A) 1/2 B) √2 C) √3/2 D) none

86) cos 2π/3 cos π/4 - sin 2π/3 sin π/4 =? 

A) (√3+1)/√2 B) -(√3+1)/√2 C) (√3+1)/2√2 D) -(√3+1)/2√2

87) sin π/12 =? 

A) (√3-1)/2√2 B) -(√3+1)/2√2 C) (√3+1)/2√2 D) none

88) sin 5π/12 =? 

A) (√3-1)/2√2 B) (√3+1)/2√2 C) - (√3-1)/2√2 D) -(√3+1)/2√2

89) If sin x =15/17 and cos y= 12/13, where x and y both lie in quadrant I, then sin(x+ y)=?

A) 171/221 B) 180/221 C) 220/221 D) 181/221

90) If sin x =3/5 and cos y= -12/13, where x and y both lie in quadrant II, then sin(x- y)=?

A) 16/65  B) -16/65 C) 33/65 D) -33/65

91) If cos x =4/5 and cos y= 12/13, where x and y both lie in quadrant IV, then cos(x+ y)=?

A) 33/65 B) -33/65 C) 16/65 D) -16/65

92) If cot x =1/2 and sec y= -5/3, where x lie in quadrant III, and y lies quadrant II then tan(x+ y)=?

A) 5/11 B) 211 C) -6/11 D) 10/11

93) cos 15 - sin 15 =  ? 

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

94) cot 105 - tan 105 =  ? 

A) √3 B)  2√3 C) √3/2 D) (√3+1)/(√3-1)

95) 2 sin 5π/12 sin π/12 =  ? 

A) √2 B)  1/√2 C) 1/2 D) √3/2

96) 2 cos 5π/12 cos π/12 =  ? 

A) √2 B)  1/√2 C) 1/2 D) √3/2

97) 2 sin 5π/12 cos π/12 =  ? 

A) √3/2 B)  (2+√3)/2 C) 1/2 D) (√3+1)/2

98) {sin(180+x) cos(90+x) tan(270-x) cot(360- x)}/{sin(360-x) cos(360+x) cosec(-x) sin(270+x)} =  ? 

A) √2 B)  √3/2 C) 3/2 D) 1

99) {sin(40+x) cos(10+x) - cos(40 +x) sin(10-+x)} =  ? 

A) 2 B) √3/2 C) 1/2 D) none

100) {cos(90+x) sec(270+x) sin(180+x)}/{cosec(-x) cos(270- x) tan(180+x)} =  ? 

A) cos x B) sec x C) cot x D) none

101) cosx+ sin(270+x) -  sin(270-x)+ cos(180+x) =  ? 

A) 2cos x B) 2sin x C) 0 D) none

102) (cos8 - sin 8)/(cos8 + sin 8) =  ? 

A) tan 8 B) tan 37 C) tan 52 D) none

103) {cos(π+x) cos(-x)}/{ cos(π-x) cos(π/2 + x)  =  ? 

A) -cot x B) cot x C) -tan x D) tan x

104) cos(π/4 +x) +  cos(π/4 -x) =  ? 

A) 2cos x B) √2 cos x C) 2 sin x D) √2 sin x

105) cos(3π/4 +x) - cos(3π/4 - x) =  ? 

A) √2 sin x B) 2 sin x C) -√2 sin x D) 1/√2 sin x

106) (sin 3x - sin x)/(cosx - cos 3x) =  ? 

A) tan 2x B) cot 2x C) - tan 2x D) - cot 2x

107) (cos 6x + cos 4x)/(sin6x - sin4x)= ? 

A) cot x B) - tan x C) - tan x D) - cot x

108) (cos4x +cos3x+ cos2x)/(sin4x + sin 3x + sin2x) =  ? 

A) tan 2x B) cot 2x C)  tan 3x D) cot 3x

109) (sin 7x - sin 5x)/(cos7x + cos5x)= ? 

A) tan x B) cot x C)  tan 2x D)  cot 2x

110) (sin²6x - sin²4x)=  ? 

A) sin10x B) sin 2x C) sin10x sin 2x D) n

111) cos20 cos40 cos80 =  ? 

A) 1/16 B) 1/8 C) √3/8 D) √3/16

112) sin 10 sin 50 sin 70 =  ? 

A) 1/16 B) 1/8 C) √3/8 D) √3/4

113) 2cos 45 cos 15  =  ? 

A) 3/2 B) (√3+1)/2 C) (√3-2)/2 D) √3/2

114) 2 sin 75 sin 15  =  ? 

A) 1/2  B) (√3+1)/2 C) √3/2 D) n

115) cos 15 - sin 15 =  ? 

A) 1/2 B) 1/√2 C) 1/2√2 D) √3/2

116) If sinx =-1/2 and x lies in quadrant III, then sin 2x =  ? 

A) 1/2 B) 1/2√3 C) √3 D) √3/2

117) If sec x =-13/12 and π/2< x<π, then cos 2x =  ? 

A) -120/169 B) 119/169 C) -120/119 D)n

118) If tan x =-3/4 and 3π/2< x<2π, then tan 2x =  ? 

A) -24/25 B) 7/25 C) 24/7 D) -24/7

119) If sin x = 1/3  then sin 3x =  ? 

A) 1 B) 1/9 C) 7/9 D) 23/27

120) If cos x = 1/2 then cos 3x =  ? 

A) 3/2 B) 1/6 C) -1  D) 2/3

121) cos² π/12 - sin²π/12 =  ? 

A) 1/2 B) √3/2 C) 3/2  D) -1/2

122) (1+sin 2x - cos2x)/(1+ sin2x+ cos2x) =  ? 

A) tan 2x B) tan x C) cot2x D) cot x

123) If cos x =-3/5 and π/2< x<π, then sin(x/2) =  ? 

A) -2/√5 B) 2/√5 C) -2/√5 D) 1/√5

124) If cos x =-3/5 and π/2< x<π, then cos(x/2) =  ? 

A) 1/√5 B) -1/√5 C) 2/√5 D) -2/√5

125) If cos x =-4/5 and π< x<3π/2,  then cos(x/2) =  ? 

A) 1/√10 B) -1/√10 C) 3/√10 D) -3/√10

126) If tan x =3/4 and π < x<3π/3, then tan(x/2) =  ? 

A) 3  B) 1/3 C) -3 D) -1/3

127) If cos x =-1/3 and x lies in quadrant III, then tan(x/2) =  ? 

A) √2 B) -√2 C) √3 D) -√3

128) If sin x =-1/2 and x lies in quadrant IV then sin(x/2) =  ? 

A) √{2 +√3}/2 B) {√3+ √2}/2 C) √(2- √3)/2 D) none

129) (1+ cosx)/((1- cosx)=?

A) tan²(x/2) B) cot²(x/2) C) sec²(x/2) D) cosec²(x/2)

130) √{(1+ sinx)/((1- sinx)} =?

A) tan(x/2) B) cot(x/2) C) tan(π/4+ x/2) D) cot(π/4 + x/2)

131) sinx/((1+ cos x) =?

A) tan(x/2) B) cot(x/2) C) tan(π/4+ x/2) D) n

132) cot(x/2) - tan(x/2) =?

A) 2tanx B) 2cot x C) 2 sin x D) 2cos x

133) sin 18°=?

A) (√5-1)/4 B)  (√5+1)/4 C)  (√3+ 1)/2 D)  (√3 -1)/2

134) cos 18°=?

A) √(10- 2√5)/4 B) √(10+2√5)/4  C)  (√5+ 1)/4 D) none

135) cos 36°=?

A) (√5-1)/4 B) (√5+1)/4  C)  (√3+ 1)/2 D) 

(√3- 1)/2

136) sin 36°=?

A) (√5 -1)/4 B) √(10+2√5)/4  C) √(10- 2√5)/4 D) none

137) sin 54°=?

A) (√5+1)/4 B) (√5-1)/4  C)  √(10+2√5)/4 D) √(10-2√5)/4

138) cos 72°=?

A) (√5+1)/4 B) (√5-1)/4  C)  (2+√5)/3 D) (2 - √5)/3

139) cos 54°=?

A) (√5+1)/4 B) (√5-1)/4  C)  √(10+2√5)/4 D) √(10-2√5)/4

140) sin 72°=?

A) (√5+1)/4 B) (√5-1)/4  C)  √(10+2√5)/4 D) √(10-2√5)/4

141) 2sin 45/2 cos 45/2 =?

A) 1 B) 1/2  C) 1/√2 D) √2

142) (2 cos² 15 - 1) =?

A) 3/2 B) √3/2  C) 2√3 D) √3/√2

143) 3sin 40- 4 sin² 40 =?

A) 3√3 B) 3/2 C) none D) √3/2

144) 8cos³20- 6 cos 20 =?

A) 5/2 B) 5/3  C) 1 D) √3/2

145) If tan x= a/b, then a sin2x+ b cos2x = ?

A) a B) b  C) a+ b D) a-- b

146) cotx - 2 cot 2x =?

A) tanx B) cosx  C) sinx D) cos2x

147) cos 2x+ 2 sin²x =?

A) 2 B) 1/2  C) 1 D) 3/2

148) sin2x/(1-  cos 2x) =?

A) tanx B) cot x  C) sec x D) cosec x

149) sin2x/(1+ cos 2x) =?

A) tanx B) cot x  C) sec x D) cosec x

150) tan 2x/(1+ sec 2x) =?

A) tanx B) cot x  C) sin x D) cos x

151) (1- cos 2x+ sinx)/(sin2x+ cosx) =?

A) tanx B) cot x  C) sec x D) cosec x

152) cosx/(1-  sinx) =?

A) tan(x/2) B) tan(π/4+ x/2)  C) tan(π/4- x/2) D) none

153) 

PERMUTATION AND COMBINATIONS 

EXERCISE -1

1) If 1/6! + 1/7! = x/8!, Then x=?

A) 32 B) 48 C) 56 D) 64

2) If ⁿ⁻¹P₃ :  ⁿP₄ = 1:9, then n=?

A) 12 B) 11 C) 9 D) 10 

3) ⁿPₙ/ⁿPₙ₋₂ =?

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

4) ¹⁵Pₙ =2730, then n=?

A) 3 B) 4 C) 5 D) 6

5) ⁷P₃ =?

A) 105 B) 140 C) 210 D) 175

6) ⁿP₅ = 20. ⁿP₃, then n=?

A) 8 B) 9 C) 10 D) 11

7) ¹⁵Pₙ₋₁ : ¹⁶Pₙ₋₂=  3:4, then n= ?

A) 8 B) 10 C) 12 D) 14

8) ⁿC₁₀ =ⁿC₁₄ , then n=?

A) 4 B) 24 C) 14 D) 10

9) If ⁿC₃ = 220, then n=?

A) 9 B) 10 C) 11 D) 12

10) ⁿCₐ + ⁿCₐ₊₁ = ⁿ⁺¹Cₓ , then x=?

A) a-1 B) a C) a+1 D) n

11) ³⁶C₃₄ = ?

A) 1224 B) 612 C) 630 D) none

12) ⁿCₐ/ⁿCₐ₋₁ =?

A) (n-r)/r B) (n-r -1)/r  C) (n-r +1)/r D) none

13) ⁿC₁₈= ⁿC₁₂ , then ¹²Cₙ=?

A) 248 B) 496 C) 992 D) none

14) ⁶⁰C₆₀ =?

A) 60! B) 1 C) 1/60 D) none

15) In how many ways can 5 persons occupy 3 seats ?

A) 15 B) 20 C) 30 D) 60 

16) In how many ways can 5 children stand in a queue ?

A) 5 B) 25 C) 60 D) 120 

17) In how many ways can 4 different books be arranged in a self ?

A) 4 B) 8 C) 24 D) 16 

18) 10 students are participating in a race. In how many different ways can the first prize be own ?

A) 30 B) 60 B) 120 D) 720

19) three different rings are to be worn in four fingers. In how many ways can this be done ?

A) 12 B) 24 C) 64 D) 81 

20) There are 6 periods on each working day of a school. In how many ways can one arrange 5 subjects such that each subject is allowed at least one period.

A) 360 B) 720 C) 3600 D) 1800

21) How many words with or without meaning can be formed by using all the letters of the word, DELHI, using each letter exactly once ?

A) 20 B) 60 C) 120 D) 5

22) It is required to seat 5 man and 4 Women in a row so that the women occupied the even places. How many such arrangements are possible ?

A) 24 B)120 C) 720 D) 2880

23) How many three-digit numbers are there ?

A) 648 B) 729 C) 900 D) 1000

24) How many 3-digit numbers are there with no digit repeated ?

A) 648 B) 720 C) 729 D) none

25) How many four-digit numbers can be formed with no digital repeated by using the digit 3,4,5,6,7,8 and 0?

A) 280 B) 560 C) 720 D) 840

26) How many 3-digits even numbers can be formed with no digit repeated by using the digit 0, 1, 2, 3, 4 and 5 ?

A) 50 B) 52 C) 54 D) 56 

27) The number of positive integers greater than 6000 less than 7000 which are divisible by 5 with no digit repeated is

A) 28 B) 56 C) 84 D) 112

28) How many 10 digit numbers can be formed by using the digit 1 and 2 ?

A) ¹⁰P₂  B) ¹⁰C₂ C) 2¹⁰ D) 10!

29) How many words beginning with T and ending with E can be made with no letter repeated out of the letters of the word TRIANGLE?

A) ⁸P₆ B) 720 C) 722 D) 1440 

30) How many words can be formed from the letters of the word DAUGHTER so that the vowels always come together ?

A) 720 B) 2160 C) 4320 D) none

31) How many words can be found from the letters of the word LAUGHTER so that the vowels are never together ?

A) 3600 B) 4320 C) 36000 D) 40320

32) In how many ways can the letters of the word MACHINE be arranged so that the vowels may occupy only odd positions ?

A) 288 B) 576 C) 5040 D) none

33) in how many ways can the letter of the word PENCIL be arranged so that N is always next to E?

A) 120 B) 240 C) 720 D) 1440 

34) In how many ways can the letters of the word APPLE be arranged?

A) 6 B) 60 C) 90 D) 120

35) How many words can be formed by using all the letter of the word ALLAHABAD?

A) 9! B) 1890 C) 3780 D) 7560

36) How many words can be formed using the letter A thrice, the letter B twice and the letter C once ?

A) 6 B) 60 C) 90 D) 120 

37) In how many ways can the 10 books be arranged in a self so that a particular pair of books shall be always together ?

A) 8! B) 9! C) 2 . 8! D) 2 . 9!

38) In how many ways can 10 books be arranged in a shelf so that a particular pair of books shall be never together ?

A) 8! B) 9! C) 2. 9! D) 8.9.

39) How many 4 digit numbers are there where a digit may be repeated any number of times in each number ?

A) 5040 B) 9000 C) 10000 D) 4500

40) How many ways can 6 boys be arranged in a row?

A) 5! B) 6! C) 6 D) 2.6! 

41) In how many ways can 6 girls be seated in a circle ?

A) 6 B) 6! C) 5! D) 2.5!

42) How many diagonals are there in a polygon of n sides?

A) n(n-1)/2 B) n(n-2)/2 C) n(n-3)/2 D) n(n+1)/2  

43) How many diagonals are there in an octagon ?

A) 28 B) 24 C) 20  D) 36 

44) A polygon as 54 diagonals. Number of sides in this polygon is.

A) 9  B) 12 C) 15 D) 16 

45) There are 10 points in a plane, out of which 4 points are collinear. The number of line segments obtained from the pairs of these points is

A) 39 B) 40 C) 41 D) 45 

46) There are 10 points in a plane, out of which 4 points are collinear. The number of triangles formed with the vertices as these points is 

A) 20 B) 120 C) 116  D) none

47) Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2 vowels can be formed?

A) 330 B) 1050 C) 6300 D) 25200

48) In how many ways can a committee of 5 member be selected from 6 men and 5 ladies, consisting of 3 men and 2 ladies?

A) 25 B) 50 C) 100  D) 200

49) Out of 5 men and 2 women, a committee of 3 is to be formed. In how many ways it be formed if at least one woman is included in each committee ?

A) 21 B) 25 C) 32 D) 50 

50) A committee of five is to be formed out of six gents and four ladies. In how many ways can this be done when each committee may have atmost 2 ladies ?

A) 120 B) 160 C) 180 D) 186 

51) How many different teams of 7 players can be chosen out of 10 players?

A) 720 B) 70 C) 120 D) none 

52) 12 persons meet in a room and each shakes hands with all the others. How many handshakes are there ?

A) 144 B) 132 C) 72 D) 66 

53) In how many ways can we select 9 balls out of 6 red balls, 5 white balls and five blue balls if 3 balls if each colour are selected 

A) 40 B) 200 C) 2000 D) 400

54) How many ways can a cricket team be chosen out of a batch of 15 players, if a particular player is always chosen?

A) 1364 B) 364 C) 1001 D) none

55) In how many ways can a cricket team be chosen out of a batch of 15 players, if a particular player is never chosen ?

A) 364  B) 1001 C) 1364 D) none 

56) For the post of 5 teachers, there are 23 applicants. 2 posts are reserved for SC candidates and there are 7 SC candidates among the applicants. In how many ways can the selection be made?

A) 5880 B) 11760 C) 3920 D) none

57) In how many ways can 5 white balls and 3 black balls be arranged in a row so that no two black balls are together 

A) 192 B) 40 C) 20 D) 120 

58) In an examination a candidate has to pass in each of the five subjects. In how many ways he fail ?

A) 5 B) 10 C) 21 D) 31

59)  An examination paper content 12 questions consisting of two parts, A and B. Part A contains 7 questions and part B contains 5 questions. A candidate is required to attempt 8 questions, selecting at least three from each part. In how many ways can the candidate select the questions   ?

A)210 B) 175 C) 420  D) none   

QUICK REVISION TEST (X) state board

MODEL TEST PAPER -5

1) ₹5000 lent out CI to be compound annually amounts to ₹5400 the end of the first year. Calculate

A) the amount at the end of the second year.

B) the sum which must be written at the end of the first year, so that the payment of ₹2700 at the end of the second year may settle the debt.

2) A a semi-circle is described on AB as diameter. Given that PC is perpendicular to AB at P. AP= 3cm, PB =9 cm.

A) prove that ∆APC/∆ACB are similar

B) calculate AC.

3) If a/b = c/d = e/f = g/h, prove that 

e/f = √{(a² + c² + e² + g²)/(b² + d² + f² + h²).

4) Calculate mean, median, mode of the following numbers : 3, 22, 7, 15, 17, 12, 10, 5, 5, 15.

5) Glass marbles of diameter 3.5cm are dropped into a cylindrical vessel, whose base radius is 7 cm. If the Marbles are fully submerged in water and water level in the vessel increases by 3.5cm. calculate the number of marbles in the vessel.

6) calculate the mean daily wage of the worker, from the following table:

Daily wages        no. of workers

40-45                       2

45-50                       3

50-55                       7

55-60                      12

60-65                       6

If the daily wages of all workers is increased by ₹8000. What will be the new mean daily wage of a worker ?

7) TV viewing of 30 persons in a locality was observed to be as follows:

 number of hours    no. Of viewers

00-02                             4

03-05                             7

06-08                             8

09-11                            8

12-14                            3

A) Draw up the community frequency table.

B)  draw the Ogive.

C) find median

MODEL TEST PAPER -4

1) The work done by (x-3) men in (2x+1) days and the work done by (2x+1) men in (x+4) days are in ratio 3:10. Find the value of x.     42

2) Find the mean of the following:
Class-interval           Frequency
00-50                              4
50-100                            8
100-150                        16
150-200                        13
200-250                         6
250-309                         3.                143

3) Prove: cos x/(1- tan x)+ sin x/(1- cot x) = cos x + sin x.

4) 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 was reduced by 10, The total number of seats increased by 300. Find
A) the number of rows in the original arrangement.      30
B) the number of seats in the Auditorium after rearrangement.     1200

5)  Draw a histogram and hence estimate the mode for the following frequency distribution :
Class           Frequency
00-10                   2
10-20                   8
20-30                  10
30-40                   5
40-50                   4
50-60                   3.                          23

6) A man standing on the bank of a river observes that the angle of elevation of a tree on the opposite Bank in 60°. When he moves 50m away from the bank, he  finds the angle of the elevation to be 30°. Calculate:
A) the width of the river.        25
B) the height of the tree.       25√3 

7) A vessel is in the form of an inverted cone. Its height is 11 cm and the radius of its top which is open, is 2.5 cm. It is filled with water up to the Rim. When lead shots, each of which is a sphere of radius 0.25cm dropped Into The vessel, 2/5 of the water flows out. Find the number of lead shots dropped into the vessel.     440

8) In a triangle PQR, L and M  are two points on the base QR , such that angle LPQ = angle QRP and angle RPM = angle RQP. Prove that:
A) ∆ PQL similar to ∆ RPM
B) QL. RM = PL. PM
C) PQ² = QR. QL.

MODEL TEST PAPER -3

1) Class:  50  60  70  80   90  100  110
Workers:   2    4    8    12   10    6      8
A) calculate the mean wage, correct to the nearest rupee.                      85
B) If the number of workers in each category is doubled, what would be the new wage.           85

2) In ∆ ABC, AD is the perpendicular on BC. Tan B= 3/4 and tanC= 5/12, BC= 56cm. Calculate the length of AD.    15

3) AB and CD chord extended to meet at X of the circle whose centre is O. Given AB= 4, BX=6, XD= 5cm, calculate the length of CD.           7

4) Given 5 Cos A - 12 sin A = 0 without using table find (Sin A + Cos A)/(2 cos A - sin A).     12/13

5) Two circles of radii 4cm and 2.5cm with their centres 9cm apart. Calculate the length of the transverse common tangent.      6.2225 cm

6) Age             number of casualties
05-15                      6
15-25                     10
25-35                     15
35-45                     13
45-55                     24
55-65                     8
65-75                     7
A) Construct the 'less than' cumulative frequency curve for the above data, using 2cm= 10 years, on x-axis and 2cm= 10 casualties on the other.
B) From your graph determine
i) the median.      43 years

7) The cross section of an ice-cream cone consisting of a cone surmounted by a hemisphere. The radius of the hemisphere is 3.5cm and height of the cone is 10.5cm. The outer shell is shaded and is not filled with ice-cream. AE= DC= 0.5 cm, AB||EF and BC||FD. Calculate the volume of icecream.   174.69

8) The distance by the road between two towns A and B 216km, and by rail it is 208km. A car travels at a speed of x km/hr, and the train travels at a speed of which is 16 km/ hour 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 on 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.
D) Hence find the speed of the train.    52

9) A total buys 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

10) A metal container in the form of a cylinder is a surmounted by a hemisphere of the same radius. The internal height of the cylinder is 7m and the internal radius 3.5m. Calculate
A) the total area of the internal surface, excluding the base.      231m²
B) the internal volume of the container in m³.           1078/3 m³

11) the surface area of a solid metallic sphere is 1256cm³. It is the melted and recast into solid right circular cone of  radius 2.5cm and height 8 cm. Calculate
A) radius of the solid sphere.   10, 80
B) the number of cones recast (π=3.14).

12) Two numbers are in the ratio 3:5. If 8 is added to each number, the ratio becomes 2:3, Find the numbers.    24,40

13) Prove: 1 - cos²x/((1+ sin x) = sin x.

14) A, B and C are three points on a circle. The triangle at C meets BA produced at T. Given that angle ATC= 36° and that the angle ACT= 48°, Calculate the angle subtended by AB at the centre of the circle.          96

15) An exhibition tent is in the form of a cylinder surmounted by a cone. The height of the tent above the ground is 85m and the height of the cylindrical part is 50m. if the diameter of the base is 168m. find the quantity of Canvas required to make the tent. Allow 20% extra for folds and for stitching. give your answer to nearest m².     60509

16) In triangle ABC, angle B= 90° and D is the midpoint of BC. Prove that: AC²= AD²+ 3CD².

17) Using the data given below construct the community frequency table and draw the Ogive. From the Ogive determine the median.
Marks         No. of students
00-10                  3
10-20                  8
20-30                 12
30-40                 14
40-50                 10
50-60                  6
60-70                  5
70-80                  2                         35

18) Prove: 1/(sinx + cos x) + 1/(sinx - cosx) =2 sin x/(1- 2 + cos²x).

19) The following table gives the weekly wages of workers in a factory:
Weekly wages in ₹     No. of workers
50-55                                   5
55-60                                  20
60-65                                  10
65-70                                  10
70-75                                   9
75-80                                   6
80-85                                   12
85-90                                    8
Calculate
A) the mean.            69
B) the modal class.              55-60
C) the number of workers getting weekly wages, below ₹80.               60
D) the number of workers getting ₹65 or more, but less than ₹85 as weekly wages.        27

20) A hollow sphere of internal and external diameter 4cm and 8 cm respectively, is melted into a cone of base diameter 8cm. Find the height of the cone.            14cm

21) An aeroplane Travelled a distance of 400k.m at an average speed of X km/ hr. On the return journey, the speed was increased by 40 km/hr. Write down an expression for the time taken for:
A) onward journey
B) the return journey.
if the return journey took 30 minutes less than the onward journey, write down an equation in x and find its value.   160

22) The shadow of a vertical Tower AB on level ground is increased by 10m, when the altitude of the sun changes from 45° to 30°, find the height of the tower and give your answer, correct to 1/10 of a metre.     13.7

23) In the right angled ∆PQR. PM is an altitude. Given that QR= 8cm and MQ = 3.5 cm, Calculate the value of PR.    6

24) the marks obtained by 120 students test in a mathematics test is given below :
Marks          No of students
00-10                  5
10-20                  9
20-30                16
30-40                22
40-50                26
50-60                18
60-70                11
70-80                 6
80-90                 4
90-100               3 Draw an Ogive for the given distribution on a graph sheet. Use a suitable scale for your Ogive. Use your Ogive to estimate:
A) the median.      43
B) the lower quartile.        30
C)  the number a students who obtained more than 75% in the test.      10
D) the number of students who did not pass in the test if the pass percentage was 40.         52

MODEL TEST PAPER -2

1) ₹8000 are lent out at C.I., calculated yearly for two years, the rate of interest for the first year being 5%p.a and that the second year being 4%p.a. The borrower returns some money at the end of the first year. Now he pays back ₹5200 at the end of the second year to settle the payment of the loan. Calculate as to what amount of a money was paid at the end of the first year.

2) Solve the correct to 2 d.p. :

 x/(x+1) - 4/(x+2) = 0.

3) In ∆ ABC , P and Q are the points on AB and BC respectively. If PQ|| AC and PQ: AC = 3:7 Calculate:

A) PB:AP

B) Given area ∆ ABC= 24.5 cm², calculate area of the trapezium APQC.

4) The Arithmetic mean for the following distribution:

 Class            frequency 

 00-10                  6

 10-20                  8

 20-30                  x

 30-40                  x

40-50                   6 is 25.5. Calculate the value of x.

5) A boat takes 1 hour longer to move 36 km up a river then to move some distance down the river. Given that the river is flowing at 1 km/ hr. Calculate the speed of the boat in still water.

6) Four solid spherical balls each of diameter 3.6cm are kept inside a right circular cylinder, the radius of whose base is 5 cm and height is 15cm and then the cylinder is filled up to top by water. Calculate the volume of water, required to fill the cylinder Use π=22/7. Give your answer to the nearest cm³.

7) 16 students scored as under in a test of 20 marks.

  4     9       18     15      12     10     6     10

14   16       15    20       12     19   10   10 Calculate

A) the median 

B) the mode

MODEL TEST PAPER (I)

1) ₹20000 are borrowed for 2 years, C. I to be compounded yrly. The rate of interest for the first year is 10% p.a and the rate of interest for the 2nd year is 8% per annum. The borrower returns ₹5000 at the end of the first year. Calculate

A) the sum of money on which interest would be calculated for the second year.

B) the amount to be paid at the end of 2 years to settle the loan.

C) the total interest paid on the loan.

2) ABCD is a rectangle AB= 7cm, and BC= 3.5cm. At the two ends of the rectangle are two equal quadrant. Calculate:

A) perimeter of the figure ABEFCDGH.

B) area of the figure ABEFCDGH (π=22/7).

3) Evaluate : 3 cos18/ sin 72  + cos 61/ Sec 29.

4) ₹ 5000 were borrowed for 2 years, at 10% p.a. CI, the interest being reckoned annually. At the end of the first year, a sum of ₹800 was returned by the borrower. Calculato

A) the amount payable at the end of two years, to settle the account.

B) compound interest paid.

5) Triangles ABC and XYZ are similar. AM and XN are perpendiculars drawn from A to BC in triangle ABC and from X to YZ in triangle XYZ. prove that:

A) angle BAM= angle YXN

B) ∆ABM and ∆ XYZ are similar.

C) AM/XN= BC/YZ

6)  Given a/b = c/d ,. prove that (2a- c)/(2a +c)/(2b - d)/(2b +d).

7) find the value sin 48/cos 42  + (sin29 + cos 6)(sin 29 - cos 61).

8) Calculate the (i) Arithmetic mean (ii) the median (iii) the mode of the following: 11, 10 ,11, 13,13, 12, 15, 17, 14 12,13, 14.

9) the bottom of a cylindrical vessel has hemispherical bottom. Given radius of the hemisphere= 3.5cm. Calculate the level if 1309/6 cm³ of water is poured in to it .

10) A bus moving at its usual speed covers distance between town X and Y, which are 550km apart, in 1 hour less than it takes to cover the same distance, when it is raining and the bus has to reduce the speed by 5 km/hr.

Calculate the time taken to cover the distance between X and Y, when it is raining.

11) The performance of 50 students in a test is recorded as under:

 Marks     Number of students

30-40            2

40-50            4

50-60           12

60-70           18

70-80            9

80-90            5

Find Mode

12) Marks obtained by 40 students were recorded as follows:

Marks.          Students

 00-08               5

 08-16               3

 16-24              10

 24-32              16

 32-40               4

 40-48               2 

Calculate Arithmetic mean.

13)  Use A + B= 60°, A= 30°, to verify

 sin (A + B)= sin A cosB + cos A sin B.

 

Test paper -2(J++)

1) In the expansion of (√x - √k/x²)¹⁰, the terms independent of x be 405: find the value of k.

2) If the equation x²+ bx+ ca=0 and x²+ cx+ ab=0 have a common root, prove that their other roots will satisfy the equation x²+ ax+ bc=0.

3) If x - 1/x = 2i sink, show that x⁴- 1/x⁴ = 2i sin 4k.

4) Prove tan6 tan42 tan66 tan78=1.

Or

Show that tan20 tan40 tan80= tan60

5) If a cosk + b sink= a cosx + b sinx, show that sin(k+ x)= 2ab/(a²+ b²)

6) Solve: sin⁸x + cos⁸x =17/32

7) If a²(1- sinx) + b²(1+ sinx) 2ab cosx, show that a/b - b/a = 2tanx.

8) y= mx is the equation of a chord of the circle x²+ y²-2ax=0 . Show that the equation of the circle on this chord as diameter is (1+ m²)(x²+ y²) - 2a(x+ my)=0

9) The points (1,3) and (5,1) are two opposite vertices of a rectangle. The other two vertices lie on the line y= 2x + c. Find c and the remaining vertices.

10) P is a variable point on the hyperbola x² - y²=a and A is the fixed point (2a,0). Show that the locus of the midpoint of the line segment AP is another hyperbola.

11) Given the ellipse 4x²+ 9y²=36, find the equation of the chord which is bisected at (2,-1).

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

f(x)= 2 - |x|/x, when -2 ≤ x ≤2

Test Paper -1 J+

1) If m,n be the roots of the equation ax²+ bx + c=0 and s, t those of the equation px²+ qx+ r=0, prove that ac/pr = b²/q², if ms= nt.

2) If n be positive integer greater than unity, then show 49ⁿ -16n -1 is divisible by 64.

3) If the sum of the first 2n terms of a GP is twice the sum of the reciprocals of the terms, then show that the continued product of the terms is equal to 2ⁿ.

4) How many numbers of four digits can be formed from the numbers 1,2,3,4 ? Find the sum of all such numbers (digits being used once only).

5) If 9x =π, find the value of sinx sin2x sin3x sin4x.

6) If tanx= (tana - tanb)/(1- tana tanb), then show sin2x= (sin2a - sin2b)/(1- sin2a sin2b).

7) Solve: cos³x cos3x+ sin³x sin3x =1/8.

8) If {m tan(x-y)}/cos²y = (n tany)/cos²(x - y), then show that y= 1/2 {x - tan⁻¹{(n - m)/(n+ m).  Tanx}

9) If lx+ my=0 be the perpendicular bisector of the segment joining the points (a,b) and (c,d), then show that (c- a)/l = (d - b)/m = 2(la+ mb)/(l²+ m²).

10) Show that the two circles x²+ y²+ 2gx+ 2fy=0 and x²+ y²+ 2g'x + 2f'y=0 will touch each other if f'g= g'f.

11) Find the equation of latus rectum of the parabola whose focus is (5,3) and vertex is (3,1).

12) If m, n be the eccentric angles of the extremities of a focal chord of the ellipse x²/a² + y²/b² =1.

Prove that, tan(m/2) tan(n/2)= (e-1)/(e+1) or (e+1)/(e -1).

13) Find dy/dx: y= ₓx² + ₐx².

14) Evaluate: lim ₓ→₀ (tan2x - x)/(3x - sinx).

15) If y= xⁿ{a cos(logx)+ bsin(logx)}, show that x² d²y/dx² + (1- 2n) x dy/dx + (1+ n²)y=0.

16) Show that the equation ₑsinx - ₑ-sinx =4 has no real solution.

17) Find dy/dx of x² cosx.

18) ∫ tdt/(1+t²) at (tanx, 1/e) + ∫dt/t(1+ t²)  at (cotx, 1/e) = 1.

19) Solve: (x+ y)² dy/dx = 2x +2y+5.

20) Evaluate:

lim ₓ→∞ 1/n[sec²(π/4n)+ sec²(2π/4n)+ ......sec²(nπ/4n].

21) ∫ (cosx - sinx)(2+ 2sin2x)/(cosx + sinx) dx.

22) Solve: d²y/dx² - 2a dy/dx + a²y=0, given y= a and dy/dx=0 when x=0.

23) Shade the region above the x-axis, included between the parabola y²= 4x and the circle (x -4)=4 cosk, y= 4sink. Find the area of the region by integration.

24) Show that the maximum value of the function x+ 1/x is less than its minimum value.

25) Show that the line lx+ my = n is a normal to the ellipse x²/a²+ y²/b²=1, if a²/l²+ b²/m² = (a²- b²)²/n².

26) Determine the sign of the expression (x-1)(x-2)(x-3)(x-4)+5 for real values of x.

27) If cotx = 2 and coty=3, then find (x+ y).

28) If y= f(x)= (x+1)/(x+2), show that, f(y)= (2x+C)/(3x+5).

29) ¹₋₁∫sin³x cos²x dx.

30) Is it possible to draw a tangent from the point (-2,-1) to the circle x²+ y²- 4x (6y -12=0 ? Give reasons.

31) If f(x)=tan(x - π/4), find f(x) . f(- x).

Revision Questions - 8

1) A student needs to score 75% or more to secure grade A in an examination. How much must the student score out of 900 get grade A?          675 or more

2) Ravi's Monthly salary is ₹16000 and Rahul's salary is 25/4% more than Ravi's salary. What is their total monthly salary.                   33000

3) In a railway accident, 84 out of 525 passengers were injured. What percent of the passangers were unhurt.       84%

4) A student has to get 33% of the full marks to pass an examination. Rajani secured 150 marks and failed by 15 marks. Find the full marks in the examination.                          500

5) Suresh spent ₹560 in the market. If the money he spent 40% of the money he had, how much money did he have?

REVISION TEST(XII) 2023/24

Day-2

1) The differential coefficient of tan⁻¹ {(sinx + cosx)/(sinx - cosx) is

a) -1  b) 0 c) 1 d) none

2) eʸ = eˣ, then dy/dx is 

a) logx b) xˣ c) log(ex) d) none

3) If y= 4cos³x - 3cosx, find d²y/dx² at x=0.

4) If f(x)= tan⁻¹{√(1+ x²) - 1}/x , then f'(0) is

a) 0 b) 1 c) 1/2 d) none 

5) If y= f(x)= (2x+3)/(3x+5), find the inverse function of f(x).

6) If xʸ = yˣ, find dy/dx

7) f(x)= logx, then f"(1) is equal to 

a) -1 b) 1 c) e d) none

8) Show that the derivative of an odd function is always an even function.

9) y= sin⁻¹(cosx), find dy/dx.

10) The differential coefficient of tan⁻¹(Secx+ tanx) is

a) 0 b) Secx - tanx c) 1/2 d) 2

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

12) If y= x - x²/2 + x³/3- x⁴/4+.....∞, show dy/dx= 1/(1+ x)

13) The derivative of f(logx) w.r.t.x where f(x)= logx, is

a) x/logx b) logx/x c) 1/(x logx) d) none

Day -1

1) Evaluate:

a) sin{2tan⁻¹(1/5) + tan⁻¹(5/12)}.

b) tan[sin⁻¹(1/3) + coa⁻¹(1/√3)}.

2) If A=1   0    2 & Adj A= 5    a     -2

            -1   1   -2                1    1      0

             0   2    1               -2   -2      b then the values of a and b.

A) a=-4, b=1 B) a=-4, b=-2 C) a= 4, b=1 D) a= 4 b= -1

3) If A= -1     0

               0     2 then the value of A³ - A² is

A) I B) A C) 2A D) 2I 

4) If A= -x     -y 

              z       t then the transpose of adj A is ?

5) Find the value of x of following determinant 

   x        2       -1

   2        5       x.    = 0

  -1         2.     x

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

6) Value of Determinant: 1+ a     1      1

                                              1      1+b    1

                                              1       1    1+ c

7) If A= 2     -1

            -1       2 then value of A² is

8) find the Inverse Matrix of 2      1

                                                 7      4

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