Math Lecturer Cadre 2026 Solved Paper – All 150 Questions with Answer Key

Math Lecturer Cadre Question Paper (LCRE-2026) – 150 MCQs with Answers and Hints

Practise the complete Mathematics Lecturer Cadre question paper (LCRE-2026, Series B) with all 150 multiple choice questions, their options, correct answers and a helpful hint for every question. This paper is useful for Punjab Lecturer Cadre recruitment, PGT Mathematics, school lecturer exams, TET and other state-level teaching exams.

The questions cover Real Analysis, Calculus, Complex Analysis, Abstract Algebra (Groups, Rings and Fields), Linear Algebra, Topology, Ordinary and Partial Differential Equations, Numerical Analysis, Integral Equations, Calculus of Variations, Classical Mechanics, Probability, Statistics, Linear Programming, Trigonometry, Vectors and 3D Geometry.

Try each question on your own first. If you are stuck, use Show hint. Then use Show answer to check your result and read the short explanation. Where the original paper has a printing error or no correct option, this is clearly marked in the answer.

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LCRE-2026 · Mathematics · Series B · 150 MCQs

Math Lecturer Cadre — Question Paper with Answers

All 150 questions from the original paper, with four options each. Try each question, use the hint if you're stuck, then reveal the answer.

1

Limits

\(\lim_{t\to 0}\left(1^{\frac{1}{\sin^2 t}}+2^{\frac{1}{\sin^2 t}}+3^{\frac{1}{\sin^2 t}}+\cdots+n^{\frac{1}{\sin^2 t}}\right)^{\sin^2 t}=\)

1. A\(n^2+n\)
2. B\(n\)
3. C\(n(n+1)\)
4. D\(n^2\)

Hint

As \(t\to0\), \(p=\frac{1}{\sin^2 t}\to\infty\). Use \(\left(\sum k^p\right)^{1/p}\to\) the largest term.

Answer  (B) \(n\)

2

Limits

\(\displaystyle\lim_{x\to 0}\frac{\sqrt{1+x}-1}{(1+x)^{1/3}-1}\) equals

1. A\(\frac{1}{2}\)
2. B\(1\)
3. C\(\frac{3}{2}\)
4. D\(5\)

Hint

Use \((1+x)^m-1\approx mx\) for small \(x\), on both the numerator and the denominator.

Answer  (C) \(\frac{3}{2}\)

3

Functions of Two Variables

Let \(f:\mathbb{R}^2\to\mathbb{R}\) be defined by \(f(x,y)=\begin{cases} xy\dfrac{x^2-y^2}{x^2+y^2}, & (x,y)\neq(0,0)\\ 0, & (x,y)=(0,0)\end{cases}\). At \((0,0)\)

1. A\(f\) is not continuous
2. B\(f\) is continuous and both \(f_x, f_y\) do not exist
3. C\(f\) is differentiable
4. DEach of \(f_x, f_y\) exists but not differentiable

Hint

Check \(|f(x,y)|\le|xy|\le\frac{x^2+y^2}{2}\). Is the error \(o(\sqrt{x^2+y^2})\)?

|f| ≤ |xy| = o(r), so f is differentiable at (0,0).

Answer  (C) \(f\) is differentiable

4

Uniform Continuity

Which of the following is not uniformly continuous on \((0,1)\)?

1. A\(e^{-1/x^2}\)
2. B\(\frac{1}{x^2}\sin x\tan x\)
3. C\(e^x\cos\frac{1}{x}\)
4. D\(x^2\cos\frac{1}{x}\)

Hint

On a bounded interval, \(f\) is uniformly continuous iff it extends continuously to the endpoints. Look at the behaviour as \(x\to0^+\).

Answer  (C) \(e^x\cos\frac{1}{x}\)

5

Differentials

The approximate value of \(\sqrt[3]{0.026}\) is

1. A0.2762
2. B0.2632
3. C0.2963
4. D0.2692

Hint

Take \(f(x)=x^{1/3}\) at \(x=0.027\) with \(\Delta x=-0.001\), and use \(f(x+\Delta x)\approx f(x)+f'(x)\Delta x\).

0.3 − 0.001/(3·0.09) ≈ 0.2963.

Answer  (C) 0.2963

6

Mean Value Theorem

If \(f(x)=1+|x-2|+|\sin x|\), then Lagrange Mean Value Theorem is applicable for \(f(x)\) in

1. A\([0,\pi)\)
2. B\([\pi,2\pi]\)
3. C\(\left[\frac{\pi}{2},\frac{3\pi}{2}\right]\)
4. D\(\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\)

Hint

\(|x-2|\) has a corner at \(x=2\), and \(|\sin x|\) has corners at \(x=k\pi\). Pick the interval whose interior avoids both.

Answer  (B) \([\pi,2\pi]\)

7

Tangents

The curve \(y=x^{1/5}\) has one of the following at \((0,0)\):

1. Avertical tangent
2. Bhorizontal tangent
3. Coblique tangent
4. Dno tangent

Hint

Find \(\frac{dy}{dx}=\frac{1}{5}x^{-4/5}\). What happens to it as \(x\to0\)?

dy/dx → ∞ at 0, so the tangent is vertical.

Answer  (A) vertical tangent

8

Continuity

The domain of continuity of \(f(x)=\dfrac{\sqrt{\sqrt{x^2}-x}}{\sin^{-1}x}\) is

1. A\((-1,0)\)
2. B\([-1,0)\cup(0,1]\)
3. C\((-1,0)\cup(0,1)\)
4. D\((0,1]\)

Hint

\(\sqrt{x^2}=|x|\). Find where \(|x|-x\ge0\) and where \(\sin^{-1}x\) is defined and non-zero.

f is defined and continuous on its whole domain [−1,0)∪(0,1] (one-sided at ±1). Some keys give C; B is the mathematically correct choice.

Answer  (B) \([-1,0)\cup(0,1]\)

9

Maxima & Minima

The difference between greatest and least values of \(f(x)=\dfrac{x^2+4x+2}{x^2+2}\) is

1. A\(4\)
2. B\(2\sqrt{2}\)
3. C\(10\)
4. D\(\sqrt{5}\)

Hint

Write \(f(x)=1+\frac{4x}{x^2+2}\) and find the max and min of \(\frac{4x}{x^2+2}\) (AM–GM).

f = 1 + 4x/(x²+2), range [1−√2, 1+√2].

Answer  (B) \(2\sqrt{2}\)

10

Limits

\(\displaystyle\lim_{x\to\pi/6}\frac{2-\sqrt{3}\cos x-\sin x}{(6x-\pi)^2}=\)

1. A\(\frac{1}{18}\)
2. B\(\frac{1}{36}\)
3. C\(\frac{5}{9}\)
4. D\(\frac{1}{54}\)

Hint

\(\sqrt3\cos x+\sin x=2\cos\left(x-\frac{\pi}{6}\right)\). Put \(h=x-\frac{\pi}{6}\).

Answer  (B) \(\frac{1}{36}\)

11

Binomial Theorem

If \((1+x^2)^4(1+ax)^3=1+3ax+2abx^2+4(a^2+b)x^3+\cdots+a^3x^{11}\), then the value of \(a\) and \(b\) are

1. A1, 2
2. B2, 4
3. C3, 5
4. D−1, 4

Hint

Compare the coefficients of \(x^2\) and \(x^3\) on both sides, then test the options.

3a²+4 = 2ab and a³+12a = 4(a²+b) give a=2, b=4.

Answer  (B) 2, 4

12

Binomial Theorem

The largest coefficient in expansion of \((1+x)^{23}\) is

1. A\(^{23}C_{13}\)
2. B\(^{24}C_{13}\)
3. C\(\frac{1}{2}\,{}^{23}C_{13}\)
4. D\(^{23}C_{12}\)

Hint

In \((1+x)^n\) with \(n\) odd, the two middle terms \(^nC_{(n-1)/2}\) and \(^nC_{(n+1)/2}\) are the largest.

Answer  (D) \(^{23}C_{12}\)

13

Number Theory

The last digit of \(23^{100}\) is

1. A10
2. B9
3. C1
4. D7

Hint

The last digit of \(23^{100}\) is the last digit of \(3^{100}\). Powers of 3 repeat with cycle 4.

Answer  (C) 1

14

Linear Algebra

The characteristic polynomial of a matrix \(A\in M_5(\mathbb{R})\) is given by \(x^5+\alpha x^4+\beta x^3\), where \(\alpha\) and \(\beta\) are non-zero real numbers. What among the following is a possible value of rank \(A\)?

1. A3
2. B1
3. C5
4. DNone of these

Hint

\(x^5+\alpha x^4+\beta x^3=x^3(x^2+\alpha x+\beta)\). How many eigenvalues are 0 and how many are non-zero?

0 is an eigenvalue (mult. 3), so rank ≤ 4; the two non-zero eigenvalues force rank ≥ 2.

Answer  (A) 3

15

Conic Sections

The pole of line \(x+5y-3\) w.r.t. ellipse \(3x^2+4y^2=12\) is

1. A\((-4/3,\,2)\)
2. B\((-3/4,\,2)\)
3. C\((4/3,\,5)\)
4. D\((-4/3,\,-5)\)

Hint

The polar of \((h,k)\) w.r.t. \(3x^2+4y^2=12\) is \(3hx+4ky=12\). Compare it with \(x+5y=3\).

Answer  (C) \((4/3,\,5)\)

16

Maxima & Minima

In the interval \(\left[0,\frac{3}{2}\right]\), the function \(2e^{2x}\cdot x^{21}\cdot(5-2x)^{16}\) has maximum value at point \(x=\)

1. A\(\frac{3}{2}\)
2. B\(\frac{1}{2}\)
3. C\(\frac{7}{2}\)
4. D\(\frac{5}{2}\)

Hint

Take the log-derivative: \(2+\frac{21}{x}-\frac{32}{5-2x}\). Check its sign on \(\left(0,\frac32\right)\).

f′>0 on (0, 3/2), so the maximum is at the endpoint x = 3/2.

Answer  (A) \(\frac{3}{2}\)

17

Complex Analysis

The conjugate of the complex number \(\dfrac{2+5i}{4-3i}\) is

1. A\(\frac{-7-26i}{25}\)
2. B\(\frac{-7+26i}{25}\)
3. C\(\frac{7-26i}{25}\)
4. D\(\frac{7+26i}{25}\)

Hint

Multiply the numerator and denominator by \(4+3i\), then change the sign of the imaginary part.

Answer  (A) \(\frac{-7-26i}{25}\)

18

Complex Analysis

If \(f(z)=\dfrac{z}{(z^3-8)(z-3)}\), \(z=x+iy\), then Residue of \(f(z)\) at \(z=2\) is

1. A\(\frac{1}{6}\)
2. B\(\frac{1}{12}\)
3. C\(-\frac{1}{12}\)
4. D\(-\frac{1}{6}\)

Hint

\(z^3-8=(z-2)(z^2+2z+4)\). Res \(=\lim_{z\to2}(z-2)f(z)\).

Res = 2/[(4+4+4)(2−3)] = −1/6.

Answer  (D) \(-\frac{1}{6}\)

19

Complex Analysis

The integral \(\displaystyle\oint_{|z|=2}\frac{3z^2+11z-1}{z-4}\,dz\) where C is the circle \(|z|=2\) travelled clockwise is

1. A\(2\pi i\)
2. B\(4\pi i\)
3. C\(0\)
4. D\(8\pi i\)

Hint

Where is the singularity \(z=4\) relative to the circle \(|z|=2\)?

The only singularity z = 4 lies outside |z| = 2, so the integral is 0 (Cauchy's theorem).

Answer  (C) \(0\)

20

Complex Analysis

Using the Cauchy–Riemann equations, determine the points at which the function \(f(z)=x^2-y^2+i(2xy)\), where \(z=x+iy\), is analytic.

1. AOnly at \((0,0)\)
2. BOnly on the x-axis
3. CAt every point in the complex plane
4. DNowhere in the complex plane

Hint

\(f(z)=z^2\). Check that \(u_x=v_y\) and \(u_y=-v_x\) hold everywhere.

Answer  (C) At every point in the complex plane

21

Complex Analysis

\(\operatorname{Arg}(-i)\) is

1. A\(\frac{\pi}{2}\)
2. B\(-\frac{\pi}{2}\)
3. C\(\frac{\pi}{3}\)
4. D\(\frac{\pi}{4}\)

Hint

\(-i\) lies on the negative imaginary axis. The principal argument lies in \((-\pi,\pi]\).

Answer  (B) \(-\frac{\pi}{2}\)

22

Number Theory

The Euler's Phi-function is:

1. AInjective
2. BMultiplicative
3. CNon-multiplicative
4. DSurjective

Hint

\(\phi(mn)=\phi(m)\phi(n)\) when \(\gcd(m,n)=1\).

Answer  (B) Multiplicative

23

Number Theory

The value of Euler's Phi-function \(\phi(18)\) is:

1. A4
2. B8
3. C10
4. D6

Hint

\(18=2\cdot3^2\), so \(\phi(18)=18\left(1-\frac12\right)\left(1-\frac13\right)\).

Answer  (D) 6

24

Number Theory

The digit in the unit place of \(2018^{2018}+2019^{2019}+2020^{2020}\) is:

1. A3
2. B4
3. C5
4. D6

Hint

Unit digits: \(8^{2018}\) (cycle 8,4,2,6), \(9^{\text{odd}}\), and \(0\). Add them.

unit digits 4 + 9 + 0 = 13 → 3.

Answer  (A) 3

25

Binomial Coefficients

If the binomial coefficients satisfy \(\binom{43}{r-6}=\binom{43}{3r+1}\), then the value of r is:

1. A10
2. B11
3. C12
4. D13

Hint

\(^nC_a={}^nC_b\Rightarrow a=b\) or \(a+b=n\).

Answer  (C) 12

26

Group Theory

Which of the following groups is cyclic?

1. A\(S_3\)
2. B\((\mathbb{Z}_6,+)\)
3. C\(S_4\)
4. D\(D_4\)

Hint

A cyclic group must be abelian. Which of these groups is abelian and generated by one element?

S₃ is non-abelian, so it cannot be cyclic.

Answer  (B) \((\mathbb{Z}_6,+)\)

27

Group Theory

Which of the following is a normal subgroup of \(S_3\)?

1. A\(\{e,(123),(132)\}\)
2. B\(\{e,(12)\}\)
3. C\(\{e,(13)\}\)
4. D\(\{e,(23)\}\)

Hint

A subgroup of index 2 is always normal.

Answer  (A) \(\{e,(123),(132)\}\)

28

Group Theory

The solution of \(25x\equiv 15 \pmod{29}\) is

1. A\(18 \pmod{29}\)
2. B\(29 \pmod{29}\)
3. C\(17 \pmod{29}\)
4. D\(21 \pmod{29}\)

Hint

Reduce: \(25\equiv-4 \pmod{29}\). Solve \(-4x\equiv15\), or check the options directly.

25·18 = 450 = 15·29 + 15.

Answer  (A) \(18 \pmod{29}\)

29

Group Theory

If G is a non-Abelian group of order 27, then the order of its centre \(Z(G)\) is:

1. A1
2. B27
3. C9
4. D3

Hint

For a non-abelian \(p\)-group, \(Z(G)\ne\{e\}\), and \(G/Z(G)\) cannot be cyclic.

Answer  (D) 3

30

Group Theory

The order of the permutation \(\sigma=(1\,2\,3)(4\,5)\) in \(S_3\) is:

1. A5
2. B4
3. C6
4. D3

Hint

The order of a product of disjoint cycles is the lcm of the cycle lengths.

Printing error: (123)(45) lives in S₅, not S₃. Order = lcm(3,2) = 6.

Answer  (C) 6

31

Ring Theory

Which of the following is an ideal of \(\mathbb{Z}\)?

1. A\(2\mathbb{Z}\)
2. B\(\{1,2,3,4,\ldots\}\)
3. CThe set of prime integers
4. DThe set of odd integers

Hint

An ideal must be an additive subgroup that absorbs multiplication by every integer.

Answer  (A) \(2\mathbb{Z}\)

32

Ring Theory

Which of the following is a maximal ideal of \(\mathbb{Z}\)?

1. A\((4)\)
2. B\((6)\)
3. C\((9)\)
4. D\((7)\)

Hint

In \(\mathbb{Z}\), \((n)\) is maximal iff \(n\) is prime.

(p) is maximal only for p prime; 7 is prime.

Answer  (D) \((7)\)

33

Ring Theory

Which of the following is a Unique Factorization Domain (UFD)?

1. A\(\mathbb{Z}\)
2. B\(\mathbb{Z}[\sqrt{-5}]\)
3. C\(\mathbb{Z}[\sqrt{-6}]\)
4. DNone of these

Hint

Recall the Fundamental Theorem of Arithmetic. In \(\mathbb{Z}[\sqrt{-5}]\), \(6=2\cdot3=(1+\sqrt{-5})(1-\sqrt{-5})\).

ℤ is a UFD (Fundamental Theorem of Arithmetic).

Answer  (A) \(\mathbb{Z}\)

34

Ring Theory

Which of the following is a Principal Ideal Domain (PID)?

1. A\(\mathbb{Z}[x]\)
2. B\(\mathbb{Z}\)
3. C\(\mathbb{Z}[i]\)
4. DBoth (B) and (C)

Hint

Euclidean domains are PIDs. In \(\mathbb{Z}[x]\), the ideal \((2,x)\) is not principal.

Answer  (D) Both (B) and (C)

35

Group Theory

Which of the following mappings is a group homomorphism from \((\mathbb{Z},+)\) to \((\mathbb{Z},+)\)?

1. A\(f(x)=2x\)
2. B\(f(x)=x^2\)
3. C\(f(x)=x+1\)
4. D\(f(x)=2x+1\)

Hint

A homomorphism must satisfy \(f(a+b)=f(a)+f(b)\), so in particular \(f(0)=0\).

f(0) = 1 ≠ 0, so x+1 is not a homomorphism.

Answer  (A) \(f(x)=2x\)

36

Topology

Let \(X=\mathbb{R}\) with the usual topology and \(A=\mathbb{Q}\). Which of the following statements is true?

1. AA is closed but not dense in X.
2. BA is open and closed in X.
3. CA is dense but not open in X.
4. DA is neither dense nor closed in X.

Hint

Every open interval contains a rational (dense). Every open interval also contains irrationals, so \(\mathbb{Q}\) is not open.

ℚ is dense in ℝ.

Answer  (C) A is dense but not open in X.

37

Topology

Which of the following collections forms a basis for the usual topology on \(\mathbb{R}\)?

1. A\(\{(a,b): a,b\in\mathbb{R}, a<b\}\)
2. B\(\{[a,b): a,b\in\mathbb{R}, a<b\}\)
3. C\(\{\{a\}: a\in\mathbb{R}\}\)
4. DNone of these

Hint

The usual topology on \(\mathbb{R}\) is generated by open intervals.

[a,b) generates the lower-limit topology.

Answer  (A) \(\{(a,b): a,b\in\mathbb{R}, a<b\}\)

38

Trigonometry

The minimum value of \(2\sin^2\theta+3\cos^2\theta\) is:

1. A1
2. B3
3. C4
4. D2

Hint

\(2\sin^2\theta+3\cos^2\theta=2+\cos^2\theta\).

2 + cos²θ has minimum 2.

Answer  (D) 2

39

Trigonometry

If \(\cos(\pi x)=x^2-x+\frac{5}{4}\), then value of \(x\) is:

1. A1
2. B\(\frac{1}{2}\)
3. C−1
4. D0

Hint

Complete the square: \(x^2-x+\frac54=\left(x-\frac12\right)^2+1\ge1\). Compare with the range of the left side.

Question has a misprint — as printed there is NO solution (RHS ≥ 1 with equality at x = ½, but cos(π/2) = 0). Intended version is sin(πx) = …, giving x = ½.

Answer  (B) \(\frac{1}{2}\)

40

Trigonometry

The value of \(\sin(30^\circ+\theta)-\cos(60^\circ-\theta)\) is:

1. A1
2. B\(2\cos\theta\)
3. C0
4. D\(2\sin\theta\)

Hint

\(\cos(60^\circ-\theta)=\sin(90^\circ-(60^\circ-\theta))\).

Answer  (C) 0

41

Trigonometry

Determine the number of solutions of the equation \(\tan x+\sec x=2\cos x\), in the interval \([0,2\pi]\).

1. A4
2. B1
3. C3
4. D2

Hint

Multiply by \(\cos x\): \(\sin x+1=2\cos^2 x\). Remember \(\cos x\ne0\).

sin x = ½ (x = π/6, 5π/6); sin x = −1 is rejected because cos x = 0.

Answer  (D) 2

42

Field Theory

Which of the following is a finite field?

1. A\(\mathbb{Z}_6\)
2. B\(\mathbb{Z}_7\)
3. C\(\mathbb{Q}\)
4. D\(\mathbb{Z}\)

Hint

\(\mathbb{Z}_n\) is a field iff \(n\) is prime.

Answer  (B) \(\mathbb{Z}_7\)

43

Trigonometry

General solution of the equation \(\cos\left(\frac{3x}{2}\right)=\frac{1}{2}\) is given by

1. A\(x=\frac{4n\pi}{3}\pm\frac{\pi}{9}\)
2. B\(x=\frac{4n\pi}{3}\pm\frac{2\pi}{9}\)
3. C\(x=\frac{2n\pi}{3}\pm\frac{2\pi}{9}\)
4. D\(x=\frac{2n\pi}{3}\pm\frac{\pi}{9}\)

Hint

\(\cos\theta=\cos\alpha\Rightarrow\theta=2n\pi\pm\alpha\), with \(\theta=\frac{3x}{2}\) and \(\alpha=\frac{\pi}{3}\).

Answer  (B) \(x=\frac{4n\pi}{3}\pm\frac{2\pi}{9}\)

44

Trigonometry

A wheel of a vehicle makes 360 revolutions in one minute. Through how many radians does it turn in 1.5 seconds?

1. A\(18\pi\)
2. B\(12\pi\)
3. C\(14\pi\)
4. D\(10\pi\)

Hint

360 rev/min = 6 rev/s. One revolution = \(2\pi\) radians.

6 rev/s × 1.5 s = 9 rev = 18π.

Answer  (A) \(18\pi\)

45

Trigonometry

If \(\tan\theta=\frac{1}{2}\), \(\tan\phi=\frac{1}{3}\), then \(\tan(\theta+\phi)=\)?

1. A\(\frac{1}{2}\)
2. B\(\frac{1}{4}\)
3. C1
4. D0

Hint

\(\tan(\theta+\phi)=\frac{\tan\theta+\tan\phi}{1-\tan\theta\tan\phi}\).

Answer  (C) 1

46

Ordinary Differential Equations

A solution of the differential equation \(y\,dx+x\,dy=0\) is given by

1. A\(x+y=C\)
2. B\(x-y=C\)
3. C\(x^2+y^2=C\)
4. D\(xy=C\)

Hint

\(y\,dx+x\,dy=d(xy)\).

Answer  (D) \(xy=C\)

47

Ordinary Differential Equations

The order of the differential equation \(\dfrac{d^3y}{dx^3}+2\left(\dfrac{d^2y}{dx^2}\right)^2+\dfrac{dy}{dx}=0\) is:

1. A4
2. B3
3. C2
4. D1

Hint

Order = the highest derivative present.

Answer  (B) 3

48

Ordinary Differential Equations

General solution of the equation \(\sin x=\frac{1}{2}\) is given by

1. A\(x=n\pi+(-1)^n\frac{\pi}{6}\)
2. B\(x=2n\pi+\frac{\pi}{6}\)
3. C\(x=n\pi+\frac{\pi}{6}\)
4. D\(x=2n\pi+\frac{5\pi}{6}\)

Hint

\(\sin x=\sin\alpha\Rightarrow x=n\pi+(-1)^n\alpha\).

Answer  (A) \(x=n\pi+(-1)^n\frac{\pi}{6}\)

49

Ordinary Differential Equations

The eigen functions corresponding to distinct eigen values of a regular Sturm–Liouville problem are:

1. AOrthogonal with respect to the weight function
2. BLinearly dependent
3. CAlways identical
4. DNone of these

Hint

This is a standard property of Sturm–Liouville eigenfunctions (with weight \(r(x)\)).

Answer  (A) Orthogonal with respect to the weight function

50

Ordinary Differential Equations

The general solution of the homogeneous differential equation \(\dfrac{dy}{dx}=\dfrac{x+y}{x-y}\) is given by

1. A\(x^2+y^2=C\)
2. B\(x-y=C\)
3. C\(\tan^{-1}\left(\frac{y}{x}\right)-\frac{1}{2}\ln(x^2+y^2)=C\)
4. D\(x+y=C\)

Hint

Put \(y=vx\), or switch to polar coordinates.

Answer  (C) \(\tan^{-1}\left(\frac{y}{x}\right)-\frac{1}{2}\ln(x^2+y^2)=C\)

51

Ordinary Differential Equations

The solution of the differential equation \(2x\dfrac{dy}{dx}=y+3\) represents a family of

1. ACircles
2. BParabolas
3. CStraight lines
4. DEllipses

Hint

Separate: \(\frac{dy}{y+3}=\frac{dx}{2x}\), then integrate.

(y+3)² = kx, a family of parabolas.

Answer  (B) Parabolas

52

Ordinary Differential Equations

The degree of the differential equation \(\sin\left(\dfrac{d^2y}{dx^2}\right)+\dfrac{dy}{dx}=0\) is

1. A0
2. B1
3. C2
4. DNot defined

Hint

The degree is defined only when the equation is a polynomial in its derivatives.

Answer  (D) Not defined

53

Partial Differential Equations

General solution of the PDE \(u_x+u_y=u\) passing through the curve \(x=t,\ y=2t,\ u=1\) is

1. A\(u=e^{x-y}\)
2. B\(u=e^{x+y}\)
3. C\(u=e^{2x-y}\)
4. D\(u=e^{2y-x}\)

Hint

Test each option in \(u_x+u_y=u\), then check that \(u=1\) on \(x=t, y=2t\).

Answer  (C) \(u=e^{2x-y}\)

54

Partial Differential Equations

Let \(u(x,y)\) be the solution of the Cauchy problem \(xu_x+u_y=1\), \(u(x,0)=2\log x\); \(x\ge 1\). Then \(u(e,1)\) is:

1. A1
2. B−1
3. C2
4. D0

Hint

Characteristics: \(\frac{dx}{x}=\frac{dy}{1}=\frac{du}{1}\).

u = 2 log x − y, so u(e,1) = 1.

Answer  (A) 1

55

Partial Differential Equations

The PDE \(u_{xx}+u_{yy}=0\) is:

1. AHyperbolic
2. BParabolic
3. CElliptic
4. DNone of these

Hint

For \(Au_{xx}+Bu_{xy}+Cu_{yy}\), look at the sign of \(B^2-4AC\).

B² − 4AC = −4 < 0 → elliptic.

Answer  (C) Elliptic

56

Partial Differential Equations

Which of the following PDEs is elliptic?

1. AHeat equation
2. BWave equation
3. C\(u_{xx}+2u_{xy}-4u_{yy}=0\)
4. DLaplace equation

Hint

Elliptic means \(B^2-4AC<0\). Classify each equation.

the wave equation is hyperbolic.

Answer  (D) Laplace equation

57

Partial Differential Equations

General solution of the differential equation \((D^2+4)y=\sin 2x\) is

1. A\(y=C_1\cos 2x+C_2\sin 2x-\frac{x}{4}\cos 2x\)
2. B\(y=C_1\cos 2x+C_2\sin 2x+\frac{x}{4}\cos 2x\)
3. C\(y=C_1e^{2x}+C_2e^{-2x}+\frac{x}{4}\cos 2x\)
4. D\(y=C_1\cos 2x+C_2\sin 2x+\frac{1}{4}\sin 2x\)

Hint

This is resonance: the particular integral of \(\frac{1}{D^2+a^2}\sin ax\) is \(-\frac{x}{2a}\cos ax\).

Answer  (A) \(y=C_1\cos 2x+C_2\sin 2x-\frac{x}{4}\cos 2x\)

58

Partial Differential Equations

The complete integral of the PDE \(pq=1\), where \(p=u_x,\ q=u_y\) is:

1. A\(u=x+y\)
2. B\(u=xy+a\)
3. C\(u=ax+\frac{y}{a}+b,\ a\neq 0\)
4. D\(u=x^2+y^2+a\)

Hint

Try \(u=ax+by+c\) with \(pq=ab=1\).

for u = xy + a, pq = xy ≠ 1.

Answer  (C) \(u=ax+\frac{y}{a}+b,\ a\neq 0\)

59

Partial Differential Equations

General solution of \((D^2-4DD'+4D'^2)u=0\), where \(D\equiv\frac{\partial}{\partial x}, D'\equiv\frac{\partial}{\partial y}\) is given by

1. A\(u=f(y+2x)+xg(y+2x)\)
2. B\(u=f(y+2x)+yg(y+2x)\)
3. C\(u=f(y-x)+xg(y-x)\)
4. D\(u=f(2y+x)+xg(2y+x)\)

Hint

\((D-2D')^2u=0\) has a repeated root \(m=2\), so the solution is \(f(y+mx)+xg(y+mx)\).

(D − 2D′)²u = 0 has repeated root m = 2.

Answer  (A) \(u=f(y+2x)+xg(y+2x)\)

60

Ordinary Differential Equations

A solution of the initial value problem \(ty'+2y=4t^2\) with initial condition \(y(1)=2\) is:

1. A\(y=t^2-\frac{1}{t^2},\ t>0\)
2. B\(y=t^2+\frac{1}{t^2},\ t>0\)
3. C\(y=2t^2+\frac{1}{t^2},\ t>0\)
4. DNone of these

Hint

Multiply by \(t\): \((t^2y)'=4t^3\).

(t²y)′ = 4t³ ⇒ y = t² + C/t², y(1)=2 ⇒ C = 1.

Answer  (B) \(y=t^2+\frac{1}{t^2},\ t>0\)

61

Ordinary Differential Equations

A particular solution of the differential equation \(y''-3y'-4y=3e^{2t}\) is:

1. A\(-2e^{2t}\)
2. B\(\frac{1}{2}e^{2t}\)
3. C\(-\frac{1}{2}e^{2t}\)
4. D\(3e^{2t}\)

Hint

Try \(y_p=Ae^{2t}\) and substitute.

Answer  (C) \(-\frac{1}{2}e^{2t}\)

62

Partial Differential Equations

Which of the following represents the Wave equation?

1. A\(u_{xx}+u_{tt}=0\)
2. B\(u_{xx}-u_{tt}=0\)
3. C\(u_{xx}+u_t=0\)
4. D\(u_{xx}-u_t=0\)

Hint

The wave equation is \(u_{tt}=c^2u_{xx}\).

Answer  (B) \(u_{xx}-u_{tt}=0\)

63

Field Theory

If \(\mathbb{Q}(\sqrt{2})\) is considered as an extension of \(\mathbb{Q}\), then the degree of \([\mathbb{Q}(\sqrt{2}):\mathbb{Q}]\) is:

1. A1
2. B2
3. C3
4. D4

Hint

The minimal polynomial of \(\sqrt2\) over \(\mathbb{Q}\) is \(x^2-2\).

Answer  (B) 2

64

Numerical Analysis

What is the order of convergence of the Secant method?

1. A1.618
2. B2.312
3. C1
4. D2

Hint

The order of the secant method is the golden ratio \(\frac{1+\sqrt5}{2}\).

Answer  (A) 1.618

65

Numerical Analysis

What is the convergence rate of bisection method?

1. AQuadratic
2. BExponential
3. CCubic
4. DLinear

Hint

Bisection halves the error at each step.

Answer  (D) Linear

66

Numerical Analysis

What is the primary purpose of trapezoidal rule?

1. AMaximizing computational efficiency
2. BEstimating the derivative of a function
3. CApproximating definite integrals
4. DSolving linear system of equations

Hint

The trapezoidal rule is a numerical integration technique.

Answer  (C) Approximating definite integrals

67

Numerical Analysis

Given the tabulated values, x: 1, 2, 3 and f(x): 2, 5, 10, using Lagrange's interpolation formula, the value of \(f(1.5)\) is:

1. A3.75
2. B3.25
3. C4.25
4. D4.75

Hint

Check whether a simple polynomial passes through all three points.

the data fit f(x) = x² + 1, so f(1.5) = 3.25.

Answer  (B) 3.25

68

Group Theory

The symmetric group \(S_3\) is:

1. AAbelian
2. BInfinite
3. CNon-Abelian
4. DCyclic

Hint

Does \((12)(13)=(13)(12)\) in \(S_3\)?

S₃ is non-abelian (hence not cyclic).

Answer  (C) Non-Abelian

69

Numerical Methods

The first Picard approximation for \(\dfrac{dy}{dx}=x+y,\ y(0)=1\) is

1. A\(y_1=1+x+\frac{x^2}{2}\)
2. B\(y_1=1+x\)
3. C\(y_1=1+x+x^2\)
4. DNone of these

Hint

\(y_1=y_0+\int_0^x f(t,y_0)\,dt\) with \(y_0=1\).

Answer  (A) \(y_1=1+x+\frac{x^2}{2}\)

70

Partial Differential Equations

The complete integral of the PDE \(z=px+qy+pq\) is given by

1. A\(z=ax+by-ab\)
2. B\(z=ax+by+ab\)
3. C\(z=ax+by\)
4. DNone of these

Hint

This is Clairaut's form \(z=px+qy+f(p,q)\). Replace \(p\to a\) and \(q\to b\).

Answer  (B) \(z=ax+by+ab\)

71

Calculus of Variations

The Euler–Lagrange equation for the functional \(J[y]=\int_a^b F(x,y,y')\,dx\) is given by

1. A\(F_y+\frac{d}{dx}F_{y'}=0\)
2. B\(F_y-\frac{d}{dx}F_{y'}=0\)
3. C\(F_x-\frac{d}{dx}F_y=0\)
4. D\(F_{y'}-\frac{d}{dx}F_y=0\)

Hint

Recall the Euler–Lagrange equation from the calculus of variations.

Answer  (B) \(F_y-\frac{d}{dx}F_{y'}=0\)

72

Calculus of Variations

The extremal of the functional \(J[y]=\int_0^1 (y')^2\,dx\) subject to \(y(0)=0,\ y(1)=1\), is:

1. A\(y=x^2\)
2. B\(y=1-x\)
3. C\(y=x\)
4. D\(y=x-1\)

Hint

\(F=y'^2\) gives \(y''=0\). Apply the boundary conditions.

y″ = 0 with the boundary conditions gives y = x.

Answer  (C) \(y=x\)

73

Numerical Methods

For the differential equation \(\dfrac{dy}{dx}=x+y,\ y(0)=1\), using Euler's method with \(h=0.1\), the approximate value of \(y(0.1)\) is:

1. A1.11
2. B1.09
3. C1.12
4. D1.10

Hint

\(y_1=y_0+hf(x_0,y_0)\).

y₁ = 1 + 0.1(0 + 1) = 1.10.

Answer  (D) 1.10

74

Integral Equations

Which of the following is a Fredholm integral equation of the second kind?

1. A\(f(x)=\int_a^b K(x,t)y(t)\,dt\)
2. B\(y(x)=\int_0^x K(x,t)y(t)\,dt\)
3. C\(y(x)=f(x)+\lambda\int_a^b K(x,t)y(t)\,dt\)
4. DNone of these

Hint

Second kind: the unknown \(y\) appears both outside and inside the integral, with fixed limits.

Answer  (C) \(y(x)=f(x)+\lambda\int_a^b K(x,t)y(t)\,dt\)

75

Integral Equations

Consider the Fredholm integral equation \(y(x)=f(x)+\lambda\int_0^1 xt\,y(t)\,dt\). For \(\lambda=1\), the resolvent kernel \(R(x,t;1)\) is:

1. A\(R(x,t;1)=xt\)
2. B\(R(x,t;1)=\dfrac{xt}{1-\frac{1}{9}}\)
3. C\(R(x,t;1)=\dfrac{xt}{1-\frac{1}{3}}\)
4. DNone of these

Hint

For \(K=xt\), \(R=\frac{xt}{1-\lambda\int_0^1 t^2\,dt}\).

R = xt/(1 − λ∫₀¹t²dt) = xt/(1 − 1/3).

Answer  (C) \(R(x,t;1)=\dfrac{xt}{1-\frac{1}{3}}\)

76

Integral Equations

Which of the following kernels is non-separable?

1. A\(K(x,t)=x+t\)
2. B\(K(x,t)=x^2t+xt^2\)
3. C\(K(x,t)=\sin x\cos t\)
4. D\(K(x,t)=e^{xt}\)

Hint

A separable (degenerate) kernel is a finite sum \(\sum a_i(x)b_i(t)\).

x + t is a finite sum of products (separable); e^{xt} is not.

Answer  (D) \(K(x,t)=e^{xt}\)

77

Integral Equations

Which of the following is a Volterra integral equation of the first kind?

1. A\(f(x)=\int_0^x K(x,t)y(t)\,dt\)
2. B\(y(x)=f(x)+\int_0^1 K(x,t)y(t)\,dt\)
3. C\(y(x)=f(x)+\lambda\int_0^x K(x,t)y(t)\,dt\)
4. D\(f(x)=y(x)+\int_0^1 K(x,t)y(t)\,dt\)

Hint

First kind: \(y\) appears only inside the integral. Volterra means a variable upper limit \(x\).

Answer  (A) \(f(x)=\int_0^x K(x,t)y(t)\,dt\)

78

Classical Mechanics

For a particle of mass m, the Lagrangian is given by \(L=\frac{1}{2}m\dot{x}^2-V(x)\). The corresponding Hamiltonian is:

1. A\(H=\frac{1}{2}m\dot{x}^2-V(x)\)
2. B\(H=m\dot{x}^2+V(x)\)
3. C\(H=\frac{1}{2}m\dot{x}^2+V(x)\)
4. D\(H=\frac{1}{2}m\dot{x}^2-2V(x)\)

Hint

\(H=p\dot{x}-L\) with \(p=m\dot{x}\).

Answer  (C) \(H=\frac{1}{2}m\dot{x}^2+V(x)\)

79

Classical Mechanics

For a mechanical system with Lagrangian \(L(q,\dot{q},t)\), the action is defined as: \(S=\int_{t_1}^{t_2}L\,dt\). If the action is stationary for the actual path, which equation follows?

1. A\(\frac{\partial L}{\partial q}=0\)
2. B\(\frac{d}{dt}\left(\frac{\partial L}{\partial\dot{q}}\right)-\frac{\partial L}{\partial q}=0\)
3. C\(\frac{\partial L}{\partial\dot{q}}=0\)
4. D\(\frac{dL}{dt}=0\)

Hint

Stationary action gives Lagrange's equations.

Answer  (B) \(\frac{d}{dt}\left(\frac{\partial L}{\partial\dot{q}}\right)-\frac{\partial L}{\partial q}=0\)

80

Ordinary Differential Equations

The integrating factor of the differential equation \((x\log x)\dfrac{dy}{dx}+y=2\log x\) is:

1. A\(\log x\)
2. B\(x\log x\)
3. C\(x\)
4. D\((\log x)^2\)

Hint

Divide by \(x\log x\). The IF is \(e^{\int P\,dx}\) with \(P=\frac{1}{x\log x}\).

Answer  (A) \(\log x\)

81

Combinatorics

A class has 37 students. Each student was born in one of the 12 months of the year. By the Pigeonhole Principle, at least how many students must have been born in the same month?

1. A1
2. B2
3. C3
4. D4

Hint

The generalized pigeonhole principle gives \(\left\lceil\frac{37}{12}\right\rceil\).

Answer  (D) 4

82

Calculus of Variations

The functional \(J[y]=\int_0^1[(y')^2+2y]\,dx\) is subjected to the boundary conditions \(y(0)=0,\ y(1)=0\). The Euler–Lagrange equation for the extremal is:

1. A\(y''=-1\)
2. B\(y''=1\)
3. C\(y''=2\)
4. D\(y''=-2\)

Hint

\(F=y'^2+2y\), so \(F_y=2\) and \(F_{y'}=2y'\).

F_y − (d/dx)F_{y′} = 2 − 2y″ = 0 ⇒ y″ = 1.

Answer  (B) \(y''=1\)

83

Statistics

10 babies are born in a hospital on the same day. All the babies weigh 2.8 kg each. The standard deviation of their weights is:

1. A2.8 kg
2. B0 kg
3. C1.4 kg
4. D0.28 kg

Hint

If there is no variation in the data, what is the SD?

identical values have zero spread.

Answer  (B) 0 kg

84

Statistics

The following table represents the distribution of marks obtained by students: Marks (x): 10, 20, 30, 40, 50; Frequency (f): 2, 3, 4, 3, 2. The mean deviation about the arithmetic mean is:

1. A10
2. B9
3. C8
4. D12

Hint

First find the mean. Then MD \(=\frac{\sum f|x-\bar{x}|}{\sum f}\).

Answer  (A) 10

85

Statistics

If V is the variance and M is the mean of the first 15 natural numbers, then \(V+M^2=\)

1. A\(\frac{224}{3}\)
2. B\(\frac{224}{8}\)
3. C\(\frac{240}{3}\)
4. D\(\frac{248}{3}\)

Hint

\(V+M^2=\frac{\sum x^2}{n}\), the mean of the squares.

Answer  (D) \(\frac{248}{3}\)

86

Classical Mechanics

For a Hamiltonian function \(H(q,p,t)\), which of the following represents Hamilton's canonical equations?

1. A\(\frac{dq}{dt}=\frac{\partial H}{\partial q},\ \frac{dp}{dt}=\frac{\partial H}{\partial p}\)
2. B\(\frac{dq}{dt}=-\frac{\partial H}{\partial p},\ \frac{dp}{dt}=-\frac{\partial H}{\partial q}\)
3. C\(\frac{dq}{dt}=\frac{\partial H}{\partial p},\ \frac{dp}{dt}=-\frac{\partial H}{\partial q}\)
4. D\(\frac{dq}{dt}=-\frac{\partial H}{\partial q},\ \frac{dp}{dt}=\frac{\partial H}{\partial p}\)

Hint

\(\dot{q}=\frac{\partial H}{\partial p}\) and \(\dot{p}=-\frac{\partial H}{\partial q}\).

Answer  (C) \(\frac{dq}{dt}=\frac{\partial H}{\partial p},\ \frac{dp}{dt}=-\frac{\partial H}{\partial q}\)

87

Probability

Let A and B be two independent events such that \(P(A)=\frac{2}{5},\ P(B)=\frac{3}{4}\). Then the probability of at least one of the events A or B occurring is:

1. A\(\frac{3}{5}\)
2. B\(\frac{7}{10}\)
3. C\(\frac{17}{20}\)
4. D\(\frac{7}{20}\)

Hint

\(P(A\cup B)=1-P(A')P(B')\) for independent events.

Answer  (C) \(\frac{17}{20}\)

88

Probability

A factory has two machines A and B producing bulbs. Machine A produces 60% of the bulbs, while machine B produces 40%. The probability that a bulb produced by A is defective is 2%, whereas for B it is 5%. If a randomly selected bulb is found to be defective, what is the probability that it was produced by machine B?

1. A\(\frac{1}{2}\)
2. B\(\frac{5}{8}\)
3. C\(\frac{1}{8}\)
4. D\(\frac{5}{32}\)

Hint

Use Bayes' theorem: \(P(B|D)=\frac{P(B)P(D|B)}{P(A)P(D|A)+P(B)P(D|B)}\).

Answer  (B) \(\frac{5}{8}\)

89

Probability

A machine produces a defective item with probability 0.1. If 6 items are selected independently, what is the probability that exactly one item is defective?

1. A0.3543
2. B0.3281
3. C0.5314
4. D0.5905

Hint

Binomial: \(^6C_1(0.1)(0.9)^5\).

6 × 0.1 × 0.9⁵ = 0.3543.

Answer  (A) 0.3543

90

Probability

A box contains 5 red balls, 4 blue balls, and 3 green balls. Two balls are drawn at random without replacement. What is the probability that both balls are of the same colour?

1. A\(\frac{19}{66}\)
2. B\(\frac{17}{66}\)
3. C\(\frac{21}{66}\)
4. DNone of these

Hint

\(\frac{{}^5C_2+{}^4C_2+{}^3C_2}{{}^{12}C_2}\).

Answer  (A) \(\frac{19}{66}\)

91

Probability

A random variable X has the following probability distribution: x: 0, 1, 2, 3, 4, 5, 6, 7, 8; p(x): a, 3a, 5a, 7a, 9a, 11a, 13a, 15a, 17a. Determine the value of a:

1. A\(\frac{1}{41}\)
2. B\(\frac{1}{51}\)
3. C\(\frac{2}{81}\)
4. D\(\frac{1}{81}\)

Hint

The probabilities sum to 1: \(a(1+3+\cdots+17)=1\).

Answer  (D) \(\frac{1}{81}\)

92

Probability

A continuous random variable X has the probability density function \(f(x)=A+Bx\), \(0\le x\le 1\), and \(f(x)=0\), otherwise. If the mean of X is \(\frac{7}{12}\), then the value of A and B are:

1. A\(A=1,\ B=0\)
2. B\(A=\frac{1}{2},\ B=1\)
3. C\(A=2,\ B=-2\)
4. D\(A=2,\ B=0\)

Hint

Use \(\int_0^1 f=1\) and \(\int_0^1 xf=\frac{7}{12}\).

Answer  (B) \(A=\frac{1}{2},\ B=1\)

93

Estimation

Let \(T_n\) be an estimator of \(\theta\). If the expected value \(E(T_n)=\theta\), then \(T_n\) is ______ estimator of \(\theta\).

1. ASufficient
2. BConsistent
3. CUnbiased
4. DEfficient

Hint

Recall the definition of bias.

Answer  (C) Unbiased

94

Estimation

For two unbiased estimators \(T_1\) and \(T_2\), if the variance V satisfies \(V(T_1)>V(T_2)\) then:

1. A\(T_1\) is more efficient than \(T_2\).
2. B\(T_2\) is more efficient than \(T_1\).
3. CBoth are equally efficient.
4. DNone of the estimators is efficient.

Hint

Among unbiased estimators, the one with smaller variance is more efficient.

smaller variance = more efficient.

Answer  (B) \(T_2\) is more efficient than \(T_1\).

95

Probability

A card is drawn at random from a standard deck of 52 playing cards. Given that the card drawn is a face card, what is the probability that it is a king?

1. A\(\frac{1}{4}\)
2. B\(\frac{1}{3}\)
3. C\(\frac{3}{13}\)
4. D\(\frac{1}{13}\)

Hint

There are 12 face cards, and 4 of them are kings.

Answer  (B) \(\frac{1}{3}\)

96

Markov Chains

Consider the following transition matrix of a Markov chain: \(P=\begin{pmatrix}0.5&0.5&0\\0.2&0.3&0.5\\0&0&1\end{pmatrix}\). Which of the following statements is correct?

1. AState 3 is recurrent and absorbing.
2. BState 3 is transient.
3. CAll three states are recurrent.
4. DStates 1 and 2 are absorbing.

Hint

A state with \(p_{ii}=1\) is absorbing.

p₃₃ = 1, so state 3 is absorbing (hence recurrent).

Answer  (A) State 3 is recurrent and absorbing.

97

Markov Chains

Consider a discrete-time Markov chain on the state space {1,2,3} with one-step transition probability matrix \(P=\begin{pmatrix}0.7&0.3&0\\0&0.6&0.4\\0&0&1\end{pmatrix}\). Which of the following statements is true?

1. AStates 1 and 3 are recurrent, and state 2 is transient.
2. BStates 1, 2, and 3 are recurrent.
3. CState 3 is recurrent, and states 1 and 2 are transient.
4. DNone of these

Hint

Can the chain return to state 1 after leaving it? To state 2?

once the chain leaves state 1 it can never return.

Answer  (C) State 3 is recurrent, and states 1 and 2 are transient.

98

Probability

According to Chebyshev's inequality, for any random variable with finite mean and variance, at least what percentage of the observations lies within two standard deviations of the mean?

1. A50%
2. B25%
3. C75%
4. D90%

Hint

Chebyshev: \(P(|X-\mu|<k\sigma)\ge1-\frac{1}{k^2}\).

1 − 1/2² = 75%.

Answer  (C) 75%

99

Statistics

The first three moments of a distribution about the mean are 0, 4, and 0, respectively. The distribution is:

1. ASymmetrical
2. BPositively skewed
3. CNegatively skewed
4. DNone of these

Hint

Skewness depends on the third central moment \(\mu_3\).

μ₃ = 0 ⇒ symmetrical.

Answer  (A) Symmetrical

100

Statistics

If the mean of a distribution is 20, the median is 16, and the standard deviation is 6, then the coefficient of skewness is:

1. A\(\frac{1}{2}\)
2. B2
3. C1
4. D0

Hint

Karl Pearson: \(S_k=\frac{3(\text{Mean}-\text{Median})}{\sigma}\).

Karl Pearson's Sk = 3(Mean − Median)/σ = 3(4)/6 = 2.

Answer  (B) 2

101

Statistics

For a distribution, the second central moment is 4 and the fourth central moment is 48. The coefficient of kurtosis \(\beta_2\) is:

1. A2
2. B4
3. C12
4. D3

Hint

\(\beta_2=\frac{\mu_4}{\mu_2^2}\).

Answer  (D) 3

102

Statistics

In a one-way ANOVA, there are 4 groups with a total of 20 observations. If the degrees of freedom for the between-group variation is 3, what is the degrees of freedom for the within-group variation?

1. A18
2. B17
3. C16
4. D15

Hint

Within-group df \(=N-k\).

Answer  (C) 16

103

Statistics

The first and second moments about arbitrary constant are −2 and 13 respectively. The standard deviation will be:

1. A−2
2. B2
3. C1
4. D3

Hint

\(\sigma^2=\mu_2'-(\mu_1')^2\).

σ² = 13 − (−2)² = 9 ⇒ σ = 3.

Answer  (D) 3

104

Sampling & Inference

A random sample of 400 observations is taken from a population. The sample mean is 52, while the hypothesized population mean is 50. The population standard deviation is known to be 10. The value of the Z-test statistic is:

1. A4
2. B2
3. C3
4. D1

Hint

\(Z=\frac{\bar{x}-\mu}{\sigma/\sqrt{n}}\).

Z = (52 − 50)/(10/√400) = 4.

Answer  (A) 4

105

Sampling & Inference

Which statistic is used in ANOVA to compare variance between groups?

1. At-statistic
2. BZ-score
3. C\(\chi^2\)-statistic
4. DF-statistic

Hint

ANOVA compares the between-group and within-group mean squares.

Answer  (D) F-statistic

106

Sampling & Inference

The sampling procedure in which a population is first divided into homogeneous groups, and then a sample is drawn from each group, is called:

1. AStratified sampling
2. BCluster sampling
3. CSystematic sampling
4. DNone of these

Hint

Homogeneous sub-groups are called strata.

Answer  (A) Stratified sampling

107

Sampling & Inference

Suppose a finite population contains 4 items and 2 items are selected at random with replacement, then all possible samples will be:

1. A8
2. B2
3. C4
4. D16

Hint

With replacement, the number of samples is \(N^n\).

with replacement, Nⁿ = 4² = 16.

Answer  (D) 16

108

Sampling & Inference

Regression analysis is mainly used to

1. ACompare two means
2. BPredict value of a dependent variable from independent variables
3. CTest normality of data
4. DNone of these

Hint

Regression models the relationship \(y=f(x)\).

Answer  (B) Predict value of a dependent variable from independent variables

109

Linear Programming

The minimum value of z for the following linear programming problem: Minimize \(Z=-3x+2y\), subject to \(0\le x\le 4\), \(1\le y\le 6\), \(x+y\le 5\), is

1. A0
2. B2
3. C10
4. D−10

Hint

Evaluate \(Z\) at the corner points of the feasible region.

Answer  (D) −10

110

Linear Programming

A variable used to convert a \(\le\) inequality into an equation is called:

1. ASurplus variable
2. BSlack variable
3. CArtificial variable
4. DBasic variable

Hint

A \(\le\) constraint needs a non-negative variable added.

Answer  (B) Slack variable

111

Linear Programming

In the Big-M method of linear programming, where M is a very large positive number, the cost assigned to which variables is M (or −M, depending on the objective function)?

1. AArtificial variables
2. BBasic variables
3. CSlack variables
4. DSurplus variables

Hint

The Big-M penalty drives the artificial variables out of the basis.

Answer  (A) Artificial variables

112

Linear Programming

Which of the following statements is correct?

1. AEvery linear programming problem has at least one optimal solution.
2. BEvery linear programming problem has a unique solution.
3. CIf a linear programming problem has two distinct optimal solutions, then it has infinitely many optimal solutions.
4. DIf the feasible region is unbounded, then the linear programming problem has no solution.

Hint

The optimal set is convex: any combination of two optimal points is also optimal.

Answer  (C) If a linear programming problem has two distinct optimal solutions, then it has infinitely many optimal solutions.

113

Sets & Real Analysis

Which of the following is not true?

1. AThe Cantor set is countable.
2. B\(\{(n,n): n\in\mathbb{N}\}\) is countable.
3. C\(\{x\in\mathbb{R}: 1\le x\le 2\}\) is uncountable.
4. DThe set of all integers is countable.

Hint

The Cantor set has the same cardinality as \([0,1]\).

Answer  (A) The Cantor set is countable.

114

Sets & Real Analysis

Let A and B be two sets, then \((A\cup B)'\cup(A'\cap B)=\)

1. A\(A'\)
2. B\(A\)
3. C\(B'\)
4. DNone of the options

Hint

\((A\cup B)'=A'\cap B'\). Then factor out \(A'\).

Answer  (A) \(A'\)

115

Sets & Real Analysis

Let \(X=\left\{\dfrac{n^2+1}{\left|n+\frac{1}{2}\right|}: n\in\mathbb{Z}\right\}\). Then the infimum and supremum of X are

1. A1, 2
2. B4/3, does not exist
3. C4/3, 2
4. DNone of these

Hint

Compute the terms for \(n=-2,-1,0,1,2,\ldots\) and see how they grow.

minimum 4/3 at n = 1; the set is unbounded above.

Answer  (B) 4/3, does not exist

116

Linear Algebra

Let \(T:V_2(\mathbb{R})\to V_2(\mathbb{R})\) be the reflection of points through the line \(y=-x\). Then the matrix of T w.r.t. standard basis is

1. A\(\begin{bmatrix}1&0\\0&-1\end{bmatrix}\)
2. B\(\begin{bmatrix}0&-1\\-1&0\end{bmatrix}\)
3. C\(\begin{bmatrix}-1&0\\0&1\end{bmatrix}\)
4. D\(\begin{bmatrix}0&1\\1&1\end{bmatrix}\)

Hint

Reflection in \(y=-x\) maps \((x,y)\to(-y,-x)\). Find the images of \(e_1\) and \(e_2\).

Answer  (B) \(\begin{bmatrix}0&-1\\-1&0\end{bmatrix}\)

117

3D Geometry

The image of a point having position vector \(\hat{i}+3\hat{j}+4\hat{k}\) in the plane \(\vec{r}\cdot(2\hat{i}-\hat{j}+\hat{k})+3=0\) is

1. A\(-\hat{i}+2\hat{j}+\hat{k}\)
2. B\(4\hat{i}-\hat{j}+2\hat{k}\)
3. C\(-3\hat{i}+5\hat{j}+2\hat{k}\)
4. D\(2\hat{i}+\hat{j}-2\hat{k}\)

Hint

Image \(=P-\frac{2(\vec{p}\cdot\vec{n}+d)}{|\vec{n}|^2}\,\vec{n}\).

image = (1,3,4) − 2·(6/6)(2,−1,1) = (−3,5,2).

Answer  (C) \(-3\hat{i}+5\hat{j}+2\hat{k}\)

118

Linear Algebra

Let \(A=\begin{bmatrix}2&0&0&0\\1&2&0&0\\0&0&2&0\\0&0&0&3\end{bmatrix}\), then geometric multiplicity of 2 is

1. A0
2. B1
3. C2
4. D3

Hint

GM \(=n-\text{rank}(A-2I)\).

rank(A − 2I) = 2 ⇒ nullity = 4 − 2 = 2.

Answer  (C) 2

119

Functional Analysis

A normed space N is a Banach space if N is

1. AInfinite dimensional
2. BEmpty
3. CCompact
4. DFinite dimensional

Hint

Every finite-dimensional normed space is complete.

Answer  (D) Finite dimensional

120

Functional Analysis

If a metric space X satisfies the Bolzano–Weierstrass property, then

1. AX is not compact.
2. BX is not sequentially compact.
3. CEvery infinite sequence \(\{x_n\}\) in X has no cluster point.
4. DEvery infinite sequence \(\{x_n\}\) in X has at least one cluster point.

Hint

The Bolzano–Weierstrass property means every sequence has a limit point.

Answer  (D) Every infinite sequence \(\{x_n\}\) in X has at least one cluster point.

121

Matrices

If \(A=\begin{bmatrix}\alpha&2\\2&\alpha\end{bmatrix}\) and \(\det A^3=125\), then \(\alpha\) equals

1. A±2
2. B±1
3. C±3
4. D±6

Hint

\(\det(A^3)=(\det A)^3=(\alpha^2-4)^3\).

Answer  (C) ±3

122

Matrices

The number of non-empty proper subsets of \(X=\{1,2,3,4,5\}\) is

1. A16
2. B31
3. C32
4. D30

Hint

Total subsets \(=2^5\). Remove \(\emptyset\) and the set itself.

2⁵ − 2 = 30 (exclude ∅ and X itself).

Answer  (D) 30

123

Matrices

The quadratic form corresponding to the matrix \(\begin{bmatrix}3&1&0\\2&2&1\\0&-1&2\end{bmatrix}\) is

1. APositive definite
2. BNegative definite
3. CPositive semidefinite
4. DNegative semidefinite

Hint

Use the symmetric part \(\frac{A+A^T}{2}\) and check its leading principal minors.

Symmetric part has leading minors 3, 3.75, 7.5 — all positive.

Answer  (A) Positive definite

124

Relations

Let \(A=\{0,1,2,3,4,5\}\). Let R be a relation on A defined by \((x,y)\in R\) if and only if \(\max\{x,y\}\in\{3,4\}\). Then among the following statements — S1: The number of elements in R is 18. S2: The relation R is symmetric but neither transitive nor reflexive.

1. ABoth are false.
2. BOnly S1 is true.
3. COnly S2 is true.
4. DBoth are true.

Hint

Count the pairs with max = 3 (\(4^2-3^2\)) and max = 4 (\(5^2-4^2\)).

|R| = 7 + 9 = 16, so S1 is false; S2 is true.

Answer  (C) Only S2 is true.

125

Metric Spaces

Let \((\mathbb{Q},d)\) be the metric space with \(d(x,y)=|x-y|\). Let \(E=\{p\in\mathbb{Q}: 2<p^2<3\}\). Then the set E is

1. AClosed but not compact
2. BNot closed but compact
3. CCompact
4. DNeither closed nor compact

Hint

The boundary points \(\pm\sqrt2,\pm\sqrt3\) are not in \(\mathbb{Q}\). Is \(E\) complete?

Answer  (A) Closed but not compact

126

Sequences

Consider the two arithmetic progressions 3, 7, 11, …, 407 and 2, 9, 16, …, 709. The number of common terms of these two progressions is

1. A0
2. B7
3. C15
4. D14

Hint

Common terms form an AP with difference lcm(4,7) = 28. Find the first common term.

common terms 23, 51, …, 387 (step 28) ⇒ 14 terms.

Answer  (D) 14

127

Inverse Trigonometry

\(\cos\left(\sin^{-1}\frac{3}{5}+\sin^{-1}\frac{5}{13}+\sin^{-1}\frac{33}{65}\right)=\)

1. A1
2. B0
3. C\(\frac{33}{65}\)
4. D\(\frac{32}{65}\)

Hint

First combine \(\sin^{-1}\frac35+\sin^{-1}\frac{5}{13}\) and find its cosine.

the first two angles sum to cos⁻¹(33/65), which with sin⁻¹(33/65) makes π/2.

Answer  (B) 0

128

Linear Algebra

Let \(S:\mathbb{R}^3\to\mathbb{R}^4\), \(T:\mathbb{R}^4\to\mathbb{R}^3\) be linear transformations. Which of the following is true?

1. Arank \(S\circ T\ge 1\)
2. Brank \(S\circ T=\) rank \(T\circ S\)
3. Crank \(S\circ T\le\min\)(rank S, rank T)
4. Drank \(S\circ T=0\) implies rank S = 0 or rank T = 0

Hint

The image of \(S\circ T\) lies in the image of \(S\), and its rank is limited by rank \(T\).

Answer  (C) rank \(S\circ T\le\min\)(rank S, rank T)

129

Functional Analysis

If \(T:H\to H\) is a bounded linear operator on a Hilbert space H, T is said to be normal if

1. A\(T^*=T\)
2. B\(T^*=T^{-1}\)
3. C\(TT^*=T^*T\)
4. DNone of these

Hint

A normal operator commutes with its adjoint.

T* = T⁻¹ defines unitary.

Answer  (C) \(TT^*=T^*T\)

130

Linear Algebra

Let \(T:\mathbb{R}^3\to\mathbb{R}^2\) be a linear map such that \(T(1,0,0)=(1,0)\), \(T(0,1,0)=(0,1)\) and \(T(0,1,1)=(1,1)\). Then \(T(1,1,1)=\)?

1. A\((2,1)\)
2. B\((1,2)\)
3. C\((-1,2)\)
4. D\((2,2)\)

Hint

Find \(T(0,0,1)=T(0,1,1)-T(0,1,0)\) using linearity.

T(0,0,1) = (1,1) − (0,1) = (1,0), so T(1,1,1) = (2,1).

Answer  (A) \((2,1)\)

131

Linear Algebra

Let V be the real vector space of all polynomials in \(\mathbb{R}[x]\) of degree less than or equal to 6. Consider \(T:V\to V\) defined by \((Tp)(x)=p(2x+1)\) for all \(p\in V\). What is trace of T?

1. A127
2. B63
3. C15
4. DNone

Hint

The matrix of \(T\) on the basis \(1,x,\ldots,x^6\) is triangular with diagonal entries \(2^k\).

diagonal entries 2⁰, 2¹, …, 2⁶ ⇒ trace 127.

Answer  (A) 127

132

Linear Algebra

Consider the space \(X=\left\{A\in M_2(\mathbb{R}) \,\middle|\, AA^t=\begin{pmatrix}2&-2\\-2&3\end{pmatrix}\right\}\). Then number of connected components of X =

1. A0
2. B1
3. C2
4. D4

Hint

If \(AA^t=M\) is positive definite, then \(A=M^{1/2}O\) with \(O\) orthogonal. How many components does \(O(2)\) have?

X = {S·O : O ∈ O(2)} ≅ O(2), which has 2 components.

Answer  (C) 2

133

Linear Algebra

Let \(A\in M_n(\mathbb{C})\) be a normal matrix. Consider the following statements: I. If all eigen values of A are real, then A is Hermitian matrix. II. If all eigen values of A have absolute value 1, then A is unitary matrix. Which of the following is correct?

1. ABoth I and II are true.
2. BI is false and II is true.
3. CI is true and II is false.
4. DBoth I and II are false.

Hint

A normal matrix is unitarily diagonalizable.

Answer  (A) Both I and II are true.

134

Linear Algebra

Let \(A\in M_n(\mathbb{C})\). Then \(\begin{pmatrix}A&A\\0&A\end{pmatrix}\) is diagonalizable if and only if

1. A\(A=0\)
2. B\(n=2\)
3. C\(A=1\)
4. DNone of these

Hint

Diagonalize \(A\) first. The block \(\begin{pmatrix}\lambda&\lambda\\0&\lambda\end{pmatrix}\) is diagonalizable only when \(\lambda=0\).

Answer  (A) \(A=0\)

135

Conic Sections

Eccentricity of hyperbola conjugate to the hyperbola \(\frac{x^2}{4}-\frac{y^2}{12}=1\) is

1. A2
2. B\(\sqrt{3}\)
3. C4
4. D\(\frac{2}{\sqrt{3}}\)

Hint

The conjugate hyperbola is \(\frac{y^2}{12}-\frac{x^2}{4}=1\). Use \(e^2=1+\frac{a^2}{b^2}\).

2 is the eccentricity of the given hyperbola; for the conjugate, e² = 1 + 4/12 ⇒ e = 2/√3.

Answer  (D) \(\frac{2}{\sqrt{3}}\)

136

Area Under Curves

The area bounded by y-axis, \(y=\sin x\), \(y=\cos x\), \(0\le x\le\frac{\pi}{2}\) is

1. A\(\sqrt{2}-1\) sq. units
2. B\(1-\sqrt{2}\) sq. units
3. C\(\sqrt{2(\sqrt{2}-1)}\) sq. units
4. D\(2(\sqrt{2}-1)\) sq. units

Hint

The curves meet at \(x=\frac{\pi}{4}\). Area \(=\int_0^{\pi/4}(\cos x-\sin x)\,dx\).

∫₀^{π/4}(cos x − sin x)dx = √2 − 1.

Answer  (A) \(\sqrt{2}-1\) sq. units

137

Inequalities

If \(\frac{1}{3}\le\dfrac{x^2-2x+4}{x^2+2x+4}<3\), then \(\dfrac{9\cdot 3^{2x}+6\cdot 3^{x}+4}{9\cdot 3^{2x}-6\cdot 3^{x}+4}\) lies between

1. A\(\frac{1}{3},\ 3\)
2. B\(\frac{1}{4},\ 5\)
3. C\(-1,\ 1\)
4. D\(1,\ -2\)

Hint

Put \(u=3^{x+1}\). The expression becomes \(\frac{u^2+2u+4}{u^2-2u+4}\), the reciprocal of the given form.

The original paper prints the numerator as 9·3ˣ; it should be 9·3²ˣ (fixed above). With u = 3ˣ⁺¹ the expression is (u²+2u+4)/(u²−2u+4), the reciprocal of the given form, so it lies between 1/3 and 3.

Answer  (A) \(\frac{1}{3},\ 3\)

138

Vectors

If \(2\hat{i}-\hat{k}\), \(\hat{j}+2\hat{k}\) are diagonals of a parallelogram, then area of the parallelogram is

1. A\(\sqrt{21}\)
2. B\(\frac{21}{2}\)
3. C\(\frac{17}{2}\)
4. D\(\frac{19}{2}\)

Hint

Area of a parallelogram from its diagonals \(=\frac12|\vec{d_1}\times\vec{d_2}|\).

d₁ × d₂ = (2,0,−1) × (0,1,2) = (1,−4,2), so |d₁ × d₂| = √21. The area of a parallelogram from its diagonals is ½|d₁ × d₂| = √21/2. Option (A) √21 would be right only if the vectors were adjacent sides.

Answer  None of the given options. The correct area is \(\frac{\sqrt{21}}{2}\).

139

Application of Derivatives

The points on curve \(9y^2=x^3\) where normal to the curve make equal intercepts with axes are

1. A\(\left(4,\pm\frac{8}{3}\right)\)
2. B\(\left(4,-\frac{8}{3}\right)\)
3. C\(\left(4,\pm\frac{3}{8}\right)\)
4. D\(\left(\pm 4,\frac{3}{8}\right)\)

Hint

The normal slope is \(-\frac{6y}{x^2}\). Equal intercepts means slope \(=-1\).

Answer  (A) \(\left(4,\pm\frac{8}{3}\right)\)

140

Vectors & 3D Geometry

The sine of angle between straight lines \(\frac{x-1}{3}=\frac{y-7}{4}=\frac{z-3}{5}\) and the plane \(2x-2y+z=10\) is

1. A\(\frac{4}{5\sqrt{2}}\)
2. B\(\frac{2}{5\sqrt{3}}\)
3. C\(\frac{7\sqrt{2}}{10}\)
4. D\(\frac{\sqrt{5}}{6}\)

Hint

\(\sin\theta=\frac{|\vec{b}\cdot\vec{n}|}{|\vec{b}||\vec{n}|}\).

Direction of the line b = (3,4,5), normal to the plane n = (2,−2,1). sin θ = |b·n| / (|b||n|) = |6 − 8 + 5| / (5√2 · 3) = 1/(5√2) = √2/10.

Answer  None of the given options. The correct value is \(\frac{\sqrt{2}}{10}\).

141

Vectors & 3D Geometry

The vectors \(\lambda\hat{i}+\hat{k}\), \(\hat{i}+\hat{j}+2\hat{k}\), \(2\hat{i}-\hat{j}+\lambda\hat{k}\) are coplanar if

1. A\(\lambda=1\)
2. B\(\lambda=0\)
3. C\(\lambda=-\frac{1}{2}\)
4. D\(\lambda=\frac{2}{3}\)

Hint

Coplanar means the scalar triple product (determinant) is 0.

Answer  (A) \(\lambda=1\)

142

Vectors & 3D Geometry

The two vectors \(\hat{i}+\hat{j}+\hat{k}\) and \(3\hat{i}-8\hat{k}\) represent two sides AB and AC respectively of triangle ABC. The length of median through A is

1. A\(\frac{1}{2}\sqrt{66}\)
2. B\(\frac{1}{2}\sqrt{34}\)
3. C\(\sqrt{56}\)
4. D7

Hint

Median \(\vec{AD}=\frac{\vec{AB}+\vec{AC}}{2}\).

Answer  (A) \(\frac{1}{2}\sqrt{66}\)

143

Sequences of Functions

Let \(f_n:[0,1]\to\mathbb{R}\) be given by \(f_n(x)=\dfrac{2x^2}{x^2+(1-2nx)^2}\), \(n=1,2,\ldots\) Then the sequence \(f_n\)

1. Adoes not converge pointwise on [0,1]
2. Bconverges uniformly on [0,1]
3. Cdoes not converge uniformly on [0,1] but has a subsequence that converges uniformly on [0,1]
4. Dconverges pointwise on [0,1] but does not have a subsequence that converges uniformly on [0,1]

Hint

Evaluate \(f_n\) at \(x=\frac{1}{2n}\).

fₙ → 0 pointwise, but fₙ(1/(2n)) = 2 for every n, so no subsequence converges uniformly.

Answer  (D) converges pointwise on [0,1] but does not have a subsequence that converges uniformly on [0,1]

144

Differentiation

If \(\sqrt{y+x}+\sqrt{y-x}=a\), then \(\dfrac{d^2y}{dx^2}=\)

1. A\(\frac{-2}{a}\)
2. B\(\frac{-a^2}{2}\)
3. C\(\frac{2}{a^2}\)
4. DNone of these

Hint

Square twice to get \(y\) as an explicit function of \(x\).

squaring gives y = x²/a² + a²/4, so y″ = 2/a².

Answer  (C) \(\frac{2}{a^2}\)

145

Binomial Theorem

The sum of series \(^{20}C_0-{}^{20}C_1+{}^{20}C_2-{}^{20}C_3+\cdots+{}^{20}C_{10}\) is

1. A\(\frac{1}{2}\,{}^{20}C_{10}\)
2. B\(^{20}C_{10}\)
3. C\(^{20}C_{11}\)
4. D\(\frac{1}{2}\,{}^{20}C_{10}-{}^{20}C_1\)

Hint

\(\sum_{k=0}^{m}(-1)^k\,{}^nC_k=(-1)^m\,{}^{n-1}C_m\).

the partial alternating sum = ¹⁹C₁₀ = ½·²⁰C₁₀.

Answer  (A) \(\frac{1}{2}\,{}^{20}C_{10}\)

146

Series

\(\dfrac{1\times 2^2+2\times 3^2+\cdots+n\times(n+1)^2}{1^2\times 2+2^2\times 3+\cdots+n^2\times(n+1)}=\)?

1. A\(\frac{5n+23}{7n+7}\)
2. B\(\frac{3n+5}{3n+1}\)
3. C\(\frac{n+7}{n+3}\)
4. D\(\frac{2n+5}{2n}\)

Hint

Expand to \(\sum(k^3+2k^2+k)\) and \(\sum(k^3+k^2)\), then factor.

Answer  (B) \(\frac{3n+5}{3n+1}\)

147

Series

If \(t_n\) denotes \(n^{th}\) term of the series \(2+3+6+11+\cdots\), then \(t_{48}\) is:

1. A2209
2. B\(1+48^2\)
3. C2300
4. D\(2+47^2\)

Hint

The differences 1, 3, 5, … are odd numbers, so \(t_n=2+(n-1)^2\).

tₙ = 2 + (n − 1)², so t₄₈ = 2 + 47².

Answer  (D) \(2+47^2\)

148

Limits

Let \(f(x)=\dfrac{x+|x|(1+x)}{x}\sin\frac{1}{x}\), \(x\neq 0\). If \(L=\lim_{x\to 0^-}f(x)\) and \(R=\lim_{x\to 0^+}f(x)\), then which of the following is true?

1. AL exists, but R does not exist.
2. BL does not exist, but R exists.
3. CBoth L and R exist.
4. DNeither L nor R exists.

Hint

Simplify separately for \(x<0\) (\(|x|=-x\)) and for \(x>0\).

for x<0, f = −x sin(1/x) → 0; for x>0, f = (2+x) sin(1/x) oscillates.

Answer  (A) L exists, but R does not exist.

149

Maxima & Minima

Let \(f(x)=|x^2+2x-3|\) for all \(x\in\mathbb{R}\). The total number of points of \(\mathbb{R}\) at which f attains a local extremum is

1. A2
2. B4
3. C1
4. D3

Hint

Sketch \(|(x+3)(x-1)|\): it has two zeros and a reflected vertex.

local minima at x = −3 and x = 1, local maximum at x = −1 ⇒ 3 points.

Answer  (D) 3

150

Definite Integrals

\(\displaystyle\int_0^{\pi/4}\frac{\sin x\cos x}{\cos^4 x+\sin^4 x}\,dx=\)

1. A\(\frac{\pi}{2}\)
2. B\(\frac{\pi}{4}\)
3. C\(\frac{\pi}{8}\)
4. D\(\frac{\pi}{12}\)

Hint

Divide by \(\cos^4x\) and put \(t=\tan^2x\).

Answer  (C) \(\frac{\pi}{8}\)