HPSC Assistant Professor Mathematics Old Paper | 2019 Answer Key

HPSC Assistant Professor Mathematics 2019 Question Paper with Answer Key

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Q.1. A closed ball in normed space \(V\) is compact then the condition is:

1. only sufficient for \(V\) to be finite dimensional
2. only necessary for \(V\) to be finite dimensional
3. necessary and sufficient for \(V\) to be finite dimensional
4. none of the above

Correct Answer: (C)

Q.2. The Banach space \(\ell^p\) is separable:

1. only for \(p = 1\)
2. only for \(p = 2\)
3. for \(1 < p = \infty\)
4. only for \(p = \frac12\)

Correct Answer: (C)

Q.3. \(L^p [0,1]\) is Hilbert space:

1. only for \(p = 1\)
2. only for \(p = 2\)
3. only for \(p = 3\)
4. only for \(p = 4\)

Correct Answer: (B)

Q.4. \(T : X \to Y\) be bounded operator on normed space then what is true?

1. \(T\) is not continuous
2. \(T\) is not continuous at \(0\)
3. \(T\) is continuous at \(0\)
4. \(T\) does not map null sequence to bounded sequence

Correct Answer: (C)

Q.5. Let \(M\) be \(R\)-module. Then \(M\) is \(R/I\) module if:

1. \(I\) is any ideal
2. \(I\) is any subring
3. \(M\) is annihilated by \(I\)
4. None of the above

Correct Answer: (C)

Q.6. In a \(2^3\)-factorial experiment:

(1) Seven factorial effects are mutually orthogonal
(2) The factorial effects are orthogonal to the general mean

1. both (1) and (2) are true
2. only (1) is true
3. only (2) is true
4. (1) is true but (2) is not necessarily true

Correct Answer: (A)

Q.7. Suppose that \(u \sim N_p(\mu, \Sigma)\; \mu, \Sigma \text{ being unknown.}\) For testing the null hypothesis \(H_0 : \mu = \mu_0\) (specified) against \(H_0 : \mu \ne \mu_0\), the test statistic used is:

1. Student’s \(t\)
2. Hotelling \(T^2\)
3. Mahalanobis \(D^2\)
4. \(\chi^2\)

Correct Answer: (A)

Q.8. If \(E(T_n) \to \theta\) and \(V(T_n) \to 0\) as \(n \to \infty\) then \(T_n\) is:

1. Consistent
2. Sufficient
3. Efficient
4. None of the above

Correct Answer: (D)

Q.9. If \(T_1\) is UMVUE and \(T_2\) is unbiased estimator of \(\theta\), then for \(\theta\), \(T = aT_1 + bT_2\) is:

1. UMVUE
2. UMVUE under certain conditions
3. Not an UMVUE
4. Always unbiased

Correct Answer: (C)

Q.10. To obtain confidence interval for variance, use:

1. \(t\)-statistic
2. \(\chi^2\)-statistic
3. \(F\)-statistic
4. None of the above

Correct Answer: (B)

Q.11. Wronskian is a:

1. difference
2. integration
3. determinant
4. differentiation

Correct Answer: (C)

Q.12. Value of \(\Gamma(1)\) is equal to:

1. 1
2. 2
3. 0
4. \(\pi\)

Correct Answer: (A)

Q.13. Reduction formula for Gamma function is:

1. \(\Gamma(n+1) = (n-1)\Gamma(n-1)\)
2. \(\Gamma(n+1) = n\Gamma(n)\)
3. \(\Gamma(n+1) = (n-1)\Gamma(n)\)
4. None of the above

Correct Answer: (B)

Q.14. Value of Riemann Zeta function at \(-1\):

1. 1
2. \(-\frac{1}{12}\)
3. 0
4. 2

Correct Answer: (B)

Q.15. Value of Riemann Zeta function at \(1\):

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

Correct Answer: (D)

Q.16. A solution of the integral equation \(\int_0^x 3^{x-t}\phi(t)\,dt = x\) is:

1. \(\phi(x) = 3x^e\)
2. \(\phi(x) = 1 - 3x^e\)
3. \(\phi(x) = 1 - x\log 3\)
4. \(\phi(x) = x\log 3\)

Correct Answer: (C)

Q.17. The eigen value of the integral equation \(u(x) = \int_0^1 \sin \pi x \cos \pi t \, u(t)\, dt\) is:

1. \(\frac{1}{\pi}\)
2. eigen value does not exist
3. \(\frac{1}{2\pi}\)
4. \(-\frac{1}{\pi}\)

Correct Answer: (B)

Q.18. Hamiltonian equation for a simple pendulum in usual notation is:

1. \(L=\frac{1}{2}ml^2\dot{\theta}^2 + mgl(1 - \cos\theta)\)
2. \(L=\frac{1}{2}ml^2\dot{\theta}^2 - mgl(1 - \cos\theta)\)
3. \(L=\frac{1}{2}ml\dot{\theta} - mg(1 - \cos\theta)\)
4. \(L=\frac{1}{2}mg^2\dot{\theta}^2 - ml(1 - \cos\theta)\)

Correct Answer: (B)

Q.19. A rigid body moving freely in space has degree of freedom:

1. 3
2. 4
3. 6
4. 9

Correct Answer: (C)

Q.20. Equation of constraints that contain time as explicit variable are referred as:

1. Rheononomous constraints
2. Holonomic constraints
3. Non-holonomic constraints
4. Schemeonous constraints

Correct Answer: (A)

Q.21. What is meaning of concatenation?

1. separating a string from another string
2. adding two strings
3. adding two functions
4. none of them

Correct Answer: (B)

Q.22. Boolean expression gives results in form of:

1. strings
2. characters
3. integers
4. true and false

Correct Answer: (D)

Q.23. The differential equation \(2y' + x^2y = 2x + 3\) is:

1. linear
2. non-linear
3. linear with fixed constants
4. undetermined to be linear or non-linear

Correct Answer: (A)

Q.24. A differential equation is considered ordinary if it has:

1. one dependent variable
2. more than one dependent variable
3. one independent variable
4. more than one independent variable

Correct Answer: (C)

Q.25. A technique by which problems in analysis, in particular differential equations, are transformed into algebraic problems is called:

1. static calculus
2. dynamic calculus
3. operational calculus
4. integral

Correct Answer: (C)

Q.26. The order of the element \((2,2)\) in \(\mathbb{Z}_4 \times \mathbb{Z}_6\) is:

1. 2
2. 4
3. 6
4. 12

Correct Answer: (C)

Q.27. For \(X=\{a,b,c,d,e\}\) the topology is defined as \(\tau=\{\varnothing,X,\{a\},\{a,b\},\)
\( \{a,c,d\},\{a,b,c,d\},\{a,b,e\}\}\). The local base at \(b\) for this topology is:

1. \(\{\{a\}\}\)
2. \(\{\{a,b\}\}\)
3. \(\{\{a,c,d\}\}\)
4. \(\{\{d,e\}\}\)

Correct Answer: (B)

Q.28. Which one of the following statement is not true?

1. An infinite discrete space is compact
2. A closed subset of a compact space is compact
3. A closed and bounded subset (subspace) of \(\mathbb{R}\) is compact
4. Every finite topological space is compact

Correct Answer: (A)

Q.29. Which of the following subset is connected?

1. \(\{(x,y): |x^2-y^2|\ge 4\}\) of \(\mathbb{R}^2\)
2. \(A=\{a,d,e\}\) as a subset of a topological space \((X,\tau)\), \(X=\{a,b,c,d,e\}\), \(\tau=\{X,\varnothing,\{a,b,c\},\{c,d,e\},\{e\}\}\)
3. Real line \(\mathbb{R}\) with usual topology
4. \(A=\{x\in\mathbb{R}:|x|>2\}\) as a subset of a real line with usual topology

Correct Answer: (C)

Q.30. Let \(G\) be a non-abelian group then its order can be:

1. 25
2. 35
3. 125
4. 49

Correct Answer: (C)

Q.31. Using Euler’s method, first approximate value of \(y\) at \(x=0.2\) from initial value problem \(\dfrac{dy}{dx}=1-x+4y,\; y(0)=1\), taking the step size \(h=0.1\), is:

1. 1.5
2. 2
3. 2.1
4. 2.5

Correct Answer: (A)

Q.32. The curve that makes the functional \(J[y(x)] = 2\pi\displaystyle\int_{x_0}^{x} y\sqrt{1+(y')^2}\,dx\) extremum, is:

1. Ellipse
2. Catenary
3. Cycloid
4. Cardioid

Correct Answer: (C)

Q.33. The extremal of the functional \(\displaystyle\int_0^1\left[y + x^2 + \frac{(y')^2}{4}\right]dx,;\)
\( y(0)=0,\;y(1)=0,\) is:

1. \(y = 2(x-x^2)\)
2. \(y = x^2 - x\)
3. \(y = 3(x-x^2)\)
4. \(y = 4(x-x^2)\)

Correct Answer: (B)

Q.34. Which of the following differential equation is satisfied by the extremals for the variational problem \(J[y(x)] = \displaystyle\int_1^2 \big[y^2 + x^2 (y')^2\big] dx\) ?

1. \(-y + 2xy' = 0\)
2. \(x^2y'' + 2xy' - y = 0\)
3. \(x^2y'' - y = 0\)
4. \(x^2y'' - 2xy' + y = 0\)

Correct Answer: (B)

Q.35. The type of integral equation:
\(f(x) = \int_{0}^{x} \frac{1}{(x-t)^{\alpha}} u(t)\, dt,\; 0 < \alpha < 1 \)

1. Fredholm linear integral equation of first kind
2. Fredholm linear integral equation of second kind
3. Singular integral equation
4. Fredholm linear integral equation of third kind

Correct Answer: (C)

Q.36. The partial differential equation
\(U_{x}^4U_{xx} + U_{yy}(U_y)^2 = 0\), is:

1. Linear and of order \(4\)
2. Quasi linear and of order \(2\)
3. Quasi linear and of order \(4\)
4. Linear and of order \(2\)

Correct Answer: (B)

Q.37. The solution of the Cauchy problem:
\(U_x + U_y = 0,\; U(x,0) = e^x\)

1. \(U(x,y) = e^{\,x+y}\)
2. \(U(x,y) = e^{\,x + 2y}\)
3. \(U(x,y) = e^{\,x-y}\)
4. \(U(x,y) = \frac{1}{2}\left[e^{\,x+y} + e^{\,x-y}\right]\)

Correct Answer: (C)

Q.38. The order of convergence in Newton–Raphson method is:

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

Correct Answer: (C)

Q.39. In the Gauss elimination method for solving a system of linear algebraic equations, triangularization leads to:

1. diagonal matrix
2. lower triangular matrix
3. upper triangular matrix
4. singular matrix

Correct Answer: (A)

Q.40. If \(f(x)\) has an isolated zero of multiplicity 4 at \(x=0\) and the iteration
\(x_{n+1} = x_n - \dfrac{4 f(x_n)}{f'(x_n)} ,\; n=0,1,2,\ldots\)
converges to \(0\), then the rate of convergence is:

1. Linear
2. Cubic
3. Quadratic
4. Faster than one but slower than two

Correct Answer: (C)

Q.41. A person appears in an examination. There are only two possibilities, either he will pass or he will fail. What is the probability that he will not pass:

1. Nothing definite can be said
2. \(\frac{1}{2}\)
3. More than \(\frac{1}{2}\)
4. Less than \(\frac{1}{2}\)

Correct Answer: (A)

Q.42. Given \(P(A)=l\), \(P(B)=m\), then:

1. \(P(A \mid B) < \dfrac{l+m-1}{m}\)
2. \(P(A \mid B) > \dfrac{l+m-1}{m}\)
3. \(P(A \mid B) \geq \dfrac{l+m-1}{m}\)
4. \(P(A \mid B) = \dfrac{l+m-1}{m}\)

Correct Answer: (A)

Q.43. The central limit theorem states that:

1. The distribution for \(\dfrac{\overline{Y} - \mu_{Y}}{\sigma_{Y}}\) becomes arbitrarily well approximated by the standard normal distribution
2. \(\overline{Y}^P \rightarrow \mu_{Y}\)
3. The probability that \(Y\) is in the range \(\mu_Y \pm c\) becomes arbitrarily close to one as \(n\) increases for any constant \(c > 0\)
4. The \(t\) distribution converges to the \(F\) distribution for approximately \(n > 30\)

Correct Answer: (A)

Q.44. In a birth/death model of a queue:

1. Time between arrivals has poisson distribution
2. The distribution of the number of customers in the system has exponential distribution
3. The arrival rate is the same for all states
4. None of the above

Q.45. A coin is tossed five times in succession. What is the probability of getting at least four heads?

1. \(\dfrac{1}{4}\)
2. \(\dfrac{3}{4}\)
3. \(\dfrac{1}{16}\)
4. \(\dfrac{3}{16}\)

Correct Answer: (C)

Q.46. If \(f : M \to N\) be module homomorphism and Tor M denotes the Torsion submodule of M which is true:

1. f (Tor M) is subset of Tor N
2. f (Tor M) is superset of Tor N
3. f (Tor M) = Tor M
4. None of the above

Correct Answer: (A)

Q.47. Which of the following is free \(\mathbb{Z}\)-module:

1. \(\mathbb{Z}_m\)
2. \(\mathbb{Q}\)
3. \(\mathbb{Z} \times \mathbb{Z}\)
4. \(\mathbb{Q} \times \mathbb{Q}\)

Correct Answer: (C)

Q.48. If M is Noetherian R-module then which is true:

1. Every sub module of M is finitely generated
2. Some submodule is not Torsion module
3. If it is \(\mathbb{Z}\)-module
4. None of the above

Correct Answer: (A)

Q.49. An artificial and informal language that helps programmers to develop algorithms, is called:

1. Instruction code
2. Algocode
3. Pseudocode
4. Control code

Correct Answer: (C)

Q.50. Variables cannot be used before they are declared, so their scopes begin:

1. Outside the function
2. Where they are declared
3. At the main function
4. None of them

Correct Answer: (B)

Q.51. With standard notations, which one of the following is not true:

1. \(\text{Cov}(X+Y, Z) = \text{Cov}(X,Z)\)
   \( + \text{Cov}(Y,Z)\)
2. \(\text{Cov}( aX + b, cY + d ) = ac \text{Cov}(X, Y)\)
3. \(\text{Cov}(aX + bY, cX + dY) = ac\sigma_X^2 \)
   \(+ bd\sigma_Y^2 + (ad+bc)\sigma_{XY}\)
4. If \(\text{Cov}(X + Y) = 0\), then \(X\) and \(Y\) are independent

Correct Answer: (D)

Q.52. From OGIVE one can find:

1. Mean
2. Median
3. Mode
4. None of the above

Correct Answer: (B)

Q.53. The distribution of test statistic used in median test is:

1. Binomial
2. Normal
3. t-test
4. \(\chi^2\)

Correct Answer: (D)

Q.54. In a Latin square design, which one does not give a biased estimate of the population variance:

1. Row Sum of Squares
2. Column Sum of Squares
3. Treatment Sum of Squares
4. Error Sum of Squares

Q.55. Let \(X\) be distributed as \(\chi^2\) on \(5\) degrees of freedom. Then \(E(X^2)\) will be:

1. 20
2. 25
3. 30
4. 35

Correct Answer: (D)

Q.56. Consider a discrete-time Markov chain with transition probability matrix:

\[ \begin{pmatrix} 0.6 & 0.4 \\ 0.3 & 0.7 \end{pmatrix} \]

If the system is initially in state \#1, the probability that the system will be in state 2 after exactly one step is:

1. 0.40
2. 0.70
3. 0.60
4. 0.52

Correct Answer: (A)

Q.57. Consider the

\[ \begin{pmatrix} 0.6 & 0.4 \\ 0.3 & 0.7 \end{pmatrix} \]

Markov chain. If the chain was initially in state \#1, the probability that the system will still be in state 1 after 2 transitions is:

1. 0.36
2. 0.30
3. 0.48
4. 0.52

Correct Answer: (C)

Q.58. It is proposed to test \(H_0 : \theta = \theta_0\) against \(H_1 : \theta = \theta_1\), given a sample of size \(n\) from \(N(\mu,1)\). The critical region of the most powerful test depends on:

1. \(\theta_0\) only
2. \(\theta_1\) only
3. \(|\theta_0 - \theta_1|\) only
4. \(\theta_0\) and the sign \(\theta_0 - \theta_1\) only

Correct Answer: (C)

Q.59. Suppose that we have estimated the regression model,
\(y_i = \beta_1 + \beta_2x_{i2} + \cdots + \beta_k x_{ik} + e_i\).
Let \(\hat{y}_i\) be the fitted value of \(y\).
Now we estimate the artificial model:
\( y_i = \beta_1 + \beta_2 x_{i2} + \cdots + \beta_k x_{ik} +\)
\(\gamma_1 \hat{y}_i + \gamma_2 \hat{y}_i^2 + v_i \) to test \(H_0 : \gamma_1 = \gamma_2 = 0\) against \(H_1 : H_0\) is wrong.

Choose the correct statement:

1. \(H_1\) can be equivalently rewritten as \(H_1 : \gamma_1 \neq \gamma_2 \neq 0\)
2. An F-test cannot be appropriate for testing \(H_0\)
3. This test is called the Augmented Dicky-Fuller test
4. Rejection of \(H_0\) suggests that there can be omitted variables

Correct Answer: (D)

60). The test statistic \(T\) in Wilcoxon single sample test is approximately (if the sample size is larger than 25) distributed as:

(A) \( N\left( \frac{n(n+1)}{2}, \sqrt{\frac{n(n+1)(2n+1)}{12}} \right) \)
(B) \( N\left( \frac{n(n+1)}{4}, \sqrt{\frac{n(n+1)(2n+1)}{24}} \right) \)
(C) \( N\left( \frac{n(n+1)}{8}, \sqrt{\frac{n(n+1)(2n+1)}{48}} \right) \)
(D) \( N\left( \frac{n(n+1)}{6}, \sqrt{\frac{n(n+1)(2n+1)}{36}} \right) \)

Q.61. If \(f\) is continuous on \(\mathbb{R}\) and satisfies \(f(x+y)=f(x)+f(y)\) for all \(x,y\in\mathbb{R}\), then:

1. \(f(x)=x+f(1)\), \(\forall x \in \mathbb{R}\)
2. \(f(x)=xf(1)\), \(\forall x \in \mathbb{R}\)
3. \(f(x)+f(y)=x + y,\ \forall x,y\in\mathbb{R}\)
4. \(f(x)=\dfrac{x}{f(1)},\ \forall x \in \mathbb{R}\)

Correct Answer: (B)

62). Consider the improper integrals:
I. \(\displaystyle \int_{0}^{1} \frac{dx}{\sqrt{1-x}}\)
II. \(\displaystyle \int_{0}^{1} \frac{\log x}{\sqrt{x}}\,dx\)
III. \(\displaystyle \int_{0}^{\infty} x \sin x \, dx\)

Which one of the following statements is true for these integrals?
(A) All are convergent
(B) All are divergent
(C) I and II are convergent whereas III is divergent
(D) I and III are convergent whereas II is divergent

Correct Answer: (C)

Q.63. The function \(f\) be defined on the interval \([0,1]\) as:

\[ f(x) = \begin{cases} 1, & x \text{ is rational} \\ 0, & x \text{ is irrational} \end{cases} \]

The correct statement is:

1. f is Riemann integrable but not Lebesgue integrable
2. f is not Riemann integrable but it is Lebesgue integrable
3. f is neither Riemann integrable nor Lebesgue integrable
4. f is both Riemann integrable and Lebesgue integrable

Correct Answer: (B)

Q.64. For the function:

\[ f(x,y) = \begin{cases} \dfrac{x^2 y}{x^4 + y^2}, & (x,y)\neq(0,0) \\ 0, & (x,y)=(0,0) \end{cases} \]

The directional derivative along \(u=(\sqrt{3},\sqrt{3})\) at \((0,0)\) is:

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

Correct Answer: (C)

Q.65. Every compact subset \(A\) of a metric space \(X\) is:

1. Closed and bounded
2. Closed and unbounded
3. Open and bounded
4. Open and unbounded

Correct Answer: (D)

Q.66. If \(\overline{x}\) and \(\overline{y} \in \mathbb{R}^n\), then:

1. \(\|\overline{x}+y\| \le \|x\| + \|\overline{y}\|\)
2. \(\|\overline{x}+\overline{y}\| < \|\overline{x}\| + \|y\|\)
3. \(\|\overline{x}+y\| > \|\overline{x}\| + \|y\|\)
4. \(\|\overline{x}+\overline{y}\| = \|\overline{x}\| + \|y\|\)

Correct Answer: (A)

Q.67. Which of the following are subspaces of vector space \(\mathbb{R}^3\)?

1. \(\{(x,y,z) : x+y=0\}\)
2. \(\{(x,y,z) : x-y=2\}\)
3. \(\{(x,y,z) : x+y=1\}\)
4. \(\{(x,y,z) : x-y=1\}\)

Correct Answer: (A)

Q.68. The dimension of the vector space of all \(4 \times 4\) real symmetric matrices is:

1. 20
2. 16
3. 5
4. 12

Correct Answer: (B)

Q.69. For a linear transformation \(T : \mathbb{R}^{10} \to \mathbb{R}^6\), the kernel has dimension \(5\), then the dimension of the image of \(T\) is:

1. 5
2. 6
3. 2
4. 1

Correct Answer: (A)

Q.70. Which of the following statement is false for the matrix \(A = \begin{pmatrix} 2 & 4 \\ 1 & 2 \end{pmatrix}?\)

1. \(A^2 - 4A = 0\)
2. \(A^3 - 4A^2 = 0\)
3. \(A^6 - 4A^5 = 0\)
4. \(A - 4I = 0\)

Correct Answer: (D)

Q.71. Let the set \(A\) and \(B\) have \(5\) and \(9\) elements respectively. What can be the minimum number of elements in \(A \cup B\):

1. 6
2. 15
3. 9
4. 5

Correct Answer: (C)

Q.72. Which of the following statement is true?

1. \(\mathbb{N} \times \mathbb{N}\) is uncountable set
2. Set of all polynomials with rational coefficients is a countable set
3. \(\Gamma(2)\) is a transcendental number
4. Every rational number is a transcendental number

Correct Answer: (B)

Q.73. If \(R^*\) is an extended real number system then the \(\inf R^*\) is:

1. \(-\infty\)
2. 0
3. \(+\infty\)
4. No Infimum

Correct Answer: (A)

Q.74. Sequence \(S = \langle S_n \rangle\), where \(S_n = \sin n\pi \theta\) and \(\theta\) is a rational number such that \(0 < \theta < 1\) then the sequence \(\langle S_n \rangle\) is

1. Not convergent
2. Convergent to \(\dfrac{1}{\sqrt{2}}\)
3. Convergent to 1
4. Convergent to 0

Correct Answer: (A)

Q.75. If the subsequences of a sequence are convergent then the sequence is:

1. Definitely convergent
2. Definitely divergent
3. Oscillatory
4. Convergent only if all subsequences converges to the same limit

Correct Answer: (D)

Q.76. For any complex number \(z\), the minimum value of \(|z| + |z - 2i|\) is:

1. 2
2. 0
3. 1
4. Cannot be determined

Correct Answer: (B)

Q.77. How many elements does the set \( \{z \in \mathbb{C} : z^{60} = -1, z^k \ne -1, \text{ for }\)
\( 0 < k < 60 \} \) have?

1. 15
2. 60
3. 32
4. 45

Correct Answer: (C)

Q.78. Consider the functions \(f(z)=x^2 + iy^2\) and \(g(z)=x^2 + y^2 + ixy\), at \(z=0\)

1. \(f\) is analytic but \(g\) is not analytic
2. \(g\) is analytic but \(f\) is not analytic
3. Neither \(f\) is analytic nor \(g\) is analytic
4. Both \(f\) and \(g\) are analytic

Correct Answer: (C)

Q.79. For the function \(f(z)=\sin \left(\dfrac{1}{z}\right)\), \(z=0\) is a:

1. Removable singularity
2. Simple pole
3. Non-isolated singularity
4. Essential singularity

Correct Answer: (D)

Q.80. The radius of convergence of the series \(\sum\limits_{n=1}^{\infty} z^{n^{2}}\) is:

1. 0
2. \(\infty\)
3. 1
4. 2

Correct Answer: (C)

Q.81. The sum of the eigenvalues of \[ \begin{pmatrix} 2 & 2 & 2 \\ 2 & 2 & 2 \\ 2 & 2 & 2 \end{pmatrix} \] is:

1. 6
2. 0
3. 2
4. 4

Correct Answer: (A)

Q.82. Which of the following matrix is not diagonalizable?

1. \(\begin{pmatrix} 1 & 1 \\ 1 & 2 \end{pmatrix}\)
2. \(\begin{pmatrix} 1 & 0 \\ 3 & 2 \end{pmatrix}\)
3. \(\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\)
4. \(\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}\)

Correct Answer: (A)

Q.83. Let \(A\) be a \(m \times n\) matrix with rank \(m\) and \(B\) be a \(p \times m\) matrix with rank \(p\). What will be the rank of \(BA\) if \((p < m < n)\)?

1. m
2. p
3. n
4. p + m

Correct Answer: (B)

Q.84. Let \(S\) be a subspace of a finite dimensional inner product space \(V\), then which of the following is not correct?

1. \((S^\perp)^\perp = S\)
2. \(\dim S^\perp = \dim V + \dim S\)
3. \(V = S \oplus S^\perp\)
4. \(S^{\perp\perp} = [S]\)

Correct Answer: (B)

Q.85. If \(n\) is the order, \(r\) is the rank and \(S\) is the signature of a real quadratic form in \(n\) variables, then the quadratic form is negative semi-definite, if:

1. \(S = r = n\)
2. \(-S = r = n\)
3. \(S = r < n\)
4. \(-S = r < n\)

Correct Answer: (D)

Q.86. The number of group morphism form \(\mathbb{Z}_2\) to \(\mathbb{Z}_8\) is:

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

Correct Answer: (C)

Q.87. The order of \(a\) and \(x\) in group are 8 and 4 respectively, then the order of \(x^{-1} a x\) be:

1. 8
2. 4
3. 6
4. 12

Correct Answer: (A)

Q.88. If \(\mathbb{R_0}\) denote the multiplicative group of non‐zero real numbers and the mapping \(f : \mathbb{R_0} \to \mathbb{R_0}\), \(f(x)=x^4\), \(\forall x \in \mathbb{R_0}\) then its kernel is:

1. \(\{1,-1\}\)
2. \(\{0\}\)
3. \(\{0,1,2\}\)
4. All non-zero real number

Correct Answer: (A)

Q.89. Which one of the following statement is not correct?

1. Every field is an integral domain
2. A finite commutative ring without zero divisors is a field
3. Every group is isomorphic to some permutation group
4. Every integral domain is a field

Correct Answer: (D)

Q.90. The number of surjective map from a set of 3 elements to a set of 4 element is:

1. 36
2. 0
3. 49
4. cannot be determined

Correct Answer: (D)

Q.91. The value of the contour integral \(\oint_C \dfrac{\cos z}{z} dz\), where \(C\) is circle \(|z|=1\), is:

1. 2\(\pi i\)
2. 0
3. 1
4. does not exist

Correct Answer: (A)

Q.92. Let \(f(z)= \dfrac{z}{8 - z^3}\), \(z = x + iy\), then \(\text{Res}_{z=2} f(z)\), is:

1. \(\dfrac{1}{8}\)
2. \(-\dfrac{1}{8}\)
3. \(-\dfrac{1}{6}\)
4. \(\dfrac{1}{6}\)

Correct Answer: (C)

Q.93. The image of imaginary axis in z‐plane under the transformation \(w = e^z\), is:

1. Unit circle
2. Any circle
3. Parabola
4. Straight line

Correct Answer: (A)

Q.94. If \(^{(n-1)}C_r = (k^2 -3)\cdot {}^nC_{r + 1}\), then \(k\) belongs to:

1. \((\sqrt{3},2)\)
2. (\(-\sqrt{3}, \sqrt{3}\))
3. \((-\infty, -2)\)
4. Cannot be determined

Correct Answer: (A)

Q.95. The remainder obtained when \(16^{2016}\) is divided by \(9\), is:

1. 1
2. 2
3. 3
4. 7

Correct Answer: (A)

Q.96. For the initial value problem \(\dfrac{dy}{dx}=y^2,\ y(0)=1\) the true statement is:

1. Solution does not Exists
2. Solution exists for all real < 1
3. solution exists for all real x > 1
4. Solution exists for all real x

Correct Answer: (B)

Q.97. The particular solution of the boundary value problem:
\( \frac{d^2 y}{dx^2} + y = \csc x,\; 0 < x < \frac{1}{2},;\)
\(y(0)=y\left(\frac{1}{2}\right)=0 \) is:

1. Convex
2. Concave
3. Convex and concave both
4. Neither convex nor concave

Correct Answer: (B)

Q.98. The eigen value of the Sturm–Liouville system
\( y'' + \lambda y = 0,\; 0 \le x \le \pi,\; y(0)=0,\)
\(; y'(\pi)=0 \) are:

1. \(\dfrac{(2n-1)^2}{4}\)
2. \(\dfrac{n^2 \pi^2}{4}\)
3. \(\dfrac{n^2}{4}\)
4. Cannot be determined

Correct Answer: (A)

Q.99. A homogenous linear differential equation with real coefficients has
\(y = x e^{-3x} \cos 2x + e^{-3x} \sin 2x\)
as one of its solution is given by:

1. \((D^2 + 6D + 13)y = 0\)
2. \((D^2 - 6D + 13)y = 0\)
3. \((D^2 - 6D + 13)^2 y = 0\)
4. \((D^2 + 6D + 13)^2 y = 0\)

Correct Answer: (D)

Q.100. Which of the following equation is elliptic:

1. Laplace equation
2. Wave equation
3. Heat equation
4. None of these

Correct Answer: (A)