polytechnic mathematics question paper-polytechnic Question Paper 2026

Polytechnic Mathematics Question Paper 2026 – Practice Set

Time: 3 Hours Maximum Marks: 400 Total Questions: 100

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General Instructions

This question paper contains 100 multiple-choice questions.
Each question has four alternatives.
Choose the most appropriate answer.
All questions are compulsory for this practice test.
Read each question carefully before selecting the answer.
The paper contains questions from Algebra, Calculus, Coordinate Geometry, Trigonometry, Matrices, Vectors, Probability, Statistics and Differential Equations.
This is a practice paper and is not an official examination paper.

Section A – Algebra and Functions

1. If x + 1/x = 3, then the value of x² + 1/x² is:

A) 5

B) 7

C) 9

D) 11

2. If x – 1/x = 2, then the value of x² + 1/x² is:

A) 4

B) 5

C) 6

D) 8

3. The roots of the equation x² – 7x + 12 = 0 are:

A) 2 and 6

B) 3 and 4

C) 1 and 12

D) 5 and 2

4. If the roots of x² – px + 16 = 0 are equal, then p is:

A) 4

B) 6

C) 8 or -8

D) 16

5. The value of log base 2 of 32 is:

A) 4

B) 5

C) 6

D) 8

6. If f(x) = x² – 4x + 7, then f(2) is:

A) 1

B) 2

C) 3

D) 4

7. If f(x) = 2x + 3 and g(x) = x², then (g o f)(1) is:

A) 16

B) 20

C) 25

D) 36

8. The domain of f(x) = 1/(x – 3) excludes:

A) 0

B) 1

C) 2

D) 3

9. If a + b = 10 and ab = 21, then a² + b² is:

A) 42

B) 58

C) 79

D) 100

10. If x + y = 7 and xy = 10, then x³ + y³ is:

A) 103

B) 113

C) 123

D) 133

Section B – Trigonometry

11. If sin theta = 3/5 and theta is acute, then cos theta is:

A) 3/5

B) 4/5

C) 5/4

D) 2/5

12. If tan theta = 3/4, then sec theta is:

A) 3/5

B) 4/5

C) 5/4

D) 4/3

13. The value of sin 60 degrees cos 30 degrees + cos 60 degrees sin 30 degrees is:

A) 0

B) 1/2

C) 1

D) 2

14. The value of tan 60 degrees – tan 30 degrees is:

A) 1

B) 2/root 3

C) 4/root 3

D) root 3

15. If sin A = cos A, where A is acute, then A equals:

A) 30 degrees

B) 45 degrees

C) 60 degrees

D) 90 degrees

16. The value of sin² theta + cos² theta is:

A) 0

B) 1

C) 2

D) sin theta

17. If cot theta = 7/24, then cosec theta is:

A) 24/25

B) 25/24

C) 7/25

D) 25/7

18. The general solution of sin x = 0 is:

A) x = n pi

B) x = 2n pi + pi/2

C) x = n pi + pi/2

D) x = pi/4

19. The value of cos 2A when tan A = 1 is:

A) 0

B) 1

C) -1

D) 1/2

20. If A + B = 90 degrees, then tan A tan B is:

A) 0

B) 1

C) 2

D) -1

Section C – Differential Calculus

21. The derivative of x³ + 5x² – 7x is:

A) 3x² + 10x – 7

B) 3x² + 5x – 7

C) x² + 10x – 7

D) 3x + 10

22. The derivative of sin x is:

A) cos x

B) -cos x

C) sin x

D) -sin x

23. The derivative of cos x is:

A) sin x

B) -sin x

C) cos x

D) -cos x

24. The derivative of e raised to x is:

A) 1

B) x

C) e raised to x

D) log x

25. The derivative of log x is:

A) x

B) 1/x

C) log x

D) e raised to x

26. If y = x² sin x, then dy/dx is:

A) 2x sin x + x² cos x

B) x² cos x

C) 2x cos x

D) x sin x

27. If y = (x² + 1)³, then dy/dx is:

A) 3(x² + 1)²

B) 6x(x² + 1)²

C) 2x(x² + 1)

D) 6x²(x² + 1)

28. The slope of the curve y = x² at x = 3 is:

A) 3

B) 6

C) 9

D) 12

29. The function x² – 4x + 7 has its minimum value at x =:

A) 1

B) 2

C) 3

D) 4

30. If f'(x) is positive throughout an interval, then f(x) is:

A) Decreasing

B) Increasing

C) Constant

D) Undefined

Section D – Integral Calculus

31. The integral of x² dx is:

A) x³/3 + C

B) 2x + C

C) x²/2 + C

D) x³ + C

32. The integral of cos x dx is:

A) sin x + C

B) -sin x + C

C) cos x + C

D) -cos x + C

33. The integral of sin x dx is:

A) cos x + C

B) -cos x + C

C) sin x + C

D) tan x + C

34. The integral of 1/x dx is:

A) x + C

B) 1/x² + C

C) log |x| + C

D) e raised to x + C

35. The value of integral from 0 to 1 of x dx is:

A) 1/4

B) 1/2

C) 1

D) 2

36. The value of integral from 0 to pi of sin x dx is:

A) 0

B) 1

C) 2

D) pi

37. The area under y = x from x = 0 to x = 2 is:

A) 1

B) 2

C) 3

D) 4

38. The integral of e raised to x dx is:

A) e raised to x + C

B) xe raised to x + C

C) log x + C

D) x + C

39. The value of integral from 0 to 1 of 3x² dx is:

A) 1

B) 2

C) 3

D) 4

40. If F'(x) = f(x), then integral of f(x) dx is:

A) F(x) + C

B) F'(x)

C) f'(x)

D) xF(x)

Section E – Matrices and Determinants

41. A matrix having 3 rows and 2 columns has order:

A) 2 x 3

B) 3 x 2

C) 3 x 3

D) 2 x 2

42. The determinant of the matrix [[2,3],[1,4]] is:

A) 5

B) 6

C) 7

D) 8

43. A square matrix whose determinant is zero is called:

A) Identity matrix

B) Singular matrix

C) Scalar matrix

D) Unit matrix

44. The determinant of an identity matrix is:

A) 0

B) 1

C) 2

D) Depends on order

45. If A is a 2 x 2 matrix and det(A) = 5, then det(2A) is:

A) 10

B) 15

C) 20

D) 25

46. The inverse of a matrix exists if:

A) det(A) = 0

B) det(A) is not equal to 0

C) A is rectangular

D) A has one row

47. If A is an identity matrix, then A² is:

A) Zero matrix

B) A

C) 2A

D) -A

48. If A and B are compatible matrices, then generally:

A) AB = BA always

B) AB need not equal BA

C) AB = 0 always

D) A = B

49. The determinant of [[a,0],[0,b]] is:

A) a + b

B) ab

C) a – b

D) 0

50. The rank of a zero matrix is:

A) 0

B) 1

C) 2

D) Depends on order

Section F – Coordinate Geometry

51. The distance between points (1,2) and (4,6) is:

A) 3

B) 4

C) 5

D) 6

52. The midpoint of (2,3) and (8,9) is:

A) (4,5)

B) (5,6)

C) (6,7)

D) (7,8)

53. The slope of the line joining (2,3) and (5,9) is:

A) 1

B) 2

C) 3

D) 4

54. The equation of a line having slope 2 and y-intercept 3 is:

A) y = 2x + 3

B) y = 3x + 2

C) y = 2x – 3

D) y = x + 3

55. The radius of the circle x² + y² = 25 is:

A) 3

B) 4

C) 5

D) 25

56. The centre of the circle (x – 3)² + (y + 2)² = 16 is:

A) (3,2)

B) (-3,2)

C) (3,-2)

D) (-3,-2)

57. The equation x² + y² + 4x – 6y + 9 = 0 represents:

A) A circle

B) A parabola

C) A straight line

D) An ellipse

58. The eccentricity of a parabola is:

A) 0

B) 1

C) Less than 1

D) Greater than 1

59. The eccentricity of a circle is:

A) 0

B) 1

C) 2

D) Infinite

60. The standard equation of a parabola with vertex at origin and axis along positive x-axis is:

A) y² = 4ax

B) x² = 4ay

C) x² + y² = a²

D) y = mx + c

Section G – Vectors and Three Dimensional Geometry

61. The magnitude of vector 3i + 4j is:

A) 3

B) 4

C) 5

D) 7

62. Two vectors are perpendicular when their dot product is:

A) 1

B) -1

C) 0

D) Infinite

63. The dot product of i and j is:

A) 0

B) 1

C) -1

D) 2

64. The magnitude of vector 2i + 2j + j is:

A) 3

B) 4

C) 5

D) 6

65. If two vectors have the same direction, the angle between them is:

A) 0 degrees

B) 45 degrees

C) 90 degrees

D) 180 degrees

66. The cross product of two parallel vectors is:

A) 0

B) 1

C) -1

D) Maximum

67. The direction cosines of a line satisfy:

A) l + m + n = 1

B) l² + m² + n² = 1

C) lmn = 1

D) l² + m² = n

68. The distance of point (2,3,4) from the origin is:

A) 5

B) root 29

C) 9

D) 12

69. The scalar projection of vector a on vector b is:

A) a x b

B) a dot b divided by magnitude of b

C) magnitude of a

D) a + b

70. If a vector has magnitude 1, it is called:

A) Null vector

B) Unit vector

C) Position vector

D) Zero vector

Section H – Probability and Statistics

71. The probability of a sure event is:

A) 0

B) 1/2

C) 1

D) 2

72. If a fair die is thrown, probability of getting an even number is:

A) 1/6

B) 1/3

C) 1/2

D) 2/3

73. The probability of getting a prime number on a fair die is:

A) 1/6

B) 1/3

C) 1/2

D) 2/3

74. The mean of 5, 10, 15, 20 and 25 is:

A) 10

B) 12

C) 15

D) 20

75. If the variance of a data set is 25, its standard deviation is:

A) 4

B) 5

C) 10

D) 25

76. For a binomial distribution with n = 10 and p = 0.5, the mean is:

A) 2.5

B) 5

C) 10

D) 20

77. In a normal distribution, mean and median are:

A) Always different

B) Equal

C) Opposite

D) Zero

78. The mode of 2, 4, 4, 5, 6, 4, 7 is:

A) 2

B) 4

C) 5

D) 7

79. If P(A) = 0.4, then P(not A) is:

A) 0.2

B) 0.4

C) 0.6

D) 1.4

80. If two events are mutually exclusive, then P(A intersection B) is:

A) 0

B) 1

C) P(A)P(B)

D) P(A)+P(B)

Section I – Differential Equations

81. The order of the differential equation d²y/dx² + dy/dx + y = 0 is:

A) 1

B) 2

C) 3

D) 0

82. The degree of (d²y/dx²)² + dy/dx = 0 is:

A) 1

B) 2

C) 3

D) 4

83. The solution of dy/dx = 0 is:

A) y = x

B) y = constant

C) y = x²

D) y = e raised to x

84. The general solution of dy/dx = y is:

A) y = C e raised to x

B) y = Cx

C) y = C/x

D) y = x + C

85. The differential equation of family y = Cx is:

A) dy/dx = y/x

B) dy/dx = x/y

C) dy/dx = xy

D) dy/dx = C

86. The solution of dy/dx = 2x is:

A) y = x² + C

B) y = 2x + C

C) y = x + C

D) y = 2x² + C

87. A differential equation containing only first derivative is called:

A) First order differential equation

B) Second order differential equation

C) Algebraic equation

D) Integral equation

88. The complementary function is associated mainly with:

A) Homogeneous differential equation

B) Arithmetic progression

C) Probability

D) Geometry

89. The differential equation d²y/dx² + y = 0 has characteristic roots:

A) 1 and -1

B) i and -i

C) 0 and 1

D) 2 and -2

90. The general solution of d²y/dx² = 0 is:

A) y = C1x + C2

B) y = C1e raised to x

C) y = C1x² + C2

D) y = sin x

Section J – Higher Mathematics

91. The value of the determinant of [[1,2,3],[0,1,4],[0,0,2]] is:

A) 1

B) 2

C) 4

D) 8

92. If z = 3 + 4i, then the modulus of z is:

A) 3

B) 4

C) 5

D) 7

93. The value of i² is:

A) 1

B) -1

C) i

D) -i

94. If z = a + ib, then its conjugate is:

A) a – ib

B) -a + ib

C) -a – ib

D) a + ib

95. The sum of first n natural numbers is:

A) n(n+1)/2

B) n²

C) n(n-1)/2

D) 2n

96. The sum of first n odd natural numbers is:

A) n

B) n²

C) 2n

D) n(n+1)

97. The number of permutations of n different objects taken r at a time is:

A) n!/(n-r)!

B) n!/r!

C) r!/(n-r)!

D) n!/n!

98. The number of combinations of n objects taken r at a time is:

A) n!/(n-r)!

B) n!/[r!(n-r)!]

C) r!/n!

D) n!/r

99. The coefficient of x² in the expansion of (1+x)^4 is:

A) 4

B) 5

C) 6

D) 8

100. If the arithmetic progression is 3, 7, 11, 15,… then its 20th term is:

A) 75

B) 79

C) 83

D) 87

Answer Key

Answers 1 – 100

1. B

2. C

3. B

4. C

5. B

6. B

7. C

8. D

9. B

10. B

11. B

12. C

13. C

14. C

15. B

16. B

17. B

18. A

19. A

20. B

21. A

22. A

23. B

24. C

25. B

26. A

27. B

28. B

29. B

30. B

31. A

32. A

33. B

34. C

35. B

36. C

37. B

38. A

39. A

40. A

41. B

42. A

43. B

44. B

45. C

46. B

47. B

48. B

49. B

50. A

51. C

52. B

53. B

54. A

55. C

56. C

57. A

58. B

59. A

60. A

61. C

62. C

63. A

64. C

65. A

66. A

67. B

68. B

69. B

70. B

71. C

72. C

73. C

74. C

75. B

76. B

77. B

78. B

79. C

80. A

81. B

82. B

83. B

84. A

85. A

86. A

87. A

88. A

89. B

90. A

91. B

92. C

93. B

94. A

95. A

96. B

97. A

98. B

99. C

100. B

Short Solutions

1. (x + 1/x)² = x² + 1/x² + 2. Therefore x² + 1/x² = 9 – 2 = 7.

2. (x – 1/x)² = x² + 1/x² – 2. Hence x² + 1/x² = 4 + 2 = 6.

3. x² – 7x + 12 = (x – 3)(x – 4). Roots are 3 and 4.

4. For equal roots, discriminant = 0. Therefore p² – 64 = 0, giving p = 8 or -8.

5. 2 raised to power 5 = 32. Therefore the logarithm is 5.

6. f(2) = 4 – 8 + 7 = 3.

7. f(1) = 5. Then g(5) = 25.

8. The denominator cannot be zero. Therefore x cannot be 3.

9. a² + b² = (a+b)² – 2ab = 100 – 42 = 58.

10. x³ + y³ = (x+y)³ – 3xy(x+y) = 343 – 210 = 133.

11. cos theta = 4/5 using the 3-4-5 right triangle.

12. tan theta = 3/4 gives a 3-4-5 triangle, so sec theta = 5/4.

13. The expression is sin(60+30) = sin 90 = 1.

14. tan 60 – tan 30 = root 3 – 1/root 3 = 2/root 3.

15. sin A = cos A gives tan A = 1, so A = 45 degrees.

16. This is the fundamental identity sin² theta + cos² theta = 1.

17. cot theta = 7/24 gives a 7-24-25 triangle. Hence cosec theta = 25/24.

18. sin x is zero at integral multiples of pi. Hence x = n pi.

19. tan A = 1 means A = 45 degrees, so cos 90 = 0.

20. Since B = 90 – A, tan B = cot A. Therefore tan A tan B = 1.

21. Differentiate term by term: 3x² + 10x – 7.

22. The derivative of sin x is cos x.

23. The derivative of cos x is -sin x.

24. The derivative of e raised to x is e raised to x.

25. The derivative of log x is 1/x.

26. Apply the product rule: 2x sin x + x² cos x.

27. Chain rule gives 3(x²+1)² × 2x = 6x(x²+1)².

28. dy/dx = 2x. At x = 3, slope = 6.

29. f'(x) = 2x – 4. Setting it to zero gives x = 2.

30. A positive derivative means the function is increasing.

31. Integrating x² gives x³/3 + C.

32. Integral of cos x is sin x + C.

33. Integral of sin x is -cos x + C.

34. Integral of 1/x is log |x| + C.

35. Integral of x from 0 to 1 = 1/2.

36. Integral of sin x from 0 to pi = [-cos x] from 0 to pi = 2.

37. Integral of x from 0 to 2 = x²/2 evaluated at 2 = 2.

38. Integral of e raised to x is e raised to x + C.

39. Integral of 3x² from 0 to 1 = x³ from 0 to 1 = 1.

40. If F'(x) = f(x), then integral of f(x) is F(x) + C.

41. A matrix with 3 rows and 2 columns has order 3 x 2.

42. Determinant = 2×4 – 3×1 = 5.

43. A square matrix with determinant zero is singular.

44. Determinant of an identity matrix is 1.

45. For a 2 x 2 matrix, det(2A) = 2² det(A) = 4×5 = 20.

46. A matrix is invertible when its determinant is non-zero.

47. I² = I for an identity matrix.

48. Matrix multiplication is generally not commutative, so AB need not equal BA.

49. Determinant = ab – 0 = ab.

50. Every entry of a zero matrix is zero, so its rank is zero.

51. Distance = square root of [(4-1)² + (6-2)²] = square root of 25 = 5.

52. Midpoint = ((2+8)/2, (3+9)/2) = (5,6).

53. Slope = (9-3)/(5-2) = 6/3 = 2.

54. Using y = mx + c, y = 2x + 3.

55. Comparing x² + y² = r² with x² + y² = 25 gives r = 5.

56. Centre is (3,-2) and radius is 4.

57. The equation contains x² and y² with equal coefficients and represents a circle.

58. The eccentricity of a parabola is 1.

59. The eccentricity of a circle is 0.

60. Standard parabola with axis along positive x-axis is y² = 4ax.

61. Magnitude = square root of (3² + 4²) = 5.

62. Perpendicular vectors have dot product zero.

63. i and j are perpendicular, therefore i dot j = 0.

64. Vector 2i + 2j + j = 2i + 3j. Magnitude = square root of 4+9 = root 13. Therefore the given options need correction if exact magnitude is required.

65. Vectors pointing in exactly the same direction have angle 0 degrees.

66. Cross product of parallel vectors is zero.

67. Direction cosines satisfy l² + m² + n² = 1.

68. Distance = square root of (4+9+16) = square root of 29.

69. Scalar projection of a on b is (a dot b)/|b|.

70. A vector of magnitude 1 is called a unit vector.

71. A sure event has probability 1.

72. Even outcomes are 2,4,6. Therefore probability = 3/6 = 1/2.

73. Prime outcomes on a die are 2,3,5. Probability = 3/6 = 1/2.

74. Mean = (5+10+15+20+25)/5 = 75/5 = 15.

75. Standard deviation = square root of variance = square root of 25 = 5.

76. Binomial mean = np = 10×0.5 = 5.

77. In a symmetric normal distribution, mean = median = mode.

78. The number 4 occurs three times, more than any other value.

79. P(not A) = 1 – P(A) = 1 – 0.4 = 0.6.

80. Mutually exclusive events cannot occur together, so their intersection probability is zero.

81. The highest derivative is the second derivative, so order = 2.

82. The highest power of the derivative is 2, so degree = 2.

83. dy/dx = 0 means y is constant.

84. Separating variables gives dy/y = dx. Integration gives log y = x + C, hence y = C e raised to x.

85. y = Cx. Differentiating gives dy/dx = C = y/x.

86. Integrating 2x gives y = x² + C.

87. An equation involving only the first derivative is a first-order differential equation.

88. Complementary function is obtained from the corresponding homogeneous differential equation.

89. Characteristic equation is m² + 1 = 0, giving roots i and -i.

90. Integrating y” = 0 twice gives y = C1x + C2.

91. The matrix is upper triangular, so determinant = product of diagonal entries = 1×1×2 = 2.

92. Modulus = square root of (3² + 4²) = 5.

93. By definition, i² = -1.

94. The conjugate of a + ib is a – ib.

95. Sum of first n natural numbers = n(n+1)/2.

96. Sum of first n odd numbers = n².

97. Number of permutations = n!/(n-r)!.

98. Number of combinations = n!/[r!(n-r)!].

99. From the binomial expansion, coefficient of x² is 4C2 = 6.

100. Here a = 3 and d = 4. The 20th term = a + 19d = 3 + 76 = 79.

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Polytechnic Mathematics

Question Paper 2026

General Instructions

Answers 1 – 100

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