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SCRA 2015- Paper 3 Key MATHS

 

Download SCRA Paper: CLick here

SCRA 2015 Paper 3 - SET D - STudent are requested to send scanned copy of question paper.

SCRA 2015

MATHEMATICS

 

 

1.   If , then the function g(x) will be of the form

      (a)                                                       (b)    

      (c)                                                          (d)    

      where a and b are non-zero constants.

 

 

2.   What is the area bounded by the curves  and ?

      (a)     e – 1                                                       (b)     e – 2

      (c)     3 – e                                                       (d)     e

 

 

 

For the next three (03) items that follow:

Consider the function

 

3.   The function attains local maximum at

      (a)     x = 0                                                       (b)     x = 1

      (c)     x = 2                                                       (d)     x = 4

 

 

4.   What is the local maximum value of the function?

      (a)     0                                                             (b)     1

      (c)     4                                                             (d)     16

 

 

5.   Consider the following statements:

      1.      The function attains local minimum value at .

      2.       is the point of inflexion.

      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 two (02) items that follow:

Consider  and

              

 

6.   For  is equal to

      (a)     x – 1                                                       (b)     1 – x

      (c)     x + 1                                                       (d)     – x – 1

 

 

7.   Consider the following statements:

      1.      For

      2.      For

      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 three (03) items that folow :

Let be defined by

 for  and .

 

8.   The function f(x) is  

      (a)     continuous and differentiable at x = 0

      (b)     nowhere continuous over R

      (c)     continuous at x = 0, but not differentiable at x = 0

      (d)     nowhere differentiable over R.

 

 

9.   The function f(x) has

      (a)     local maximum at x = 0

      (b)     local minimum at x = 0 but it has no absolute minimum at x = 0

      (c)     absolute minimum at x = 0

      (d)     absolute maximum at x = 0.

 

 

10. Let  and  where n N. the derivative of the function f(x) attains

      (a)     positive value at x1 and negative value at x2.

      (b)     positive value at x1 and positive value at x2.

      (c)     negative value at x1 and positive value at x2.

      (d)     negative value at x1 and negative value at x2.

 

 

1 d

2 c

3b

4a

5c

6a

7c

8a

9c

10b

11. How many integral points are there within the graph of |x| + |y| < 3?

      (a)     13                                                           (b)     15

      (c)     21                                                           (d)     24

 

 

12. The distance of the point (4, 5) from the straight line joining the points (1, 2) and (–2, 3) measured parallel to the line x + y + 1 = 0 is

      (a)     4 units                                                    (b)     4  units

      (c)     6 units                                                    (d)     6  units

 

 

13. A double ordinate of the parabola y2 = 4ax is of length 8a. What is the angle between the lines from the vertex to its ends?

      (a)     30o                                                                                                                    (b)     45o

      (c)     60o                                                          (d)     90o

 

 

14. For how many values of k, the line 3x – 4y = k may touch the circle x2 + y2 – 8y – 5 = 0?

      (a)     1                                                             (b)     2

      (c)     3                                                             (d)     None of the values of k

 

 

15. What is one of the angles between the straight lines

     

      (a)                                                                  (b)    

      (c)                                                               (d)    

 

 

16. Let S be any set and P(S) be its power set. We define a relation R on P(S) by ARB to mean  for all A, B P(S). Consider the following in respect of the relation R :

      1.      R is a reflexive relation.

      2.      R is an anti-symmetric relation.

      3.      R is a symmetric relation.

      4.      R is a transitive relation.

      Which of the above are correct?

      (a)     1, 3 and 4                                               (b)     3 and 4 only

      (c)     1, 2 and 4                                               (d)     1 and 2 only

 

 

17. What is the real part of , where  ?

      (a)                                                     (b)    

      (c)                                                         (d)    

 

 

18. If  is devided by 13, then the remainder is

      (a)     1                                                             (b)     5

      (c)     8                                                             (d)     11

 

 

19. The number of consecutive odd integers whose sum can be expressed as 502 – 132 is

      (a)     33                                                           (b)     35

      (c)     37                                                           (d)     39

 

 

20. A group of order 4 is

      (a)     always cyclic                                          (b)     always non-cyclic

      (c)     abelian and may not be cyclic               (d)     always non-cyclic

 

21. If a and b rational and (b2 + 1) is not a perfect square, then the quadratic equation with rational coefficients whose one root is  is

      (a)     x2 – 2abx – a2 = 0                                   (b)     4x2 – 4abx – a2 = 0

      (c)     x2 – abx – a2 = 0                                     (d)     x2 – abx + a2 = 0

 

 

22. If 2|z – 1| = |z – 2| and 3(x2 + y2) = kx, then what is k equal to?

      (a)     2/3                                                          (b)     4/3

      (c)     4                                                             (d)     1

 

 

23. Let a1, a2, a3, .... be a sequence of real numbers such that  for  and a­­1 = 0. If A denotes the arithmetic mean of a1, a2, a3, ...., an then which one of the following is correct?

      (a)                                         (b)    

      (c)                                  (d)    

 

 

24. If A is a non-singular matrix of order 3, then what is adj(adj A) equal to?

      (a)                                                           (b)    

      (c)                                                            (d)    

 

 

25. If A, B and C are the angles of an isosceles triangle, then what is

       equal to?

      (a)     0                                                             (b)     1

      (c)     sin A . sin B . sin C                                  (d)     None of the above

 

 

26. Let A and B be two 3  3 matrices whose determenants are 2 and  4 respectively. What is det(adj(A-1B)) equal to ?

      (a)     |A|                                                         (b)     |B|

      (c)     4|A|                                                       (d)     4|B|

 

 

27. Let S be the set S = {2, 4, 6, 8, …, 20}. Define the operation pnq as remainder when pq is divided by n. Then the inverse of the element 2 in (S, 22) is

      (a)     12                                                           (b)     8

      (c)     6                                                             (d)     4

 

 

28. If |z - 25|  15 where  = , then what is |max amp (z) – min amp (z)| equal to?

      (a)                                                    (b)    

      (c)                                             (d)    

 

 

29. If the quadratic equation

      Where p is real, has its real roots greater than p, then p lies in the intervel

      (a)                                                         (b)    

      (c)                                                        (d)    

 

 

30. What is the sum of the 10 terms of the series ?

      (a)                                (b)    

      (c)                                (d)    

 

 

31. If  and

               , then what is  equal to ?

      (a)     4                                                             (b)    

      (c)     1                                                             (d)     0

 

 

 

32. Let F(x) be a twice difference function with F''(x) = – F(x) and F'(x) = G(x). If H(x) = {F(x)}2 + {G(x)}2 and H(5) = 5, then what is H(0) equal to?

      (a)     0                                                             (b)     5

      (c)     9                                                             (d)

 

 

33. f(a) = 2, f'(a) = 1, g(a) = –1, g'(a) = 2, then what is  equal to?

      (a)                                                               (b)    

      (c)                                                                  (d)    

34. The function f(x) = ex (1 – x2) is

      (a)     increasing for x >

      (b)     decreasing for x <

      (c)     increasing for |x – 1| <

      (d)     increasing for |x + 1| <

 

 

35. Consider the function f : [0, ]  [0, 1] defined by f(x) = sin. The function f is

      (a)     one-one                                                  (b)     onto

      (c)     both one-one and onto                          (d)     neither one-one nor onto

 

 

36. What is  equal to?

      (a)                                                              (b)    

      (c)                                                              (d)    

 

 

37. If f(x) is a second order polymial (or quadratic expression in x) and

      ,

      then f(x) will be of the form

      (a)                                               (b)    

      (c)                                                   (d)    

 

 

38. If  and , then which one of the following is correct?

      (a)                                                (b)    

      (c)                                                 (d)    

39. If a > 1, b > 1 then the minimum value of logab + logba is

      (a)     0                                                             (b)     2

      (c)     1                                                             (d)     None of the above

 

 

40. If , then which of the following is/are correct?

      1.       for

      2.       for

      3.       for

      Select the correct answer using the code given below:

      (a)     1 only                                                     (b)     1 and 2 only

      (c)     2 and 3 only                                           (d)     2 and 3 only

 

 

41. The equation of the curve passing through the point (0, 1) and having x3y–3 as the slope of the tangent to the curve at any point (x, y) is

      (a)     x4 – y4 + 1 = 0                                         (b)     x4 + y4 – 1 = 0

      (c)     x3 + y3 – 1 = 0                                         (d)     x3 – y3 + 1 = 0

 

 

42. If  is the integrating factor of the differential equation  and  is the integrating factor of the differential equation , then which one of the following is not correct?

      (a)                                                       (b)    

      (c)                                                     (d)      for

 

 

43. Consider the following differential equations:

      1.     

      2.     

      3.     

      4.     

      How many of the above are homogeneous?

      (a)     One                                                        (b)     Two

      (c)     Three                                                      (d)     Four

 

 

44. If  where x > 0 and , then  is

      (a)     less than 2                                              (b)     greater than 2

      (c)     greater than or equal to 2                     (d)     equal 2

 

 

45. What is , where [·] denotes the greatest integer function, equal to?

      (a)     4                                                             (b)     3

      (c)     2                                                             (d)     1

 

 

46. A straight line passes through a fixed point (h, g). The locus of the foot of the perpendicular on it drawn from the origin is

      (a)     a straight line                                         (b)     an ellipse

      (c)     a parpabola                                            (d)     a circle

 

 

11a

12d

13d

14b

15b

16c

17d

18c

19c

20c

21b

22c

23a

24c

25a

26b

27a

28b

29a

30a

31d

32b

33c

34d

35a

36c

37d

38c

39b

40d

41a

42c

43c

44b

45d

46d

47. If the three distinct points  for  are collinear, then the sum of the abscissa of the points is

      (a)     –1                                                           (b)     0

      (c)     1                                                             (d)     3

 

 

48. Let (a, b) and (c, d) be two points in a plane. Any point on the line joining these points has coordinates

      (a)     (a + kc, b + kd)

      (b)     (ka + c, kb + d)

      (c)     ((1 – k) a + kc, (1 – k) b + kd)

      (d)     (a + (1 – k) c, b + (1 – k) d)

      where k is any real number.

 

 

49. The equation

     

      represents a sphere of radius

      (a)     2 units                                                    (b)     3 units

      (c)     4 units                                                    (d)     5 units

 

 

50. What is , where , equal to?

      (a)                                                    (b)    

      (c)                                                        (d)     None of thje above

 

 

For the next three (03) items that follow:

Consider a point A(–2, 3 0) above the line PQ. The line PQ passes through P(–3, 5, 2) and makes equal angles with the coordinate axes.

 

51. What are the coordinates of the foot of the perpendicular from A on the line PQ?

      (a)     (–4, 4, 1)                                                 (b)     (4, 4, 1)

      (c)     (– 2, 2, 1)                                                (d)     (2, 2, 1)

 

 

52. What are the direction ratios of the line perpendicular to the line PQ?

      (a)     < 2, 1, – 1 >                                            (b)     < – 2, 1, 1 >

      (c)     < 4, 1, 1 >                                               (d)     < 1, 1, 1 >

 

 

53. What is the square of the perpendicular distance of the point A from the line PQ?

      (a)     4                                                             (b)     5

      (c)     6                                                             (d)     9

 

For the next two (02) items that follow:

A variable plane  at unit distance from the origin cuts the coordinate axes at A, B and C respectively. The centroid of the triangle ABC satisfies the equation .

 

54. The centroid of the triangle is at

      (a)                                                       (b)    

      (c)                                                  (d)    

 

 

55. The value of k is

      (a)                                                                   (b)    

      (c)     3                                                             (d)     9

 

 

56. if ABCDEF is a regular hexagon with   and  , then what is   equal to?

      (a)                                                           (b)    

      (c)                                                         (d)    

 

 

57. If (0, 1) and (1, 0) are mid-points of the sides of a right-angled triangle, then consider the following statements:

      1.      (0, 0) can be the orthocentre of the triangle.

      2.      (1, 1) can be the orthocentre of the triangle.

      Which of the above statements is/are correct?

      (a)     1 only                                                     (b)     2 only

      (c)     Either 1 and 2                                        (d)     Neither 1 nor 2

 

 

 

For the next three (03) items that follow :

The vector  and  are orthogonal and a vector  makes an abtuse angle with z-axis.

 

58. What is/are the permissible value(s) of ?

      (a)     – 2 only                                                  (b)     3 only

      (c)     Both – 2 and 3                                        (d)     Neither –2 nor 3

 

 

59. in which quadrant does  lie?

      (a)     First quadrant                                        (b)     Second quadrant

      (c)     Third quadrant                                       (d)     Fourth quadrant

 

 

60. What is  equal to?

      (a)    

      (b)    

      (c)     ;

      (d)     None of the above

      where n is an integer.

 

 

47b

48c

49c

50d

51a

52b

53c

54b

55c

56b

57c

58a

59b

60c

61. A square matrix of third order is said to be skew-symetric if

      (a)     All elements of leading diagonal are zero

      (b)    

      (c)     All elements of leading diagonal are 1

      (d)    

 

 

62. The equations

      , ,  has no solution if

      (a)      only                                              (b)    

      (c)      or                                            (d)      only

 

 

63. What is the value of

     

      where

      (a)                                                               (b)    

      (c)     2                                                             (d)     3

 

 

64. The function

     

      is differentiable at x = 0

      (a)     only when  

      (b)     only when

      (c)     only when

      (d)     for any values of  and

 

 

65. If the complex numbers  are in AP, they lie on

      (a)     a circle                                                   (b)     a line

      (c)     a parabola                                              (d)     an ellipse

 

 

66. Which one of the following binary operations is associative on the set of real numbers?

      (a)     a ∗ b = ab                                              (b)     a ∗ b = a + b – 1

      (c)     a ∗ b = , b  0                                     (d)     a b = a – b

 

 

67. All the fourth roots of unity are

      (a)                                                 (b)    

      (c)                                                  (d)    

      where .

 

 

68. Consider the following in respect of the equation (x + 2)2 – 3|x + 2| = 0:

      1.      The sum of all possible rootsof the equation is – 8.

      2.      The product of all possible roots of the equation is 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

 

 

69. Three straight lines  are parallel and lie on the same plane. 5 points are taken on line , 6 points are taken on line  and 7 points are taken on line . What is the maximum number of triangles formed with vertices at these points?

      (a)     620                                                         (b)     746

      (c)     751                                                         (d)     781

 

 

70. Consider the following statements in respect of the expansion  :

      1.      Independent term does not exist in the expansion.

      2.      The coeficient of x is equal to cofficient of  in the expansion.

      Which of the above statements is/are correct?

      (a)     1 only                                                     (b)     2 only

      (c)     Both 1 and 2                                           (d)     neither 1 nor 2

71. If , how many solutions of  are possible?

      (a)     Only one                                                 (b)     Two

      (c)     Four                                                        (d)     No solution is possible

 

 

72. Consider the following statements

      1.       for any , where

      2.       for all , where

      Which of the above statements is/are correct?

      (a)     1 only                                                     (b)     2 only

      (c)     Both 1 and 2                                           (d)     neither 1 nor 2

 

 

73. Let  is not of the form  for any

      Consider the following statements:

      Statement      :    

There exist one  for which there exists no

      Statement     :    

                                    For any , .

      Which one of the following is correct in respect of the above statements?

      (a)     Both the statements are true and Statement  is the correct explanation of Statement

      (b)     Both the statements are true but Statement is not the correct explanation of Statement

      (c)     Statement  is true, but Statement  is false

      (d)     Statement  is false, but Statement  is true

 

 

74. Consider the following statements:

      Statement      :    

There exist no triangle ABC satisfying , where R is the circum-radius of the triangle ABC.

      Statement     :    

                              If ABC is an isosceles triangle satisfying , then .

      Which one of the following is correct in respect of the above statements?

      (a)     Both the statements are true and Statement is the correct explanation of Statement

      (b)     Both the statements are true but Statement is not the correct explanation of Statement

      (c)     Statement  is true, but Statement  is false

      (d)     Statement  is false, but Statement  is true

 

 

75. Let ABC be a triangle with .

      Statement      :    

If , then .

      Statement     :    

.

      Which one of the following is correct in respect of the above statements?

      (a)     Both the statements are true and Statement is the correct explanation of Statement

      (b)     Both the statements are true but Statement is not the correct explanation of Statement

      (c)     Statement  is true, but Statement  is false

      (d)     Statement  is false, but Statement  is true

 

 

76. The number 1, 2, 3, 4, 5, 6, 7, 8 are arranged in a random order. The probability that the digits 1, 2, 3, 4 appear as neighbours in that order is

      (a)                                                                   (b)    

 

      (c)                                                                (d)    

 

 

77. The average marks of 10 students in a class was 60 with a standard deviation 4, while the average marks of other 10 students was 40 with a standard deviation 6. If all the 20 students are taken together, their standard deviation will be

      (a)     5.0                                                          (b)     7.5

      (c)     9.8                                                          (d)     11.2

 

 

78. The two lines of regression of y on x and x on y are 5y + 4x = 37 and y + 5x 20 respectively. The correlation between x and y will be

      (a)                                                                   (b)    

 

      (c)                                                                   (d)    

 

 

79. Correlation between two variable X and Y is given to be 0.6. These variables are transformed to new variables u = –2X + 3 and v = 5Y – 2. What will be the correlation between u and v?

      (a)     0.6                                                          (b)     –0.6

      (c)     0.2                                                          (d)     Information is insufficient

 

 

80. If A and B are any two events with P(A) = 0.6, P(B) = 0.3 and P(A  B) = 0.2, what will be P(Ac|Bc), where Ac is the complementary event of A?

      (a)                                                                   (b)    

 

      (c)                                                                   (d)    

81. A point is chosen at random inside a rectangle measuring 5 inches. What is the probability that the point chosen at random inside the rectangle is at least one inch from the edge?

      (a)                                                                   (b)    

 

      (c)                                                                   (d)    

 

 

82. A box contains three types of seeds : 50% of type A; 20% of type B and rest of type C. It is known that 20% of A, 30% of B and 30% of C germinate. A seed is drown randomly from the box. What is its probability to germinate?

      (a)     0.25                                                        (b)     0.50

      (c)     0.80                                                        (d)     1

 

 

83. A box contains a fair coin and a two-headed coin B. A coin is selected at random from the box and tossed twice. If head comes both the times, the probability that it is by the two-headed coin is

      (a)     1/4                                                          (b)     1/2

      (c)     4/5                                                          (d)     5/8

 

 

84. Some urns contain 4 white and 6 black balls, while one urn contains 5 white and 5 black balls. One urn is chosen at random from these and 2 balls are drawn from it, and both are found to be black. The probability that 5 white and 3 black balls remain in the chosen urn is 1/7. The total number of urns is

      (a)     4                                                             (b)     5

      (c)     6                                                             (d)     7

 

 

85. n observations on a variable  and  for  where A, B are real constants. The mean of the observations is

      (a)                                                (b)    

      (c)                                              (d)    

 

 

86. It is given that . Which one of the following is not correct?

      (a)    

      (b)    

      (c)    

      (d)    

 

87. Consider the following statements:

      1.      If ,  are supplementary angles and , then

      2.      if ,  are complementary angles and , then

      Which of the above statements is/are correct?

      (a)     1 only                                                     (b)     2 only

      (c)     Both 1 and 2                                           (d)     neither 1 nor 2

 

 

88. Consider the following statements:

      1.      If ,  are acute angles and  and , then .

      2.      If ,  are the angles in the second quadrant and , then

      Which of the above statements is/are correct?

      (a)     1 only                                                     (b)     2 only

      (c)     Both 1 and 2                                           (d)     neither 1 nor 2

 

 

 

89. What is the maximum value of

     

      (a)     11                                                           (b)     10

      (c)     5                                                             (d)     1

 

 

90. Consider the following statements:

      1.      If , then

      2.      If , then

      Which of the above statements is/are correct?

      (a)     1 only                                                     (b)     2 only

      (c)     Both 1 and 2                                           (d)     neither 1 nor 2

 

 

91. For how many distinct values of A between  and  is the expression  undefined?

      (a)     2                                                             (b)     4

      (c)     6                                                             (d)     8

 

 

92. In a triangle ABC if  are in HP, what is the value of

      (a)                                                               (b)    

      (c)     1                                                             (d)     2

 

 

93. If the function f(x) = sin x + cos (xa) is periodic then ‘a’ is

      (a)     always a natural number

      (b)     always an integer

      (c)     an irrational number

      (d)     a rational number

 

 

94. At how many points do y = x and y = tan x intersect?

      (a)     Zero                                                        (b)     Only one

      (c)     Two                                                        (d)     Infinite

 

 

95. ABCDEFG is a 7-sided polygon which is not regular. If its angles are in AP, then which one of the following is correct?

      (a)     Exactly three of its angles are greater than 125o.

      (b)     Exactly four of its angles are greater than or equal to the angle of a regular polygon of 7-sides.

      (c)     Exactly three of its angles are less than or equal to  radian.

      (d)     The sum of the greatest angle and the least angle is greater than  radian.

 

 

61d

62d

63c

64c

65a

66b

67c

68c

69c

70b

71b

72b

73a

74d

75b

76b

77d

78d

79a

80a

81a

82a

83a

84c

85d

86d

87c

88a

89b

90b

91c

92a

93d

94d

95b

96. If  then what is  equal to ?

      (a)                                                   (b)    

      (c)                                                                (d)    

      where dashes denotes the derivative with respect to t.

 

 

97. Let f(x) = sin x, g(x) = x2 and h(x) =  x be functions of real variable x > 0. Suppose fog(x) means f[g(x)]. If F(x) = [(hog)of](x), what is F’’(x) equal to ?

      (a)     2 cosec2 x                                               (b)     2 sec2 x

      (c)     – 2 cosec2 x                                            (d)     None of the above

 

 

98. If f(x) = a  |x| + bx2 + x has its extreme valeus at x = – 1 and x = 2, then what is the value of ‘a’?

      (a)     1                                                             (b)     2

      (c)     – 1                                                          (d)     – 2

 

 

99. If g(x) = x3 and 3 f(x) = 4x3 – 12x where 0  x  2, then g[f(x)]will attain its greatest value at

      (a)     x = 2                                                       (b)     x = 0

      (c)     x = 1                                                       (d)     x =

 

 

100.  If 5y = - 3[x] + 4[tan x] + 3|y| where [.] is the greatest integer function, then y as a function of x is

         (a)     not continuous at x = 0

         (b)     continuous at x = 0

         (c)     differentiable at x = 0

            (d)        continuous at x = 0 but

 

96b

97c

98b

99a

100a

 

 

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