This page contains the HPSC 2017 Assistant Professor Mathematics Recruitment Paper, carefully formatted and shared to support candidates preparing for competitive examinations. The questions presented here are from the Haryana Public Service Commission (HPSC) examination conducted for the post of Assistant Professor in Mathematics. This content is published purely for educational purposes to help HPSC Assistant Professor aspirants understand the exam pattern, difficulty level, and important Mathematics topics asked in previous years. All rights and ownership of the original examination paper belong to HPSC; we have reproduced the questions only to help students and aspirants in their preparation and academic practice.
Answer key is not official for this Paper, we apologise for inconveince.Q.1. Let \(f: X \to Y\) be a one–one map. Which of the following is not correct?
- \(X\) may be a subset of \(Y\)
- \(Y\) may be a subset of \(X\)
- \(X\) should be equal to \(Y\)
- Cardinality of \(X\) should be equal to cardinality of \(Y\)
Q.2. If mean is equal to 8, then standard deviation of exponential probability distribution is:
- 0.425
- 0.125
- 0.225
- 0.325
Q.3. Classify the differential equation \( \frac{dy}{dx} + 3y = x^2 y \):
- Separable and not linear
- Linear and not separable
- Both separable and linear
- Neither separable nor linear
Q.4. Probabilistic models are also known as:
- Deterministic models
- Stochastic models
- Dynamic models
- Static models
Q.5. Consider the linear differential equation \( \frac{dy}{dx} + \left(\frac{x}{1+x}\right)y = 1 + x \). The integrating factor is:
- \(e^x\)
- \(\frac{e^x}{1+x}\)
- \(e^x(1+x)\)
- \(e^x + \frac{x^2}{2}\)
Q.6. The solution of the equation \( \frac{d^2y}{dx^2} + y = 0 \), satisfying \(y(0)=1,\; y(\frac{\pi}{2})=2\), is:
- \(\cos x - 2\sin x\)
- \(\cos x + \sin x\)
- \(2\cos x - \sin x\)
- None of these
Q.7. The convergence of which of the following methods is sensitive to starting value?
- False position method
- Gauss–Seidel method
- Newton–Raphson method
- All of these
Q.8. If \(A\) is an upper triangular matrix with all diagonal entries zero, then \(I+A\) is:
- Invertible
- Idempotent
- Singular
- Nilpotent
Q.9. If \(\tau\) is a topology on any set \(X\), then arbitrary _____ of members of \(\tau\) belongs to \(\tau\).
- Union
- Intersection
- Product
- None of these
Q.10. If \(G = \{1, -1, i, -i\}\) is a group under multiplication, then inverse of \(i\) is:
- 1
- -1
- \(-i\)
- None of these
Q.11. In a compact metric space:
- Every sequence is a Cauchy sequence
- Every Cauchy sequence converges
- Every sequence has a Cauchy subsequence
- Every open cover has a finite subcover
Q.12. Let \(A\) and \(B\) be square matrices such that \(AB = I\). Then zero is an eigenvalue of:
- A but not of B
- B but not of A
- Both A and B
- Neither A nor B
Q.13. Let \(T\) be the identity transformation of a finite dimensional vector space \(V\). Then nullity of \(T\) is:
- \(\dim V\)
- 0
- 1
- \(\dim V - 1\)
Q.14. The value of \(x\) that makes \(x^2 + 6x + 13\) minimum is:
- 6
- -3
- 3
- None of these
Q.15. A real sequence is Cauchy iff
- Bounded
- Divergent
- Unbounded
- Not bounded
Q.16. Compact subsets of metric space are:
- Convex
- Open
- Closed
- Connected
Q.17. Column in simplex initial table used to represent new basic variable is classified as:
- Column variable
- Key column
- Key row
- Row variable
Q.18. Jacobi’s method is also known as:
- Displacement method
- Simultaneous displacement method
- Simultaneous method
- Diagonal method
Q.19. When fluid properties do not change with time, flow is called:
- Steady
- Unsteady
- Viscous
- Non-viscous
Q.20. The system of linear equations \((4d - 1)x + y + z = 0,\; -y + z = 0,\; (4d-1)z = 0\) has a non-trivial solution if determinant equals:
- \(\frac{1}{2}\)
- \(\frac{1}{4}\)
- \(\frac{3}{4}\)
- 1
Q.22. "Mathematical expectation of the product of two random variables is equal to the product of their expectations" is true for :
- Variables are dependent
- Variables are independent
- Covariance is non-zero
- Variance is zero
Q.23. Two samples of sizes 25 and 35 are independently drawn from two normal populations, where the unkonwn variances are assumed to be equal. The number of degrees of freedom for equal variance t-test is:
- 58
- 60
- 62
- 57
Q.24. For linear inequalities, solution set of a group of inequalities is classified as:
- Concave set
- Convex set
- Loss set
- Profit set
Q.25. Given that \(y'(x) =x-1\) is one solution of \(2x^2y''+3xy'-y=0,\; x>0\). The second linearly independent solution is:
- \(x^{-1}\)
- \(x^2\)
- \(x^{-1/2}\)
- None of these
Q.26. Which of the following p-values will lead us to reject the null hypothesis if the significance level of the test is 5%?
- 0.15
- 0.025
- 0.06
- 0.20
Q.27. To perform a Chi-square test:
- Data must conform to a normal distribution
- Data must be measured on a nominal scale
- Each cell has equal frequencies
- All of these
Q.28. \(A\) is a \(5 \times 5\) matrix, all of whose entries are 1. Then:
- \(A\) is not diagonalizable
- \(A\) is idempotent
- \(A\) is nilpotent
- Minimal polynomial and characteristic polynomial of \(A\) are equal
Q.29. For a rectangular uniform distribution, value length of interval a is 7 and value of interval b is 8, then mean of distribution is:
- 1.33
- 3.33
- 2.33
- 4.33
Q.30. Which of the following statements applies to the bisection method used for finding roots of functions?
- Converges within a few iterations
- Guaranteed to work for continuous functions
- Is faster than Newton–Raphson method
- Requires derivative of the function
Q.31. In a set of observations, unusual lower and higher values are called:
- Outliers
- Free liners
- Central liners
- Median liners
Q.32. The solution of the partial differential equation \(xzp+ yzq = xy\) is:
- \(\phi(x+y,y+z)=c_1\)
- \(\phi(xy+z)=c_2\)
- \(\phi(xz-y)=c_3\)
- None of these
Q.33. Which of the following equations is an exact differential equation?
- \((x^2+1)dx-xy\,dy=0\)
- \(x\,dy+(3x-2y)dx=0\)
- \(2xy\,dx+(2+x^2)dy=0\)
- \(x^2y\,dy-y\,dx=0\)
Q.34. LP model is based on the assumptions of:
- Proportionality
- Additivity
- Certainty
- All of the above
Q.35. Let \(f:\mathbb{Q}\to\mathbb{R}\) be continuous and \(f(x)=\sqrt{2}\), \(x \in \mathbb{Q}\). Then \(f(\sqrt{2})\) equals:
- \(\sqrt{2}\)
- 0
- Neither \(\sqrt{2}\) nor 0
- None of these
Q.36. Which of the following statements is not correct concerning the probability distribution of a continuous random variable?
- The vertical coordinate is the probability density function
- The range of the random variable is found on the x-axis
- The total area under the curve equals 1
- The area under the curve between two points represents probability
Q.37. A convergent sequence has only ____ limit(s).
- One
- Two
- Three
- None
Q.38. Let A be a set. What does it mean for A to be uncountable?
- There is no way to assign a distinct element of A to each natural number.
- There exist elements of A which cannot be assigned to any natural number at all.
- There is no way to assign a distinct natural number to each element of A.
- There is a bijection \(f\) from A to the real numbers \(\mathbb{R}\).
Q.39. If a random variable \(X\) follows normal distribution with mean \(\mu\) and variance \(\alpha^2\), then the random variable \(Z = X - \mu\alpha\) follows normal distribution with:
- Mean = 1, variance = 0
- Mean = 0, variance = 1
- Mean = \(\mu^2\), variance = \(\alpha^2\)
- None of these
Q.40. \(\{3n : n \in \mathbb{Z}\}\) is an abelian group under:
- Subtraction
- Division
- Multiplication
- Addition
Q.41. Find the differential equation of the family of lines passing through the origin:
- \(y\,dx-x\,dy=0\)
- \(x\,dy-y\,dx=0\)
- \(x\,dx+y\,dy=0\)
- \(ydx+xdy=0\)
Q.42. \(n\) different objects can be arranged taking all at a time in:
- \((n-1)!\)
- \((n+1)!\)
- \(n!\)
- \((2n)!\)
Q.43. Which of the following indicates that a dataset is not bell-shaped?
- Range is equal to 5 standard deviations
- Range is larger than inter-quartile range
- Mean is much smaller than median
- No outliers
Q.44. The solution of the partial differential equation \(yzp + zxq = xy\) is given by:
- \(x^2 + y^2 = c_1\) and \(x^2 + z^2 = c_2\)
- \(x^2 - y^2 = c_1\) and \(x^2 - z^2 = c_2\)
- \(x^2 + y^2 = c_1\) and \(x^2 - z^2 = c_2\)
- \(x^2 - y^2 = c_1\) and \(x^2 + z^2 = c_2\)
Q.45. Let \(u,v,w\) be three non-zero vectors which are linearly independent. Which is correct?
- \(u\) is linear combination of \(v\) and \(w\)
- \(v\) is linear combination of \(u\) and \(w\)
- \(w\) is linear combination of \(u\) and \(v\)
- None of these
Q.46. Let \(f:\mathbb{R} \to [-1,1]\) be onto. Then:
- \(f\) is not continuous
- \(f\) is continuous
- \(f\) is differentiable everywhere
- \(f\) is continuous, but not differentiable anywhere
Q.47. Since the mode is the most frequently occurring data value, it:
- Can never be larger than mean
- Is always larger than median
- Is always larger than mean
- None of the above
Q.48. Let \(f:\mathbb{R} \to \mathbb{R}\) and \(g:\mathbb{R} \to \mathbb{R}\) be continuous and \(f(x)=g(x)\) for all \(x \in \mathbb{Q}\). Then:
- \(f(x)=g(x)\) for some \(x \in \mathbb{R}/\mathbb{Q}\)
- \(f(x)=g(x)\) for all \(x \in \mathbb{R}\)
- \(f(x)\neq g(x)\) for some \(x \in \mathbb{R}/\mathbb{Q}\)
- \(f(x)\neq g(x)\) for all \(x \in \mathbb{R}/\mathbb{Q}\)
Q.49. Result of a logical or relational expression in C is:
- True or False
- 0 or 1
- 0 if false and any positive number if true
- None of these
Q.50. There is a unique function \(f:(0,\infty)\to(0,\infty)\) such that \(\log f(x)\) is convex and \(f(1)=1,\; f(x+1)=xf(x)\). Then:
- (i) is true
- (ii) is true
- (i) and (ii) both true
- None of the above
Q.51. A Type I error occurs when we:
- Reject a false null hypothesis
- Reject a true null hypothesis
- Do not reject a false null hypothesis
- Do not reject a true null hypothesis
Q.52. Let \((M,d)\) be a metric space, \(A \subset M\) be closed and \(B \subset M\) be open. Then:
- \(A/B\) is open
- \(A \cap B\) is closed
- \(A \cup B\) is closed
- None of the above
Q.53. If \(\tau\) is topology on non-empty set \(X\), then arbitrary ______ of members of \(\tau\) belongs to \(\tau\).
- Union
- Intersection
- Complement
- Difference
Q.54. Sample space for an experiment in which two coins are tossed is:
- 4
- 8
- 2
- 10
Q.55. Let \(f:[2,4]\to\mathbb{R}\) be a continuous function such that \(f(2)=3\) and \(f(4)=6\). The most we can say about the set \(f([2,4])\) is:
- It is a set which contains \([3,6]\)
- It is a closed interval
- It is a set which contains 3 and 6
- It is a closed interval which contains \([3,6]\)
Q.56. A monoid is always a:
- Groupoid
- Commutative group
- Non-abelian group
- Group
Q.57. Does logical operators in C language are evaluated with short circuit?
- True
- False
- Depends on compiler
- Depends on standard
Q.58. Suppose that a function has partial derivatives \(\partial f/\partial x = x^2 - y - 1\), \(\partial f/\partial y = y - x + 1\). Which of the following points is a critical point of \(f(x,y)\)?
- \((-1,0)\)
- \((1,-1)\)
- \((0,-1)\)
- None of these
Q.59. Boundary condition which includes derivative of boundary value is:
- Dirichlet boundary condition
- Neumann boundary condition
- Forced boundary condition
- Discrete boundary condition
Q.60. In simplex algorithm, which method is used to deal with the situation where an infeasible starting basic solution is given ?
- Slack variable
- Simplex method
- M-method
- None of the above
Q.61. Classify the differential equation \( \frac{dz}{dt} = 1 + z + t + zt \):
- Separable and not linear
- Linear and not separable
- Neither separable nor linear
- Both separable and linear
Q.62. The differential equation \( \left(\frac{dx}{dy}\right)^2 + 5^{\frac{1}{3}} = x \) is:
- Linear of degree 2
- Nonlinear of order 1 and degree 2
- Nonlinear of order 1 and degree 6
- None of these
As soon as we get those questions, we will update this section.
Q.75. In a random experiment, observations of a random variable are classified as:
- Events
- Composition
- Trials
- Functions
Q.76. The Lebesgue measure of the set \(\{ \pi \}\) is:
- 1
- 2
- \(\{0\}\)
- None of these
Q.77. Two equations that have no values to satisfy both equations simultaneously is called:
- Consistent system
- Inconsistent system
- Solution system
- Constant system
Q.78. The ring \(\mathbb{Z}[x]\) is:
- A PID
- A UFD
- A PID and a UFD
- None of these
Q.79. According to algebra of simplex method, slack variables are assigned zero coefficients because:
- No contribution in objective function
- High contribution in objective function
- Divisor contribution in objective function
- Base contribution in objective function
Q.80. If \(G\) is a cyclic group of order 24 and \(a^{2002}=e\), then value of \(n\) is:
- 4
- 10
- 8
- 6
Q.81. One use of a regression line is:
- To determine if any x-values are outliers
- To determine if any y-values are outliers
- To determine if a change in x causes a change in y
- To estimate the change in y for a given change in x
Q.82. A is a unitary matrix. Then eigenvalues of A are:
- 1, −1
- 1, −i
- i, −i
- −1, i
Q.83. Meshes which only require element size on lines and surfaces that define geometry as input are:
- Structured mesh
- Unstructured mesh
- None of these
- Hybrid mesh
Q.84. Simplify \(a^2+b^2\):
- \((a+b)(a-b)\)
- \((a+ib)(a-ib)\)
- \((a+b)(a-ib)\)
- \((a+ib)(a-b)\)
Q.85. Unary operation is one in which yields another number when performed on:
- Two numbers
- A single number
- Three numbers
- Four numbers
Q.86. Find the general solution of \(y' = y \sec x \left( \frac{dy}{dx} = y \sec x \right)\):
- \(y = C(\sec x + \tan x)\)
- \(y = C(\sec x - \tan x)\)
- \(y = C(\sec x \cdot \tan x)\)
- \(y = C(\sec^2 x + \tan x)\)
Q.87. A set \(A\) is said to be countable if there exists a function \(f:A\to\mathbb{N}\) such that \(f\) is:
- Bijective
- Injective
- Identity map
- None of these
Q.88. In binomial probability distribution, dependents of standard deviations must include:
- Probability of q
- Probability of p
- Trials
- All of the above
Q.89. If a hypothesis is rejected at 0.025 level of significance, it:
- May or may not be rejected at 0.01 level
- Must be rejected at 0.01 level
- Must not be rejected at 0.01 level
- Must not be rejected at any other level
Q.90. For linear programming equations, convex set of equations is included in region of:
- Feasible solutions
- Disposed solutions
- Profit solutions
- Loss solutions
Q.91. A large Reynolds number is indication of:
- Laminar flow
- Streamline flow
- Steady flow
- Turbulent flow
Q.92. Indicate which of the statements below does not correctly apply to normal probability distributions:
- They are all unimodal (i.e have single mode)
- They are all symmetrical
- They all have the same mean and standard deviation
- The area under the probability curve is always equal to 1
Q.93. In systems of equations, equations are linearly independent if:
- \(A^{-2}\) exists
- \(A^{-1}\) does not exist
- \(A^{-3}\) does not exist
- \(A^{-4}\) must exist
Q.94. Considering mean, mode and skewness of data, value of skewness will be negative if:
- Mean > Mode
- Mean < Mode
- Mean < Median
- Mean > Median
Q.95. If \(dy = x^2 dx\), what is the equation of y in terms of x if the curve passes through \((1,1)\)?
- \(x^3 - 3y + 3 = 0\)
- \(x^3 - 3y + 2 = 0\)
- \(x^3 + 3y^2 + 2 = 0\)
- \(2y + x^3 + 2 = 0\)
Q.96. If \(\tau_1\) and \(\tau_2\) are two topologies on a non-empty set \(X\), then ______ is a topological space :
- \(\tau_1 \cup \tau_2\)
- \(\tau_1 \cap \tau_2\)
- \(\tau_1 \ \tau_2\)
- \(\tau_2 \cup \tau_1\)
Q.97. Multiplicative property of order of real numbers is that for all \(a, b, c \in \mathbb{R}\):
- \(a > b \land c > 0\) implies \(ac < bc\)
- \(a > b \land c > 0\) implies \(ac > bc\)
- \(a > b \land c > 0\) implies \(ac \leq bc\)
- \(a > b \land c > 0\) implies \(ac = bc\)
Q.98. In classification of probability distributions, ‘Erlang distribution’ is also called:
- Alpha distribution
- Beta distribution
- Gamma distribution
- Exponential distribution
Q.99. How many elements of order 5 are there in \(S_7\)?
- 5
- 7
- 12
- None of the above
Q.100. Which of the following statements is false?
- The set of rational numbers is an abelian group under addition
- The set of rational integers is an abelian group under addition
- The set of rational numbers form an abelian group under multiplication
- None of these
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