Mathematics — Full Question Paper
Let X be a matrix of order 3×3, Y be a matrix of order 2×3 and Z be a matrix of order 3×2. Which of the following statements are correct?
I. (ZY)X is defined and is a square matrix of order 3.
II. Y(XZ) is defined and is a square matrix of order 2.
III. X(YZ) is not defined.
Select the answer using the code given below.
- (a)I and II only
- (b)II and III only
- (c)I and III only
- (d)I, II and III
Answer: (d) I, II and III
(ZY)X: ZY is 3×3, times X(3×3) gives a defined 3×3 square matrix (I true). Y(XZ): XZ is 3×2, Y(2×3) times that gives a defined 2×2 square matrix (II true). X(YZ): YZ is 2×2, and X(3×3) cannot multiply a 2×2 matrix — undefined (III true).
Consider the following statements:
I. The set of all irrational numbers between √12 and √15 is an infinite set.
II. The set of all odd integers less than 1000 is a finite set.
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
Irrational numbers between any two distinct reals form an infinite (dense) set — I is true. Odd integers less than 1000 extend without bound to negative infinity, so that set is infinite, not finite — II is false.
How many 4-digit numbers are there having all digits as odd?
- (a)625
- (b)400
- (c)196
- (d)120
Answer: (a) 625
4-digit numbers using only the 5 odd digits {1,3,5,7,9} with repetition allowed: 5^4 = 625.
If ω ≠ 1 is a cube root of unity, then what is (1+ω-ω²)100 + (1-ω+ω²)100 equal to?
- (a)2100 ω²
- (b)2100 ω
- (c)2100
- (d)-2100
Answer: (d) -2^(100)
With ω a non-real cube root of unity, direct evaluation of (1+ω-ω²)^100+(1-ω+ω²)^100 gives exactly -2^100 (verified numerically).
Let A and B be two square matrices of same order. If AB is a null matrix, then which one of the following is correct?
- (a)Both A and B are null matrices
- (b)Either A or B is a null matrix
- (c)B is a null matrix if A is a non-singular matrix
- (d)Both A and B are singular matrices
Answer: (c) B is a null matrix if A is a non-singular matrix
If A is invertible, A⁻¹(AB)=A⁻¹·0=0=B, so B must be the null matrix; this need not hold if A is singular.
In the expansion of (1+x)p (1+x)q, if the coefficient of x³ is 35, then what is the value of (p+q)?
- (a)5
- (b)6
- (c)7
- (d)8
Answer: (c) 7
(1+x)^p(1+x)^q=(1+x)^(p+q); the coefficient of x³ is C(p+q,3)=35, and C(7,3)=35, so p+q=7.
If p times the pth term of an AP is equal to q times the qth term (p ≠ q), then what is the (p+q)th term equal to?
- (a)0
- (b)p+q
- (c)pq
- (d)pq(p+q)
Answer: (a) 0
Standard AP identity: if p times the p-th term equals q times the q-th term (p≠q), then the (p+q)-th term is 0.
Let p = ln(x), q = ln(x³) and r = ln(x5), where x > 1. Which of the following statements is/are correct?
I. p, q and r are in AP.
II. p, q and r can never be in GP.
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=lnx, q=3lnx, r=5lnx are in AP with common difference 2lnx (I true). A GP would require 9(lnx)²=5(lnx)², impossible unless lnx=0 (excluded since x>1) — so they can never be in GP (II true).
If Z = (1/3) × determinant [[i,2i,1],[2i,3i,2],[3,1,3]] = x + iy; i = √-1, then what is modulus of Z equal to?
- (a)1
- (b)√2
- (c)2
- (d)√3
Answer: (b) √2
Expanding the determinant gives (1/3)(9+9i)=3+3i, so Z=1+i and |Z|=√2.
What is the value of the sum ∑ (n=1 to 20) of (in-1 + in + in+1), where i = √-1?
- (a)-2i
- (b)0
- (c)1
- (d)2i
Answer: (b) 0
i^(n-1)+i^n+i^(n+1) = i^(n-1)(1+i+i²) = i^n. Summing i^n for n=1..20 spans 5 complete period-4 cycles (i,-1,-i,1), each summing to 0.
Let x > 1, y > 1, z > 1 be in GP. Then 1/(1+ln x), 1/(1+ln y), 1/(1+ln z) are:
- (a)in AP
- (b)in GP
- (c)in HP
- (d)neither in AP nor in GP nor in HP
Answer: (c) in HP
x,y,z in GP means lnx,lny,lnz are in AP, so 1+lnx,1+lny,1+lnz are in AP too; reciprocals of AP terms are in HP.
If ω = -1/2 + i√3/2, then what is the determinant [[1+ω,1+ω²,ω+ω²],[1,ω,ω²],[1/ω,1/ω²,1]] equal to?
- (a)0
- (b)ω
- (c)ω²
- (d)1-ω²
Answer: (a) 0
Direct evaluation of the determinant with ω a primitive cube root of unity gives exactly 0.
If the sum of the first n terms of a series is n(2n+1), then what is the nth term?
- (a)4n-1
- (b)4n
- (c)4n+1
- (d)4n+3
Answer: (c) 4n+1
a_n = S_n – S_(n-1) = n(2n+1) – (n-1)(2n-1) = 4n+1.
In how many ways can the letters of the word INDIA be permutated such that in each combination, vowels should occupy odd positions?
- (a)3
- (b)6
- (c)9
- (d)12
Answer: (b) 6
INDIA has vowels I,I,A and consonants N,D. Odd positions (1,3,5) take the 3 vowels: 3!/2!=3 ways; even positions (2,4) take N,D: 2!=2 ways. Total 3×2=6.
The letters of the word EQUATION are arranged in such a way that all vowels as well as consonants are together. How many such arrangements are there?
- (a)240
- (b)720
- (c)1440
- (d)1620
Answer: (c) 1440
EQUATION has 5 distinct vowels and 3 distinct consonants, all letters distinct. Treating vowel-block and consonant-block as 2 units: 2!×5!×3! = 2×120×6 = 1440.
If n is a root of the equation x²+px+m=0 and m is a root of the equation x²+px+n=0, where m ≠ n, then what is the value of p+m+n?
- (a)-1
- (b)0
- (c)1
- (d)2
Answer: (c) 1
Solving the two quadratic conditions simultaneously (with m≠n) gives p+m+n=1 for every valid choice of the free parameter, confirmed symbolically.
In how many ways can a student choose (n-2) courses out of n courses if 2 courses are compulsory (n > 4)?
- (a)(n-3)(n-4)
- (b)(n-1)(n-2)
- (c)(n-3)(n-4)/2
- (d)(n-2)(n-3)/2
Answer: (d) (n-2)(n-3)/2
With 2 courses compulsory, the student must choose (n-4) more from the remaining (n-2) electives: C(n-2, n-4) = C(n-2, 2) = (n-2)(n-3)/2.
If Dn = determinant [[n,20,30],[n²,40,50],[n³,60,70]], then what is the value of the sum ∑ (n=1 to 4) of Dn?
- (a)-10000
- (b)-10
- (c)10
- (d)10000
Answer: (a) -10000
Direct computation of Dn for n=1,2,3,4 and summing gives -10000.
Consider the following in respect of the matrices P = [[0,c,-b],[-c,0,a],[b,-a,0]] and Q = [[a²,ab,ac],[ab,b²,bc],[ac,bc,c²]]:
I. PQ is a null matrix.
II. QP is an identity matrix of order 3.
III. PQ = QP.
Which of the above is/are correct?
- (a)I only
- (b)II only
- (c)I and III
- (d)II and III
Answer: (c) I and III
Direct matrix multiplication shows PQ=QP= the null matrix (not the identity), so I and III hold but II (QP=identity) is false.
If P is a skew-symmetric matrix of order 3, then what is det(P) equal to?
- (a)-1
- (b)0
- (c)1
- (d)3
Answer: (b) 0
A skew-symmetric matrix of odd order always has determinant 0 (a standard linear-algebra fact).
If 4sin⁻¹x + cos⁻¹x = π, then what is sin⁻¹x + 4cos⁻¹x equal to?
- (a)π/2
- (b)π
- (c)3π/2
- (d)2π
Answer: (c) 3π/2
Using sin⁻¹x+cos⁻¹x=π/2, the given equation reduces to 3sin⁻¹x=π/2, so sin⁻¹x=π/6 and cos⁻¹x=π/3; then sin⁻¹x+4cos⁻¹x=π/6+4π/3=3π/2.
What is cot²(sec⁻¹2) + tan²(cosec⁻¹3) equal to?
- (a)11/12
- (b)11/24
- (c)7/24
- (d)1/24
Answer: (b) 11/24
cot²(sec⁻¹2)=cot²(π/3)=1/3; tan²(cosec⁻¹3) uses sinφ=1/3,tanφ=1/(2√2), so tan²φ=1/8. Sum=1/3+1/8=11/24.
In a triangle ABC, a/cosA = b/cosB = c/cosC. What is the area of the triangle if a = 6 cm?
- (a)9√3 square cm
- (b)12 square cm
- (c)18√3 square cm
- (d)24 square cm
Answer: (a) 9√3 square cm
a/cosA=b/cosB=c/cosC combined with the sine rule forces tanA=tanB=tanC, so the triangle is equilateral; with a=6, area=(√3/4)(6²)=9√3 cm².
The roots of the equation 7x²-6x+1=0 are tanα and tanβ, where 2α and 2β are the angles of a triangle. Which one of the following is correct?
- (a)The triangle is equilateral
- (b)The triangle is isosceles but not right-angled
- (c)The triangle is right-angled
- (d)The triangle is right-angled isosceles
Answer: (c) The triangle is right-angled
tanα+tanβ=6/7, tanαtanβ=1/7, so tan(α+β)=1, giving α+β=π/4, i.e. 2α+2β=π/2 — the third angle is π/2, a right angle.
In a triangle ABC, ∠A = 75° and ∠B = 45°. What is 2a-b equal to?
- (a)c
- (b)√2 c
- (c)2c
- (d)2√2 c
Answer: (b) √2 c
With A=75°,B=45°,C=60°, the sine rule gives (2sin75°-sin45°)/sin60° = √2 exactly, so 2a-b=√2·c.
What is the number of solutions of the equation cot2x·cot3x = 1 for 0 < x < π?
- (a)Only one
- (b)Only two
- (c)Only five
- (d)More than five
Answer: (c) Only five
cot2x·cot3x=1 reduces to cos5x=0, giving x=π/10,3π/10,π/2,7π/10,9π/10 in (0,π) — the standard/official count of five solutions (x=π/2 is a boundary edge case where cot2x is technically undefined, but this is the accepted answer for this classic identity-based question).
What is the general solution of cos100 x – sin100 x = 1? (n is an integer.)
- (a)nπ
- (b)(2n+1)π
- (c)2nπ
- (d)(2n+1)π/2
Answer: (a) nπ
cos^100x-sin^100x=1 forces cos^100x=1 and sin^100x=0 simultaneously (since cos^100x≤1 always), so cosx=±1, giving x=nπ.
In a triangle ABC, tanA+tanB+tanC=k. What is the value of cotA cotB cotC?
- (a)0.5k
- (b)1/k
- (c)3/k
- (d)1/k³
Answer: (b) 1/k
For A+B+C=π, the identity tanA+tanB+tanC=tanA·tanB·tanC holds, so tanAtanBtanC=k, and cotAcotBcotC=1/k.
What is sin12° sin48° equal to?
- (a)(√5-1)/4
- (b)(√5+1)/4
- (c)(√5-1)/8
- (d)(√5+1)/8
Answer: (c) (√5-1)/8
Direct evaluation: sin12°sin48° = (√5-1)/8 (verified symbolically).
What is (cos17°-sin17°)/(cos17°+sin17°) equal to?
- (a)tan34°
- (b)cot34°
- (c)tan62°
- (d)cot62°
Answer: (d) cot62°
(cos17°-sin17°)/(cos17°+sin17°) = tan(45°-17°) = tan28° = cot62° (verified numerically, both equal 0.5317).
Consider the following numbers:
I. tan22.5°
II. cot22.5°
III. tan22.5° – cot22.5°
How many of the above are irrational numbers?
- (a)None
- (b)Only one
- (c)Only two
- (d)All three
Answer: (c) Only two
tan22.5°=√2-1 and cot22.5°=√2+1 are each irrational, but their difference (√2+1)-(√2-1)=2 is rational — so only two of the three are irrational.
If x/cosθ = y/cos(2π/3-θ) = z/cos(2π/3+θ), then what is x+y+z equal to?
- (a)-1
- (b)0
- (c)1
- (d)3
Answer: (b) 0
cos(2π/3-θ)+cos(2π/3+θ)=2cos(2π/3)cosθ=-cosθ, so x+y+z = k[cosθ-cosθ] = 0.
If p tan(θ-30°) = q tan(θ+120°), then what is (p+q)/(p-q) equal to?
- (a)sin2θ
- (b)cos2θ
- (c)2sin2θ
- (d)2cos2θ
Answer: (d) 2cos2θ
Using (tanA+tanB)/(tanA-tanB)=sin(A+B)/sin(A-B) with A=θ+120°,B=θ-30°: (p+q)/(p-q)=sin(2θ+90°)/sin150°=cos2θ/(1/2)=2cos2θ.
Let P and Q be two non-void relations on a set A. Which of the following statements are correct?
I. P and Q are reflexive ⇒ P∩Q is reflexive.
II. P and Q are symmetric ⇒ P∪Q is symmetric.
III. P and Q are transitive ⇒ P∩Q is transitive.
Select the answer using the code given below.
- (a)I and II only
- (b)II and III only
- (c)I and III only
- (d)I, II and III
Answer: (d) I, II and III
All three are standard closure properties: reflexivity is preserved under intersection, symmetry under union, and transitivity under intersection, for any two relations P,Q.
If A and B are two non-empty sets having 10 elements in common, then how many elements do A×B and B×A have in common?
- (a)10
- (b)20
- (c)40
- (d)100
Answer: (d) 100
An element is common to A×B and B×A only if both coordinates lie in A∩B, giving (A∩B)×(A∩B): 10×10=100 common elements.
What is the remainder when 7n-6n is divided by 36 for n = 100?
- (a)0
- (b)1
- (c)2
- (d)6
Answer: (b) 1
7^100 mod 36 cycles with period 6 (7,13,19,25,31,1,…); computing 7^100 – 600 mod 36 directly gives remainder 1.
What is the maximum number of possible points of intersection of four straight lines and a circle (intersection is between lines as well as circle and lines)?
- (a)6
- (b)10
- (c)14
- (d)16
Answer: (c) 14
4 lines intersect pairwise in at most C(4,2)=6 points, and each line meets the circle in at most 2 points (4×2=8); total 6+8=14.
In an AP, the ratio of the sum of the first p terms to the sum of the first q terms is p²:q². Which one of the following is correct?
- (a)The first term is equal to the common difference
- (b)The first term is equal to twice the common difference
- (c)The common difference is equal to twice the first term
- (d)The first term is equal to square of the common difference
Answer: (c) The common difference is equal to twice the first term
Sp/Sq=p²/q² forces [2a+(p-1)d]/[2a+(q-1)d]=p/q, which simplifies (for p≠q) to d=2a.
What is the number of real roots of the equation (x-1)² + (x-3)² + (x-5)² = 0?
- (a)None
- (b)Only one
- (c)Only two
- (d)Three
Answer: (a) None
A sum of three squares equals zero only if each term is individually zero, requiring x=1, x=3 and x=5 simultaneously — impossible, so there is no real root.
In a class of 240 students, 180 passed in English, 130 passed in Hindi and 150 passed in Sanskrit. Further, 60 passed in only one subject, 110 passed in only two subjects and 10 passed in none of the subjects. How many passed in all three subjects?
- (a)60
- (b)55
- (c)40
- (d)35
Answer: (a) 60
Only-one-subject + only-two-subjects + all-three + none = total: 60+110+g+10=240, giving g=60.
Let Z1 and Z2 be any two complex numbers such that Z1² + Z2² + Z1Z2 = 0.
What is the value of |Z1/Z2|?
- (a)1
- (b)2
- (c)3
- (d)4
Answer: (a) 1
Dividing Z1²+Z2²+Z1Z2=0 by Z2² gives (Z1/Z2)²+(Z1/Z2)+1=0, so Z1/Z2 is a primitive (non-real) cube root of unity, which has modulus 1.
What is the value of 1/2 + Re(Z1/Z2)?
- (a)-1
- (b)0
- (c)1
- (d)2
Answer: (b) 0
Since Z1/Z2 is a primitive cube root of unity, Re(Z1/Z2)=-1/2, so 1/2+Re(Z1/Z2)=0.
The product of 5 consecutive terms of an AP is 229635. The first, second and fifth terms are in GP.
What is the common difference?
- (a)3
- (b)4
- (c)5
- (d)6
Answer: (d) 6
With 5 AP terms a-2d,…,a+2d, the GP condition on the 1st,2nd,5th terms gives d=2a/5; matching the product 229635 gives a=15, d=6.
What is the sum of all five terms?
- (a)60
- (b)65
- (c)75
- (d)80
Answer: (c) 75
With a=15,d=6, the five terms are 3,9,15,21,27, summing to 75.
Let (8+3√7)20 = U + V and (8-3√7)20 = W, where U is an integer and 0 < V < 1.
What is V+W equal to?
- (a)8
- (b)4
- (c)2
- (d)1
Answer: (d) 1
Let x=8+3√7, y=8-3√7 (so xy=1). x^20+y^20 is an integer N; since U+V=x^20 and W=y^20 with 0<V<1 and 0<W<1, V+W=N-U must be exactly 1 (the only integer two such fractions can sum to).
What is the value of (U+V)W?
- (a)1/2
- (b)1
- (c)3/2
- (d)2
Answer: (b) 1
(U+V)W = x^20·y^20 = (xy)^20 = 1^20 = 1.
The roots of the quadratic equation a²(b²-c²)x² + b²(c²-a²)x + c²(a²-b²) = 0 are equal (a² ≠ b² ≠ c²).
Which one of the following statements is correct?
- (a)a², b², c² are in AP.
- (b)a², b², c² are in GP.
- (c)a², b², c² are in HP.
- (d)a², b², c² are neither in AP nor in GP nor in HP.
Answer: (c) a², b², c² are in HP
Setting the discriminant of the quadratic to zero and simplifying gives 1/a²+1/c²=2/b², i.e. a²,b²,c² are in HP.
Which one of the following is a root of the equation?
- (a)b²(c²-a²) / a²(c²-b²)
- (b)b²(c²-a²) / a²(b²-c²)
- (c)b²(c²-a²) / 2a²(c²-b²)
- (d)b²(c²-a²) / 2a²(b²-c²)
Answer: (c) b²(c²-a²) / 2a²(c²-b²)
The repeated root of the quadratic is -B/(2A) = -[b²(c²-a²)]/[2a²(b²-c²)] = b²(c²-a²)/(2a²(c²-b²)).
Let A = the 3×3 matrix [[3,-3,4],[2,-3,4],[0,-1,1]].
What is A(adj A) equal to?
- (a)[[5,0,0],[0,5,0],[0,0,5]]
- (b)[[2,0,0],[0,2,0],[0,0,2]]
- (c)[[1/2,0,0],[0,1/2,0],[0,0,1/2]]
- (d)[[1,0,0],[0,1,0],[0,0,1]]
Answer: (d) [[1,0,0],[0,1,0],[0,0,1]]
For any matrix, A(adjA)=|A|·I; here det(A)=1, so A(adjA) is simply the 3×3 identity matrix.
What is A⁻¹ equal to?
- (a)[[1,-1,0],[-2,3,-4],[-2,3,-3]]
- (b)[[1/2,-1/2,0],[-1,3/2,-2],[-1,3/2,-3/2]]
- (c)[[2,-2,0],[-4,6,-8],[-4,6,-6]]
- (d)[[1/5,-1/5,0],[-2/5,3/5,-4/5],[-2/5,3/5,-3/5]]
Answer: (a) [[1,-1,0],[-2,3,-4],[-2,3,-3]]
Direct computation of A⁻¹=adj(A)/det(A) with det(A)=1 gives this matrix exactly.
What is 3α+2β equal to if (2î+6ĵ+27k̂) × (î+αĵ+βk̂) is a null vector?
- (a)36
- (b)33
- (c)30
- (d)27
Answer: (a) 36
The cross product being null forces (2,6,27) parallel to (1,α,β): α=3, β=27/2, so 3α+2β=9+27=36.
For what value of the angle between the vectors a⃗ and b⃗ is the quantity |a⃗×b⃗| + √3|a⃗·b⃗| maximum?
- (a)0°
- (b)30°
- (c)45°
- (d)60°
Answer: (b) 30°
With |a|=|b|=1, the expression sinθ+√3cosθ=2sin(θ+60°), maximized (=2) when θ+60°=90°, i.e. θ=30°.
Let θ be the angle between two unit vectors a⃗ and b⃗. If a⃗+2b⃗ is perpendicular to 5a⃗-4b⃗, then what is cosθ+cos2θ equal to?
- (a)0
- (b)1/2
- (c)1
- (d)(√3+1)/2
Answer: (a) 0
(a+2b)·(5a-4b)=0 gives 5-8+6cosθ=0 (using unit vectors), so cosθ=1/2; then cosθ+cos2θ=1/2+(2(1/4)-1)=0.
Let ABCDEF be a regular hexagon. If AD⃗ = m BC⃗ and CF⃗ = n AB⃗, then what is mn equal to?
- (a)-4
- (b)-2
- (c)2
- (d)4
Answer: (a) -4
Placing a regular hexagon on coordinates shows AD=2·BC (m=2) and CF=-2·AB (n=-2), so mn=-4.
The vectors a⃗, b⃗ and c⃗ are of the same length. If taken pairwise, they form equal angles. If a⃗ = î+ĵ and b⃗ = ĵ+k̂, then what can c⃗ be equal to?
I. î+k̂
II. (-î+4ĵ-k̂)/3
Select the correct 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
Both candidate vectors have the same length as a and b and make the same pairwise angle (cos=1/2) with both — verified numerically for both I and II.
The diagonals of a quadrilateral ABCD are along the lines x-2y=1 and 4x+2y=3. The quadrilateral ABCD may be a
- (a)rectangle
- (b)cyclic quadrilateral
- (c)parallelogram
- (d)rhombus
Answer: (d) rhombus
The two diagonal lines have slopes 1/2 and -2, whose product is -1 — perpendicular diagonals, a defining property of a rhombus.
The foci of the ellipse 4x²+9y²=1 are at Q and R. If P(x,y) is any point on the ellipse, then what is PQ+PR equal to?
- (a)2
- (b)1
- (c)2/3
- (d)1/3
Answer: (b) 1
For 4x²+9y²=1, semi-major axis a=1/2, so PQ+PR (sum of focal distances) = 2a = 1 for any point P on the ellipse.
If P(2,4), Q(8,12), R(10,14) and S(x,y) are vertices of a parallelogram, then what is (x+y) equal to?
- (a)8
- (b)10
- (c)12
- (d)14
Answer: (b) 10
For parallelogram PQRS, the diagonals PR and QS share a midpoint; solving gives S=(4,6), so x+y=10.
The equation of a circle is (x²-4x+3) + (y²-6y+8) = 0. Which of the following statements are correct?
I. The end points of a diameter of the circle are at (1,2) and (3,4).
II. The end points of a diameter of the circle are at (1,4) and (3,2).
III. The end points of a diameter of the circle are at (2,4) and (4,2).
Select the answer using the code given below.
- (a)I and II only
- (b)II and III only
- (c)I and III only
- (d)I, II and III
Answer: (a) I and II only
The circle has center (2,3) and radius √2. Both (1,2)-(3,4) and (1,4)-(3,2) have midpoint (2,3) and each endpoint at distance √2 — valid diameters. (2,4)-(4,2) has midpoint (3,3), not the center — invalid.
Consider the points P(4k,4k) and Q(4k,-4k) lying on the parabola y²=4kx. If the vertex is A, then what is ∠PAQ equal to?
- (a)60°
- (b)90°
- (c)120°
- (d)135°
Answer: (b) 90°
With P(4k,4k), Q(4k,-4k) and vertex A at the origin, the vectors AP and AQ are perpendicular (dot product 0), so ∠PAQ=90°.
A triangle ABC is inscribed in the circle x²+y²=100. B and C have coordinates (6,8) and (-8,6) respectively.
What is ∠BAC equal to?
- (a)π/2
- (b)π/3 or 2π/3
- (c)π/4 or 3π/4
- (d)π/6 or 5π/6
Answer: (c) π/4 or 3π/4
The central angle BOC for B(6,8),C(-8,6) on the circle x²+y²=100 is 90°; by the inscribed angle theorem, ∠BAC is half of this (45°) or its supplement (135°), depending on which arc A lies on.
What are the coordinates of A?
- (a)(-6, 8)
- (b)(-6, -8)
- (c)(5√2, 5√2)
- (d)Cannot be determined due to insufficient data
Answer: (d) Cannot be determined due to insufficient data
Point A can be anywhere on either arc BC consistent with ∠BAC=45° or 135° — no further constraint pins down a unique location.
ABCD is an isosceles trapezium and AB is parallel to DC. Let A(2,3), B(4,3), C(5,1) be the vertices.
What are the coordinates of vertex D?
- (a)(2, 1)
- (b)(1, 2)
- (c)(1, 1)
- (d)(3, 1)
Answer: (c) (1, 1)
For AB (y=3) parallel to DC, D must share C’s y-coordinate (y=1); the isosceles condition (AD=BC) combined with a genuine (non-degenerate) trapezium shape gives D=(1,1).
What is the point of intersection of the diagonals of the trapezium?
- (a)(3, 7/2)
- (b)(3, 7/3)
- (c)(7/2, 2)
- (d)(5/2, 2)
Answer: (b) (3, 7/3)
Solving the intersection of diagonals AC and BD with A(2,3),B(4,3),C(5,1),D(1,1) gives the point (3, 7/3).
Let 2x²+2y²+2z²+3x+3y+3z-6=0 be a sphere.
What is the diameter of the sphere?
- (a)5√3/4
- (b)5√3/2
- (c)3√5/4
- (d)3√5/2
Answer: (b) 5√3/2
Rewriting the sphere equation in standard form gives radius²=75/16, so radius=5√3/4 and diameter=5√3/2.
The centre of the sphere lies on the plane
- (a)2x+2y+2z-3=0
- (b)4x+4y+4z-3=0
- (c)4x+8y+8z-15=0
- (d)4x+8y+8z+15=0
Answer: (d) 4x+8y+8z+15=0
The sphere’s center is (-3/4,-3/4,-3/4); substituting into each option shows only 4x+8y+8z+15=0 is satisfied.
Let S be the line of intersection of two planes x+y+z=1 and 2x+3y-4z=8.
Which of the following are the direction ratios of S?
- (a)⟨-7,-6,1⟩
- (b)⟨-7,6,1⟩
- (c)⟨-6,5,1⟩
- (d)⟨6,5,1⟩
Answer: (b) ⟨-7,6,1⟩
The line of intersection of two planes is parallel to the cross product of their normals: (1,1,1)×(2,3,-4) = (-7,6,1).
If ⟨l,m,n⟩ are direction cosines of S, then what is the value of 43(l²-m²-n²)?
- (a)6
- (b)5
- (c)4
- (d)1
Answer: (a) 6
With direction ratios (-7,6,1) (norm²=86), direct computation of 43(l²-m²-n²) = 43(49-36-1)/86 = 6.
Let L: x+y+z+4=0=2x-y-z+8 be a line and P: x+2y+3z+1=0 be a plane.
What are the direction ratios of the line?
- (a)⟨2,1,-1⟩
- (b)⟨0,-1,2⟩
- (c)⟨0,1,-1⟩
- (d)⟨2,3,-3⟩
Answer: (c) ⟨0,1,-1⟩
Line L is the intersection of x+y+z+4=0 and 2x-y-z+8=0; the cross product of normals (1,1,1)×(2,-1,-1) gives direction (0,3,-3), i.e. ⟨0,1,-1⟩.
What is the point of intersection of L and P?
- (a)(4, 3, -3)
- (b)(4, -3, 3)
- (c)(-4, -3, -3)
- (d)(-4, -3, 3)
Answer: (d) (-4, -3, 3)
Parametrizing L from the point (-4,0,0) with direction (0,1,-1) and substituting into the plane x+2y+3z+1=0 gives the intersection point (-4,-3,3).
Let z = [y] and y = [x]-x, where [·] is the greatest integer function. If x is not an integer but positive, then what is the value of z?
- (a)-1
- (b)0
- (c)1
- (d)2
Answer: (a) -1
With x positive and non-integer, y=[x]-x is the negative fractional part, lying strictly in (-1,0); its floor z=[y] is therefore -1.
If f(x) = 4x+1 and g(x) = kx+2 such that f∘g(x) = g∘f(x), then what is the value of k?
- (a)7
- (b)5
- (c)4
- (d)3
Answer: (a) 7
f∘g(x)=4(kx+2)+1=4kx+9 and g∘f(x)=k(4x+1)+2=4kx+k+2; equating constants gives k=7.
What is the minimum value of the function f(x) = log10(x²+2x+11)?
- (a)0
- (b)1
- (c)2
- (d)10
Answer: (b) 1
x²+2x+11=(x+1)²+10 has minimum value 10 at x=-1, so log10 of that minimum is log10(10)=1.
Which one of the following is correct regarding lim (x→3) |x-3|/(x-3)?
- (a)Limit exists and is equal to 1
- (b)Limit exists and is equal to 0
- (c)Limit exists and is equal to -1
- (d)Limit does not exist
Answer: (d) Limit does not exist
The left-hand limit of |x-3|/(x-3) as x→3 is -1 and the right-hand limit is +1 — they disagree, so the limit does not exist.
What is the maximum value of a cosx + b sinx + c?
- (a)√(a²+b²+c)
- (b)√(a²+b²) + c
- (c)√(a²+b²) – c
- (d)√(a²+b²)
Answer: (b) √(a²+b²) + c
acosx+bsinx has maximum value √(a²+b²); adding the constant c gives the overall maximum √(a²+b²)+c.
If f(2x) = 4x²+1, then for how many real values of x will f(2x) be the GM of f(x) and f(4x)?
- (a)Four
- (b)Two
- (c)One
- (d)None
Answer: (c) One
With f(t)=t²+1 (deduced from f(2x)=4x²+1), setting f(2x)² = f(x)f(4x) (the GM condition) reduces to 9x²=0, giving exactly one real value, x=0.
If f(x) = [x]² – 30[x] + 221 = 0, where [x] is the greatest integer function, then what is the sum of all integer solutions?
- (a)13
- (b)17
- (c)27
- (d)30
Answer: (d) 30
Solving [x]²-30[x]+221=0 for the integer [x] gives roots 13 and 17; their sum is 30.
If f(x) = 9x – 8√x such that g(x) = f(x)-1, then which one of the following is correct?
- (a)g(x) = 0 has no real roots
- (b)g(x) = 0 has only one real root which is an integer
- (c)g(x) = 0 has two real roots which are integers
- (d)g(x) = 0 has only one real root which is not an integer
Answer: (b) g(x) = 0 has only one real root which is an integer
Substituting t=√x into 9t²-8t-1=0 gives t=1 (t=-1/9 rejected as negative), so x=1 — a single integer root.
What is lim (x→π/2) (secθ-tanθ) equal to?
- (a)-1
- (b)0
- (c)1/2
- (d)1
Answer: (b) 0
secx-tanx=(1-sinx)/cosx → 0 as x→π/2, a standard limit.
Let f(x)f(y) = f(xy) for all real x, y. If f(2)=4, then what is the value of f(1/2)?
- (a)1/4
- (b)1/2
- (c)1
- (d)4
Answer: (a) 1/4
f(x)f(y)=f(xy) with f(2)=4 fits f(x)=x²; then f(1/2)=1/4.
Let f∘g(x) = cos²√x and g∘f(x) = |cosx|.
Which one of the following is f(x)?
- (a)cosx
- (b)cosx²
- (c)cos²x
- (d)cos|x|
Answer: (c) cos²x
With f(x)=cos²x and g(x)=√x: f∘g(x)=cos²(√x) matches the given fog, and g∘f(x)=√(cos²x)=|cosx| matches the given gof.
Which one of the following is g(x)?
- (a)√x
- (b)|x|
- (c)x²
- (d)x|x|
Answer: (a) √x
See q81 — g(x)=√x is the matching function making both composite conditions hold.
Let f(x) = |x²-x-2|.
What is f(0.999) + f(1.001) equal to?
- (a)-1
- (b)0
- (c)1
- (d)2
Answer: (c) 1
The site’s stored passage here duplicates the (unrelated) |x²-x-2| passage used correctly for Q91-92 — a genuine content-linking defect. Cross-checked against the original exam text, this pair concerns f(x)=[x] (the greatest integer function): f(0.999)=0, f(1.001)=1, so f(0.999)+f(1.001)=1.
Consider the following statements:
I. f(x) is continuous at x = 0.
II. f(x) is continuous 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
Answer: (d) Neither I nor II
With f(x)=[x] (see Q83’s note), the greatest integer function has a jump discontinuity at every integer, so f is discontinuous at both x=0 and x=1 — neither statement holds.
Let f(x) = cos2x + x on [-π/2, π/2].
What is the greatest value of f(x)?
- (a)√3/2 – π/12
- (b)√3/2 + π/12
- (c)√3/2 + π/9
- (d)√3/2 + π/6
Answer: (b) √3/2 + π/12
f(x)=cos2x+x on [-π/2,π/2] has its maximum at the critical point x=π/12, giving f(π/12)=√3/2+π/12.
What is the least value of f(x)?
- (a)-(1+π/2)
- (b)-(1/2+π/2)
- (c)-(1+π/4)
- (d)-2(1/2-π/4)
Answer: (a) -(1+π/2)
The least value occurs at the left endpoint x=-π/2, giving f(-π/2)=-1-π/2=-(1+π/2).
The area bounded by the parabola y²=kx and the line x=k, where k > 0, is 4/3 square units.
What is the value of k?
- (a)1/2
- (b)1
- (c)√2
- (d)2
Answer: (b) 1
The area between y²=kx and x=k works out to 4k²/3; setting this equal to 4/3 gives k=1.
What is the area of the parabola bounded by the latus rectum?
- (a)1/6 square unit
- (b)2/3 square unit
- (c)1 square unit
- (d)4/3 square units
Answer: (a) 1/6 square unit
For y²=x (k=1), the latus rectum is at x=a=1/4; the area bounded by the parabola and this line is 1/6 square unit.
Let y dx + (x-y³) dy = 0 be a differential equation.
What are the order and degree respectively of the differential equation?
- (a)1 and 1
- (b)1 and 2
- (c)2 and 1
- (d)1 and 3
Answer: (a) 1 and 1
y dx+(x-y³)dy=0 involves only first derivatives with no powers beyond the first — order 1, degree 1.
What is the solution of the differential equation?
- (a)y⁴+2x=c
- (b)y⁴+3x=c
- (c)2xy⁴+x=c
- (d)4xy-y⁴=c
Answer: (d) 4xy-y⁴=c
Treating x as a function of y, this is a linear ODE dx/dy+x/y=y²; solving with integrating factor y gives xy=y⁴/4+C, i.e. 4xy-y⁴=constant.
Let f(x) = |x²-x-2|.
What is the integral from 0 to 2 of f(x) dx equal to?
- (a)0
- (b)1
- (c)5/3
- (d)10/3
Answer: (d) 10/3
∫[0,2]|x²-x-2|dx, splitting at the root x=2 where the sign changes within [0,2]… direct integration gives 10/3.
What is the integral from 1 to 3 of f(x) dx equal to?
- (a)2
- (b)3
- (c)4
- (d)5
Answer: (b) 3
∫[1,3]|x²-x-2|dx, splitting at the internal root x=2, direct integration gives 3.
Let f(t) = ln(t+√(1+t²)) and g(t) = tan(f(t)).
Consider the following statements:
I. f(t) is an odd function.
II. g(t) is an odd function.
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(-t)=ln(-t+√(1+t²))=-ln(t+√(1+t²))=-f(t), so f is odd (I true); g(-t)=tan(f(-t))=tan(-f(t))=-g(t), so g is odd too (II true).
What is the integral from -π to π of g(t) dt equal to?
- (a)-1
- (b)0
- (c)1/2
- (d)1
Answer: (b) 0
Since g(t) is an odd function (from Q93), its integral over the symmetric interval [-π,π] is 0.
Let f: (-1,1) → R be a differentiable function with f(0) = -1 and f'(0) = 1. Let h(x) = f(2f(x)+2) and g(x) = (h(x))².
What is h'(0) equal to?
- (a)-2
- (b)-1
- (c)0
- (d)2
Answer: (d) 2
h(x)=f(2f(x)+2); h'(x)=2f'(x)f'(2f(x)+2). At x=0: h'(0)=2·f'(0)·f'(2f(0)+2)=2·1·f'(0)=2·1·1=2.
What is g'(0) equal to?
- (a)-4
- (b)-2
- (c)0
- (d)4
Answer: (a) -4
g(x)=(h(x))², so g'(0)=2h(0)h'(0). h(0)=f(2f(0)+2)=f(0)=-1, so g'(0)=2(-1)(2)=-4.
Let I = integral from 0 to π/2 of f(x)/g(x) dx, where f(x) = sinx and g(x) = sinx+cosx+1.
What is the integral from 0 to π/2 of dx/g(x) equal to?
- (a)ln2/2
- (b)ln2/4
- (c)ln2
- (d)2ln2
Answer: (c) ln2
∫[0,π/2] dx/(sinx+cosx+1) evaluates exactly to ln2 (verified symbolically via the tan-half-angle substitution).
What is I equal to?
- (a)π/4 + ln2
- (b)π/4 – ln2
- (c)π/4 – ln2/2
- (d)π/4 + ln2/2
Answer: (c) π/4 – ln2/2
I=∫[0,π/2] sinx/(sinx+cosx+1)dx evaluates exactly to π/4-ln2/2.
Let 2×∫(x²-1)/√(x²+1) dx = U(x)V(x) – 3ln{U(x)+V(x)} + c.
What is |U²(x) – V²(x)| equal to?
- (a)0
- (b)1
- (c)2
- (d)3
Answer: (b) 1
With U(x)=x, V(x)=√(x²+1) (matching the given antiderivative form), U²(x)-V²(x)=x²-(x²+1)=-1, so |U²-V²|=1.
What is U(x)V(x) equal to?
- (a)√(x²+x⁴)
- (b)√(x+x³)
- (c)√(x²+x⁴)/2
- (d)2√(x²+x⁴)
Answer: (a) √(x²+x⁴)
U(x)V(x)=x√(x²+1)=√(x²(x²+1))=√(x²+x⁴).
Let x-3y+4=0 and 2x-7y+8=0 be two lines of regression computed from some bivariate data. If byx and bxy are regression coefficients of lines of regression of y on x and x on y respectively, then what is the value of bxy + 7byx?
- (a)-2
- (b)1
- (c)2
- (d)5
Answer: (d) 5
Testing both regression assignments, only byx=2/7 (from 2x-7y+8=0 as y-on-x) and bxy=3 (from x-3y+4=0 as x-on-y) gives byx·bxy=6/7≤1; then bxy+7byx=3+2=5.
The mean of n observations 1, 4, 9, 16, …, n² is 130. What is the value of n?
- (a)18
- (b)19
- (c)20
- (d)21
Answer: (b) 19
Mean of 1²,2²,…,n² is (n+1)(2n+1)/6; setting this to 130 gives (n+1)(2n+1)=780, solved by n=19.
Three distinct natural numbers are chosen at random from 1 to 10. What is the probability that they are consecutive?
- (a)1/12
- (b)3/40
- (c)1/15
- (d)7/120
Answer: (c) 1/15
Of the C(10,3)=120 ways to choose 3 distinct numbers from 1-10, exactly 8 triples are consecutive, giving probability 8/120=1/15.
A, B, C are three mutually exclusive and exhaustive events associated with a random experiment. If 3P(B)=4P(A) and 3P(C)=2P(B), then what is P(A) equal to?
- (a)7/29
- (b)8/29
- (c)9/29
- (d)10/29
Answer: (c) 9/29
Solving 3P(B)=4P(A), 3P(C)=2P(B) and P(A)+P(B)+P(C)=1 gives P(A)=9/29.
A die has two faces with number 4, three faces with number 5 and one face with number 6. If the die is rolled once, then what is the probability of getting 4 or 5?
- (a)1/3
- (b)2/3
- (c)5/6
- (d)1/2
Answer: (c) 5/6
P(4 or 5) = (2+3)/6 = 5/6.
A box contains 2 black, 4 yellow and 6 white balls. Three balls are drawn in succession with replacement. What is the probability that all three are of the same colour?
- (a)1/6
- (b)1/36
- (c)1/12
- (d)5/12
Answer: (a) 1/6
P(all black)+P(all yellow)+P(all white) = (2/12)³+(4/12)³+(6/12)³ = 1/6.
A can hit a target 5 times in 6 shots, B can hit 4 times in 5 shots and C can hit 3 times in 4 shots. What is the probability that A and C may hit but B may lose?
- (a)1/8
- (b)1/6
- (c)1/4
- (d)1/3
Answer: (a) 1/8
P(A hits)·P(B misses)·P(C hits) = (5/6)(1/5)(3/4) = 1/8.
The letters of the word ZOOLOGY are arranged in all possible ways. What is the probability that the consonants and vowels occur alternatively?
- (a)6/35
- (b)3/35
- (c)2/35
- (d)1/35
Answer: (d) 1/35
ZOOLOGY has 4 consonants (Z,L,G,Y) and 3 identical vowels (O,O,O); an alternating pattern must be C-V-C-V-C-V-C, giving 4! arrangements out of 7!/3! total, i.e. probability 1/35.
A natural number x is chosen at random from the first 100 natural numbers. What is the probability that x²+x > 50?
- (a)93/100
- (b)47/50
- (c)24/25
- (d)23/25
Answer: (b) 47/50
Of the numbers 1 to 100, x²+x>50 fails only for x=1..6 (x²+x≤50); so 94 values satisfy it, giving probability 94/100=47/50.
What is the mean deviation of the first 10 natural numbers?
- (a)2
- (b)2.5
- (c)3
- (d)3.5
Answer: (b) 2.5
For the numbers 1-10 (mean 5.5), the mean absolute deviation from the mean is 2.5.
Let the sum from i=1 to 9 of xi² = 855. If M is the mean and σ is the standard deviation of x1, x2, …, x9, then what is the value of M²+σ²?
- (a)100
- (b)95
- (c)90
- (d)85
Answer: (b) 95
M²+σ² = (1/n)Σxi² for any dataset (a standard identity), so here it equals 855/9=95.
The mean of the series x1, x2, …, xn is x̄. If xn is replaced by k, then what is the new mean?
- (a)x̄ – xn + k
- (b)(nx̄ – x̄ + k)/n
- (c)(x̄ – xn – k)/n
- (d)(nx̄ – xn + k)/n
Answer: (d) (nx̄ – xn + k)/n
Replacing xn with k changes the sum by (k-xn); the new mean is x̄+(k-xn)/n = (nx̄-xn+k)/n.
A fair coin is tossed till two heads occur in succession. What is the probability that the number of tosses required is less than 6?
- (a)5/64
- (b)15/32
- (c)31/64
- (d)19/32
Answer: (d) 19/32
Enumerating all coin-toss sequences where two heads first occur in succession within fewer than 6 tosses gives total probability 19/32.
Urn A contains 2 white and 2 black balls while urn B contains 3 white and 2 black balls. One ball is transferred from urn A to urn B and then a ball is drawn out of urn B. What is the probability that the ball is white?
- (a)11/20
- (b)7/12
- (c)3/5
- (d)1
Answer: (b) 7/12
P(white)=P(transfer white)·P(white|transfer white)+P(transfer black)·P(white|transfer black)=(1/2)(4/6)+(1/2)(3/6)=7/12.
For two events A and B, P(A) = P(A|B) = 0.25 and P(B|A) = 0.5. Which of the following are correct?
I. A and B are independent.
II. P(Aᵕ∪Bᵕ) = 0.875
III. P(Aᵕ∩Bᵕ) = 0.375
Select the answer using the code given below.
- (a)I and II only
- (b)II and III only
- (c)I and III only
- (d)I, II and III
Answer: (d) I, II and III
P(A)=P(A|B) confirms independence (I). With P(B)=P(B|A)=0.5 (from independence) and P(A∩B)=0.125, direct computation gives P(A’∪B’)=0.875 (II) and P(A’∩B’)=0.375 (III) — both match exactly.
Two perfect dice are thrown. What is the probability that the sum of the numbers on the faces is neither 9 nor 10?
- (a)1/36
- (b)5/36
- (c)7/36
- (d)29/36
Answer: (d) 29/36
Of the 36 outcomes of two dice, 7 give a sum of 9 or 10; so 29 give neither, probability 29/36.
The occurrence of a disease in an industry is such that the workers have 20% chance of suffering from it. What is the probability that out of 6 workers chosen at random, 4 or more will suffer from the disease?
- (a)53/3125
- (b)63/3125
- (c)73/3125
- (d)83/3125
Answer: (a) 53/3125
Binomial P(X≥4) with n=6,p=0.2 evaluates to 53/3125.
Three perfect dice are rolled. Under the condition that no two show the same face, what is the probability that one of the faces shown is an ace (one)?
- (a)5/9
- (b)2/3
- (c)1/3
- (d)1/2
Answer: (d) 1/2
Of all ordered outcomes of 3 dice with no repeated face, exactly half include a 1 among the three faces shown, giving probability 1/2.
Three perfect dice D1, D2 and D3 are rolled. Let x, y and z represent the numbers on D1, D2 and D3 respectively. What is the number of possible outcomes such that x < y < z?
- (a)20
- (b)18
- (c)14
- (d)10
Answer: (a) 20
Direct enumeration of dice outcomes with x<y<z gives exactly 20 favorable outcomes.
In a binomial distribution, if the mean is 6 and the standard deviation is √2, then what are the values of the parameters n and p respectively?
- (a)18 and 1/3
- (b)9 and 1/3
- (c)18 and 2/3
- (d)9 and 2/3
English-language questions transcribed from the official N.D.A. & N.A. Examination-(II), 2024 question booklet (SURN-B-MTH), Mathematics. Hindi text omitted. Answer key not included.
Answer: (d) 9 and 2/3
np=6, npq=2 give q=1/3, p=2/3, and n=9.
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