**SCRA 2014 Mathematics Paper 3 – Set C Answer Key/Solutions**

*SCRA 2014 Paper 3 – Set C*

Mathematics Paper – 3

Some maths and chemistry figures/numbers are not clear. but answers are correct. Best of luck.

1. In four throws of a fair die, what is the probability of getting a score of more than 4 at least once ?

(a) 65/81

(b) 80/81

(c ) 7/9

(d) None of the above

Ans: a

For the next 03(three) items that follow:

A die is rolled so that the probability of face m is proportional to m, where m=1,2,3,4,5,6 .

2. What is the proportionality constant ?

(a) 1/6

(b) 1/14

(c ) 1/21

(d) 1/36

Ans: c

3. What is the probability of getting an even number ?

(a) 1/2

(b) 1/7

(c ) 4/7

(d) 1/4

Ans: c

4. What is the probability of getting a multiple of 3 ?

(a) 3/7

(b) 2/21

(c ) 2/3

(d) None of the above

Ans: a

5. The probability of a shooter hitting a target is 2/3. What is the minimum number of times that the shooter must fire so that the probability of hitting the target at least once is more than 0.99 ?

(a) 4

(b) 5

(c ) 6

(d) None of the above

Ans: c

6. Consider the following statements about the random variables X and Y on the same sample space S∶

1. (X+Y)(s)=X(s)+Y(s)

2. (XY)(s)=X(s)Y(s)

Which of the above statements is/are correct ?

(a) 1 only

(b) 2 only

(c ) Both 2 and 2

(d) Neither 1 nor 2

Ans: d

7. Consider the following statements :

1. Area under a histogram gives total frequency .

2. Width of the tallest vertical bar of the histogram gives modal class.

Which of the above statements is/are correct ?

(a) 1 only

(b) 2 only

(c ) Both 1 and 2

(d) Neither 1 nor 2

Ans: c

8. Consider the following statements related to measure of central tendency of 50 positive numbers :

1. The median is not influenced by extreme values in the set of numbers.

2. The harmonic mean is unreliable if one or more of the numbers is near zero.

Which of the above statements is/are correct ?

(a) 1 only

(b) 2 only

(c ) Both 1 and 2

(d) Neither 1 nor 2

Ans: c

9. A fair coin is tossed 6 times ; call heads a success. This is a binomial experiment with n=6 and p=q=1/2 . What is the probability of getting at least 4 heads ?

(a) 1/2

(b) 1

(c ) 11/32

(d) None of the above

Ans: c

10. If A and B are two events such that P(A∪B)=3/4,P(A∩B)=1/4,P(not A)=2/3 then what is P(B) equal to ?

(a) 1/3

(b) 2/3

(c ) 1/9

(d) 2/9

Ans: b

11. Consider the following statements :

1. 1/(1-sinA )>2sinA+1/(1+sinA )

2. 1/(1+cosA ) ≤2- 1/(1-cosA )

Where 0^0 <A<〖90〗^0

Which of the above statements is/are correct ?

(a) 1 only

(b) 2 only

(c ) Both 1and 2

(d) Neither 1 nor 2

Ans: a

12. Consider the following statements :

1. If 0< tan〖A<1,〗 then

1/(1-tanA )+cotA/cot〖A-1〗 =cotA/cot〖A+1〗 +1/(1+tanA )

2. If tan〖A>1 ,〗 then

1/(1-tanA )+1/(1+tanA )<0

Which of the above statements is/are correct ?

(a) 1 only

(b) 2 only

(c ) Both 1 and 2

(d) Neither 1 nor 2

For the next 02 (two) items that follow :

Consider S= ∑_(r=2)^n▒〖 sin〖(rα)〗 〗

Ans: b

13. What is S if (n+2)α=2π?

(a) 0

(b) 1

(c ) 1/√2

(d) 1/2

Ans: a

14. What is S of (n-1)α=2π ?

(a) 2

(b) 1

(c ) 1/2

(d) 0

For the next 02 (two) items that follow:

Consider sin〖5θ=5 sin〖θ-20 sin^3〖θ+k sin^5θ 〗 〗 〗

Ans: d

15. What is the value of k ?

(a) 5

(b) 11

(c ) 16

(d) -16

Ans: c

16. What is

40 sin^3θ -32 sin^5〖θ-10 sin〖θ+2 sin5θ 〗 〗 equal to ?

(a) 0

(b) 1

(c ) 2

(d) None of the above

Ans: a

For the next 02 (two) items that follow

Consider

f(x)=2tan^(-1)〖x+sin^(-1)〖 (2x/(1+x^2 )) ,x>1 〗 〗

17. What is f(x) equal to ?

(a) sec^(-1)x

(b) cosec^(-1) x

(c ) π

(d) π/2

Ans: c

18. what is f(5) equal to ?

(a) 5π

(b) π

(c ) π/2

(d) 2π

Ans: b

For the next 02 (two) items that follow:

Let x=(cos〖θ+i sin〖θ)(cos〖2θ+i sin〖2θ)(cos〖3θ+i sin〖3θ)〗 〗 〗 〗 〗 〗

Where θ∈R.

19. If z is real, then which one of the following is correct ?

(a) θ∈(kπ/3 ∶k is an integer } ∪{(2r+1)π/6 ∶ r is an integer }

(b) θ ∈{kπ/3:k is an integer }only

(c ) θ ∈{kπ/2:k is an integer}

(d) None of the above

Ans: d

20. If z is purely imaginary, then which one of the following is correct ?

(a) θ∈ {(4k+1)π/12:k is an integer }

(b) θ ∈{(2k+1)π/12:k is an integer }

(c ) θ∈{kπ/12:k is an integer }

(d) None of the above

Ans: b

For the next 02 (two) items that follow:

The pth , qth , rth terms of an HP are a,b,c respectively.

21. What is |■(bc&ca&ab@p&q&r@1&1&1)| equal to ?

(a) 0

(b) 1

(c ) abc

(d) (abc)^(-1)

Ans: a

22. What is

|■(b^2 c^2+c^2 a^2+a^2 b^2&pbc+qca+rab&bc+ca+ab@pbc+qca+rab&p^2+q^2+r^2&p+q+r@bc+ca+ab&p+q+r&3)|

Equal to ?

(a) (abc)^(-2)

(b) (abc)^2

(c ) 1

(d) 0

Ans: d

For the next 02 (two) items that follow:

Consider the system of equations

x+y+z=1

x+2y+4z=k

x+4y+10z=k^2

23. What is /are the value (s) of k which make(s) the system of equations to possess the solution ?

(a) 0

(b) 1 or 2

(c ) 3 or 4

(d) None of the above

Ans: b

24. Consider the following statements :

1. The system of equations can have infinite solutions for some value of k.

2. The system of equations can have unique solution for some value of k.

Which of the above statements is/are correct ?

(a) 1 only

(b) 2 only

(c ) Both 1 and 2

(d) Neither 1 nor 2

Ans: a

25. For what value (s) of n ≥1 , where n is a natural numebr, A^n-nA+nI=I, where I is the identity matrix and A=[■(1&0@1&1)]?

(a) n=1 only

(b) n=2 only

(c ) For all values of n

(d) None of the values of n

Ans: b

26. Let A and B be two points on x-axis and y-axis respectively, O being the origin. If the equal sides OA and OB, each equal to a, are produced to P and Q respectively such that AP.BQ=OA.OB, then the line PQ always passes through the fixed point

(a) (a/4,a/4)

(b) (a/3,a/3)

(c ) (a/2,a/2)

(d) (a,a)

Ans: d

27. The new position of the point (1, 2) under rotation through an angle of 〖90〗^0 about the origin in anticlockwise direction is

(a) (-2,1)

(b) (2,-1)

(c ) (1,-2)

(d) (-1,2)

Ans: a

28. What is the area of the triangle with vertices at (0,0,0),(2,0,0) and (0,-2,0)?

(a) 1/2 square unit

(b) 1 square unit

(c ) 2 square units

(d) 4 square units

Ans: c

29. Consider two circles

C_1=x^2+y^2=a^2

C_2=(x-α)^2+(y-β)^2=b^2

With C_2 lying inside C_1.A circle C lying inside C_1 touches C_1 internally and C_2 externally. Then the locus of the centre of the circle C is

(a) a circle of radius a-b

(b) a parabola of semilatus rectum a+b

(c ) an ellipse of major axis a+b

(d) None of the above

Ans: c

30. The shortest distance of a point from the x-axis, y-axis and z-axis respectively are 2, 3, 6 . What is the distance of the point from the origin ?

(a) 7/√2

(b) 7

(c ) 11

(d) 49/2

Ans: b

For the next 02 (two) items that follow :

Consider a plane parallel to x-axis and passing through the points (0 ,1, 3) and (2, 4, 5) .

31. What are the direction ratios of normal to the plane ?

(a) (1,2,-3)

(b) (4,-6,0)

(c ) (1,2,3)

(d) None of the above

Ans: d

32. What is the equation to the plane ?

(a) 2y-3z+7=0

(b) x+2y-3z+5=0

(c ) 2y-3x+9=0

(d) None of the above

Ans: a

For the next 03 (three) items that follow :

Consider a unit cube.

33. What is the perpendicular distance of a corner to the diagonal not passing through that corner ?

(a) 2/3

(b) √3/2

(c ) √(2/3)

(d) None of the above

Ans: d

34. What is the sum of squares of direction cosines of all the four diagonals of the cube ?

(a) 1

(b) 2

(c ) 4

(d) Cannot be determined as the data is inadequate

Ans: c

35. If θ is the acute angle between any two diagonals of the cube, then what is tan^2θ equal to ?

(a) 1

(b) 2

(c ) 4

(d) None of the above

Ans: d

36. If □(→┬a ), □(→┬b ) are two vectors inclined at angle θ such that □(→┬a+□(→┬b )) is a unit vector, then what is θ equal to ?

(a) π/3

(b) π/4

(c ) π/6

(d) 2π/3

Ans: d

37. If □(→┬a ) is a non-zero vector of magnitude a, then m□(→┬a ) is a unit vector if

(a) m=±1

(b) a=1/|m|

(c ) a=|m|

(d) a=m

Ans: b

For the next 02 (two) items that follow:

The vectors □(→┬a ) ,□(→┬(b ) ) ,□(→┬c ) are of same length and equally inclined to each other. Let □(→┬a )=i+j and □(→┬b=j+k.)

38. What is the angle between □(→┬b ) and □(→┬c ) ?

(a) π/3

(b) π/4

(c ) π/6

(d) π/2

Ans: a

39. What can be the direction ratios of □(→┬c )?

(a) (1,2,-3)

(b) (-1,2,-1)

(c ) (-1,4,-1)

(d) None of the above

Ans: d

40. If A and B are two matrices such that AB=B and BA=A, then what is A(A-1)+B(B-1) equal to ?

(a) AB

(b) 2AB

(c ) zero matrix

(d) Identity matrix

Ans: c

41. If

I=∫▒〖(e^x dx)/(1+x^2 )^2 ,I_2=〗 ∫▒(xe^x dx)/(1+x^2 )^2 ,I_3=∫▒(x^2 e^x dx)/(1+x^2 )^2

Then what is I_1-2I_2+I_3 equal to ?

(a) e^x/(1+x^2 )+c

(b) e^x/(1+x^2 )^2 +c

(c ) (2e^x)/(1+x^2 )+c

(d) None of the above

Where c is the constant of integration.

Ans: a

42. What is

∑_(r=1)^n▒〖 ∫_0^(π/2)▒〖 (r+sinθ )^2 cos〖θ dθ〗 〗〗

equal to ?

(a) n(n^2+3n+3)/3

(b) n(n+1)(n+2)/3

(c ) n(n+1)(2n+1)/3

(d) (n+1)(n+2)/3

Ans: a

43 What is lim┬(x→1)〖 √(1-cos(2x-2) )/(x-1)〗

Equal to ?

(a) √2

(b) -√2

(c ) 0

(d) Limit does not exist

Ans: d

44. If

I=∫_0^(π/2)▒〖cos〖x dx 〗/(1+cos〖x+sinx 〗 ) 〗

Then what is

∫_0^(π/2)▒〖 dx/(1+cos〖x+sinx 〗 ) 〗

Equal to ?

(a) I/2

(b) I

(c ) π/2-2I

(d) None of the above

Ans: c

45. What is

∫_(-1)^1▒〖xdx/(x^4+x^2+1) 〗

Equal to ?

(a) 0

(b) 1

(c ) 2

(d) None of the above

Ans: a

46. The differential equation

y dy/dx+x=a

Where a is a constant, represents

(a) a set of circles having centre on the y-axis

(b) a set of parabolas

(c ) a set of circles having centre on the x-axis

(d) a set of straight lines

Ans: c

47. What is the degree of the differential equation

((d^3 y)/(dx^3 ))^(2/3)+4-3 (d^2 y)/(dx^2 )+5 dy/dx=0 ?

(a) 1

(b) 2

(c ) 3

(d) 2/3

Ans: d

48. Consider the following statements :

1. ∫_0^a▒〖 f(x)dx-∫_0^a▒〖 f(a-x)dx=0 〗〗

2. 2 ∫_0^π▒〖 xf (cos^2x )dx –π ∫_0^π▒〖 f(cos^2x )dx=0 〗〗

Which of the above statements is/are correct ?

(a) 1 only

(b) 2 only

(c ) Both 1 and 2

(d) Neither 1 nor 2

Ans: a

49. The solution of the differential equation

y[2x sec^2〖(y^2 ) dy/dx+y^3 〗 ]=ln(x^2 e^(y^4 ) ) is

(a) sec^3〖 y^2=(lnx )^2+c〗

(b) sec^3〖 y^2=12 (lnx )+c〗

(c ) tan〖y^2=(lnx )^2+c〗

(d) None of the above

Where c is an arbitrary constant.

Ans: c

50. What is

lim┬(x→0+)〖 [1/x^2 ] ln(cosx ) 〗

Where [. ] denotes the greatest integer function ?

(a) -1/2

(b) -1/3

(c ) 0

(d) Limit does not exist

Ans: c

51. If I_1=∫▒〖 e^2x sin〖(π/3-x) cos〖x dx 〗 〗 〗

I_2=∫▒〖e^2x cos〖(π/3-x) sin〖x dx 〗 〗 〗

Then what is I_1+I_2 equal to

(a) (√3 e^2x sinx)/2+c

(b) (e^2x cosx)/2+c

(c ) (√3 e^2x)/4+c

(d) e^2x/4+c

Where c is the constant of integration.

Ans: c

52. What is the general solution of the equation

dy/dx=(3x-4y+1)/(4x+3y+1) ?

(a) (x+3y)(y-3x )+2(y-x)=c

(b) (x-3y)(y+3x)+2(y-x)=c

(c ) (3y-x)(y+3x )+y(y-x)=c

(d) None of the above

Where c is an arbitrary constant .

Ans: c

53. What is the equation of straight line parallel to the line 3x+2y+7=0 and which is such that the sum of its intercepts on the axes is 10 ?

(a) 3x+2y-12=0

(b) 3x+2y+10=0

(c ) 2y+3y-12=0

(d) 2x-3y-12=0

Ans: a

54. A straight line through P(1 ,2) is such that its intercept between the axes is bisected at P. Its equation is

(a) x+2y=4

(b) 2x-y=4

(c ) 2x+y=4

(d) x-2y=4

Ans: c

55. If the line y=mx meets the lines x+2y-1=0 and 2x-y+3=0 at the same point, then m is equal to

(a) 1

(b) 2

(c ) -2

(d) -1

Ans: d

For the next 02 (two) items that follow:

Consider the function

f(x) ={■(x when x is rational @1-x when x is irrational )┤

On the interval I={0 ,1}.

56. The function is continuous at

(a) x=0.5 only

(b) every point in I

(c ) every rational point in I

(d) every irrational point in I

Ans: a

57. Consider the following statements :

1. f(x) has its own inverse in I.

2. f(x) is differentiable at x=0.5.

Which of the above statements is/are correct ?

(a) 1 only

(b) 2 only

(c) Both 1 and 2

(d) Neither 1 nor 2

Ans: d

58. The function

f(x)=(k sin〖x+2 cosx 〗)/sin〖x+cosx 〗

Is increasing for

(a) k<0

(b) 0<k<1

(c ) 1<k<2

(d) k>2

Ans: d

For the next 02 (two) items that follow:

Consider the function

f(x)=(x-2)^3 (x-1)^2

59. Consider the following statements :

1. The function is neither increasing nor decreasing in the interval [1, 2] .

2. The function has neither relative maximum nor relative minimum at x=2.

Which of the above statements is/are correct ?

(a) 1 only

(b) 2 only

(c ) Both 1 and 2

(d) Neither 1 nor 2

Ans: c

60. Consider the following statements :

1. The function attains relative maximum at x=1.

2. The function attains relative minimum at x=7/5.

Which of the above statements is/are correct ?

(a) 1 only

(b) 2 only

(c ) Both 1 and 2

(d) Neither 1 nor 2

Ans: c

For the next 02 (two) items that follow:

Consider

a_n=∫_0^π▒〖sin(2n-1)x/sinx dx〗

Where n is a natural number.

61. What is a_100-a_99 equal to ?

(a) 0

(b) 1

(c ) π/2

(d) π

Ans: a

62. a_1,a_2,a_3,….,a_n are

(a) in AP only

(b) in GP only

(c ) both in AP and GP

(d) neither in AP nor in GP

Ans: c

For the next 02 (two) items that follow:

Let f(x)=x-ln〖|2x+1 |〗 be defined for

x ∈(-100,1/2)-{-1/2}

63. The function f(x) is monotonically decreasing in the interval

(a) (-1/2,1/2)

(b) (-100,-1/2)

(c ) (1/2,100)

(d) (1/2,1)

Ans: a

64. The function f(x) is monotonically increasing in the interval

(a) (-1/2,1/2)

(b) (-1/2,0)

(c ) (0,1/2)

(d) (-100,-1/2)

Ans: d

For the next 02 (two) items that follow :

Let f(x)=(1-x)^n, where n is a non –negative integer.

65. What is the coefficient of x^n in (1-x)^n?

(a) n

(b) -n

(c ) (-1)^n

(d) None of the above

Ans: c

66. What is

f(0)+f^’ (0)+f”(0)/2!+⋯…+(f^n (0))/n!

Equal to ?

(a) 2^n

(b) 0

(c ) 1

(d) -1

Ans: b

For the next 02 (two) items that follow:

Consider the ellipses 4x^2+y^2=1 and x^2+4y^2=1.

67. What is the area common to both the ellipses ?

(a) tan^(-1)〖 2〗 square units

(b) 2 tan^(-1)2 square units

(c ) 4 tan^(-1)〖2 〗 square units

(d) None of the above

Ans: d

68. What is the bounded area not common to both the ellipses ?

(a) (π-tan^(-1)2 ) square units

(b) (2π-tan^(-1)2 ) square units

(c ) (π-2 tan^(-1)2) square units

(d) None of the above

Ans: d

For the next 02 (two) items that follow :

Consider the functions f(x)=x^2,g(x)=2x+1 and h(x)=x-1/2 on the interval I=[0 ,1].

69. Consider the following statements :

1. The function (fg)(x) is always increasing on I.

2. The function (fh)(x) is always increasing on I.

Which of the statements given above is/are correct ?

(a) 1 only

(b) 2 only

(c ) Both 1 and 2

(d) Neither 1 nor 2

Ans: a

70. Consider the following statements :

1. The function (gh)(x) is always increasing on I.

2. The function (f+g)(x) is always increasing on I.

Which of the above statements is/are correct ?

(a) 1 only

(b) 2 only

(c ) Both 1 and 2

(d) Neither 1 nor 2

Ans: a

71. Addition is not a binary operation on the set

(a) N of natural numbers

(b) {x∶x is a real number and |x|=1}

(c ) Q of rational numbers

(d) R of real numbers

Ans: b

72. The locus of the point of intersection of the straight lines

x/a+y/b=λ and x/a-y/b=1/λ

Where λ is a variable, is

(a) a circle

(b) a parabola

(c ) an ellipse

(d) a hyperbola

Ans: d

73. If the product of n positive numbers is unity, then their sum is

(a) a positive integer

(b) divisible by n

(c ) equal to (n^2+1)/n

(d) never less than n

Ans: d

74. The number of numbers between 1 and 〖10〗^10, which contain the digit 1, is

(a) 〖10〗^10-9^10-1

(b) 9^10

(c ) 〖10〗^10 -9^10

(d) None of the above

Ans: c

75. If a,b,c are any three consecutive terms in an AP, then the line ax+by=c=0

(a) has a fixed direction

(b) passes through the origin (c ≠0)

(c ) always passes through a fixed point

(d) None of the above

Ans: c

76. A five –digit number divisible by 3 is to be formed using the numbers 0, 1, 2, 3, 4, 5 without repetition. The total number of ways in which this can be done is

(a) 216

(b) 240

(c ) 600

(d) 3125

Ans: a

77. If α,β are the roots of the equation

ax^2+3x+2=0 (a<0 )

Then α^2/β+ β^2/α is greater than

(a) 1

(b) 2

(c ) 3

(d) None of the above

Ans: d

78. The number of terms in the expansion of (2x+3y-4z)^n, which n is a positive integer, is

(a) n+1

(b) (n+1)(n+2)/2

(c ) n(n+1)/2

(d) (n-1)(n-2)/2

Ans: b

79. If sin〖(x-y),sinx 〗 and sin(x+y) are in HP, then [sin〖x sec〖 (y/2〗)] 〗 is equal to

(a) ±√2

(b) ±1

(c ) ±3

(d) ±2

Ans: a

80. If tanA=(1-cosB)/sinB

Then what is tan2A equal to ?

(a) tan〖B 〗

(b) tan〖2B 〗

(c ) sinB

(d) cosB

Ans: a

81. If an angle α is divided into two parts A and B such that

A-B=x and tanA/tanB =k

Then what is sinx equal to ?

(a) (k+1)/(k-1) sinα

(b) (k+1)/((k-1) sinα )

(c ) (k-1)/((k+1) sinα )

(d) (k-1)/(k+1) sinα

Ans: d

82. What is the sum of the first 30 terms of the series 1×2+2×3+3×4+⋯?

(a) 21010

(b) 8920

(c ) 22100

(d) 9920

Ans: d

83. The total number of ways of selecting tow numbers from the set {1, 2, 3, …, 30} , so that their sum is divisible by 3, is

(a) 95

(b) 145

(c ) 190

(d) None of the above

Ans: b

84. The sum of n terms of the series

1+(1+x)+(1+x+x^2 )=(1+x+x^2+x^3 )+⋯…

Where x<1 , is

(a) 1/(1-x)

(b) n/(1-x)

(c ) n/(1-x) – n(1-x^n )/(1-x)^2

(d) n/(1-x) – n(1-x^(n+1) )/(1-x)^2

Ans: c

85. The 5th term from the end in the expansion of (x-1/x)^3n in increasing power of x, is [n is a positive integer ]

(a) x^(8-3n)

(b) x^(7-3n)

(c ) x^(3n-4)

(d) None of the above

Ans: d

86. The function f:R →R defined by

f(x)=(x-a)(x-b)(x-c)

Where a,b,c ∈R, is

(a) not one –one but onto

(b) one –one but not onto

(c ) both one –one and onto

(d) neither one –one nor onto

Ans: a

87. If A={1,2,3,4}, then which of the following is/are the function(s) from A to itself ?

I. f_1={(x,y) ┤|x+y=5}

II. f_2={(x,y) ┤|y<x }

Select the correct answer using the code given below.

(a) I only

(b) II only

(c ) Both I and II

(d) Neither I nor II

Ans: a

88. If f:R →R be given by

y=f(x)=(x+1)^2-1

Then f(x) is invertible if

(a) y ≥-1

(b) -2≤y<-1

(c ) -3≤y<-2

(d) None of the above

Ans: d

89. The complex numbers z satisfying z^2+|z|=0 are

(a) 0 ,i-i

(b) 0 ,1,i ,-i

(c ) 0 ,1,-1,i,-i

(d) 0,-1

Ans: a

90. If z_1,z_2,z_3 are complex numbers such that

|z_1 |=|z_2 |=|z_3 | =|1/z_1 +1/z_2 +1/z_3 |=1

Then what is |z_1+z_2+z_3 | equal to ?

(a) Less than 1

(b) Lies between 1 and 3

(c ) 1

(d) 3

Ans: c

91. IF the lines x+2ay+a=0 ,x+3by+b=0 and x+4cy+c=0 are concurrent, then a,b,c are in

(a) HP

(b) AP

(c ) GP

(d) None of the above

Ans: a

92. Let (α,β),(β,γ) and (γ,α) be the roots of the equations x^2+px+qr=0 ,x^2+qx+rp=0, x^2+rx+pq=0 respectively. Then the product of their common roots (αβγ) is equal to

(a) pqr

(d) 2 pqr

(c ) 2 pqr

(d) p^2 q^2 r^2

Ans: a

93. The roots of the equation

qx^2-px+(0.5 p-0.25 q)=0

When p<q, where p,q are real numbers, are always

(a) irrational

(b) real

(c ) complex

(d) rational

Ans: b

94. If z is a complex number, then the common roots of the equations

z^1985+z^100+1=0

z^3+2z^2+2z+1=0 are

(a) ω ,ω^2

(b) 1,ω ,ω^2

(c ) -1,ω,ω^2

(d) -ω ,-ω^2

Ans: a

95. The range of θ in the interval (0,π) such that the points (3, 5) and (sin〖θ,cosθ 〗 ) lie on the same side of the line x+y-1=0 , is

(a) (0,π/4)

(b) (0,π/2)

(c ) (π/2,π)

(d) (0,π)

Ans: b

96. A line passing through the point (2, 2) encloses an area λ with the axes. The intercepts on the axes made by the line are given by the two roots of

(a) x^2+2|λ|x+|λ|=0

(b) x^2-2 |λ|x+|λ|=0

(c ) x^2+ | λ |x+2| λ |=0

(d) x^2-| λ |x+2|λ|=0

Ans: d

97. The area bounded by the curve y=2x^4-x^2, the x-axis and the two ordinates corresponding to minimal of the function is

(a) 1/40 square unit

(b) 7/120 square unit

(c ) 1/24 squre unit

(d) None of the above

Ans: b

98. If a ≤3 cos〖x+5 sin〖(x-π/6)≤b 〗 〗

Holds good for all x, then a and b are respectively

(a) -4,4

(b) -√19,√19

(c ) -√29,√29

(d) -8 ,8

Ans: b

99. In a triangle ABC

sin〖A sin〖B sinC 〗 〗=(3+√3)/8

cos〖A cos〖B cos〖C=(√3-1)/8〗 〗 〗

Then what is the value of tan〖a+tan〖B+tan〖C 〗 〗 ? 〗

(a) √3 (2-√3)

(b) 2+√3

(c ) 2-√3

(d) √3 (2+√3)

Ans: d

100. The graph of the function

y=cos〖x cos〖(x+2)-cos(x^2+1) 〗 〗 is a

(a) straight line passing through the point (0,sin^21 ) and parallel to x-axis

(b) straight line passing through the origin

(c ) parabola with vertex (0,sin^21 )

(d) None of the above

Ans: a

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