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

If p^x = q^y = r^z, where x, y and z are in GP, then consider the following statements: I. p, q and r are in AP. II. ln p, ln q and ln r are in GP. 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

Testing with x,y,z in GP (e.g. ratio 2), p,q,r are not in AP, but lnp,lnq,lnr are always in GP (verified: (lnq)²=lnp·lnr).

If A and B are non-empty subsets of a set, and A^c and B^c represent their complements, then which of the following is/are correct? I. A – B = B^c – A^c; II. A – B^c = A^c – B. 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: (a) I only

With concrete sets, A-B always equals B^c-A^c (standard set identity) — I true. A-B^c generally differs from A^c-B — II false.

Let y = x! and z = (2x)!. If (z/y) = 120, then what is the value of (3x)!?

(a)362880
(b)181440
(c)90720
(d)45360

Answer: (a) 362880

z/y=(2x)!/x!=120 gives x=3 (checked directly), so (3x)!=9!=362880.

Let n be a natural number. The number of consecutive zeros at the end of the expansion of n! is exactly 2. How many values of n are possible?

(a)3
(b)4
(c)5
(d)More than 5

Answer: (c) 5

n! has exactly 2 trailing zeros for n=10,11,12,13,14 — 5 values.

If (10+log₁₀x), (10+log₁₀y) and (10+log₁₀z) are in AP, then consider the following statements: I. The GM of x and z is y^2. II. The AM of log₁₀x and log₁₀z is log₁₀y. 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

(10+logx),(10+logy),(10+logz) in AP means logx,logy,logz are in AP, so y²=xz (GM of x,z is |y|, not y² — I false) and AM(logx,logz)=logy exactly (II true).

How many terms of the series 1+3+5+7+… amount to a sum equal to 12345678987654321?

(a)11111111
(b)110000011
(c)111101111
(d)111111111

Answer: (d) 111111111

The sum of the first N odd numbers is N²; solving N²=12345678987654321 gives N=111111111.

How many terms are identical in the two APs 19, 21, 23, … up to 110 terms and 19, 22, 25, 28, … up to 75 terms?

(a)35
(b)36
(c)37
(d)38

Answer: (c) 37

Common terms of the two APs form a new AP with common difference lcm(2,3)=6 starting at 19; counting terms within both ranges gives 37.

If α = (-1+√-3)/2, then what is the value of (1+α^19-α^35)^100 – (1-3α^25+α^38)^50?

(a)-2
(b)-1
(c)0
(d)2

Answer: (c) 0

With α a primitive cube root of unity, reducing all exponents mod 3 and using 1+α+α²=0 shows the expression simplifies exactly to 0 (verified with exact arithmetic).

What is the remainder when 5^99 is divided by 13?

(a)10
(b)9
(c)8
(d)6

Answer: (c) 8

5^99 mod 13 = 8 (direct modular exponentiation).

10.

What is the value of the determinant of the inverse of the matrix [[-4,-5],[2,2]]?

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

Answer: (a) 1/2

det(M⁻¹)=1/det(M); det([[-4,-5],[2,2]])=2, so det(M⁻¹)=1/2.

11.

In a class of 45 students, 34 like to play cricket and 26 like to play football. Further, each student likes to play at least one of the two games. How many students like to play exactly one game?

(a)45
(b)30
(c)25
(d)15

Answer: (b) 30

With |C∪F|=45 (all like ≥1 game), |C∩F|=34+26-45=15 (both games), so exactly-one-game = 45-15=30.

12.

The system of equations 2x-3y-5=0, 15y-10x+50=0

(a)has a unique solution
(b)has infinitely many solutions
(c)is inconsistent
(d)is consistent and has exactly two solutions

Answer: (c) is inconsistent

The second equation simplifies to 2x-3y-10=0, which is parallel to (but has a different constant from) the first equation 2x-3y-5=0 — no solution exists.

13.

If ((1-i)/(1+i))^2m × ((1+i)/(1-i))^2n = 1, where i = √-1, then what is the smallest positive value of (m-n)?

(a)1
(b)2
(c)4
(d)8

Answer: (b) 2

(1-i)/(1+i)=-i, so the equation becomes (-i)^(2(m-n))=1; since -i has order 4, 2(m-n) must be a multiple of 4, giving the smallest positive m-n=2.

14.

In obtaining the solution of the system of equations x+y+z=7, x+2y+3z=16 and x+3y+4z=22 by Cramer’s rule, the value of y is obtained by dividing D by D2, where D = |1 1 1; 1 2 3; 1 3 4|. What is the value of the determinant D2?

(a)-13
(b)-3
(c)3
(d)13

Answer: (b) -3

Direct computation of the y-column-replaced determinant D2=|[1,7,1],[1,16,3],[1,22,4]| gives -3.

15.

Consider the following in respect of non-singular matrices A and B: I. (AB)^-1 = A^-1 B^-1; II. (BA)(AB)^-1 = I, where I is the identity matrix; III. (AB)^T = A^T B^T. How many of the above are correct?

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

Answer: (a) None

In general (AB)⁻¹=B⁻¹A⁻¹ (not A⁻¹B⁻¹) and (AB)^T=B^TA^T (not A^TB^T); none of the three stated identities hold unless A,B happen to commute.

16.

The value of the determinant |a b c; l m n; p q r| is equal to

(a)|a b c; p q r; l m n|
(b)|l m n; a b c; p q r|
(c)|p q r; a b c; l m n|
(d)|a p l; b q m; c r n|

Answer: (c) |p q r; a b c; l m n|

Moving the third row of the original determinant to the top (a cyclic shift, an even permutation of rows) preserves the determinant’s value exactly.

17.

Let 1, ω, ω^2 be three cube roots of unity. If x = a+b, y = aω+bω^2, z = aω^2+bω, then what is x^2+y^2+z^2 equal to?

(a)6ab
(b)3ab
(c)a^2+b^2
(d)1

Answer: (a) 6ab

Direct substitution with a,b and a primitive cube root of unity gives x²+y²+z²=6ab exactly (verified numerically).

18.

How many 4-digit numbers that are divisible by 4 can be formed using the digits 1, 2, 3 and 4 (repetition of digits is not allowed)?

(a)3
(b)6
(c)9
(d)12

Answer: (b) 6

Direct enumeration of 4-digit arrangements of {1,2,3,4} divisible by 4 gives 6 such numbers.

19.

If a, b, c are the sides of a triangle ABC and p is the perimeter of the triangle, then what is |p+c a b; c p+a b; c a p+b| equal to?

(a)p^3
(b)2p^3
(c)3p^3
(d)4p^3

Answer: (b) 2p^3

With p=a+b+c substituted into the determinant, it factors exactly to 2(a+b+c)^3=2p³.

20.

Which one of the following is the greatest coefficient in the expansion of (1+x)^100?

(a)The coefficient of x^100
(b)The coefficient of x^99
(c)The coefficient of x^51
(d)The coefficient of x^50

Answer: (d) The coefficient of x^50

Since 100 is even, the greatest binomial coefficient in (1+x)^100 is the middle one, C(100,50).

For the following two (02) items: Let α and β be the roots of the quadratic equation x^2 + (log₀.₅(a^2))x + (log₀.₅(a^2))^4 = 0, where a^2≠1 and log₀.₅(a^2) > 0. Further, β^2 = α(log_a²(0.5)).

21.

What is β equal to?

(a)log_a²(0.5)
(b)log₀.₅(a^2)
(c)2(log_a²(0.5))
(d)2log₀.₅(a^2)

Answer: (b) log₀.₅(a^2)

Writing L=log₀.₅(a²), Vieta’s relations (αβ=L⁴) combined with β²=α/L give β³=L³, so β=L.

22.

What is the relation between α and β?

(a)α = 2β
(b)2α = β
(c)α = -2β
(d)2α = -β

Answer: (c) α = -2β

Using α+β=-L together with β=L (from Q21) gives α=-2β.

For the following two (02) items: Let p = Σ(j=1 to n) log₁₀2^j and q = Σ(j=1 to n) log₁₀5^j.

23.

If p+q = 66, then which one of the following is correct?

(a)n < 7
(b)7 < n < 9
(c)9 < n < 12
(d)n > 12

Answer: (c) 9 < n < 12

p+q=n(n+1)/2·log10(10)=n(n+1)/2; setting this to 66 gives n(n+1)=132, i.e. n=11, which lies in (9,12).

24.

If p+q = 15, then what is q-p equal to?

(a)log₁₀2.5
(b)5log₁₀2.5
(c)10log₁₀2.5
(d)15log₁₀2.5

Answer: (d) 15log₁₀2.5

With n(n+1)/2=15 (from p+q=15), q-p=(n(n+1)/2)·log10(5/2)=15log10(2.5).

For the following two (02) items: Let sin A + sin B = p and cos A + cos B = q.

25.

What is p/q equal to?

(a)tan((A-B)/2)
(b)cot((A-B)/2)
(c)tan((A+B)/2)
(d)cot((A+B)/2)

Answer: (c) tan((A+B)/2)

By the sum-to-product identities, (sinA+sinB)/(cosA+cosB) simplifies exactly to tan((A+B)/2).

26.

What is (p^2-q^2)/(p^2+q^2) equal to?

(a)cos(A+B)
(b)cos(A-B)
(c)cos(π/2 – A – B)
(d)cos(π – A – B)

Answer: (d) cos(π – A – B)

Direct numeric and symbolic verification shows (p²-q²)/(p²+q²) equals cos(π-A-B).

For the following two (02) items: Let p = cosec 20° and q = cosec 70°.

27.

What is (√3p/4 – q/4) equal to?

(a)-1
(b)0
(c)1
(d)2

Answer: (c) 1

Direct evaluation of √3·csc20°/4 – csc70°/4 gives exactly 1.

28.

What is (p^2+q^2)/(p^2q^2) equal to?

(a)1/2
(b)1
(c)3/2
(d)2

Answer: (b) 1

Direct evaluation of (csc²20°+csc²70°)/(csc²20°·csc²70°) gives exactly 1.

For the following two (02) items: Let cos(2x+3y) = 1/2 and cos(3x+2y) = √3/2, where -π < (2x+3y) < π and -π < (3x+2y) < π.

29.

How many values does (x+y) have?

(a)Two
(b)Three
(c)Four
(d)More than four

Answer: (c) Four

Solving cos(2x+3y)=1/2 and cos(3x+2y)=√3/2 over all sign combinations within the given range yields 4 distinct values of x+y.

30.

How many values does (y-x) have?

(a)Two
(b)Three
(c)Four
(d)More than four

Answer: (c) Four

The same system yields 4 distinct values of y-x.

For the following two (02) items: Consider the equation abx^2 + bcx + ca = cax^2 + abx + bc.

31.

If the roots of the equation are equal, then which one of the following is correct?

(a)ac = b^2
(b)a+c = 2b
(c)1/a + 1/c = 1/(2b)
(d)1/a + 1/c = 2/b

Answer: (d) 1/a + 1/c = 2/b

Rearranging the equation to (ab-ca)x²+(bc-ab)x+(ca-bc)=0 and setting its discriminant to zero simplifies to ab+bc=2ac, i.e. 1/a+1/c=2/b.

32.

If the roots of the equation are equal, then a, b, c are in

(a)AP
(b)GP
(c)HP
(d)None of the above

Answer: (c) HP

The condition 1/a+1/c=2/b is exactly the definition of a,b,c being in Harmonic Progression.

For the following two (02) items: Let (6+10+14+… up to m terms) = (1+3+5+7+… up to n terms), where m < 25 and n < 25.

33.

What is the relation between m and n?

(a)n^2 = m(m+1)
(b)n^2 = m(m+2)
(c)n^2 = 2m(m+1)
(d)n^2 = 2m(m+2)

Answer: (d) n^2 = 2m(m+2)

Sum of m terms of 6,10,14,…=2m(m+2); sum of n odd numbers=n²; equating gives n²=2m(m+2).

34.

How many values of m are possible?

(a)None
(b)One
(c)Two
(d)More than two

Answer: (c) Two

Searching m,n<25 for n²=2m(m+2) being a perfect square gives exactly two solutions: (m,n)=(2,4) and (16,24).

For the following two (02) items: There are 8 points on a plane out of which 4 points are collinear.

35.

How many triangles can be formed by joining these points?

(a)56
(b)54
(c)53
(d)52

Answer: (d) 52

Triangles = C(8,3) – C(4,3) [subtracting the degenerate collinear triples] = 56-4=52.

36.

How many quadrilaterals can be formed by joining these points?

(a)70
(b)69
(c)53
(d)None of the above

Answer: (c) 53

Quadrilaterals = C(8,4) minus selections with 3 or 4 collinear points = 70-16-1=53.

For the following two (02) items: Let f(x) = ax^2+bx+c be a quadratic polynomial such that f(1)=f(4)=2. Further, 2 is a root of f(x)=0.

37.

What is the other root of f(x) = 0?

(a)1
(b)2
(c)3
(d)Cannot be determined

Answer: (c) 3

Since f(1)=f(4), the axis of symmetry is at x=2.5; the root symmetric to 2 about 2.5 is 3.

38.

What is (a+b+c) equal to?

(a)0
(b)1
(c)2
(d)Cannot be determined

Answer: (c) 2

a+b+c = f(1), which is given directly as 2.

For the following two (02) items: Let A = [[cosθ, sinθ], [-sinθ, cosθ]].

39.

What is the value of the determinant of the matrix A^4?

(a)0
(b)1
(c)cos4θ – sin4θ
(d)cos^2 4θ – sin^2 4θ

Answer: (b) 1

det(A)=cos²θ+sin²θ=1, so det(A^4)=det(A)^4=1.

40.

What is [adj A]^-1 equal to?

(a)-A
(b)-A^T
(c)A
(d)A^T

Answer: (c) A

Since det(A)=1, A⁻¹=adj(A), so [adj(A)]⁻¹=[A⁻¹]⁻¹=A.

41.

What is the sum of the binary numbers (101101101)₂ and (100011)₂?

(a)(110010000)₂
(b)(110001000)₂
(c)(110000100)₂
(d)(100100000)₂

Answer: (a) (110010000)₂

Direct binary addition of 101101101 and 100011 gives 110010000.

42.

Set X contains 3n elements and set Y contains 2n elements, and they have n elements in common. How many elements does (X-Y)×(Y-X) have?

(a)5n^2
(b)4n^2
(c)3n^2
(d)2n^2

Answer: (d) 2n^2

|X-Y|=2n, |Y-X|=n; the Cartesian product (X-Y)×(Y-X) has 2n·n=2n² elements.

43.

Let A = {-3,-2,-1,0,1,2,3} and B = {0,1,4,9}. How many elements does the subset of A×B corresponding to the relation R = {(x,y): |x| < y} have, where x∈A and y∈B?

(a)9
(b)12
(c)15
(d)16

Answer: (c) 15

Counting pairs (x,y) with |x|<y for y=0,1,4,9 gives 0+1+7+7=15.

44.

Statement-I: If X is an n×n matrix, then det(mX) = m^n det(X), where m is a scalar.
Statement-II: If Y is a matrix obtained from X by multiplying any row or column by a scalar m, then det(Y) = m det(X).

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

(a)Both Statement-I and Statement-II are correct and Statement-II explains Statement-I
(b)Both Statement-I and Statement-II are correct but Statement-II does not explain Statement-I
(c)Statement-I is correct but Statement-II is not correct
(d)Statement-I is not correct but Statement-II is correct

Answer: (a) Both Statement-I and Statement-II are correct and Statement-II explains Statement-I

Both are standard determinant facts, and applying Statement-II once per row/column (n times) directly derives Statement-I.

45.

Statement-I: The inverse of the matrix M = [[71,23,48],[57,28,29],[65,17,48]] does not exist.
Statement-II: M is non-singular.

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

(a)Both Statement-I and Statement-II are correct and Statement-II explains Statement-I
(b)Both Statement-I and Statement-II are correct but Statement-II does not explain Statement-I
(c)Statement-I is correct but Statement-II is not correct
(d)Statement-I is not correct but Statement-II is correct

Answer: (c) Statement-I is correct but Statement-II is not correct

Direct computation shows det(M)=0, so M is singular (inverse doesn’t exist, confirming I) — meaning Statement-II (non-singular) is false.

46.

What is cot⁻¹9 + cosec⁻¹(√41/4) equal to?

(a)π/4
(b)π/3
(c)π/2
(d)π

Answer: (a) π/4

Direct numeric evaluation of cot⁻¹9+cosec⁻¹(√41/4) gives exactly π/4.

47.

How many values of θ, where -π < θ < π, satisfy both the equations cotθ = -√3 and cosecθ = -2 simultaneously?

(a)4
(b)2
(c)1
(d)None

Answer: (c) 1

cotθ=-√3 and cscθ=-2 together force sinθ=-1/2, cosθ=√3/2, i.e. θ=-π/6 — the only value in (-π,π) satisfying both.

48.

If x + 1/x = 2cosθ, then what is x^3 + 1/x^3 equal to?

(a)cos^3θ
(b)cos3θ
(c)2cos3θ
(d)3cos3θ

Answer: (c) 2cos3θ

With x=e^(iθ) (from x+1/x=2cosθ), x³+1/x³=2cos3θ by the standard identity.

49.

If 0 ≤ x ≤ π/2, then what is the number of values of x satisfying the equation tan x + sec x = 2cos x?

(a)0
(b)1
(c)2
(d)3

Answer: (b) 1

Solving tanx+secx=2cosx gives x=π/6 within [0,π/2] (the other algebraic root 5π/6 lies outside this range) — exactly one solution.

50.

What is the value of tan[(1/2)sec⁻¹(2/√3)]?

(a)2-√3
(b)2+√3
(c)√3-1
(d)√3+1

Answer: (a) 2-√3

sec⁻¹(2/√3) corresponds to angle π/6; tan(π/12)=2-√3.

For the following two (02) items: A plane P is parallel to the line having direction ratios ⟨1,3,2⟩ and contains the line of intersection of the planes 6x+4y-5z=2 and x-2y+3z=0.

51.

Which of the following are the direction ratios of the line of intersection of the given planes?

(a)⟨2, 23, 16⟩
(b)⟨2, -23, -16⟩
(c)⟨2, 3, 2⟩
(d)⟨-1, 3, -2⟩

Answer: (b) ⟨2, -23, -16⟩

The line of intersection is parallel to the cross product of the two plane normals: (6,4,-5)×(1,-2,3)=(2,-23,-16).

52.

What is the equation of the plane P?

(a)2x-20y+29z+2=0
(b)2x-20y+29z-2=0
(c)2x+3y+2z-4=0
(d)x-3y+2z+5=0

Answer: (a) 2x-20y+29z+2=0

Finding the member of the family of planes through the intersection line that is parallel to (1,3,2) gives 2x-20y+29z+2=0.

For the following two (02) items: Suppose S is the sphere with the smallest radius that passes through the points A(1,0,0), B(0,1,0) and C(0,0,1).

53.

What is the radius of S?

(a)√(1/3)
(b)√(2/3)
(c)1/3
(d)1

Answer: (b) √(2/3)

The smallest sphere through A,B,C has its center at the circumcenter of triangle ABC, giving radius √6/3=√(2/3).

54.

On which one of the following planes does the centre of S lie?

(a)x+y+z-1=0
(b)x+y+z+1=0
(c)3x+3y+3z-1=0
(d)3x+3y+3z+1=0

Answer: (a) x+y+z-1=0

The circumcenter (1/3,1/3,1/3) satisfies x+y+z-1=0 exactly.

For the following two (02) items: Let A(1,-1,0), B(-2,1,8) and C(-1,2,7) are three consecutive vertices of a parallelogram ABCD.

55.

What is the fourth vertex D?

(a)(0,-2,1)
(b)(2,0,-1)
(c)(1,0,1)
(d)(1,2,0)

Answer: (b) (2,0,-1)

For parallelogram ABCD, D=A+C-B=(2,0,-1).

56.

If angle BCD is θ, then what is cos^2θ equal to?

(a)26/77
(b)27/77
(c)82/237
(d)83/237

Answer: (b) 27/77

Direct computation of cos²(angle BCD) at vertex C gives 27/77.

57.

For different values of m, the equation 4y = mx – m + 2 represents

(a)parallel lines
(b)concurrent lines
(c)lines at a fixed distance from the origin of coordinates
(d)the same line

Answer: (b) concurrent lines

4y=m(x-1)+2 is a family of lines all passing through the fixed point (1, 1/2) for every m — concurrent lines.

58.

The equation of the locus of a point equidistant from the points (a,b) and (c,d) is (a-c)x+(b-d)y+k=0. What is the value of k?

(a)a^2-c^2+b^2-d^2
(b)c^2+d^2-a^2-b^2
(c)(a^2-c^2+b^2-d^2)/2
(d)(c^2+d^2-a^2-b^2)/2

Answer: (d) (c^2+d^2-a^2-b^2)/2

Deriving the perpendicular-bisector equation directly from the equidistance condition gives k=(c²+d²-a²-b²)/2.

59.

Consider the following statements in respect of the equation x^2+3y=0: I. The equation represents the equation to parabola that opens upwards. II. The axis of the parabola is x=0. III. The equation of the latus rectum is 4y-3=0. How many of the statements given above are correct?

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

Answer: (b) One

x²=-3y opens downward (not upward, so I is false); its axis is x=0 (II true); its latus rectum is y=-3/4, i.e. 4y+3=0 not 4y-3=0 (III false) — only one statement correct.

60.

What is the sum of the intercepts of the line x/a^2 + y/b^2 = 2/(a^2+b^2) on the coordinate axes?

(a)2
(b)1
(c)1/2
(d)a^2+b^2

Answer: (a) 2

Setting y=0 and x=0 in turn gives x-intercept=2a²/(a²+b²) and y-intercept=2b²/(a²+b²), summing to exactly 2.

61.

Which one of the following is the perpendicular form of the straight line √3x+2y=7?

(a)y = -(√3/2)x + 7/2
(b)x/(7/√3) + y/(7/2) = 1
(c)(√3/√7)x + (2/√7)y = √7
(d)(√3/√7)x + (2/√7)y = 7

Answer: (c) (√3/√7)x + (2/√7)y = √7

Dividing √3x+2y=7 by √(3+4)=√7 gives the normal (perpendicular) form.

62.

If the vertices B and D of a square ABCD are (2,3) and (4,1) respectively, then what is the area of the square?

(a)2 square units
(b)3 square units
(c)4 square units
(d)8 square units

Answer: (c) 4 square units

BD is a diagonal of the square with length √8=2√2; area = diagonal²/2 = 8/2 = 4.

63.

What is the value of sinθ if θ is the acute angle between the lines whose equations are px+qy=p+q and p(x-y)+q(x+y)=2q?

(a)√3/2
(b)3/4
(c)1/2
(d)1/√2

Answer: (d) 1/√2

Rewriting both lines in normal form and computing the angle between their direction vectors gives sinθ=1/√2.

64.

The circle x^2+y^2-2kx-2ky+k^2=0 touches the x-axis at P and y-axis at Q. What is PQ equal to?

(a)√2k
(b)2k
(c)2√2k
(d)4k

Answer: (a) √2 k

The circle has center (k,k) and radius |k|, touching the axes at P(k,0) and Q(0,k); PQ=√(k²+k²)=√2 k.

65.

What is the distance between the foci of the hyperbola x^2-4y^2=1?

(a)√3
(b)√5
(c)2√3
(d)2√5

Answer: (b) √5

For x²-4y²=1, a²=1, b²=1/4, so c²=5/4 and the foci distance 2c=√5.

66.

(a)√3
(b)√6
(c)2√3
(d)4√3

Answer: (d) 4√3

p×q=(a-b)×(a+b)=2(a×b); with |a|=|b|=2 and a·b=2 (giving a 60° angle), |a×b|=2√3, so |p×q|=4√3.

67.

How many of the following can be a vector perpendicular to both the vectors 2i-j+k and i+j+3k? I. 4i+5j-3k; II. -8i-10j+6k; III. (1/50)(-4i-5j+3k). Select the correct answer.

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

Answer: (d) All three

All three candidate vectors are scalar multiples of (2,-1,1)×(1,1,3)=(-4,-5,3), hence perpendicular to both given vectors.

68.

What is the area of the parallelogram whose sides are represented by the vectors i+2j+3k and 2i+j+2k?

(a)(1/2)√26 square units
(b)(1/2)√27 square units
(c)√26 square units
(d)√27 square units

Answer: (c) √26 square units

The cross product of (1,2,3) and (2,1,2) has magnitude √26, the parallelogram’s area.

69.

The position vectors of the vertices A, B, C and D of a quadrilateral ABCD are given by 3i+4j-2k, 4i-4j-3k, 2i-3j+2k and 6i-2j+k respectively. What is the angle between the diagonals AC and BD of the quadrilateral?

(a)90°
(b)75°
(c)60°
(d)45°

Answer: (a) 90°

The diagonal vectors AC and BD have a dot product of exactly 0, confirming a 90° angle between them.

70.

A force vector F = 2i-λj+5k is applied at the point A(1,2,5). If its moment about the point B(-1,-2,3) is 16i-6j+2λk, then what is the value of λ?

(a)-2
(b)0
(c)1
(d)2

Answer: (a) -2

Computing the moment (A-B)×F and matching each component to the given moment vector gives λ=-2 consistently.

For the following two (02) items: Let f(x) = (1-cos2x)/x^2 for x<0; f(x) = 9 for x=0; f(x) = √x / (√(16+√x) – 4) for x>0.

71.

What is lim(x→0-) f(x) equal to?

(a)2
(b)4
(c)6
(d)8

Answer: (a) 2

(1-cos2x)/x² = 2sin²x/x² → 2 as x→0 from either side of this piece.

72.

What is lim(x→0+) f(x) equal to?

(a)6
(b)7
(c)8
(d)9

Answer: (c) 8

Rationalizing √x/(√(16+√x)-4) as x→0+ gives the limit √16+4=8.

For the following three (03) items: Consider the function f(x) = x|x|.

73.

What is lim(x→-1) f(x) equal to?

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

Answer: (a) -1

f(x)=x|x| is continuous everywhere; f(-1)=(-1)(1)=-1.

74.

What is the area bounded by the curve f(x), the x-axis and the lines x=-2 and x=1?

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

Answer: (d) 3

Splitting the integral of |x|x|| at x=0: ∫[-2,0]x²dx+∫[0,1]x²dx = 8/3+1/3 = 3.

75.

Consider the following statements: I. The function is increasing in the interval (-∞, ∞). II. The function is differentiable 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

Answer: (c) Both I and II

f'(x)=2|x|≥0 everywhere (so f is monotonically increasing overall), and the left/right derivatives at 0 both equal 0, so f is differentiable there too.

For the following two (02) items: Consider the function f(x) = x/(1-x) (x>0, x≠1).

76.

What is f(x)/f(x+1) equal to?

(a)-f(x^2)
(b)-f(√x)
(c)f(x^2)
(d)f(x-1)

Answer: (a) -f(x^2)

Direct algebraic simplification of f(x)/f(x+1) with f(t)=t/(1-t) gives exactly -x²/(1-x²)=-f(x²).

77.

What is (1-x)f(√x) + xf(√x+1) equal to?

(a)-f(x)
(b)f(x)
(c)x
(d)0

Answer: (d) 0

Direct algebraic expansion of (1-x)f(√x)+xf(√x+1) simplifies term-by-term to exactly 0.

For the following three (03) items: Let y = f(x) = (x sin⁻¹x)/√(1-x^2) + ln√(1-x^2).

78.

What is the slope of the tangent to the curve y=f(x) at x=0.5?

(a)4π√3/27
(b)8π√3/27
(c)4π
(d)8π

Answer: (a) 4π√3/27

Differentiating y=(x·asin(x))/√(1-x²)+ln√(1-x²) gives dy/dx=asin(x)/(1-x²)^(3/2); evaluated at x=0.5 this is 4π√3/27.

79.

What is d^2y/dx^2 at x=0 equal to?

(a)0
(b)0.5
(c)1
(d)1.5

Answer: (c) 1

Evaluating the second derivative of y at x=0 (via limit) gives exactly 1.

80.

If x = sinθ, then what is dy/dx equal to?

(a)θsecθ
(b)θsec^2θ
(c)θsec^3θ
(d)2tanθ + θsec^2θ

Answer: (c) θsec^3θ

With x=sinθ, dy/dx=asin(sinθ)/(1-sin²θ)^(3/2)=θ/cos³θ=θsec³θ.

For the following two (02) items: Consider the function f(x) = 1 – ∛((x-1)^2).

81.

What is the domain of the function?

(a)(1, ∞)
(b)(-∞, ∞)
(c)(0, ∞)
(d)(-∞, ∞)\{1}

Answer: (b) (-∞, ∞)

f(x)=1-∛((x-1)²) is defined for every real x (odd-index cube root of a square is always defined), including x=1 where f(1)=1.

82.

The function has

(a)a minimum at x=1
(b)a maximum at x=1
(c)neither maximum nor minimum at x=1
(d)no extremum

Answer: (b) a maximum at x=1

(x-1)^(2/3)≥0 with equality only at x=1, so f(x)=1-(x-1)^(2/3) is maximized (not minimized) at x=1.

For the following two (02) items: Consider the function f(x) = 4(5^x) for x<0; f(x) = 8k+x for x≥0.

83.

If the function is continuous, then what is the value of k?

(a)0.5
(b)1
(c)1.5
(d)2

Answer: (a) 0.5

Continuity at 0 requires 4·5^0=8k, i.e. 4=8k, so k=0.5.

84.

What is f′(-1) equal to?

(a)(2/5)ln5
(b)(3/5)ln5
(c)(4/5)ln5
(d)20ln5

Answer: (c) (4/5)ln5

For x<0, f'(x)=4·5^x·ln5; at x=-1 this is (4/5)ln5.

For the following two (02) items: Let u = ∫e^x cos x dx and v = ∫e^x sin x dx.

85.

What is u+v equal to?

(a)-du/dx
(b)-dv/dx
(c)du/dx
(d)dv/dx

Answer: (d) dv/dx

u+v simplifies (via the standard e^x(sinx±cosx)/2 antiderivative forms) to exactly e^x sinx, which is dv/dx by definition of v.

86.

Consider the following: I. du/dx = -v; II. dv/dx = -u. Which of the 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

du/dx=e^x cosx and dv/dx=e^x sinx are fixed by definition (as the original integrands); neither equals -v nor -u identically.

For the following two (02) items: Let the function f(x) = |x-3| + |x-4| be defined on the interval [0,5].

87.

What is dy/dx at x=3.5 equal to?

(a)0
(b)1
(c)2
(d)3.5

Answer: (a) 0

On [3,4], f(x)=|x-3|+|x-4|=(x-3)+(4-x)=1, a constant, so its derivative at x=3.5 is 0.

88.

Consider the following statements: I. The function is differentiable at x=3. II. The function is differentiable at x=4. 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

At x=3, the left derivative is -2 and the right (constant-region) derivative is 0 — not differentiable; at x=4, the left derivative is 0 and the right is 2 — also not differentiable.

For the following two (02) items: Consider the function f(x) = (10^x – 10^-x)/(10^x + 10^-x).

89.

What is f°f°f°f°f(0) equal to?

(a)0
(b)1
(c)5
(d)10

Answer: (a) 0

f(0)=0 is a fixed point of f, so repeated composition f∘f∘f∘f∘f(0) stays at 0.

90.

What is the inverse of the function?

(a)log₁₀(2x-1)
(b)(1/2)log₁₀(2x-1)
(c)(1/4)log₁₀(2x/(2-x))
(d)(1/2)log₁₀((1+x)/(1-x))

Answer: (d) (1/2)log₁₀((1+x)/(1-x))

Solving y=(t²-1)/(t²+1) for t=10^x and back-substituting gives the inverse (1/2)log10((1+x)/(1-x)).

91.

What is the degree of the differential equation ((d^2y/dx^2))^(3/2) = (dy/dx)^(5/2)?

(a)3
(b)2
(c)5/2
(d)3/2

Answer: (a) 3

Squaring both sides to clear the 3/2 and 5/2 powers gives (d²y/dx²)^3=(dy/dx)^5, making the degree (power of the highest derivative) 3.

92.

What is ∫(n to n+1) (x-[x]) dx, where [.] is the greatest integer function and n is natural number?

(a)(4n+1)/2
(b)(2n+1)/2
(c)1/2
(d)1

Answer: (c) 1/2

On [n,n+1), x-[x]=x-n is the fractional part, and ∫[n,n+1](x-n)dx=∫[0,1]u du=1/2.

93.

Consider the following statements: I. y = xe^2x is the solution of dy/dx = y(2+1/x). II. y = x ln|x| + cx is the solution of dy/dx = (x+y)/x. 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

Direct differentiation confirms both proposed solutions satisfy their respective differential equations exactly.

94.

If k is an arbitrary constant, then what is the general solution of the equation (x+y)^2 dy/dx = k^2?

(a)y+x = tan(x+c)+k
(b)x+y = k tan((y-c)/k)
(c)x-y = k tan((y-c)/k)
(d)y-x = tan(x+c)+k

Answer: (b) x+y = k tan((y-c)/k)

Solving the separable ODE (via substitution v=x+y) and rearranging the resulting arctangent relation gives x+y=k·tan((y-c)/k).

95.

What is ∫ dx/(10^x+10^-x) equal to?

(a)tan⁻¹(10^x)+c
(b)(ln10)tan⁻¹(10^x)+c
(c)(1/ln10)tan⁻¹(10^x)+c
(d)ln(10^x+10^-x)+c

Answer: (c) (1/ln10)tan⁻¹(10^x)+c

Substituting t=10^x reduces the integral to a standard arctangent form scaled by 1/ln10.

96.

A wire of length 20 cm is to be bent into a rectangle. Which of the following statements is/are correct? I. The rectangle of the largest area is the square. II. It is possible to form a rectangle of an area of 27 cm^2. 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: (a) I only

The maximum-area rectangle for fixed perimeter is always the square (area 25 here) — I true; since 27 exceeds this maximum, it’s unattainable — II false.

97.

If I1 = ∫(e to e^2) dx/lnx and I2 = ∫(1 to 2) e^x/x dx, then which one of the following is correct?

(a)I1-I2 = 0
(b)I1+I2 = 0
(c)I1-2I2 = 0
(d)2I1-I2 = 0

Answer: (a) I1-I2 = 0

Substituting x=e^t in I1 transforms it into exactly the same integral as I2 (verified numerically, both ≈3.0591).

98.

What is the area of the region bounded by |x|≤2k and |y|≤k, where k is a positive real number?

(a)2k^2
(b)4k^2
(c)5k^2
(d)8k^2

Answer: (d) 8k^2

The region is a rectangle of width 4k and height 2k, giving area 8k².

99.

Statement-I: f(x) is decreasing on the intervals x<5 and x>5, where f(x) = 1/(x-5).
Statement-II: f′(x) > 0 for all x≠5.

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

(a)Both Statement-I and Statement-II are correct and Statement-II explains Statement-I
(b)Both Statement-I and Statement-II are correct but Statement-II does not explain Statement-I
(c)Statement-I is correct but Statement-II is not correct
(d)Statement-I is not correct but Statement-II is correct

Answer: (c) Statement-I is correct but Statement-II is not correct

f'(x)=-1/(x-5)² is always negative (never positive), so f is indeed decreasing on both sides (I true) but Statement-II’s claim of f’>0 is false.

100.

Statement-I: The function f(x) = (x^3+128)/x has a minimum value 48 at x=4.
Statement-II: As x increases through 4, f′(x) changes sign from positive to negative.

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

(a)Both Statement-I and Statement-II are correct and Statement-II explains Statement-I
(b)Both Statement-I and Statement-II are correct but Statement-II does not explain Statement-I
(c)Statement-I is correct but Statement-II is not correct
(d)Statement-I is not correct but Statement-II is correct

Answer: (c) Statement-I is correct but Statement-II is not correct

f(4)=48 is indeed a minimum (f”(4)=6>0, confirming Statement-I), but at a minimum f’ changes from negative to positive (not positive to negative as Statement-II claims).

101.

What is the harmonic mean of the numbers C(10,3), C(10,4), C(10,5), C(10,6) and C(10,7)?

(a)3150/19
(b)4000/19
(c)252
(d)225

Answer: (a) 3150/19

The harmonic mean of C(10,3..7)=120,210,252,210,120 computes exactly to 3150/19.

102.

In a sample survey of a village, the probability that a farmer is in debt is 0.60. What is the probability that three randomly selected farmers are all in debt (assume independence of events)?

(a)0.000216
(b)0.064
(c)0.216
(d)0.512

Answer: (c) 0.216

P(all three in debt) = 0.6³ = 0.216 (independence assumed).

103.

The probability that a family owns a laptop is 0.68; that it also owns a desktop is 0.56. If the probability that it owns both is 0.48, then what is the probability that a randomly selected family owns a laptop or a desktop?

(a)0.80
(b)0.76
(c)0.36
(d)0.28

Answer: (b) 0.76

P(laptop or desktop) = 0.68+0.56-0.48 = 0.76.

104.

An urn contains 10 white and 5 red balls. If two balls are drawn at random, then what is the probability that both the balls are red?

(a)2/21
(b)1/7
(c)4/21
(d)3/7

Answer: (a) 2/21

P(both red) = C(5,2)/C(15,2) = 10/105 = 2/21.

105.

An urn contains 5 white, 6 red and 4 blue balls. Three balls are drawn at random. What is the probability that a white ball, a red ball and a blue ball are drawn?

(a)28/91
(b)2/7
(c)24/91
(d)23/91

Answer: (c) 24/91

P(one of each color) = (5·6·4)/C(15,3) = 120/455 = 24/91.

106.

Under which of the following conditions may binomial distribution be used? I. The number of trials is infinite and not fixed. II. The trials are independent. III. Each trial has two possible outcomes. Select the correct answer using the code given below.

(a)II only
(b)III only
(c)I and II
(d)II and III

Answer: (d) II and III

Binomial distribution requires a fixed (not infinite) number of trials, independent trials, and exactly two outcomes per trial — so II and III are the valid conditions (I is actually false, as trials must be finite/fixed).

107.

A person X speaks the truth 4 out of 5 times and person Y speaks the truth 5 out of 6 times. What is the probability that they will contradict each other in stating the fact?

(a)3/10
(b)1/15
(c)1/6
(d)7/10

Answer: (a) 3/10

P(contradict) = P(X true,Y false)+P(X false,Y true) = (4/5)(1/6)+(1/5)(5/6) = 3/10.

108.

The probability that a student passes Physics test is 2/3 and the probability that he passes both Physics test and English test is 11/15. The probability that he passes at least one test is 4/5. What is the probability that he passes English test?

(a)11/15
(b)13/15
(c)14/15
(d)1

Answer: (b) 13/15

P(English) = P(at least one)+P(both)-P(Physics) = 4/5+11/15-2/3 = 13/15.

109.

An event X can happen with probability p and event Y can happen with probability q. Further, X and Y are independent events. Which of the following statements is/are correct? I. The probability that exactly one of the events happens is p+q-pq. II. The probability that at least one of the events happens is p+q-2pq. 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

The correct formulas are swapped: P(exactly one)=p+q-2pq (not p+q-pq as claimed in I) and P(at least one)=p+q-pq (not p+q-2pq as claimed in II) — both statements as worded are false.

110.

Three faces of a die are black, two faces are white and one face is red. The die is tossed three times. What is the probability that the colours black, white and red appear in the first, second and third tosses respectively?

(a)1/36
(b)1/6
(c)7/36
(d)5/36

Answer: (a) 1/36

P(black)·P(white)·P(red) in that order = (3/6)(2/6)(1/6) = 1/36.

111.

A fair coin is tossed 4 times. What is the probability that two heads do not occur consecutively?

(a)1/8
(b)3/8
(c)7/16
(d)1/2

Answer: (d) 1/2

Direct enumeration of all 16 four-toss sequences shows exactly 8 have no consecutive heads, giving probability 1/2.

112.

In a throw of three dice, what is the probability of getting one prime number, one composite number and one number which is neither prime nor composite?

(a)1/2
(b)1/3
(c)1/4
(d)1/6

Answer: (d) 1/6

Direct enumeration of all 216 three-dice outcomes shows 36 give one prime, one composite, and one neither (i.e. a 1) — probability 1/6.

113.

An integer is chosen at random from the first 50 integers. What is the probability that the integer is neither divisible by 5 nor 9?

(a)7/10
(b)18/25
(c)37/50
(d)19/25

Answer: (b) 18/25

Of 1-50, exactly 18 numbers are divisible by neither 5 nor 9, giving probability 18/50=18/25… (computed directly: 36/50=18/25).

114.

Out of 50 consecutive natural numbers, two integers are chosen at random. What is the probability that their sum is odd?

(a)1/2
(b)24/49
(c)1/4
(d)25/49

Answer: (d) 25/49

Among any 50 consecutive integers, 25 are even and 25 odd; P(sum odd) = (25×25)/C(50,2) = 625/1225 = 25/49.

115.

The standard deviation of 100 observations is 10. If 20 is added to each observation, then what will be the new standard deviation?

(a)10
(b)15
(c)20
(d)25

Answer: (a) 10

Adding a constant to every observation does not change the standard deviation — it remains 10.

116.

Let X be a random variable following binomial distribution with parameters n=5 and p=k. Further, P(X=1)=0.4096 and P(X=2)=0.2048. What is the value of k?

(a)0.2
(b)0.25
(c)0.3
(d)0.35

Answer: (a) 0.2

P(X=2)/P(X=1) = 2p/q = 0.5 gives p=0.2 (with q=0.8), verified against the given probabilities.

The frequency distribution of the marks obtained by students in a Science examination is given below:

Marks	5-15	15-25	25-35	35-45
Number of students	20	30	30	20

117.

What is the arithmetic mean?

(a)20
(b)25
(c)30
(d)35

Answer: (b) 25

Using midpoints 10,20,30,40 with frequencies 20,30,30,20 (total 100), the mean is 2500/100=25.

118.

If P(A)=0.3, P(B)=0.4 and P(A|B)=0.5, then what is the value of P(B|A)?

(a)0.325
(b)0.333
(c)0.375
(d)0.667

Answer: (d) 0.667

P(A∩B)=P(A|B)P(B)=0.5×0.4=0.2; P(B|A)=P(A∩B)/P(A)=0.2/0.3=0.667. (This question is self-contained and doesn’t actually need the frequency-table passage the site attaches to it — a minor content-linking quirk that doesn’t affect the answer.)

119.

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

(a)7/12
(b)2/3
(c)3/4
(d)11/12

Answer: (d) 11/12

P(Ā∩B)=P(B)-P(A∩B)=1/2-1/4=1/4; P(Ā∪B)=P(Ā)+P(B)-P(Ā∩B)=2/3+1/2-1/4=11/12.

120.

Consider the following statements: I. Mean and variance have the same unit of measurement. II. Mean deviation and standard deviation have the same unit of measurement. 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

Questions transcribed from the official National Defence Academy & Naval Academy Examination-(II), 2025 question booklet (TODC-O-MTH/44A), Mathematics paper. Answer key not included.

Answer: (b) II only

Variance carries squared units of the original data (not the same unit as the mean), so I is false; mean deviation and standard deviation both carry the same unit as the original data, so II is true.

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