PPSC Assistant Professor Mathematics 2021 Question Paper with Answers

PPSC Assistant Professor Mathematics 2021 Question Paper with Answers | 20 November Exam

The Punjab Public Service Commission (PPSC) conducted the Assistant Professor Mathematics Exam on 20 November 2021. This post features the complete set of original questions from that examination in a clean, digital format. Aspirants preparing for upcoming PPSC Assistant Professor Mathematics exams can use this paper to understand the actual question pattern, marking style, and difficulty level. The exam covered essential topics such as Linear Algebra, Real Analysis, Differential Equations, Complex Analysis, and Abstract Algebra — all fundamental areas of advanced mathematics.

Solving PPSC previous year papers is an effective strategy to improve conceptual clarity and speed during competitive exams. These authentic questions from the PPSC 2021 Mathematics Question Paper will help candidates revise key concepts and evaluate their preparation. This resource is ideal for students aiming for assistant professor or lecturer positions in Mathematics under the Punjab government.

Disclaimer: The questions are sourced from the official PPSC Question Paper (20 November 2021). While every effort has been made to ensure accuracy, Learn4Math is not responsible for any inadvertent errors or omissions. All rights to the original examination content belong to the Punjab Public Service Commission (PPSC).

PPSC Assistant Professor Mathematics – Practice Set 1

Q1.

If \(u = \cos nx\), \(v = \sin nx\), then Wronskian of \(u\) and \(v\) is:

1. 0
2. 1
3. \(n\)
4. \(n^2\)

✅ Correct Answer: C.

Q2.

If the function \(f(x,y) = x^2 + y^2\) is defined over the rectangle \(|x| \le a, |y| \le b\), then the Lipschitz constant for \(f\) will be:

1. \(a-b\)
2. \(a+b\)
3. \(2b\)
4. \(a^2 + b^2\)

✅ Correct Answer: C.

Q3.

The roots of the auxiliary equation \[x^2y_2 + xy_1 - 4y = 0\] are:

1. 1, -2
2. 2, -2
3. -2, 4
4. 1, 4

✅ Correct Answer: B.

Q4.

The PDE \((1+x)u_{xx} - (x+2)u_{xy} + (x+3)u_{yy} \)
\(= \cos(x - 2y)\) is:

1. Elliptic
2. Parabolic
3. Hyperbolic
4. None of these

✅ Correct Answer: C.

Q5.

The singular solution of \(4p^2 = 9x\) is:

1. \(x=0\)
2. \(x=y\)
3. \(y=1\)
4. Does not exist

✅ Correct Answer: D.

Q6.

Solution of \(\frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} = \sin x\) is:

1. \(\phi(y-x) + \phi_2(y+x) = k\)
2. \(\phi(x,y) = \sin x\)
3. \(\phi(y-x) - \cos x\)
4. \(p+q=k\)

✅ Correct Answer: C.

Q7.

If \(z = ax + by + ab\), then the corresponding PDE will be:

1. \(z = px + qy\)
2. \(z = qy\)
3. \(z = px + qy + pq\)
4. None of these

✅ Correct Answer: C.

Q8.

The solution of PDE \(p^3 - q^3 = 0\) is:

1. \(z = x + y\)
2. \(z = ax + ay + c\)
3. \(z = x^2 - y^2 + c\)
4. None of these

✅ Correct Answer: B.

Q9.

If \(\Delta, E\) denote the forward operator and shift operator respectively, then \((\frac{\Delta^2}{E})x^3\) is:

1. \(3x\)
2. \(6x\)
3. 0
4. \(x^6\)

✅ Correct Answer: B.

Q10.

If ∇ denotes the backward difference operator and E the shift operator, then ∇E =:

1. \(1 - E\)
2. \(1 - E^{-1}\)
3. \(1 + E + E^2\)
4. \(1 + E^{-1}\)

✅ Correct Answer: B.

Q11.

If T be the kinetic energy and V be the potential energy of a system, then Lagrangian of the system is:

1. \(2TV\)
2. \(2V + T\)
3. \(T + V\)
4. \(T - V\)

✅ Correct Answer: D.

Q12.

The shortest curve joining two fixed points is a:

1. Parabola
2. Straight line
3. Circle
4. None of these

✅ Correct Answer: B.

Q13.

If \(A\) be the action of a dynamical system such that \(A = \int_{t_1}^{t_2} mv^2 dt\), the \(\delta A\) is equal to:

1. 0
2. \(T - 2\)
3. \(T + V\)
4. None of these

✅ Correct Answer: A.

Q14.

The equation \(\int_0^b K(x,t)f(t)dt = \phi(x)\) is called:

1. Volterra equation of first kind
2. Fredholm equation of first kind
3. Fredholm equation of second kind
4. Volterra equation of second kind

✅ Correct Answer: B.

Q15.

The solution of the integral equation \(1 + \int_0^x xy\phi(y)dy, \; \phi_{0}(y)= 1\) is:

1. \(\sin x\)
2. \(\sinh x\)
3. \(\cos x\)
4. \(\cosh x\)

✅ Correct Answer: D.

Q16.

Sum of Lagrangian functions \(\sum_{i=1}^{n} L_i(x)\) is:

1. 0
2. 1
3. \(n+1\)
4. \(n-1\)

✅ Correct Answer: B.

Q17.

Kinetic energy is a quadratic function of:

1. generalised coordinates
2. velocities
3. forces
4. None of these

✅ Correct Answer: B.

Q18.

The mean and variance of first \( n \) natural numbers are respectively:

1. \(\frac{n+1}{2}\) and \(\frac{n^2 - 1}{12}\)
2. \(\frac{n^2-1}{12}\) and \(\frac{n+1}{2}\)
3. \(\frac{n-1}{2}\) and \(\frac{n^2 + 1}{12}\)
4. \(\frac{n^2+1}{12}\) and \(\frac{n - 1}{2}\)

✅ Correct Answer: A.

Q19.

If 10 is the mean of a set of 7 observations and 5 is the mean of another set of 3 observations, then the mean of these two sets, taking together, is:

1. 15
2. 10
3. 8.5
4. 7.5

✅ Correct Answer: C.

Q20.

If the events S and T have equal probabilities and are independent with \( P(S \cap T) = p > 0 \), then \( P(S) \) equals:

1. \(\sqrt{p}\)
2. \(p^2\)
3. \(p\)
4. none of these

✅ Correct Answer: A.

Q21.

In case of tossing an ordinary die, the set of events {1,2,3,4,5,6} is:

1. exhaustive
2. mutually exclusive
3. both (a) and (b)
4. neither (a) nor (b)

✅ Correct Answer: C.

Q22.

Die A has four red and two white faces whereas die B has two red and four white faces. A single coin is flipped once. If it falls head, the game starts with throwing of die A, and if it falls tail, die B is thrown. The probability of getting a red face at any throw of any die is:

1. \(\frac{1}{2}\)
2. \(\frac{1}{3}\)
3. \(\frac{1}{4}\)
4. none of these

✅ Correct Answer: A.

Q23.

The joint probability density function of \( X \) and \( Y \) is given by:

\[ f(x, y) = \begin{cases} c(2x + y), & 0 < x < 1,\\ &\; 0 < y < 2, \\ 0, & \text{otherwise.} \end{cases} \]

Then the value of constant \( c \) is:

1. \(\frac{1}{3}\)
2. \(\frac{1}{4}\)
3. \(\frac{1}{5}\)
4. none of these

✅ Correct Answer: B.

Q24.

The probability of random variables \(X\) and \(Y\) is given by:

\(P(X = x, Y = y)\)\[ = \begin{cases} \dfrac{x + y}{30}, & x = 0, 1, 2, 3; \\ &\; y = 0, 1, 2, \\[6pt] 0, & \text{otherwise.} \end{cases} \]

1. \(\frac{2}{5}\)
2. \(\frac{1}{6}\)
3. \(\frac{4}{15}\)
4. none of these

✅ Correct Answer: C.

Q25.

A sequence of random variables \( X_1, X_2, ..., X_n, ... \) is said to converge in probability to a constant \( A \) if for any \( \varepsilon > 0 \), we have:

1. \(\lim_{n \to \infty} P(|X_n - A| < \varepsilon) = 0\)
2. \(\lim_{n \to \infty} P(|X_n - A| < \varepsilon) = 1\)
3. \(\lim_{n \to \infty} P(X_n - A < \varepsilon) = 0\)
4. \(\lim_{n \to \infty} P(X_n - A < \varepsilon) = 1\)

✅ Correct Answer: B.

Q26.

If \( X \) is a random variable with mean \( \mu \) and variance \( \sigma^2 \), then for any positive number \( k \), the Chebyshev’s inequality is given by:

1. \(P(|X - \mu| \ge k\sigma) \le \frac{1}{k^2}\)
2. \(P(|X - \mu| \ge k\sigma) \ge \frac{1}{k^2}\)
3. \(P(|X - \mu| \le k\sigma) \le \frac{1}{k^2}\)
4. none of these

✅ Correct Answer: A.

Q27.

A sequence \( (X_n) \) is said to be a Markov Chain if for all \( i_0, i_1, i_2, ..., i_{n+1} \in I \) and \( \forall n \):

1. \(P(X_{n+1} = i_{n+1} | X_0 = i_0, X_1 = i_1, \) \(..., X_n = i_n = P(X_{n+1} = i_{n+1} | X_n = i_n)\)
2. \(P(X_{n+1} = i_{n+1} | X_0 = i_0, X_1 = i_1, \) \(..., X_n = i_n)= P(X_{n+1} = i_{n+1})\)
3. \(P(X_{n+1} = i_{n+1} | X_0 = i_0, X_1 = i_1,\) \(..., X_n = i_n)= P(X_n = i_n)\)
4. none of these

✅ Correct Answer: A.

Q28.

The coefficient of dispersion of Poisson distribution with mean 4 is:

1. \(\frac{1}{4}\)
2. \(\frac{1}{2}\)
3. 4
4. 2

✅ Correct Answer: B.

Q29.

The mean and variance of chi–square distribution with n degrees of freedom are respectively:

1. \(2n\) and \(n\)
2. \(n^2\) and \(\sqrt{n}\)
3. \(\sqrt{n}\) and \(n^2\)
4. \(n\) and \(2n\)

✅ Correct Answer: D.

Q30.

For a normal distribution, the area to the right-hand side of the point \(x_1\) is 0.6 and to the left-hand side of the point \(x_2\) is 0.7, then we have:

1. \(x_1 > x_2\)
2. \(x_1 < x_2\)
3. \(x_1 = x_2\)
4. none of these

✅ Correct Answer: B.

Q31.

Let \(T_n\) be an estimator, based on a sample \(x_1, x_2, ..., x_n\), of the parameter \(\theta\). Then \(T_n\) is consistent estimator of \(\theta\) if:

1. \(P(|T_n - \theta| > \varepsilon) = 0\) ∀ ε > 0
2. \(P(|T_n - \theta| < \varepsilon) = 0\) ∀ ε > 0
3. \(\lim_{n→∞} P(|T_n - \theta| > \varepsilon) = 0\) ∀ ε > 0
4. \(\lim_{n→∞} P(|T_n - \theta| < \varepsilon) = 0\) ∀ ε > 0

✅ Correct Answer: C.

Q32.

The Neyman–Pearson lemma provides the best critical region for testing [1] null hypothesis against [2] alternative hypothesis. Here:

1. [1] simple, [2] simple
2. [1] simple, [2] composite
3. [1] composite, [2] simple
4. none of these

✅ Correct Answer: A.

Q33.

For the validity of F-test in the analysis of variance, the following assumption is/are made:

1. The observations are independent
2. The parent population from which observations are taken is normal
3. The various treatment and environment effects are additive in nature
4. all of these

✅ Correct Answer: D.

Q34.

If \(X_{p\times1} \sim N_p(\mu, \Sigma)\), then \(AX\) follows, where A is a matrix of rank \(q \le p\):

1. \(N_p(A\mu, A\Sigma A')\)
2. \(N_q(A\mu, A\Sigma A')\)
3. \(N_p(A\mu, A'\Sigma A)\)
4. \(N_q(A\mu, A'\Sigma A)\)

✅ Correct Answer: B.

Q35.

In the case of simple random sampling without replacement, the probability of two specified units in the population of size N to be included in the sample of size n is:

1. \(\frac{n(n-1)}{N(N-1)}\)
2. \(\frac{n}{N}\)
3. \(\frac{(n-1)}{(N-1)}\)
4. none of these

✅ Correct Answer: A.

Q36.

In a Latin Square Design with m treatments, the degrees of freedom of error sum of squares for a fixed effect model is:

1. \((m-1)(m-2)\)
2. \(m-1\)
3. \(m-2\)
4. none of these

✅ Correct Answer: A.

Q37.

In a \(2^3\) factorial design of experiment, the number of treatments are:

1. 7
2. 6
3. 5
4. 8

✅ Correct Answer: D.

Q38.

For a balanced incomplete block design with parameters \(v, b, r, k, \lambda\), we have:

1. \(v r = b k\)
2. \(\lambda (v - 1) = r (k - 1)\)
3. \(b \ge r\)
4. all of these

✅ Correct Answer: D.

Q39.

If \(p_i\) denote the reliability of the i-th component; \(i = 1,2,...,n\), then the reliability of parallel system is given by:

1. \(\prod_{i=1}^n p_i\)
2. \(1 - \prod_{i=1}^n (1 - p_i)\)
3. \(\prod_{i=1}^n (1 - p_i)\)
4. \(1 - \prod_{i=1}^n p_i\)

✅ Correct Answer: B.

Q40.

Let S be a convex subset of the plane, bounded by lines in the plane. Then a linear function \(z = c_1x_1 + c_2x_2\) \(\; \forall x_1, x_2 \in S\;\) where \(c_1, c_2\) are scalers, attains its optimum value at:

1. The origin only
2. any points
3. the vertices only
4. none of these

✅ Correct Answer: C.

Q41.

Given a set of vectors \(\{x_1, x_2, ..., x_k\}\), a linear combination \(x = \lambda_1 x_1 + \lambda_2 x_2 + ... + \lambda_k x_k\) is called a convex combination of the given vectors if:

1. \(\lambda_1, \lambda_2, ..., \lambda_k \ge 0\) and \(\sum_{i=1}^k \lambda_i = 1\)
2. \(\lambda_1, \lambda_2, ..., \lambda_k \ge 0\) and \(\sum_{i=1}^k \lambda_i \ne 1\)
3. \(\forall \lambda_i^s\) and \(\sum_{i=1}^k \lambda_i = 1\)
4. none of these

✅ Correct Answer: A.

Q42.

A queuing system M/G/1 has:

1. a single channel
2. an exponential inter-arrival time distribution
3. arbitrary service time distribution
4. all of these

✅ Correct Answer: D.

Q43.

Which one of the following statements is not true?

1. The set of natural numbers is ordered complete
2. The set of integers is ordered complete
3. The set of rational numbers is ordered complete
4. The set of real numbers is ordered complete

✅ Correct Answer: C.

Q44.

If \(a_n = (-1)^n (1 + \frac{1}{n})\), \(n\) is a natural number, then:

1. \(\lim \inf a_n = 0\)
2. \(\lim \inf a_n = 1\)
3. \(\lim \inf a_n = -1\)
4. \(\lim \sup a_n = 0\)

✅ Correct Answer: C.

Q45.

On the real line \( R \) (with usual metric), the set \( Q \) of rational numbers is:

1. open
2. closed
3. open as well as closed
4. neither open nor closed

✅ Correct Answer: D.

Q46.

Which one of the following functions is not analytic?

1. \(e^z\)
2. \(\cos z\)
3. \(\sinh z\)
4. \(|z|^2\)

✅ Correct Answer: D.

Q47.

The function \( f \) defined by \( f(x) = \begin{cases} -x, & x \text{ is rational} \\ x, & x \text{ is irrational} \end{cases} \) is:

1. continuous for every real \( x \)
2. continuous when \( x \) is rational
3. continuous when \( x \) is irrational
4. continuous at \( x = 0 \) only

✅ Correct Answer: D.

Q48.

The improper integral \(\int_0^{\pi/2} \frac{\sin x}{x^p} dx\) converges for:

1. \(p > 2\)
2. \(p \le 2\)
3. \(p < 2\)
4. \(0 < p < 4\)

✅ Correct Answer: C.

Q49.

The function \( f: [0,1] \to R \) defined by \( f(x) = \begin{cases} x^{\alpha} \sin \frac{1}{x^{\beta}}, & 0 < x \le 1 \\ 0, & x = 0 \end{cases} \) is of bounded variation if:

1. \(\alpha > \beta\)
2. \(\alpha < \beta\)
3. \(\alpha \leq \beta\)
4. \(\alpha = \beta\)

✅ Correct Answer: A.

Q50.

Which one of the following statements is not true?

1. Any metric induced by a norm is always bounded
2. Every set of positive measure contains a non-measurable set
3. The function \(f(x,y) = |x| + |y|\) is continuous but not differentiable at the origin
4. In a normed space, interior of a closed ball is the corresponding open ball

✅ Correct Answer: A.

Q51.

If the outer measure of set A is zero, then the set A is always:

1. an empty set
2. a finite set
3. a countable set
4. a Lebesgue measurable set

✅ Correct Answer: D.

Q52.

If \(f\) is a Riemann integrable function on [a,b], then:

1. \(f\) is differentiable in [a,b]
2. \(f\) is continuous on [a,b]
3. \(f\) is monotonic on [a,b]
4. \(f\) is continuous almost everywhere on [a,b]

✅ Correct Answer: D.

Q53.

The set \(A = \{1/n : n \text{ is a natural number}\}\) is:

1. compact
2. closed but not bounded
3. bounded but not closed
4. neither bounded nor closed

✅ Correct Answer: C.

Q54.

The open interval \((a,b)\) on the real line R is:

1. compact
2. connected
3. compact and connected
4. neither compact nor connected

✅ Correct Answer: B.

Q55.

The sequence \(\frac{nx}{1+n^3x^2}\) converges uniformly to zero:

1. only for \(0 < x < 1\)
2. only for \(0 \le x < 1\)
3. for \(0 < x \le 1\)
4. for \(0 \le x \le 1\)

✅ Correct Answer: D.

Q56.

The set of algebraic numbers is:

1. finite
2. countable
3. uncountable
4. neither finite nor countable

✅ Correct Answer: B.

Q57.

Choose the incorrect statement:

1. Every Riemann integrable function is Lebesgue integrable
2. Every non-empty set of real numbers which is bounded below has infimum
3. The series \(\frac{1}{1.2.3} + \frac{3}{2.3.4} + \frac{5}{3.4.5} + \dots\) converges
4. \(\frac{\sin x}{x} < 1\) for \(0 < x \le \frac{\pi}{2}\)

✅ Correct Answer: D.

Q58.

Which one of the following defines a norm in \(IR^2\) (the real plane)?

1. \(||x||_1 = |x_1|,\; x = (x_1,x_2)\)
2. \(||x||_2 = |x_2 - x_1|\)
3. \(||x||_3 = |x_1| + |x_2|\)
4. \(||x||_4 = \frac{x_1^2}{a^2} + \frac{x_2^2}{b^2}\)

✅ Correct Answer: C.

Q59.

Which one of the following is not true?

1. Every normed linear space is a metric space
2. Every inner product space is a normed linear space
3. Every linearly independent set on a finite dimensional inner product space can be converted into an orthonormal set
4. Every orthonormal basis in a Hilbert space is a basis for the Hilbert space

✅ Correct Answer: D.

Q60.

Which one of the following is true?

1. For each real number t, there is a natural number n such that n > t
2. Every infinite subset of real numbers has a limit point
3. Every closed subset of real numbers is compact
4. Monotonic functions have discontinuities of the second kind

✅ Correct Answer: A.

Q61.

\(\lim_{n \to \infty} \frac{(1 + \frac{1}{2} + \frac{1}{3} + ... + \frac{1}{n})}{\log n} =\)

1. 0
2. 1
3. e
4. ∞

✅ Correct Answer: B.

Q62.

The function \(f(x) = \frac{1}{x}\) is:

1. uniformly continuous on (0,1]
2. bounded on [0,1]
3. continuous but not uniformly continuous on (0,1)
4. neither continuous nor uniformly continuous on (0,1)

✅ Correct Answer: C.

Q63.

\(\lim_{n \to \infty} (1 - \frac{1}{n})^{-n}\) is equal to:

1. e
2. \(\frac{1}{e}\)
3. 1
4. -1

✅ Correct Answer: A.

Q64.

Identify the correct statement:

1. The union of two subspaces of a vector space is never a subspace
2. The dimension of the vector space R of all real numbers over the field Q of rational numbers is one
3. \(\int_0^1 x^{m-1} (1-x)^{n-1} dx\) exists if and only if \(m ≥ 0, n ≥ 0\)
4. Every continuous function on [a,b] is Lebesgue measurable on [a,b]

✅ Correct Answer: D.

Q65.

If every minor of order r of a matrix A is zero, then rank A is:

1. greater than r
2. less than r
3. equal to r
4. less than or equal to r

✅ Correct Answer: B.

Q66.

Which one of the following statements is false?

1. Any square matrix A and its transpose have same eigen values
2. Eigen values of an idempotent matrix are either zero or one
3. Every matrix satisfies its characteristic equation
4. If A is a skew symmetric matrix of odd order, then determinant A is zero

✅ Correct Answer: C.

Q67.

A real quadratic form \(x' A x\) is negative definite if:

1. rank A is less than number of variables of the quadratic form
2. all eigen values of A > 0
3. all eigen values of A ≤ 0
4. all eigen values of A < 0

✅ Correct Answer: D.

Q68.

If C is the contour \(|z| = 1\), the value of \(\oint_C (x^2 - y^2 + 2ixy) dz\) is:

1. 0
2. 1
3. 2
4. 8

✅ Correct Answer: A.

Q69.

The set \(S = \{0, 1, 2, 3\}\) with addition and multiplication modulo 4 is:

1. a field
2. a ring with zero divisors
3. a ring without zero divisors but not a field
4. a division ring

✅ Correct Answer: B.

Q70.

In the ring of integers, every ideal is:

1. prime
2. maximal
3. principal
4. prime as well as maximal

✅ Correct Answer: C.

Q71.

If \(u + iv = \log(x + iy)\), then \(u\) is equal to:

1. \(\tan(x/y)\)
2. \(\frac{1}{2} \log(x^2 + y^2)\)
3. \(x^2 \)
4. \(y^2\)

✅ Correct Answer: B.

Q72.

The Galois group of \(x^n - 1\) over the field of rational numbers is always:

1. Abelian
2. non-Abelian
3. cyclic
4. non-cyclic

✅ Correct Answer: A.

Q73.

Let \(T: \mathbb{R}^3 \to \mathbb{R}^2\) be given by \(T(a,b,c) = (a + b, b + c)\), then rank T is:

1. 0
2. 1
3. 2
4. 3

✅ Correct Answer: D.

Q74.

Which one of the following statements is true?

1. The radius of convergence of the power series \(\sum (3 + 4i)^n z^n\) is 1
2. If G is a group of order 72, then G always has one 3–Sylow subgroup
3. Every solvable group is nilpotent
4. The Riemann Zeta function \(\sum_{k=1}^{\infty} \frac{1}{k^Z}\) has a simple pole at \(z = 1\) with residue 1

✅ Correct Answer: D.

Q75.

If A and B are ideals of a ring R, then \(\frac{A + B}{A}\) is isomorphic to:

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

✅ Correct Answer: B.

Q76.

The invariant points of the transformation \(w = \frac{2z - 5}{z + 4}\) are:

1. \(5, -4\)
2. \(3, -4\)
3. \(-1 + 2i, -1 - 2i\)
4. \(1 + 2i, 3 + 4i\)

✅ Correct Answer: C.

Q77.

For the function \(\frac{\sin z}{z}\), the point \(z = 0\) is:

1. a pole
2. an isolated singularity
3. a removable singularity
4. an essential singularity

✅ Correct Answer: C.

Q78.

The harmonic conjugate of \(e^x \cos y\) is:

1. \(e^x \sin y\)
2. \(\sin y\)
3. \(\cos y\)
4. \(e^x \tan y\)

✅ Correct Answer: A.

Q79.

Every discrete topological space is:

1. compact
2. connected
3. first countable
4. second countable

✅ Correct Answer: C.

Q80.

Which one of the following statements is incorrect?

1. Every compact Hausdorff space is normal
2. Every metric space is Hausdorff
3. Every \(T_2\) space is regular
4. Every completely regular space is regular

✅ Correct Answer: C.

Q81.

Which one of the following mappings is linear?

1. \(T: \mathbb{R}^2 \to \mathbb{R}\) defined by \(T(x,y) = xy\)
2. \(T: \mathbb{R}^2 \to \mathbb{R}^3\) defined by \(T(x,y) = (x + 1, 2y, xy)\)
3. \(T: \mathbb{R}^2 \to \mathbb{R}^2\) defined by \(T(x,y) = (x + y, x)\)
4. \(T: \mathbb{R}^3 \to \mathbb{R}^2\) defined by \(T(x,y,z) = (|x|, yz)\)

✅ Correct Answer: C.

Q82.

The relation \(|3 - z| + |3 + z| = 5\) represents:

1. a circle
2. a parabola
3. an ellipse
4. a straight line

✅ Correct Answer: C.

Q83.

If two eigenvalues of \(\begin{bmatrix} 8 & -6 & 2 \\ -6 & 7 & -4 \\ 2 & -4 & 3 \end{bmatrix}\) are 3 and 15, then the third eigenvalue is:

1. 0
2. 12
3. 8
4. 7

✅ Correct Answer: A.

Q84.

The set of integers with operation \(a \oplus b = a + b + 1\) is given to be a group; the identity of the group is:

1. 1
2. 0
3. -1
4. 2

✅ Correct Answer: C.

Q85.

A subset A of a topological space X is open as well as closed if and only if:

1. Frontier A ≠ \(\phi\)
2. Frontier A = \(\phi\)
3. Frontier A = A
4. Frontier A ≠ A

✅ Correct Answer: B.

Q86.

The factoring of any integer n into primes is unique apart from the order of prime factors. This is:

1. Chinese remainder theorem
2. Cauchy theorem
3. Fundamental theorem of Arithmetic
4. Pigeon-hole principle

✅ Correct Answer: C.

Q87.

For the Euler φ function, φ(4) =

1. 1
2. 3
3. 4
4. 2

✅ Correct Answer: D.

Q88.

Every principal ideal domain is a:

1. field
2. Euclidean domain
3. division ring
4. unique factorization domain

✅ Correct Answer: D.

Q89.

The number of primitive roots of 13 are:

1. 1
2. 2
3. 4
4. 3

✅ Correct Answer: C.

Q90.

If G is a group such that order (G) = \(p^2\), where p is a prime number, then:

1. G is always cyclic
2. G is always non-cyclic
3. G is always Abelian
4. G is always non-Abelian

✅ Correct Answer: C.

Q91.

The order of the permutation \(\begin{pmatrix} 1 & 2 & 3 & 4 \\ 1 & 3 & 4 & 2 \end{pmatrix}\) is:

1. 0
2. 1
3. 2
4. 3

✅ Correct Answer: D.

Q92.

Choose the incorrect statement:

1. \(\oint_C \frac{dz}{z - a} = 2\pi i\), when C is the circle \(|z - a| = r\)
2. If f(z) is analytic and bounded in the whole complex plane, then f(z) is a constant
3. \(Q[\sqrt{2}] = \{a + b\sqrt{2} : a,b \in Q\}\) is a finite field extension of Q
4. Any polynomial of degree 1 over any integral domain R is reducible polynomial over R

✅ Correct Answer: D.

Q93.

When \(f(x, y, y')\) is independent of y, then Euler–Lagrange equation reduces to:

1. \(\frac{\partial f}{\partial y'} = \text{constant}\)
2. \(\frac{\partial f}{\partial y} = 0\)
3. \(y\frac{\partial f}{\partial y'} = 0\)
4. \(x\frac{\partial f}{\partial y} = k\)

✅ Correct Answer: A.

Q94.

The necessary condition for \( I = \int_{x_1}^{x_2} f(x, y, y', y'') dx \) to be an extremum is:

1. \( f_y - \frac{d}{dx}(f_{y'}) + \frac{d^2}{dx^2}(f_{y''}) = 0 \)
2. \( f_x - \frac{d}{dx}(f_{y'}) + \frac{d^2}{dx^2}(f_{y''}) = 0 \)
3. \(f_x + \frac{d}{dx}(f_{y'}) - \frac{d^2}{dx^2}(f_{y''}) = 0 \)
4. \( f_{xy} - \frac{d}{dx}(f_{y'}) - \frac{d^2}{dx^2}(f_{y''}) = 0 \)

✅ Correct Answer: A.

Q95.

The extremal of the functional \( \int_{x_1}^{x_2} (x + y') y' dx \) will be:

1. \( y = x^2 + c_1x \)
2. \( y = -x^3 + c_1x + c_2 \)
3. \( y = \frac{x^2}{4} + c_1x \)
4. \( y = \frac{-x^2}{4} + c_1x + c_2 \)

✅ Correct Answer: D.

Q96.

For a function \( f(x) \), first order backward difference is defined to be:

1. \( f(x + h) - f(x) \)
2. \( f(x - h) + f(x) \)
3. \( f(x) - f(x + h) \)
4. \( f(x) - f(x - h)\)

✅ Correct Answer: D.

Q97.

The rate of convergence of Newton–Raphson method is:

1. Linear
2. Quadratic
3. Bi-quadratic
4. Zero

✅ Correct Answer: B.

Q98.

For finding the root of the equation \( 2x = \cos x + 3 \), by iteration method, the necessary condition to be satisfied is:

1. \(|\sin x| > 3\)
2. \(|\frac{\sin x}{2}| < 1\)
3. \(|\frac{\cos x}{2}| < 1\)
4. \(|\cos x| < 3\)

✅ Correct Answer: B.

Q99.

For the function \( y = f(x) \), if \( y_1 = 4, y_3 = 12, y_4 = 19 \) and \( y_x = 7 \), then by Lagrange’s interpolation formula, \( x \) (approx.) is:

1. 1.86
2. 3.42
3. 0
4. 4.93

✅ Correct Answer: A.

Q100.

If \( y' = x^2 + y \) with \( y(0) = 0 \), then by Picard’s method, the first approximation \( y^{(1)}(x) \) will be:

1. \(2x\)
2. \(1\)
3. \(\frac{x^3}{3}\)
4. \(\frac{x^3}{3} + \frac{x^4}{12}\)

✅ Correct Answer: C.