Mathematics
Let p, q and r be three unequal numbers such that p, q and r are in AP. If (q – p), (r – q) and p are in GP, then (p + q) : (q + r) : (r + p) equals
- (a) 1 : 2 : 3
- (b) 3 : 4 : 5
- (c) 3 : 5 : 4
- (d) 1 : 3 : 2
Answer: (c) 3 : 5 : 4
p,q,r in AP (q=p+d,r=p+2d). (q-p),(r-q),p in GP forces d=p (d=0 rejected, p,q,r must be unequal). Then p+q:q+r:r+p = 3p:5p:4p = 3:5:4.
If p, g1, g2 and q are in GP and m is the arithmetic mean of p and q, then g12g2 + g22g1 is equal to
- (a) m
- (b) 2m
- (c) 1
- (d) 1/2
Answer: (b) 2m
With p,g1,g2,q in GP (ratio t): g1=pt,g2=pt^2,q=pt^3. g1^2/g2+g2^2/g1 = p(t^3+1) = p+q = 2m, since m=(p+q)/2.
Consider the following inequalities:
- I.1 + 4i > 3 + 2i
- II.2 + 3i < 3 + 4i
- III.4 + 3i > 3 + 4i
where i = √-1 How many of the above are valid?
- (a) None
- (b) One
- (c) Two
- (d) All the three
Answer: (a) None
Complex numbers with non-zero imaginary parts cannot be compared using real inequalities (> or <) — order is undefined on complex numbers, so none of the three statements is valid.
Let Z1 and Z2 be complex numbers such that 3Z14Z2 is purely imaginary. What is |Z1+Z2Z1-Z2| equal to?
- (a) 2
- (b) 3/2
- (c) 5/4
- (d) 1
Answer: (d) 1
3Z1/(4Z2) purely imaginary means Z1/Z2 = (4/3)it for real t. Then |(Z1+Z2)/(Z1-Z2)| = |(ratio+1)/(ratio-1)| = 1 for any purely-imaginary ratio.
If α, β, γ are cube roots of -8, then what is α2p2 + β2q2 + γ2r2β2p2 + γ2q2 + α2r2 equal to?
- (a) γ/α
- (b) γ/β
- (c) 2γ/α
- (d) 2γ/β
Answer: (b) γ/β
With α,β,γ the cube roots of -8 (i.e. -2,-2ω,-2ω²), direct substitution shows the given ratio equals γ/β for all p,q,r.
The sum of the first n terms of an AP is 3n2 + 5n. If the m-th term of the AP is 68, then what is the value of m?
- (a) 9
- (b) 10
- (c) 11
- (d) 12
Answer: (c) 11
Sn=3n²+5n so a_m = Sm-S(m-1) = 6m+2. Setting 6m+2=68 gives m=11.
A set S contains (2n + 1) elements. If the number of subsets of S which contain at most n elements is 1024, then what is the value of n?
- (a) 10
- (b) 8
- (c) 6
- (d) 5
Answer: (d) 5
Subsets of a (2n+1)-element set with at most n elements number 2^(2n) by the symmetry k <-> (2n+1-k). 2^(2n)=1024=2^10 gives n=5.
What is the maximum number of points of intersection of 5 circles?
- (a) 10
- (b) 15
- (c) 20
- (d) 25
Answer: (c) 20
Maximum intersection points of 5 circles = 2*C(5,2) = 20 (each pair of circles meets in at most 2 points).
What is the greatest value of r satisfying the inequality 15Cr+1 > 2 × 15Cr?
- (a) 2
- (b) 3
- (c) 4
- (d) 5
Answer: (c) 4
15C(r+1) > 2*15C(r) holds for r=0,1,2,3,4 (checked directly); the greatest such r is 4.
If n = mC2 then what is nC2 equal to?
- (a) m+1C4
- (b) 2 × m+1C4
- (c) 3 × m+1C4
- (d) m+2C4
Answer: (c) 3 × ^(m+1)C_(4)
n=mC2=m(m-1)/2. Direct expansion of nC2=n(n-1)/2 in terms of m equals 3*(m+1)C4 after simplification.
If the highest degree coefficient is equal to 1, then what is the total number of quadratic equations which are unchanged on squaring their roots?
- (a) 6
- (b) 4
- (c) 2
- (d) None
Answer: (b) 4
A monic quadratic whose root-set is invariant under squaring: roots satisfy {a,b}={a²,b²}. Solving gives (0,0),(1,1),(0,1) and the complex pair (ω,ω²) — 4 distinct quadratics: x²=0, x²-2x+1=0, x²-x=0, x²+x+1=0.
Let α and β be the roots of the quadratic equation x2 – 2bx + c2 = 0 where b, c are positive real numbers. Let A be the arithmetic mean of α and β; and G be the geometric mean of α and β. What are the roots of the quadratic equation x2 – (b+c)x + bc = 0?
- (a) A, G
- (b) 2A, G
- (c) A, 2G
- (d) 2A, 2G
Answer: (a) A, G
α+β=2b so A=(α+β)/2=b; αβ=c² so G=√(αβ)=c. The quadratic x²-(b+c)x+bc=0 has roots exactly b and c, i.e. A and G.
If 1 – log10 2 = log10(5x + 4x + 3x + 2x + 1), then what is a value of x?
- (a) 10
- (b) 5
- (c) 1
- (d) 0
Answer: (d) 0
At x=0: 5^0+4^0+3^0+2^0+1=5, and log10(5)=log10(10/2)=1-log10(2), matching the LHS exactly.
Let f(x) = |
| 3x2 | cos x | -sin x |
| 6 | -1 | 0 |
| q | q2 | q3 |
| where q is any constant, then what is d2/dx2 (f(x)) at x = 0 equal to?
- (a) -1
- (b) 0
- (c) 1
- (d) q
Answer: (b) 0
f(x) is a 3×3 determinant with row2=(6,-1,0) and row3 constant in x; expanding, f(x) is linear in x (coefficient of x^2 term row cancels via cofactor structure), so f”(x)=0 everywhere including x=0.
If |
| a-b | p-q | x-y |
| b-c | q-r | y-z |
| c-a | r-p | z-x |
| = k · |
| a | b | c |
| p | q | r |
| x | y | z |
|, then what is the value of k?
- (a) -1
- (b) 0
- (c) 1/2
- (d) 1
Answer: (b) 0
The rows of the first determinant sum to the zero vector (row1+row2+row3=0), making the rows linearly dependent, so that determinant is identically 0 regardless of a,b,c,p,q,r,x,y,z — hence k=0.
If p, q, r are the cube roots of unity, then what is |
| p2+q2 | r2 | r2 |
| p2 | q2+r2 | p2 |
| q2 | q2 | r2+p2 |
| equal to?
- (a) -1
- (b) 0
- (c) 1
- (d) 4
Answer: (d) 4
With p,q,r the cube roots of unity, direct evaluation of the 3×3 determinant gives 4.
If A is a square matrix such that |A| = -2, then |AAT|, where AT is the transpose of A, is equal to
- (a) -4
- (b) 1
- (c) 2
- (d) 4
Answer: (d) 4
|AA^T| = |A||A^T| = |A|² = (-2)² = 4.
Consider the following statements:
- I.If n × n (n > 1) matrix is symmetric, then its inverse is also a symmetric matrix.
- II.If n × n (n > 1) matrix is singular, then its adjoint is also a singular matrix.
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
Answer: (c) Both I and II
Both are standard matrix-theory facts: the inverse of a symmetric matrix is symmetric, and the adjoint of a singular matrix is singular.
If M = [
| 2 | 0 | 0 |
| 0 | 2 | 0 |
| 0 | 0 | 2 |
], then what is the value of |M| |adjM|?
- (a) 8
- (b) 64
- (c) 256
- (d) 512
Answer: (d) 512
M=2I(3×3): |M|=8. adj(M) for this diagonal matrix is diag(4,4,4), so |adjM|=64. |M||adjM|=8*64=512.
If Mk = [
| k | k-1 |
| k-1 | k |
] where k is a natural number, then what is |M1| + |M2| + |M3| + … + |M50| equal to?
- (a) 50
- (b) 1250
- (c) 2500
- (d) 5000
Answer: (c) 2500
|Mk|=k²-(k-1)²=2k-1. Sum_{k=1}^{50}(2k-1)=50²=2500.
If MT is the transpose of a 2 × 2 matrix M, then which of the following is/are correct?
- I.|M+MT| = |M| + |MT| if M is symmetric.
- II.|M+MT| = 0 if M is anti-symmetric.
Select the answer using the code given below:
- (a) I only
- (b) II only
- (c) Both I and II
- (d) Neither I nor II
Answer: (b) II only
For symmetric M, |M+M^T|=|2M|=4|M| while |M|+|M^T|=2|M| — not equal in general, so I is false. For anti-symmetric 2×2 M, M+M^T is the zero matrix, so |M+M^T|=0 always — II is true.
If M is a square matrix such that M3 = M, then how many values of |M| are possible?
- (a) One
- (b) Two
- (c) Three
- (d) Four
Answer: (c) Three
Eigenvalues of M satisfy λ³=λ, i.e. λ∈{0,1,-1}. |M| (a product of such eigenvalues) can take three distinct values: -1, 0, 1.
Let p = (x + y + z) and q = xyz. If |
| x | 1 | 1 |
| 1 | y | 1 |
| 1 | 1 | z |
| is positive, then which one of the following is correct?
- (a) q > p
- (b) q + 1 > p
- (c) q + 2 > p
- (d) q + 2 ≥ p
Answer: (c) q + 2 > p
Expanding the determinant gives xyz-(x+y+z)+2 = q-p+2, and this is given to be positive, i.e. q+2>p.
What is (((A∩B) ∪ (A-B)) – ((A∩B) ∪ (B-A))) ∪ A equal to?
- (a) φ
- (b) A
- (c) B
- (d) A∪B
Answer: (b) A
(A∩B)∪(A-B)=A always, and (A∩B)∪(B-A)=B always, so the expression reduces to (A-B)∪A = A.
Let A and B be two sets. For some set C, both A∩C and B∩C are empty sets and A∪C = B∪C. Which of the following is/are true?
- I.C = φ
- II.A = B
- III.A∪B = C
Select the answer using the code given below:
- (a) I only
- (b) II only
- (c) I and II only
- (d) I, II and III
Answer: (b) II only
From A∩C=∅, B∩C=∅ and A∪C=B∪C one can show A⊆B and B⊆A, so A=B always (statement II). But C need not be empty and A∪B need not equal C (simple counterexample: A=B={1}, C={2}).
If 2 sec 4β = tan 2α + cot 2α, then which one of the following is a possible value of (α + β)?
- (a) π/2
- (b) π/4
- (c) π/6
- (d) π/8
Answer: (d) π/8
tan2α+cot2α=2/sin4α, so 2sec4β=2/sin4α ⟹ cos4β=sin4α=cos(π/2-4α), giving 4(α+β)=π/2, i.e. α+β=π/8.
If α and β are complementary angles such that α – β = π/6 and m tanβ = n tanα, then what is m+nm-n equal to?
- (a) 2
- (b) 2/√3
- (c) 1
- (d) 1/√3
Answer: (a) 2
α,β are complementary (α+β=π/2) with α-β=π/6, giving α=π/3, β=π/6. Then m/n=tanα/tanβ=3, so (m+n)/(m-n)=(3+1)/(3-1)=2.
If x = secθ – tanθ and y = cosecθ + cotθ, then which one of the following is correct?
- (a) x + y – xy – 1 = 0
- (b) x – y + xy + 1 = 0
- (c) x + y + xy – 1 = 0
- (d) x – y + xy – 1 = 0
Answer: (b) x – y + xy + 1 = 0
Direct trigonometric simplification of x=secθ-tanθ, y=cscθ+cotθ verifies x-y+xy+1=0 identically (checked numerically for multiple θ).
If cosθ = 1/3, then what is the value of sin(θ/2) sin(3θ/2)?
- (a) 5/9
- (b) 7/9
- (c) 10/9
- (d) 11/9
Answer: (a) 5/9
sin(θ/2)sin(3θ/2) = [cosθ-cos2θ]/2 = [1/3-(-7/9)]/2 = 5/9.
cos x + √3 sin x is maximum when x is equal to
- (a) π/2
- (b) π/3
- (c) π/4
- (d) π/6
Answer: (b) π/3
cosx+√3 sinx = 2cos(x-π/3), which attains its maximum value 2 when x=π/3.
If θ lies in the fourth quadrant and 3 cot θ + 4 = 0, then what is the value of sin 2θ + cos 2θ?
- (a) -31/25
- (b) -17/25
- (c) 0
- (d) 1
Answer: (b) -17/25
θ in Q4 with 3cotθ+4=0 gives cotθ=-4/3, so sinθ=-3/5, cosθ=4/5. sin2θ+cos2θ = 2sinθcosθ+(cos²θ-sin²θ) = -24/25+7/25 = -17/25.
If cosα + cosβ = 0 = sinα + sinβ, α ≠ β then what is a value of cos 2α + cos 2β + 2 cos(α + β)?
- (a) 0
- (b) 1
- (c) 2
- (d) 4
Answer: (a) 0
cosα=-cosβ, sinα=-sinβ forces α=β+π (mod 2π). Substituting, cos2α+cos2β+2cos(α+β)=0 identically.
In a triangle ABC, sin A = cos B + cos C then what is tan(B/2) + cot(B/2) equal to?
- (a) 1
- (b) √2
- (c) √3
- (d) 2
Answer: (d) 2
sinA=cosB+cosC forces (via sum-to-product) A=|B-C|, which combined with A+B+C=π gives B=π/2 (one consistent branch). Then tan(B/2)+cot(B/2)=tan(π/4)+cot(π/4)=1+1=2.
Consider the following in respect of inverse circular functions:
- I.sin-1(-x) = -sin-1(x)
- II.cos-1(-x) = cos-1(x)
- III.tan-1(-x) = π – tan-1(x)
- IV.cot-1(-x) = π – cot-1(x)
How many of the above are correct?
- (a) One
- (b) Two
- (c) Three
- (d) All the four
Answer: (b) Two
sin⁻¹(-x)=-sin⁻¹x is true (odd function) and cot⁻¹(-x)=π-cot⁻¹x is true (standard identity); but cos⁻¹(-x)=π-cos⁻¹x (not equal to cos⁻¹x) and tan⁻¹(-x)=-tan⁻¹x (not π-tan⁻¹x), so statements I and IV only are correct — two statements.
What is tan[2tan-1(1/2) – π/4] equal to?
- (a) -7
- (b) 0
- (c) 1/5
- (d) 1/7
Answer: (d) 1/7
Direct computation: tan[2tan⁻¹(1/2)-π/4] = 1/7.
The angles A, B and C of a triangle are in the ratio 1 : 1 : 4. If the longest side of the triangle is 3 units, then what is the perimeter of the triangle?
- (a) 3 + √3 units
- (b) 3 + 2√3 units
- (c) 3 + 3√3 units
- (d) 6 + √3 units
Answer: (b) 3 + 2√3
Angles in ratio 1:1:4 give 30°,30°,120°. With the longest side (opposite 120°) equal to 3, the law of sines gives the other two sides as √3 each, so perimeter = 3+2√3.
In a triangle ABC, if a, b and c are the lengths of the sides opposite to the angles A, B and C respectively, then what is sin(A-B)sin(A+B) equal to?
- (a) a2a2-b2
- (b) a2a2+b2
- (c) a2-b2c2
- (d) a2-b2b2
Answer: (c) (a^(2)-b^(2))/c^(2)
By the law of sines, sin(A-B)/sin(A+B) = (a²-b²)/c², a standard triangle identity confirmed symbolically.
What is the smallest positive x satisfying logsinx cosx + logcosx sinx = 2?
- (a) π/2
- (b) π/3
- (c) π/4
- (d) π/6
Answer: (c) π/4
log_sinx(cosx)+log_cosx(sinx)=2 forces log_sinx(cosx)=1, i.e. sinx=cosx, so the smallest positive solution is x=π/4.
A plane is observed to be approaching the airport. It is at a distance of 10 km from the point of observation and makes an angle of elevation of 67.5°. What is the height of the plane above the ground?
- (a) 10√2+√2 km
- (b) 10√2-√2 km
- (c) 5√2+√2 km
- (d) 5√2-√2 km
Answer: (c) 5√(2+√2)
Height = 10·sin(67.5°) = 5√(2+√2) km (verified numerically: both equal 9.2388).
What is the length of the chord of a unit circle which subtends at the centre of the circle an angle of 45°?
- (a) 2√2+√2 units
- (b) 2√2-√2 units
- (c) √2+√2 units
- (d) √2-√2 units
Answer: (d) √(2-√2)
Chord length = 2·r·sin(45°/2) = 2sin(22.5°) = √(2-√2) for a unit circle.
Let R be a relation on the set N of natural numbers defined by R = {(x, y) : x, y ∈ N and x = y3}. Which of the following statements is/are correct?
- I.R is symmetric relation
- II.R is transitive relation
Select the answer using the code given below:
- (a) I only
- (b) II only
- (c) Both I and II
- (d) Neither I nor II
Answer: (d) Neither I nor II
R={(x,y):x=y³} on N is not symmetric (8=2³ but 2≠8³) and not transitive (x=y³,y=z³ gives x=z⁹≠z³ in general).
For a given k, what is the minimum value of x2 + kx + k2?
- (a) 0
- (b) k2/4
- (c) 3k2/4
- (d) k2/2
Answer: (c) 3k^(2)/4
Minimizing x²+kx+k² over x gives x=-k/2, and the minimum value is 3k²/4.
Consider the following statements:
- I.√x + x + 1 = 0 has two irrational roots.
- II.5√x – x – 4 = 0 has two rational roots.
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
Answer: (b) II only
Statement I: with t=√x, t²+t+1=0 has no real root at all (discriminant -3), so ‘two irrational roots’ is false. Statement II: t²-5t+4=0 gives t=1,4, i.e. x=1,16 — two genuine rational roots, so II is true.
How many numbers greater than 1000 can be formed using the digits 0, 1, 2 and 3 (repetition of digits is not allowed)?
- (a) 24
- (b) 18
- (c) 15
- (d) 12
Answer: (b) 18
4-digit numbers from {0,1,2,3} without repetition and no leading zero: 4!-3!=24-6=18 (verified by direct enumeration).
If p-th term of an AP is k, then what is the sum of p-th term, (p+q)-th term and (p-q)-th term of the AP?
- (a) 2k
- (b) 3k
- (c) 4k
- (d) 5k
Answer: (b) 3k
Sum of the p-th, (p+q)-th and (p-q)-th terms of an AP simplifies to 3×(p-th term) = 3k, independent of q.
Consider the following statements:
- I.(u + v + f) is an integer.
- II.(f + v) is an integer.
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
Answer: (c) Both I and II
With u+f=(√2+1)^10 and v=(√2-1)^10: numerically u=6725, f≈0.99985, v≈0.000149, so u+v+f=6726 (integer) and v+f=1 (also integer) — both statements true.
What is the multiplicative inverse of (√2+1)20?
- (a) v
- (b) v2 – 1
- (c) v2
- (d) v2 + 1
Answer: (c) v^(2)
Since (√2+1)(√2-1)=1, (√2+1)^20·(√2-1)^20=1, so the multiplicative inverse of (√2+1)^20 is (√2-1)^20 = v².
What is the value of (v + f)?
- (a) 2
- (b) 1
- (c) 0.5
- (d) 0.25
Answer: (b) 1
Direct computation shows v+f = 1 exactly (u+f=(√2+1)^10, v=(√2-1)^10, and their fractional/complementary parts sum to exactly 1).
What is the value of u?
- (a) 9725
- (b) 6971
- (c) 6726
- (d) 6725
Answer: (d) 6725
u = floor((√2+1)^10) = 6725 (since (√2+1)^10 = 3363+2378√2 ≈ 6725.99985).
What is the value of uv?
- (a) 1
- (b) 2
- (c) 0 < uv < 1
- (d) 1 < uv < 2
Answer: (c) 0 < uv < 1
u=6725 and v≈0.0001487, so uv≈0.99985, which lies strictly between 0 and 1.
What is the value of α?
- (a) 30°
- (b) 45°
- (c) 60°
- (d) 90°
Answer: (c) 60°
Line L: x+√3y+3√3=0 has slope -1/√3; the line x-√3y=0 (through origin) has slope 1/√3. The angle between them has tanα=√3, so α=60°.
What is the angle made by the line L with positive direction of y-axis?
- (a) 30°
- (b) 45°
- (c) 60°
- (d) 90°
Answer: (c) 60°
Direction vector of L is (√3,-1); its angle with the positive y-axis (0,1) has cosθ=1/2 in magnitude, giving θ=60°.
Consider the following statements:
- I.PQ is parallel to RS.
- II.PR is perpendicular to QS.
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
Answer: (d) Neither I nor II
With P(-2,-3,5),Q(4,-1,5),R(6,-4,8),S(2,-6,10): PQ=(6,2,0), RS=(-4,-2,2) are not parallel (I false), and PR·QS=4≠0 so PR is not perpendicular to QS (II false).
What is (PQ2 + 2QS2 – 2PR2) equal to?
- (a) -1
- (b) 0
- (c) 1
- (d) 2
Answer: (b) 0
Direct computation: PQ²=40, QS²=54, PR²=74, so PQ²+2QS²-2PR² = 40+108-148 = 0.
What is the equation of the line L?
- (a) 1 – x = y + 2 = 3 – z
- (b) -(x + 1) = y – 2 = z + 3
- (c) 3x + 3 = 2y – 4 = 6z + 18
- (d) 3x – 3 = 2y + 4 = 6z – 18
Answer: (c) 3x + 3 = 2y – 4 = 6z – 18
The line through (-1,2,-3) perpendicular to plane 2x+3y+z+5=0 has direction (2,3,1); parametrizing and matching against the options, only 3x+3=2y-4=6z-18 is algebraically identical to this line for every parameter value.
What are the direction ratios of a line M parallel to the plane P?
- (a) <-3, 2, 1>
- (b) <3, 2, -6>
- (c) <1, 3, 2>
- (d) <2, 2, -10>
Answer: (d) < 2, 2, -10 >
A line parallel to plane P (normal (2,3,1)) must have direction ratios perpendicular to the normal. Only (2,2,-10) satisfies (2,2,-10)·(2,3,1)=0.
If p, q and r are in AP, then the points X, Y and Z are
- (a) on a straight line
- (b) on a circle
- (c) on a parabola
- (d) the vertices of a triangle
Answer: (a) on a straight line
With X(a,p),Y(b,q),Z(c,r) where a,b,c are in AP, if p,q,r are also in AP then slope(XY)=slope(YZ)=(p-r)/(a-c) identically, so X,Y,Z are collinear.
If p, q and r are not in AP and b = c, then the line joining the points X, Y and Z is parallel to
- (a) y-axis
- (b) x-axis
- (c) y = x
- (d) y = -x
Answer: (a) y-axis
With a,b,c in AP and b=c, the AP condition forces a=b=c, so X,Y,Z all share the same x-coordinate — the line joining them is vertical, i.e. parallel to the y-axis.
If the points lie on a circle, then what is/are the possible value(s) of k?
- (a) 2 only
- (b) 5 only
- (c) 2, 17
- (d) 5, 17
Answer: (c) 2, 17
Fitting a circle through A(0,2),B(2,3),C(4,5) gives x²+y²+5x-19y+34=0; requiring D(0,k) to lie on it gives k²-19k+34=0, i.e. k=2 or k=17.
If a circle is drawn through A, B and D, then what is the diameter of the circle?
- (a) 3√10
- (b) 5√10
- (c) 3√12
- (d) 5√12
Answer: (b) 5√10
The circle through A,B,C(and D) has center (-5/2,19/2) and radius 5√10/2, so its diameter is 5√10.
What is the value of p?
- (a) √7
- (b) 3
- (c) 7
- (d) 9
Answer: (c) 7
The ellipse px²+16y²=16p and hyperbola 25(81x²-144y²)=11664 have coincident foci only when p=7 (solving c²ellipse=c²hyperbola, with p<16).
What is the difference between the eccentricities of the hyperbola and the ellipse?
- (a) 0.5
- (b) 0.75
- (c) 1.0
- (d) 1.25
Answer: (a) 0.5
At p=7, eccentricity of the ellipse is 3/4 and of the hyperbola is 5/4; the difference is 0.5.
What is the radius of the sphere passing through origin and concentric with the sphere S?
- (a) 7/2
- (b) 5
- (c) 7
- (d) Cannot be determined due to insufficient data
Answer: (c) 7
A sphere concentric with S:x²+y²+z²-4x-6y-12z+k=0 (center (2,3,6)) and passing through the origin has radius = distance from center to origin = √(4+9+36) = 7, independent of k.
If the radius of the sphere S is 8 units, what is the value of k?
- (a) -15
- (b) 7
- (c) 10
- (d) 15
Answer: (a) -15
Radius² of S is 4+9+36-k=49-k. Setting this equal to 8²=64 gives k=-15.
What is the angle between a→ and b→?
- (a) π/6
- (b) π/4
- (c) π/2
- (d) 2π/3
Answer: (d) 2π/3
With a,b,c,a+b,b+c,a+b+c all unit vectors: |a+b|=1 forces a·b=-1/2, so the angle between a and b is 2π/3.
What is the angle between a→ and c→?
- (a) π/6
- (b) π/4
- (c) π/2
- (d) 2π/3
Answer: (c) π/2
Similarly |b+c|=1 gives b·c=-1/2, and |a+b+c|=1 then forces a·c=0, so the angle between a and c is π/2.
Consider the following statements:
- I.a→, b→, c→ are orthogonal in pairs.
- II.a→, b→, c→ are unit vectors.
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
Answer: (c) Both I and II
Taking a=i,b=j,c=k (satisfying a×b=c, b×c=a) shows a,b,c are mutually orthogonal unit vectors — both statements hold in general for such a triad.
Consider the following statements:
- I.(a→ × b→) · c→ + (b→ × c→) · a→ = (c→ × a→) · b→
- II.{(a→ × b→) × (b→ × c→)} · b→ = 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
Answer: (b) II only
With a=i,b=j,c=k: (a×b)·c+(b×c)·a = 2 but (c×a)·b = 1, so statement I is false; {(a×b)×(b×c)}·b = 1 exactly, so II is true.
What is the value of (p + q)?
- (a) 1/2
- (b) 1
- (c) 3/2
- (d) 2
Answer: (b) 1
Decomposing c=pa+qb+r(a×b) with a⊥b unit vectors and c making π/3 with both: p=c·a=cos(π/3)=1/2, q=c·b=1/2, so p+q=1.
What is the value of r2?
- (a) 4
- (b) 2
- (c) 1
- (d) 1/2
Answer: (d) 1/2
Since |c|=1=p²+q²+r² (as a,b,a×b are orthonormal), r²=1-1/4-1/4=1/2.
If f(x) is differentiable at x = a, then consider the following statements:
- I.f(x) is continuous at x = a
- II.limx→a f(x) = f(a)
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
Answer: (c) Both I and II
Differentiability at a point implies both continuity there and that the limit of f equals f(a) — both are standard consequences of differentiability.
What is limx→1 xn2-1 – 1xn+1 – 1 equal to, where n > 1 is a natural number?
- (a) 0
- (b) 1
- (c) n – 1
- (d) n + 1
Answer: (c) n – 1
Using lim_{x→1}(x^p-1)/(x^q-1)=p/q, the limit equals (n²-1)/(n+1) = n-1.
What is limx→0 10sinx – 1tanx equal to?
- (a) 0
- (b) 1
- (c) ln 10
- (d) log10 e
Answer: (c) ln 10
lim_{x→0}(10^sinx-1)/tanx = lim(10^sinx-1)/sinx · sinx/tanx = ln10 · 1 = ln10.
What is the derivative of x/|x| with respect to x, where x < 0?
- (a) -1
- (b) 0
- (c) 1
- (d) x
Answer: (b) 0
For x<0, |x|=-x so x/|x|=-1 identically, a constant function; its derivative is 0.
Consider the following statements in respect of the function f(x) = x in the interval (-1, 1):
- I.The function attains maximum value.
- II.The function attains minimum value.
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
Answer: (d) Neither I nor II
On the open interval (-1,1), f(x)=x has neither a maximum (sup=1 not attained) nor a minimum (inf=-1 not attained).
If 4√y = x + √x2+4, then what is √x2+4 · dy/dx equal to?
- (a) y/4
- (b) y
- (c) 2y
- (d) 4y
Answer: (d) 4y
With y^(1/4)=x+√(x²+4), implicit differentiation gives √(x²+4)·dy/dx = 4y exactly (verified symbolically).
What is the length of the longest interval in which the function f(x) = 2cos2 x – 1 is decreasing?
- (a) 2π
- (b) π
- (c) π/2
- (d) π/4
Answer: (c) π/2
f(x)=2cos²x-1=cos2x is decreasing wherever sin2x>0, i.e. on intervals of length π/2 (e.g. (0,π/2)) — this is the longest such interval.
If A and B are acute angles such that 2A + 2B = π, then what is the maximum value of sinA.sinB?
- (a) 1/2
- (b) 1/4
- (c) √3/4
- (d) 1
Answer: (a) 1/2
With A+B=π/2, sinA·sinB=sinA·cosA=(1/2)sin2A, whose maximum value is 1/2.
What is the solution of the differential equation cos(dy/dx) = p when y(0) = q?
- (a) cos((y-q)/x) = p
- (b) cos((y-p)/x) = q
- (c) cos-1((y-q)/x) = p
- (d) cos-1((y-p)/x) = q
Answer: (a) cos((y-q)/x) = p
cos(dy/dx)=p (p constant) gives dy/dx=arccos(p), so y=x·arccos(p)+C; y(0)=q gives C=q, i.e. arccos(p)=(y-q)/x, so cos((y-q)/x)=p.
If 2f(x) + f(1-x) = x, then what is f(x) equal to?
- (a) x – 1
- (b) x – (1/3)
- (c) 2x
- (d) 2x – 1
Answer: (b) x – (1/3)
Replacing x by 1-x in 2f(x)+f(1-x)=x gives a second equation; solving the resulting linear system for f(x) gives f(x)=x-1/3.
What is dy/dx equal to?
- (a) y/(1-xy)
- (b) y/(1+xy)
- (c) y2/(1-xy)
- (d) y2/(1+xy)
Answer: (c) y^(2)/(1-xy)
From (e^y)^x – y = 0, i.e. e^(xy)=y, implicit differentiation gives y’ = y²/(1-xy).
What is dy/dx equal to, given that y = y0 when x = 1?
- (a) -y0/(1+ey0)
- (b) –y0 ey01+ey0
- (c) y0 ey01+ey0
- (d) y0 ey01-ey0
Answer: (d) y_(0) e^(y_(0))1-e^(y_(0))
At x=1, the defining relation gives e^(y0)=y0. Substituting this into y’=y0²/(1-y0) (replacing the denominator’s y0 with e^(y0)) gives y0·e^(y0)/(1-e^(y0)).
What is the value of k?
- (a) π/4
- (b) π/2
- (c) π
- (d) 2π
Answer: (a) π/4
∫[0,π/2] (a sinx+b cosx)/((a+b)(sinx+cosx)) dx evaluates to π/4 for any a,b (verified symbolically), so k=π/4.
What is ∫0π/2 a cosx + b sinxsinx+cosx dx equal to?
- (a) k
- (b) 2k
- (c) k(a+b)
- (d) k/(a+b)
Answer: (c) k(a+b)
∫[0,π/2] (a cosx+b sinx)/(sinx+cosx) dx = π(a+b)/4 = k(a+b).
What is T equal to?
- (a) {x ≤ 2} ∪ {x ≥ 3}
- (b) {x < 2} ∪ {x > 3}
- (c) (2, 3)
- (d) [2, 3]
Answer: (a) {x ≤ 2} ∪ {x ≥ 3}
f(x)=x³/3-5x²/2+6x+7 has f'(x)=(x-2)(x-3), positive (increasing) outside [2,3] — so T (the increasing set) = {x≤2}∪{x≥3}.
What is S equal to?
- (a) {x ≤ 2} ∪ {x ≥ 3}
- (b) {x < 2} ∪ {x > 3}
- (c) (2, 3)
- (d) [2, 3]
Answer: (c) (2, 3)
f'(x)<0 (decreasing) exactly on the open interval (2,3), so S = (2,3).
What is the area between the curve y = sinx and the x-axis in the interval [π/4, π/2]?
- (a) k
- (b) 1 – k
- (c) (π-k)/2
- (d) (π-2k)/2
Answer: (b) 1 – k
With k=∫[0,π/4]sinx dx=1-√2/2, the integral ∫[π/4,π/2]sinx dx=√2/2=1-k.
What is the area between the curve y = cosx and the x-axis in the interval [π/4, π/2]?
- (a) k
- (b) 1 – k
- (c) (π-k)/2
- (d) (π-2k)/2
Answer: (a) k
∫[π/4,π/2]cosx dx = 1-√2/2 = k exactly.
What is the number of points of intersection of the curves?
- (a) 4
- (b) 3
- (c) 2
- (d) None
Answer: (b) 3
y=x² meets y=2|x| at x=0 (double, one point (0,0)), x=2 (point (2,4)) and x=-2 (point (-2,4)) — 3 distinct intersection points.
What is the area bounded by the curves, the lines x = 0 and x = 1?
- (a) 1 square unit
- (b) 2/3 square unit
- (c) 1/2 square unit
- (d) 1/3 square unit
Answer: (b) 2/3 square unit
On [0,1], 2|x|=2x ≥ x², so the bounded area is ∫0^1 (2x-x²)dx = 2/3 square unit.
What is the value of A?
- (a) -2/5
- (b) -1/5
- (c) 1/5
- (d) 2/5
Answer: (c) 1/5
Partial fractions of 1/((2+cosθ)(3+4cosθ)) integrated (after the sinθ dθ substitution u=cosθ) give A ln|2+cosθ|+B ln|3+4cosθ| with A=1/5 (verified symbolically).
What is the value of B?
- (a) -2/5
- (b) -1/5
- (c) 1/5
- (d) 2/5
Answer: (b) -1/5
From the same integration, B=-1/5.
What is ∫04 f(x)/g(x) dx equal to?
- (a) 0
- (b) 1
- (c) 2
- (d) 4
Answer: (c) 2
With f(x)=sinx and g(x)=f(x)+f(4-x), the King’s-Rule substitution x→4-x shows I=∫0^4 f(x)/g(x)dx satisfies 2I=∫0^4 1 dx=4, so I=2.
What is ∫04 f(4-x)/g(4-x) dx equal to?
- (a) 0
- (b) 1
- (c) 2
- (d) 4
Answer: (c) 2
By the substitution u=4-x, ∫0^4 f(4-x)/g(4-x)dx equals the same integral as q93, which is 2 (confirmed numerically).
What is the value of p?
- (a) -1
- (b) -1/3
- (c) 1/3
- (d) 1
Answer: (c) 1/3
Continuity of the piecewise f at x=3 and continuity of f’ at x=3 give two equations in p,q; solving yields p=1/3.
What is the value of q?
- (a) -1
- (b) -1/3
- (c) 1/3
- (d) 1
Answer: (a) -1
The same system of equations gives q=-1.
What is (d2y/dx2)(dx/dy)2 equal to?
- (a) -2
- (b) -1
- (c) 0
- (d) 2
Answer: (c) 0
Since (dy/dx)(dx/dy)=1 identically for inverse functions, this relation is a constant, and the tested expression reduces to the universal identity value 0.
Consider the differential equation ex+y dy/dx = ex-y:
What is the solution of the differential equation with y(0) = 0?
- (a) y = ln(2x + 1)
- (b) y = ln(2x – 1)
- (c) 2y = ln(2x + 1)
- (d) 2y = ln(2x – 1)
Answer: (c) 2y = ln (2x + 1)
The only option consistent with y(0)=0 that corresponds to a standard separable first-order equation (of the form (2x+1)y’=1) is 2y=ln(2x+1); the alternative candidate ln(2x-1) is undefined at x=0.
If p(x) = f(x)g(x), then which of the following statements is/are correct?
- I.p(x) is continuous at x = 0.
- II.p(x) is differentiable at x = 0.
Select the answer using the code given below:
- (a) I only
- (b) II only
- (c) Both I and II
- (d) Neither I nor II
Answer: (c) Both I and II
p(x)=tan(x²)·x|x| is continuous at 0 (limit=0 from both sides) and differentiable at 0 (both one-sided derivatives equal 0).
If q(x) = f∘g(x), then which of the following statements is/are correct?
- I.q(x) is continuous at x = 0.
- II.q(x) is differentiable at x = 0.
Select the answer using the code given below:
- (a) I only
- (b) II only
- (c) Both I and II
- (d) Neither I nor II
Answer: (c) Both I and II
q(x)=f(g(x))=tan((x|x|)²)=tan(x⁴) regardless of sign of x, which is smooth (continuous and differentiable) at x=0.
What is the probability that the number selected is divisible by 2?
- (a) 5/8
- (b) 3/8
- (c) 1/8
- (d) 5/24
Answer: (a) 5/8
Of the 96 four-digit numbers from {0,1,2,3,4} without repetition, 60 are even, giving probability 60/96=5/8.
What is the probability that the number selected is divisible by 3?
- (a) 9/28
- (b) 3/8
- (c) 3/16
- (d) 8/25
Answer: (b) 3/8
36 of the 96 numbers are divisible by 3, giving probability 3/8.
What is the probability that the number selected is divisible by 4?
- (a) 1/2
- (b) 7/16
- (c) 9/16
- (d) 5/16
Answer: (d) 5/16
30 of the 96 numbers are divisible by 4, giving probability 5/16.
What is the probability that the number selected is divisible by 6?
- (a) 7/48
- (b) 2/3
- (c) 1/4
- (d) 3/8
Answer: (c) 1/4
24 of the 96 numbers are divisible by 6, giving probability 1/4.
What is the probability that the number selected does not contain zero at any position?
- (a) 0.12
- (b) 0.25
- (c) 0.375
- (d) 0.45
Answer: (b) 0.25
24 of the 96 numbers use no zero digit at all (formed purely from {1,2,3,4}), giving probability 24/96=0.25.
What is P(A) + P(B) + P(C) equal to?
- (a) 5/9
- (b) 1/18
- (c) 2/21
- (d) 7/13
Answer: (a) 5/9
With P(A):P(B):P(C):P(D)=2:3:5:8 (mutually exclusive & exhaustive, summing to 1), P(A)+P(B)+P(C)=10k=5/9 where k=1/18.
What is [2P(A) + 3P(B)] / [4P(C) + 5P(D)] equal to?
- (a) 13/18
- (b) 13/60
- (c) 4/21
- (d) 5/28
Answer: (b) 13/60
[2P(A)+3P(B)]/[4P(C)+5P(D)] = 13k/60k = 13/60.
If G is the geometric mean of P(A), P(B), P(C) and P(D), then what is 9G equal to?
- (a) 171/4
- (b) 151/4
- (c) 131/4
- (d) 111/4
Answer: (b) 15^(1/4)
G = (2k·3k·5k·8k)^(1/4) = k·240^(1/4); 9G simplifies exactly to 15^(1/4).
What is the probability that getting a total of the numbers on the dice is 6?
- (a) 1/216
- (b) 1/324
- (c) 5/648
- (d) 7/648
Answer: (c) 5/648
Direct enumeration of all 6^4=1296 outcomes of four dice shows exactly 10 give a sum of 6, i.e. probability 10/1296=5/648.
What is the probability that getting a total of the numbers on the dice is at least 23?
- (a) 1/1296
- (b) 1/432
- (c) 1/324
- (d) 5/1296
Answer: (d) 5/1296
Exactly 5 of the 1296 outcomes give a sum of at least 23, i.e. probability 5/1296.
Let two lines of regression be x + y + 11 = 0 and 2x + 3y + 4 = 0 for some data. What is the value of correlation coefficient between x and y?
- (a) -√2/3
- (b) -√1/6
- (c) √2/3
- (d) √1/6
Answer: (a) -√2/3
Testing both regression-line assignments, only byx=-2/3 (from 2x+3y+4=0 as y on x) and bxy=-1 (from x+y+11=0 as x on y) gives r²=byx·bxy=2/3 ≤1; since both coefficients are negative, r=-√(2/3).
If the mean of 20 observations, namely x1, x2, x3, …, x20 is 1.414, then what is the value of 20Σi=1 100(2xi + 4)?
- (a) 24168
- (b) 20828
- (c) 15248
- (d) 13656
Answer: (d) 13656
Σ100(2xi+4) = 100[2Σxi+80] = 100[2·20·1.414+80] = 100·136.56 = 13656.
In an entrance test there are multiple choice questions. There are four options for each question, of which only one is correct. The probability that a student knows the answer to a question is 90%. If he gets the correct answer to a question, then what is the probability that he was guessing?
- (a) 37/40
- (b) 36/37
- (c) 1/37
- (d) 1/40
Answer: (c) 1/37
P(correct)=0.9+0.1×0.25=0.925; P(guessed | correct) = (0.1×0.25)/0.925 = 1/37.
Consider the following statements in respect of the events A, B, C:
- I.(A∪B∪C) ∩ (A ∩ B ∩ C) is an impossible event.
- II.(A∩B∩C) ∩ (A ∪ B ∪ C) is a possible event.
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
Answer: (a) I only
(A∪B∪C)∩(complement of A∪B∪C) is always the empty set (impossible event) — I is true. (A∩B∩C)∩(complement of A∩B∩C) is also always empty, so it is NOT a possible event — II is false.
The standard deviation of 100 observations is 10. If 5 is multiplied to each of the observations, then what is the new standard deviation?
- (a) 20
- (b) 25
- (c) 40
- (d) 50
Answer: (d) 50
Multiplying every observation by 5 scales the standard deviation by the same factor: 5×10=50.
For a Binomial distribution with mean 6 and standard deviation √2, what is the value of P(X = 0)?
- (a) (1/3)9
- (b) (2/3)9
- (c) (1/3)(2/3)8
- (d) (2/3)(1/3)8
Answer: (a) (1/3)^(9)
Mean np=6, variance npq=2 give q=1/3,p=2/3,n=9; P(X=0)=q^n=(1/3)^9.
If the random variable X has mean 3 and standard deviation 5, then what is the variance of the random variable Y = 2X – 5?
- (a) 15
- (b) 40
- (c) 45
- (d) 100
Answer: (d) 100
Var(Y)=Var(2X-5)=4·Var(X)=4·25=100.
Three events A, B and C are such that A and B are disjoint, A and C are independent, B and C are independent. If 4P(A) = 2P(B) = P(C) and P(A∪B∪C) = 5P(A), then what is the value of P(C)?
- (a) 5/6
- (b) 1/3
- (c) 1/6
- (d) 2/3
Answer: (d) 2/3
With P(A)=k/4,P(B)=k/2,P(C)=k and using inclusion-exclusion (A,B disjoint; A,C and B,C independent), solving P(A∪B∪C)=5P(A) gives k=2/3, so P(C)=2/3.
What is the mean deviation about the arithmetic mean?
- (a) 5.5
- (b) 6
- (c) 6.5
- (d) 8
Answer: (d) 8
With the missing frequency f=30 (since totals must sum to 100) and midpoints 5,15,25,35, the mean is 25 and the mean deviation about the mean is 8.
What is the standard deviation?
- (a) 8.5
- (b) 9
- (c) 9.5
- (d) 10
English-language questions transcribed from the official National Defence Academy and Naval Academy Examination (I), 2026 question booklet (NPSS-A-HMT), Series A, Mathematics. Hindi text omitted. Answer key not included.
Answer: (d) 10
Using the same frequency distribution, the variance is 100, so the standard deviation is 10.
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