NDA & NA (I) 2025 — Mathematics (Full Question Paper)

The sum of the first k terms of a series S 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 common ratio 10/7.
• (d) The terms of S form a geometric progression with common ratio 11/4.

The sum of the first 8 terms of a GP is five 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 r?

• (a) One
• (b) Two
• (c) Three
• (d) More than three

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

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

If 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

If p, 1, q are in AP and p, 2, q 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

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

If Δ = |

| and A, B, C, D, G are the cofactors of the elements a, b, c, d, g respectively, then what is bB + cC – dD – gG equal to?

• (a) 0
• (b) 1
• (c) Δ
• (d) -Δ

Consider the following statements in respect of the determinant Δ = |

|:

• 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

If |

| = A + iB, where i = √-1, then what is A + B equal to?

• (a) -10
• (b) -6
• (c) 0
• (d) 6

If A² + B² + C² = 0, then what is the value of the following?
|

|

• (a) -1
• (b) 0
• (c) 1
• (d) 2

If ω is a non-real cube root of unity, then what is a root of the following equation?
|

| = 0

• (a) x = 0
• (b) x = 1
• (c) x = ω
• (d) x = ω²

What is (√3+i√3-i)³ equal to?

• (a) -1
• (b) 0
• (c) 1
• (d) 3

If x² – x + 1 = 0, then what is (x – 1/x)² + (x – 1/x)⁴ + (x – 1/x)⁸ equal to?

• (a) 81
• (b) 85
• (c) 87
• (d) 90

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

If z ≠ 0 is a complex number, then what is amp(z) + amp(z) equal to?

• (a) 0
• (b) π/2
• (c) π
• (d) 2π

How many sides are there in a polygon which has 20 diagonals?

• (a) 6
• (b) 7
• (c) 8
• (d) 10

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

What is the number of positive integer solutions of x+y+z = 5?

• (a) 3
• (b) 5
• (c) 6
• (d) 9

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

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

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

If [

] × [

] × [

] = [

], then which one of the following is a value of x?

• (a) -2
• (b) -1
• (c) 0
• (d) 1

If A = [

] where x, y, z are integers, is an orthogonal matrix, then what is the value of x²+y²+z²?

• (a) 0
• (b) 1
• (c) 4
• (d) 14

Consider the following in respect of a non-singular matrix M:

• I.|M²| = |M|²
• II.|M| = |M^-1|
• III.|M| = |M^T|

How many of the above are correct?

• (a) None
• (b) One
• (c) Two
• (d) All three

If f(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

  ]

If A = [

] then what is A² – 4A equal to?

• (a) -5I₃
• (b) -I₃
• (c) I₃
• (d) 5I₃

where I₃ is the identity matrix of order 3.

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

What is the number of selections of at most 3 things from 6 different things?

• (a) 20
• (b) 22
• (c) 41
• (d) 42

If A = [

] where x, y, z are integers, is an orthogonal matrix, then what is A² equal to?

• (a) Null matrix
• (b) Identity matrix
• (c) A
• (d) -A

What is (p+q+r) equal to?

• (a) -1
• (b) 0
• (c) 2sin5°
• (d) 2cos5°

What is (pq+qr+rp) equal to?

• (a) -3/4
• (b) 0
• (c) 1/4
• (d) 3/4

What is (p²+q²+r²) equal to?

• (a) 1/2
• (b) 1
• (c) 3/2
• (d) 2

What is the minimum value of p?

• (a) 0
• (b) 1/2
• (c) 1/√2
• (d) 1

What is the maximum value of p?

• (a) 1
• (b) √2
• (c) √3
• (d) 2

Consider the following statements:

• I.The triangle is 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

What is ∠B equal to?

• (a) 60°
• (b) 105°
• (c) 120°
• (d) 150°

What is the area of the triangle?

• (a) 15√3/4 square cm
• (b) 15√3/2 square cm
• (c) 15√3 square cm
• (d) 30√3 square cm

What is PN equal to?

• (a) 3-√32 · a
• (b) 3+√32 · a
• (c) 3-√34 · a
• (d) 3+√34 · a

What is MN equal to?

• (a) 3+√32 · b
• (b) 3-√32 · b
• (c) 3-√34 · b
• (d) 3+√34 · b

What is (p/q) equal to?

• (a) -tanα · tan2α
• (b) -cotα · cot2α
• (c) tanα · tan2α
• (d) cotα · cot2α

What is (p+q) equal to?

• (a) sec4α
• (b) cosec4α
• (c) 2sec4α
• (d) 2cosec4α

What is tan² α equal to?

• (a) pqp+q
• (b) p+2qp
• (c) pp+2q
• (d) p2p+q

What is tanα equal to?

• (a) 1/2
• (b) 1
• (c) 3/4
• (d) 2

What is 2sin2α + cos2α equal to?

• (a) 11/10
• (b) 11/5
• (c) 12/5
• (d) 13/5

One of the angles of the triangle is

• (a) 15°
• (b) 30°
• (c) 45°
• (d) 75°

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

A man at M, standing 100 m away from the base (P) of a chimney of height 50 m, 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)

If k is a root of x² – 4x + 1 = 0, then what is tan^-1 k + tan^-1(1/k) equal to?

• (a) -π/2
• (b) 0
• (c) π/4
• (d) π/2

If tan^-1 k + tan^-1(1/2) = π/4, then what is the value of k?

• (a) 1
• (b) 1/2
• (c) 1/3
• (d) 1/4

Under what condition will the lines m²x + ny – 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

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

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)

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 √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)

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 = 0
• (d) 3x-4y+30 = 0

What is the equation of the circle whose diameter is 10 cm and the equations of two of its diameters are x+y=0 and x-y=0?

• (a) x²+y² = 1
• (b) x²+y² = 25
• (c) x²+y² = 100
• (d) x²+y²-2x-2y-23 = 0

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

A tangent to the parabola y² = 4x is inclined at an angle 45° with the positive direction of x-axis. What is the point of contact of the tangent and the parabola?

• (a) (1,1)
• (b) (2, 2√2)
• (c) (1/2, 1/√2)
• (d) (1,2)

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

If any point on an ellipse is (3sinα, 5cosα), then what is the eccentricity of the ellipse?

• (a) 4/3
• (b) 4/5
• (c) 3/4
• (d) 1/2

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) cos² γ
• (b) -cos² γ
• (c) sin² γ
• (d) -sin² γ

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

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

If a line (x+1)/p = (y-1)/q = (z-2)/r where p=2q=3r, makes an angle θ with the positive direction of y-axis, then what is cos2θ equal to?

• (a) -31/49
• (b) -37/49
• (c) 31/49
• (d) 37/49

What is the equation of the plane passing through the point (1,1,1) and perpendicular to the line whose direction ratios are <3,2,1>?

• (a) x+2y+3z = 6
• (b) 3x+2y+z = 6
• (c) x+y+z = 3
• (d) 3x+2y+z = 0

A line makes angles α, β and γ with the positive directions of the coordinate axes. If a→ = (sin² α)i + (sin² β)j + (sin² γ)k and b→ = i+j+k, then what is a→ · b→ equal to?

• (a) -2
• (b) -1
• (c) 1
• (d) 2

Consider the following statements in respect of a vector d→ = (a→ × b→) × c→:

• I.d→ is coplanar with a→ and b→.
• II.d→ is perpendicular to c→.

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

The position vectors of three points A, B and C are a→, b→ and c→ respectively such that 3·a→ – 4·b→ + c→ = 0. What is AB:BC equal to?

• (a) 3:1
• (b) 1:3
• (c) 3:4
• (d) 1:4

The position vectors of three points A, B and C are a→, b→ and c→ respectively, where c→ = (cos² θ)a→ + (sin² θ)b→. What is (a→ × b→) + (b→ × c→) + (c→ × a→) equal to?

• (a) 0 (zero vector)
• (b) 2·c→
• (c) 3·c→
• (d) Unit vector

Let a→, b→, (a→ × b→) be unit vectors. What is (a→ · b→) equal to?

• (a) 0
• (b) 1/2
• (c) 1
• (d) 3

What is (dy/dx)² equal to?

• (a) 4y²+4x²+4
• (b) 4y²-4x²-4
• (c) 16y²+4x²+4
• (d) 16y²-4x²-4

What is x²+4y²+4 · dy/dx · [(x²+4) d²y/dx² – 16y] equal to?

• (a) 16x
• (b) 16y
• (c) -16x
• (d) -16y

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

What is the maximum area of the triangle?

• (a) √3/2 square unit
• (b) √3 square units
• (c) √6/2 square units
• (d) √6 square units

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

If p+q = 10, then what is dy/dx equal to?

• (a) y/x
• (b) xy
• (c) x¹⁰ y¹⁰
• (d) (y/x)¹⁰

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)

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

What is limx→0 f'(x) equal to?

• (a) 2
• (b) 1
• (c) 0
• (d) Limit does not exist

Consider the following statements:

• I.The function is continuous at x = -1.
• II.The function is differentiable at x = 1.

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

What is the range of the function?

• (a) [0, ∞)
• (b) [0.5, ∞)
• (c) [1, ∞)
• (d) (-∞, 0.5]

What is ∫y dx equal to?

• (a) -tan(x/2) + c
• (b) -cot(x/2) + c
• (c) tan(x/2) + c
• (d) cot(x/2) + c

where c is the constant of integration.

What is limx→0 {f(x)g(x)} equal to?

• (a) -1
• (b) 0
• (c) 1
• (d) Limit does not exist

What is limx→0 f(x)/g(x) equal to?

• (a) -sin1
• (b) sin1
• (c) 0
• (d) Limit does not exist

What is the domain of the function f(x)?

• (a) (0, ∞)
• (b) (3, ∞)
• (c) (-∞, ∞)
• (d) (-∞, ∞) {3}

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

If f(x) = px+q, then what is the value of (p+q)?

• (a) -1
• (b) 0
• (c) 1
• (d) 5

Consider the following statements:

• I.f is 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

What is limx→1 {f∘f(x)} equal to?

• (a) -1
• (b) 0
• (c) 1
• (d) 2

What is the area bounded by the function f(x) and the x-axis?

• (a) 1/3 square unit
• (b) 2/3 square unit
• (c) 4/3 square units
• (d) 2 square units

What is y equal to?

• (a) sin^-1 x
• (b) sin^-1(x/3)
• (c) 3sin^-1 x
• (d) 3sin^-1(x/3)

What is dy/dx equal to?

• (a) 1/√9-x²
• (b) 1/√3-x²
• (c) 3/√9-x²
• (d) 9/√9-x²

What is limx→0 √f(x) – 3√f(x)+7 – 4 equal to?

• (a) 2/3
• (b) 1
• (c) 4/3
• (d) 2

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

What is f(16) equal to?

• (a) 18
• (b) 27
• (c) 54
• (d) 81

What is f(1)f(4) equal to?

• (a) 4
• (b) 8
• (c) 9
• (d) 18

What is f(0) equal to?

• (a) 0
• (b) 1
• (c) 5
• (d) 10

What is f(20) + f(-20) equal to?

• (a) 0
• (b) 10
• (c) 20
• (d) 40

What is ∫_√2^√3 f(x) dx equal to?

• (a) √3 – √2
• (b) 2(√3 – √2)
• (c) 3 – √2
• (d) 1

What is ∫_√2² f(x) dx equal to?

• (a) 6 – √3 – 2√2
• (b) 6 – √3 – √2
• (c) 6 – √3 + 2√2
• (d) 6 + √3 – 2√2

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

What is the median height of the class?

• (a) 162.41 cm
• (b) 163.41 cm
• (c) 164.41 cm
• (d) 165.41 cm

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

The most appropriate graphical representation of the given frequency distribution is

• (a) bar chart
• (b) percentage bar chart
• (c) histogram
• (d) pie chart

Which one 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

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 same as coefficient of variation of Y.
• (d) Coefficient of variation cannot be determined from the given data.

What is the value of k?

• (a) 1/2
• (b) 1/3
• (c) 1/4
• (d) 1/5

What is the value of P(X=3)?

• (a) 135/1024
• (b) 5/128
• (c) 45/1024
• (d) 70/1024

What is the probability that the committee includes exactly 3 gentlemen?

• (a) 10/33
• (b) 30/77
• (c) 100/231
• (d) 5/11

What is the probability that the committee includes at least 2 ladies?

• (a) 41/66
• (b) 47/66
• (c) 49/66
• (d) 53/66

What is the probability that the bonus scheme will be introduced?

• (a) 17/45
• (b) 19/45
• (c) 23/45
• (d) 26/45

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

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

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

If P(A) = 1/3, P(B) = 1/2 and P(A∩B) = 1/4, then what is the value of P(B|A^c)?

• (a) 1/8
• (b) 3/8
• (c) 5/8
• (d) 7/8

If P(A) = 1/3, P(B) = 1/2 and P(A∩B) = 1/4, then what is the value of P(A^c ∩ B^c)?

• (a) 1/4
• (b) 5/12
• (c) 7/12
• (d) 11/12

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

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)(4/5)⁷
• (c) 1 – (11/5)(4/5)⁶
• (d) 1 – (11/5)(4/5)⁷

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

What is the arithmetic mean of 8², 9², 10², …, 15²?

• (a) 133.5
• (b) 135.5
• (c) 137.5
• (d) 139.5