1. If the sum of binomial coefficients in the expansion of (x + y)ⁿ is 256, then the greatest binomial coefficient occurs in which one of the following terms?
(a) Third
(b) Fourth
(c) Fifth
(d) Ninth
2. If k < (√2 + 1)³ < k + 2, where k is a natural number, then what is the value of k?
(a) 11
(b) 13
(c) 15
(d) 17
3. If [x 1 1] [1 2 3 ; 4 5 6 ; 7 8 9] [1] = [45]
then which one of the following is a value of x?
(a) -2
(b) -1
(c) 0
(d) 1
4. If A = [ y z x ; z x y ; x y z ]
Where x, y and z are integers, is an orthogonal matrix, then what is the value of x² + y² + z²?
(a) 0
(b) 1
(c) 4
(d) 14
5. Consider the following in respect of a nonsingular matrix M:
I. |M²| = |M|²
II. |M| = |M⁻¹|
III. |M| = |Mᵀ|
How many of the above are correct?
(a) None
(b) One
(c) Two
(d) All three
6. If f(θ) = [ cos θ sin θ ; -sin θ cos θ ] then what is (f(π))² equal to?
(a) [ -1 0 ; 0 -1 ]
(b) [ 1 1 ; 1 1 ]
(c) [ -1 0 ; 0 1 ]
(d) [ 1 0 ; 0 1 ]
7. If A = [ 1 2 2 ; 2 1 2 ; 2 2 1 ] then what is A² - 4A equal to?
(a) -5I₃
(b) -I₃
(c) I₃
(d) 5I₃
Where I₃ is the identity matrix of order 3.
8. If the number of selections of r as well as (n + r) things from 5n different things are equal, then what is the value of r?
(a) n
(b) 2n
(c) 3n
(d) 4n
9. What is the number of selections of at most 3 things from 6 different things?
(a) 20
(b) 22
(c) 41
(d) 42
10. If A = [ x y z ; y z x ; z x y ] where x, y and z are integers is an orthogonal matrix, then what is A² equal to?
(a) Null matrix
(b) Identity matrix
(c) A
(d) -A
Direction for Questions (11-13): Consider the following for the three (03) items that follow:
Let p = tan 2α - tan α and q = cot α - cot 2α
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11. What is (p/q) equal to?
(a) -tan α·tan 2α
(b) cot α·cot 2α
(c) tan α·tan 2α
(d) cot α·cot 2α
12. What is (p + q) equal to?
(a) sec 4α
(b) cosec 4α
(c) 2sec 4α
(d) 2cosec 4α
13. What is tan²α equal to?
(a) (pq)/(p + q)
(b) (p + 2q)/p
(c) p/(p + 2q)
(d) p/(2p + q)
Direction for Questions (14-15): Consider the following for the two (02) items that follow:
Let 2sin α + cos α = 2, where 0 < α < 90°
14. What is tan α equal to?
(a) 1/2
(b) 1
(c) 3/4
(d) 2
15. What is 2 sin 2α + cos 2α equal to?
(a) 11/10
(b) 11/5
(c) 12/5
(d) 13/5
Direction for Questions (16-17): Consider the following for the two (02) items that follow:
In a triangle ABC, two sides BC and CA are in the ratio 2:1, and their opposite corresponding angles are in the ratio 3:1.
16. One of the angles of the triangle is
(a) 15°
(b) 30°
(c) 45°
(d) 75°
17. Consider the following statements:
I. The triangle is right-angled.
II. One of the sides of the triangle is 3 times the other.
III. The angles A, C and B of the triangle are in AP.
Which of the statements given above is/are correct?
(a) I only
(b) II and III only
(c) I and III only
(d) I, II and III
18. A man at M standing 100 m away from the base (P) of a chimney of height 50 m. He observes the angle of elevation of the highest point (Q) of the smoke to be 45°. The highest point of the chimney is at R. Further P, R and Q are in a straight line and the straight line is perpendicular to PM. What is the angle RMQ equal to?
(a) tan⁻¹(1/2)
(b) tan⁻¹(1/3)
(c) tan⁻¹(2/3)
(d) tan⁻¹(3/4)
19. If k is a root of x² - 4x + 1 = 0, then what is tan⁻¹k + tan⁻¹(1/k) equal to?
(a) -π/2
(b) 0
(c) π/4
(d) π/2
20. If tan⁻¹k + tan⁻¹(1/2) = π/4
(a) 1
(b) 1/2
(c) 1/3
(d) 1/4
21. If a line in 3 dimensions makes angles α, β and γ with the positive directions of the coordinate axes, then what is cos(α + β) cos(α - β) equal to?
(a) cos²γ
(b) -cos²γ
(c) sin²γ
(d) -sin²γ
22. A(1, 2, -1), B(2, 5, -2) and C(4, 4, -3) are three vertices of a rectangle. What is the area of the rectangle?
(a) 8 square units
(b) 9 square units
(c) √66 square units
(d) √68 square units
23. ABC is a triangle right-angled at B. If A(k, 1, -1), B(2k, 0, 2) and C(2 + 2k, k, 1) are the vertices of the triangle, then what is the value of k?
(a) -3
(b) -1
(c) 1
(d) 3
24. If a line (x + 1)/p = (y - 1)/q = (z - 2)/r
Where p = 2q = 3r makes an angle with the position direction of the y-axis then what is cos 29 equal to?
(a) -31/49
(b) -37/49
(c) 31/49
(d) 37/49
25. What is the equation of the plane passing through the point (1, 1, 1) and perpendicular to the line whose direction ratio is (3, 2, 1)?
(a) x + 2y + 3z = 6
(b) 3x + 2y + z = 6
(c) x + y + z = 3
(d) 3x + 2y + z = 0
26. A line makes angles α, β and γ with the positive directions of the coordinate axes.
If a = (sin²α)i + (sin²β)j + (sin²γ)k and
b = i + j + k then what is a·b equal to?
(a) -2
(b) -1
(c) 1
(d) 2
27. Consider the following statements with respect to a vector a = (a × b) × c
I. a is coplanar with a and a
II. a is perpendicular to c
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36. If p, 1 and q are in AP and p and 2q are in GP then which of the following statements is/are correct?
Which of the statements given above is/are correct?
(a) I only
(b) II only
(c) Both I and II
(d) Neither I nor II
28. The position vectors of three points A, B and C are a, b and c respectively, such that 3a - 4b + c = 0. What is AB:BC equal to?
(a) 3:1
(b) 1:3
(c) 3:4
(d) 1:4
29. The position vectors of three points A, B and C are a, b and c respectively, where c = (cos²θ)a + (sin²θ)b. What is (a × b) + (b × c) + (c × a) equal to?
(a) 0
(b) 2c
(c) 3c
(d) unit vector
30. Let a, b and (a × b) be unit vectors. What is (a, b) equal to?
(a) 0
(b) 1/2
(c) 1
(d) 3
31. The sum of the first k terms of a series is 3k² + 5k. Which one of the following is correct?
(a) The terms of S form an arithmetic progression with common difference 14.
(b) The terms of S form an arithmetic progression with common difference 6.
(c) The terms of S form a geometric progression with a common ratio 10/7.
(d) The terms of S form a geometric progression with a common ratio 11/4.
32. The sum of the first 8 terms of a GP is 5 times the sum of its first 4 terms. If r ≠ 1 is the common ratio, then what is the number of possible real values of k?
(a) One
(b) Two
(c) Three
(d) More than three
33. If one root of the equation x² - kx + k = 0 exceeds the other by 2√3 then which one of the following is a value of k?
(a) 3
(b) 6
(c) 9
(d) 12
34. If x + 5/y = 4 and y + 5/x = -4, then what is (x + y) equal to?
(a) 0
(b) 1
(c) 4
(d) 5
35. If the 5th, 7th and 13th terms of an AP are in GP, then what is the ratio of its first term to its common difference?
(a) -3
(b) -2
(c) 2
(d) 3
36. If p, 1 and q are in AP and p and 2q are in GP then which of the following statements is/are correct?
I. p, 4, q are in HP.
II. (1/p), 1/4, (1/q) are in AP.
Select the answer using the code given below:
(a) I only
(b) II only
(c) Both I and II
(d) Neither I nor II
37. If x = (1111)₂, y = (1001)₂ and z = (110)₂ then what is x³ - y³ - z³ - 3xyz equal to?
(a) (1111001)₂
(b) (1001111)₂
(c) (1)₂
(d) (0)₂
38. If Δ = | a b c ; d e f ; g h i | and A, B, C, D and G are the cofactors of the elements a, b, c, d and g respectively, then what is bB + cC - dD - gG equal to?
(a) 0
(b) 1
(c) Δ
(d) -A
39. Consider the following statements in respect of the determinant Δ = | k(k + 2) 2k + 1 1 ; 2k + 1 k + 2 1 ; 3 3 1 |
(a) 0
(b) 1
(c) Δ
(d) -A
39. Consider the following statements in respect of the determinant Δ = | k(k + 2) 2k + 1 1 ; 2k + 1 k + 2 1 ; 3 3 1 |
I. Δ is positive if k > 0
II. Δ is negative if k < 0
III. Δ is zero if k = 0
How many of the statements given above are correct?
(a) None
(b) One
(c) Two
(d) All three
40. If | 3 - i 0 i - 1 ; -1 -1 - i 1 | = A + iB where i = √-1 then what is A + B equal to?
(a) -10
(b) -6
(c) 0
(d) 6
Direction for Questions (41-43): Consider the following for the three (03) items that follow:
Let p = sin 35°, q = sin 25° and r = sin(-95°)
41. What is (p + q + r) equal to?
(a) -1
(b) 0
(c) 2sin 5°
(d) 2cos 5°
42. What is (pq + qr + rp) equal to?
(a) -3
(b) 0
(c) 1/4
(d) 3/4
43. What is (p² + q² + r²) equal to?
(a) 1/2
(b) 1
(c) 3/2
(d) 2
Direction for Questions (44-45): Consider the following for the two (02) items that follow:
Let p = [sin α - sin(α - 90°)]
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44. What is the minimum value of p?
(a) 0
(b) 1/2
(c) 1/√2
(d) 1
45. What is the maximum value of p?
(a) 1
(b) √2
(c) √3
(d) 2
Direction for Questions (46-48): Consider the following for the three (03) items that follow:
The sides of a triangle ABC are AB = 3 cm, BC = 5 cm and CA = 7 cm.
46. Consider the following statements:
I. The triangle is an obtuse-angled triangle.
II. The sum of acute angles of the triangle is also acute.
Which of the statements given above is/are correct?
(a) I only
(b) II only
(c) Both I and II
(d) Neither I nor II
47. What is ∠B equal to?
(a) 60°
(b) 105°
(c) 120°
(d) 150°
48. What is the area of the triangle?
(a) 15√3 square cm
(b) 15√3 square cm
(c) 15√3 square cm
(d) 30√3 square cm
Direction for Questions (49-50): Consider the following for the two (02) items that follow:
The top (M) of a tower is observed from three points P, Q and R lying in a horizontal straight line which passes directly along the foot (N) of the tower. The angles of elevations of M from P, Q and R are 30°, 45° and 60°, respectively. Let PQ = a and QR = b.
49. What is PN equal to?
(a) ((3 - √3)/2)a
(b) ((3 + √3)/2)a
(c) ((3 - √3)/4)a
(d) ((3 + √3)/4)a
50. What is MN equal to?
(a) ((3 + √3)/2)b
(b) ((3 - √3)/2)b
(c) ((3 - √3)/4)b
(d) ((3 + √3)/4)b
Direction for Questions (51-52): Consider the following for the two (02) items that follow:
The probabilities that A, B and C become managers are 3/10, 1/2 and 4/5, respectively. The probabilities that the bonus scheme will be introduced if A, B and C become managers are 4/9, 2/9 and 1/3, respectively.
51. What is the probability that the bonus scheme will be introduced?
(a) 17/45
(b) 19/45
(c) 23/45
(d) 26/45
52. If the bonus scheme has been introduced, then what is the probability that the manager appointed was B?
(a) 5/23
(b) 6/23
(c) 7/23
(d) 8/23
53. The arithmetic mean of 100 observations is 50. If 5 is subtracted from each observation and then divided by 20, then what is the new arithmetic mean?
(a) 2.25
(b) 3.5
(c) 4.25
(d) 5.5
54. The standard deviation of 100 observations is 10. If 5 is added to each observation and then divided by 20, then what will be the new standard deviation?
(a) 0.25
(b) 0.5
(c) 0.75
(d) 1.00
55. If P(A) = 1/3, P(B) = 1/2 and P(A ∩ B) = 1/4, then what is the value of P(B | Aᶜ)?
(a) 1/8
(b) 3/8
(c) 5/8
(d) 7/8
56. If P(A) = 1/3, P(B) = 1/2 and P(A ∩ B) = 1/4, then what is the value of P(Aᶜ ∩ Bᶜ)?
(a) 1/4
(b) 5/12
(c) 7/12
(d) 11/12
57. If two fair dice are tossed then what is the probability that the sum of the numbers on the faces of the dice is strictly greater than 7?
(a) 1/3
(b) 5/12
(c) 7/12
(d) 3/4
58. The probability of a man hitting a target is 1/5. If the man fires 7 times, then what is the probability that he hits the target at least twice?
(a) 1 - (3/5)(4/5)⁶
(b) 1 - (3/5)(5/4)⁷
(c) 1 - (11/5)(4/5)⁶
(d) 1 - (11/5)(5/4)⁷
59. Let X be a random variable following binomial distribution whose mean and variance are 200 and 160, respectively. What is the value of the number of trials (n)?
(a) 500
(b) 1000
(c) 1500
(d) 2000
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60. What is the arithmetic mean of 8², 9², 10², 15?
(a) 133.5
(b) 135.5
(c) 137.5
(d) 139.5
Direction for Questions (61-62): Consider the following for the two (02) items
Let y = sin⁻¹(x - 4x³/27)
61. What is y equal to?
(a) sin⁻¹x
(b) sin⁻¹(x/3)
(c) 3sin⁻¹x
(d) 3sin⁻¹(x/3)
62. What is dy/dx equal to?
(a) 1/√(9 - x²)
(b) 1/√(3 - x²)
(c) 3/√(9 - x²)
(d) 9/√(9 - x²)
Direction for Questions (63-64): Consider the following for the two (02) items that follow:
Let the function f(x) = x² + 9
63. What is limₓ→₀ (√f(x) - 3)/(√f(x) + 7 - 4) equal to?
(a) 2/3
(b) 2
(c) 4/3
(d) 2
64. Consider the following statements:
I. f(x) is an increasing function.
II. f(x) has local maximum at x = 0
Which of the statements given above is/are correct?
(a) I only
(b) II only
(c) Both I and II
(d) Neither I nor II
Direction for Questions (65-66): Consider the following for the two (02) items that follow:
The function f(x) satisfies f(x/y) = f(x)/f(y) for all positive real values of x and y and f(2) = 3
65. What is f(16) equal to?
(a) 18
(b) 27
(c) 54
(d) 81
66. What is f(1) × f(4) equal to?
(a) 4
(b) 8
(c) 9
(d) 18
Direction for Questions (67-68): Consider the following for the two (02) items that follow:
A function f is such that f(xy) = f(x + y) for all real values of x and y and f(5) = 10
67. What is f(0) equal to?
(a) 0
(b) 1
(c) 5
(d) 10
68. What is f(20) + f(-20) equal to?
(a) 0
(b) 10
(c) 20
(d) 40
Direction for Questions (69-70): Consider the following for the two (02) items that follow:
Let f(x) = [x²], where [.] is the greater integer function.
69. What is ∫ from √2 to √3 f(x) equal to?
(a) √3 - √2
(b) 2(√3 - √2)
(c) 3 - √2
(d) 1
70. What is ∫ from √2 to √3 f(x) dx equal to?
(a) 6 - √3 - 2√2
(b) 6 - √3 - √2
(c) 6 - √3 + 2√2
(d) 6 + √3 - 2√2
71. If A² + B² + C² = 0, then what is the value of the following?
(a) -1
(b) 0
(c) 1
(d) 2
72. If ω is a non-real cube root of unity, then what is a root of the following equation?
(a) x = 0
(b) x = 1
(c) x = ω
(d) x = ω²
73. What is ((√3 + i)/(√3 - i))³ equal to?
(a) -1
(b) 0
(c) 1
(d) 3
74. If x² - x + 1 = 0, then what is
(x - 1/x)² + (x - 1/x)⁴ + (x - 1/x)⁸
(a) 81
(b) 85
(c) 87
(d) 90
75. How many 7-letter words (with or without meaning) can be constructed using all the letters of the word CAPITAL so that all consonants come together in each word?
(a) 360
(b) 300
(c) 288
(d) 240
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76. If z ≠ 0 is a complex number, then what is amp(z) + amp(z̄) equal to?
(a) 0
(b) π/2
(c) π
(d) 2π
77. How many sides are there in a polygon which has 20 diagonals?
(a) 6
(b) 7
(c) 8
(d) 10
78. In how many ways can the letters of the word DELHI be arranged keeping the positions of vowels and consonants unchanged?
(a) 6
(b) 9
(c) 12
(d) 24
79. What is the number of positive integer solutions of x + y + z = 5?
(a) 3
(b) 5
(c) 6
(d) 9
80. What is the number of rational terms in the expansion of (3^(1/2) + 5^(1/4))¹²?
(a) 2
(b) 3
(c) 4
(d) 6
81. Under what condition will the lines m²x + my - 1 = 0 and n²x - my + 2 = 0 be perpendicular?
(a) mn - 1 = 0
(b) mn + 1 = 0
(c) m + n = 0
(d) m - n = 0
82. If p and q are real numbers between 0 and 1 such that the points (p, 1), (1, q) and (0, 0) form an equilateral triangle, then what is (p + q) equal to?
(a) √2
(b) √2 - 1
(c) 2 - √3
(d) 4 - 2√3
83. The vertices of a triangle are A(1, 1), B(0, 0) and C(2, 0). The angular bisectors of the triangle meet at P. What are the coordinates of P?
(a) (1, √2 - 1)
(b) (1, √3 - 1)
(c) (1, 1/2)
(d) (1/2, √2 - 1)
84. Let A(3, -1) and B(1, 1) be the end points of line segment AB. Let P be the middle point of the line segment AB. Let Q be the point situated at a distance of √2 units from P on the perpendicular bisector line of AB. What are the possible coordinates of Q?
(a) (2, 1)
(b) (3, 1)
(c) (2, 2)
(d) (1, 3)
85. ABC is an equilateral triangle and AD is the altitude on BC. If the coordinates of A are (1, 2) and that of D are (-2, 6) then what is the equation of BC?
(a) 3x + 4y - 18 = 0
(b) 4x + 3y - 1 = 0
(c) 4x - 3y = 26
(d) 3x - 4y + 30 = 0
86. What is the equation of the circle whose diameter is 10 cm and the equation of two of its diameters are x + y = 0 and x - y = 0?
(a) x² + y² = 0
(b) x² + y² = 25
(c) x² + y² = 100
(d) x² + y² - 2x - 2y - 23 = 0
87. A square is inscribed in a circle x² + y² + 2x + 2y + 1 = 0 and its sides are parallel to coordinate axes. Which one of the following is a vertex of the square?
(a) (-2, 2)
(b) (-2, -2)
(c) (-1 + 1/√2, -1 - 1/√2)
(d) None of the above
88. A tangent to the parabola y² = 4x is inclined at an angle of 45° with the positive direction of x-axis. What is point of contact of the tangent and the parabola?
(a) (1, 1)
(b) (2, √2)
(c) (1/2, 1/√2)
(d) (1, 2)
89. What is the distance between the two foci of the hyperbola 25x² - 75y² = 225?
(a) 2√3 units
(b) 4√3 units
(c) √6 units
(d) 2√6 units
90. If any point on an ellipse is (3 sin α, 5 cos α), then what is the eccentricity of the ellipse?
(a) 4/3
(b) 4/5
(c) 3/4
(d) 1/2
Direction for Questions (91-94): Consider the following for the four (04) items that follow:
The frequency distribution of height of students of a class is given below:
Height (in cm) | Number of students
160-162 | 12
162-164 | 15
164-166 | 24
166-168 | 13
91. What is the total number of students whose height is less than or equal to 165 cm?
(a) 15
(b) 39
(c) 51
(d) None of the above
92. What is the median height of the class?
(a) 162.41 cm
(b) 163.41 cm
(c) 164.41 cm
(d) 165.41 cm
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93. The height which occurs most frequently in the class is:
(a) 163.5 cm
(b) 163.9 cm
(c) 164.5 cm
(d) 164.9 cm
94. The most appropriate graphical representation of the given frequency distribution is:
(a) bar chart
(b) percentage bar chart
(c) histogram
(d) pie chart
Direction for Questions (95-96): Consider the following for the two (02) items that follow:
The sum and the sum of squares of the observation corresponding to length X (in cm) and weight Y (in gm) of 50 tropical tubers are given as ΣX = 200, ΣY = 250, ΣX² = 900 and ΣY² = 1400
95. Which of the following is correct?
(a) Variance(X) > Variance(Y)
(b) Variance(X) < Variance(Y)
(c) Variance(X) = Variance(Y)
(d) Cannot be determined from the given data
96. Which one of the following statements is correct?
(a) Coefficient of variation of X is strictly more than coefficient of variation of Y
(b) Coefficient of variation of X is strictly less than coefficient of variation of Y
(c) Coefficient of variation of X is the same as coefficient of variation of Y
(d) Coefficient of variation cannot be determined from the given data.
Direction for Questions (97-98): Consider the following for the two (02) items that follow:
Let X be a random variable following binomial distribution with parameters n = 6 and p = k. Further, 9P(X = 4) = P(X = 2)
97. What is the value of k?
(a) 1/2
(b) 1/3
(c) 1/4
(d) 1/5
98. What is the value of P(X = 3)?
(a) 135/1024
(b) 5/128
(c) 45/1024
(d) 70/1024
Direction for Questions (99-100): Consider the following for the two (02) items that follow:
A committee of 6 members is formed from a group of 7 gentlemen and 4 ladies.
99. What is the probability that the committee includes exactly 3 gentlemen?
(a) 10/33
(b) 30/77
(c) 100/231
(d) 5/11
100. What is the probability that the committee includes at least 2 ladies?
(a) 41/66
(b) 47/66
(c) 49/66
(d) 53/66
Direction for Questions (101-102): Consider the following for the two (02) items that follow:
Let the function y = (1 - cos x)⁻¹, where x ≠ 2nπ and n is an integer.
101. What is the range of the function?
(a) [0, ∞)
(b) (0.5, ∞)
(c) [1, ∞)
(d) (-∞, 0.5]
102. What is ∫ y dx equal to?
where c is the constant of integration.
Direction for Questions (103-104): Consider the following for the two (02) items that follow:
Let the function f(x) = sin [x], where [.] is the greatest integer function and g(x) = [x].
103. What is limₓ→₀ (f(x)g(x)) equal to?
(a) -1
(b) 0
(c) 1
(d) Limit does not exist
104. What is limₓ→₀ f(x)/g(x) equal to?
(a) -sin 1
(b) sin 1
(c) 0
(d) Limit does not exist
Direction for Questions (105-106): Consider the following for the two (02) items that follow:
Let the curve f(x) = |x - 3|.
105. What is the domain of the function f(x)?
(a) (0, ∞)
(b) (3, ∞)
(c) (-∞, ∞)
(d) (-∞, ∞) - {3}
106. What is the area bounded by the curve f(x) and y = 3?
(a) 3 square units
(b) 4.5 square units
(c) 7.5 square units
(d) 9 square units
Direction for Questions (107-108): Consider the following for the two (02) items that follow:
Let f = {(1,1), (2,4), (3,7), (4,10)}
107. If f(x) = px + q, then what is the value of (p + q)?
(a) -1
(b) 0
(c) 1
(d) 5
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108. Consider the following statements:
I. f is an one-one function.
II. f is onto function if the codomain is the set of natural numbers.
Which of the statements given above is/are correct?
(a) I only
(b) II only
(c) Both I and II
(d) Neither I nor II
Direction for Questions (109-110): Consider the following for the two (02) items that follow:
Let the function f(x) = x² - 1.
109. What is limₓ→₁ {f(x)} equal to?
(a) -1
(b) 0
(c) 1
(d) 2
110. What is the area bounded by the function f(x) and the x-axis?
(a) 1/3 square units
(b) 2/3 square unit
(c) 4/3 square units
(d) 2 square units
Direction for Questions (111-112): Consider the following for the two (02) items that follow:
Let x = sec θ - cos θ and y = sec⁴θ - cos⁴θ
111. What is (dy/dx)² equal to?
(a) 4(y² + 4)/(x² + 4)
(b) 4(y² - 4)/(x² - 4)
(c) 16(y² + 4)/(x² + 4)
(d) 16(y² - 4)/(x² - 4)
112. What is ((x² + 4)/(y² + 4))(dy/dx)[(x² + 4)d²y/dx² - 16y] equal to?
(a) 16x
(b) 16y
(c) -16x
(d) -16y
Direction for Questions (113-114): Consider the following for the two (02) items that follow:
Let ABC be a triangle right-angled at B and AB + AC = 3 units.
113. What is ∠A equal to if the area of the triangle is maximum?
(a) π/6
(b) π/4
(c) π/3
(d) 5π/12
114. What is the maximum area of the triangle?
(a) √3/2 square unit
(b) √3 square unit
(c) √6/2 square unit
(d) √6 square unit
Direction for Questions (115-116): Consider the following for the two (02) items that follow:
Let (x + y)ᵖ + q = xᵖyᵖ, where p, q are positive integers.
115. The derivative of y with respect to x:
(a) depends on p only
(b) depends on q only
(c) depends on both p and q
(d) is independent of both p and q
116. If p + q = 10, then what is dy/dx equal to?
(a) y/x
(b) xy
(c) x¹⁰y¹⁰
(d) (y/x)¹⁰
Direction for Questions (117-118): Consider the following for the two (02) items that follow:
The slope of the tangent of the curve y = f(x) at (x, f(x)) is 4 for every real number x and the curve passes through the origin.
117. What is the nature of the curve?
(a) A straight line passing through (1, 4)
(b) A straight line passing through (-1, 4)
(c) A parabola with vertex at origin and focus at (2, 0)
(d) A parabola with vertex at origin and focus at (1, 0)
118. What is the area bounded by the curve, the x-axis and the line x = 4?
(a) 8 square units
(b) 16 square units
(c) 32 square units
(d) 64 square units
Direction for Questions (119-120): Consider the following for the two (02) items that follow:
119. What is limₓ→₀ f'(x) equal to?
(a) 2
(b) 1
(c) 0
(d) Limit does not exist
120. Consider the following statements:
I. The function is continuous at x = -1
II. The function is differentiable at x = 1
(a) I only
(b) II only
(c) Both I and II
(d) Neither I nor II
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Answer Key
1 31 61 d 91 c
2 b 32 b 62 c 92 c
3 d 33 a 63 c 93 d
4 b 34 a 64 d 94 c
5 c 35 b 65 d 95 b
6 d 36 c 66 c 96 b
7 d 37 d 67 d 97 c
8 b 38 a 68 c 98 a
9 c 39 a 69 b 99 a
10 b 40 b 70 a 100 d
11 c 41 b 71 b 101 b
12 d 42 a 72 a 102 b
13 c 43 c 73 a 103 b
14 b 44 a 74 c 104 d
15 b 45 b 75 c 105 c
16 b 46 c 76 a 106 d
17 c 47 c 77 c 107 c
18 a 48 a 78 c 108 a
19 d 49 b 79 c 109 a
20 c 50 a 80 c 110 c
21 b 51 c 81 a 111 c
22 c 52 b 82 d 112 c
23 d 53 a 83 a 113 c
24 a 54 b 84 b 114 a
25 b 55 b 85 d 115 d
26 d 56 b 86 b 116 a
27 c 57 b 87 c 117 a
28 b 58 c 88 d 118 c
29 a 59 b 89 b 119 c
30 a 60 c 90 a 120 d
NDA previous Year Question Paper 2025 -Free PDF download with Solution
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