# PPSC Paper 2011 (Lecturer in Mathematics)

On this page, we have given question from old (past) paper of Lecturer in Mathematics conducted in year 2011. This is a MCQs paper and answers are given at the end of the paper. At the end of the PDF is also given to download. This paper is provided by Ms. Iqra Liaqat. We are very thankful to her for providing this paper.

- A ring $R$ is a Boolean Ring if for all $x\in R$.

- $x^2=x$
- $x^2=-x$
- $x^2=0$
- $x^2=1$

- The group of Quaterminons is a non-abelian group of order ———

- $6$
- $8$
- $10$
- $4$

- Every group of prime order is ——-

- an abelian but not cyclic
- an abelian group
- a non-abelian group
- a cyclic group

- Any two conjugate subgroup of a group $G$ are

- Equivalent
- Similar
- Isomorphic
- None of these

- If $H$ is a subgroup of index —— then $H$ is a normal subgroup of $G$

- $2$
- $4$
- Prime number
- None of these

- $nZ$ is a maximal ideal of a ring $Z$ if and only if $n$ is ——

- Prime number
- Composite number
- Natural number
- None of these

- Let $G$ be a cyclic group of order $24$ generated by $a$ then order of $a^{10}$ is ——

- $2$
- $12$
- $10$
- None of these

- If a vector space $V$ has a basis of $n$ vectors, then every basis of $V$ must consist of exactly —– vectors.

- $n+1$
- $n$
- $n-1$
- None of these

- An indexed set of a vectors $v_1,v_2,v_3,....,v_r$ in $R^n$ is said to be —— if the vector equation $x_1v_1+x_2v_2+.....+x_pv_p=0$ has only trivial solution.

- Linearly independent
- Basis
- Linearly dependent
- None of these

- The set $C_n$ of all, $nth$ roots of unity for a fixed positive integer $n$ is a group under —–

- Addition
- Addition modulo $n$
- Multiplication
- Multiplication modulo $n$

- Intersection of any collection of normal subgroups of a group $G$ ——

- is normal subgroup
- may not be normal subgroup
- is cyclic subgroup
- is abelian subgroup

- $\mathbb{Z}/ 2\mathbb{Z}$ is a quotient group of order ——-

- $1$
- $2$
- infinite
- none of these

- A group $G$ having order ———–, where $p$ is prime, is always abelian.

- $p^4$
- $p^2$
- $2p$
- $p^3$

- The number of conjugacy classes of symmetric group of degree $3$ is ————

- $6$
- $2$
- $3$
- $4$

- ———— is a set of all those elements of a group $G$ which commutes with all other elements of $G$

- commutator subgroup
- centre of group
- automorphism of $G$
- None of these

- What are zero divisors in the ring of integers modulo $6$

- $\bar{1},\bar{2},\bar{4}$
- $\bar{0},\bar{2},\bar{3}$
- $\bar{0},\bar{2},\bar{4}$
- $\bar{2},\bar{3},\bar{4}$

- If $H$ is a normal subgroup of $G$, then $Na(H)=$ ————

- $H$
- $G$
- $\{e\}$
- None of these

- An $n\times n $ matrix with $n$ distinct eigenvalues is ————-

- Diagonalization
- Similar matrix
- Not diagonalizable
- None of these

- Let $T:U\longrightarrow V$ be a linear transformation from an $n$ dimensional vector space $U(F)$ to a vector space $V(F)$ then

- $\dim N(T)+\dim R(T)=0$
- $\dim N(T)+\dim R(T)=2n$

- $\dim N(T)+\dim R(T)=n^2$
- $\dim N(T)+\dim R(T)=n$

- The dimension of the row space or column space of a matrix is called the ——- of the matrix.

- Basis
- Null space
- Rank
- None of those

- $\underline{a}\times (\underline{b}\times\underline{c})$ is a vector lying in the plane containing vectors

- $\underline{a},\underline{b}$ and $\underline{c}$
- $\underline{a}$ and $\underline{c}$
- $\underline{b}$ and $\underline{c}$
- $\underline{b}$ and $\underline{a}$

- The square matrix $A$ and its transpose have the ——– eigenvalues.

- Same
- Different
- Unique
- None of these

- The set $S=\left\{ \left[\begin{array}{c} 1 \\ 2 \end{array}\right], \left[\begin{array}{c}2\\ 2\end{array}\right],\left[\begin{array}{c}0\\ 0\end{array}\right] \right\}$ of vectors in $\mathbb{R}^2$ is ————

- linearly independent
- linearly dependent
- basis of $\mathbb{R}^2$
- none of these

- Let $X$ and $Y$ vectors spaces over the field \(F\) with \(\dim X=m\) and \(\dim Y=n\) then the \(\dim Hom(X,Y)=\)

- \(mn\)
- \(n\)
- \(n^m\)
- \(m^2\)

- All subgroups of an abelian group are ———– subgroups.

- cyclic
- normal
- characteristic
- None of these

- The set of all solutions to the homogenous equation \(Ax=0\) when \(A\) is an \(m \times n\) matrix is ———

- Null space
- Column space
- Rank
- None of these

- If \(7\) cards are dealt from an ordinary deck of \(52\) playing cards, what is the probability that at least \(1\) of them will be a queen?

- \(0.4773\)
- \(0.4774\)
- \(0.4775\)
- \(0.4776\)

- Let \(G\) be an abelian group. Then which one of the following is not true.

- every commutator of \(G\) is identity

- iF \(m\) is divisor of order of \(G\) then \(G\) must have subgroup of order \(m\)

- centre of \(G\) is \(G\) itself

- every subgroup of \(G\) is cyclic

- Every group of order \(\leq5\) is

- cyclic
- abelian
- non abelian
- none of these

- Number of non-isomorphic groups of order \(8\) is ——

- $4$
- $2$
- $3$
- $5$

- Centre of the group of quaternions \(Q_8\) is of order

- $1$
- $2$
- $8$
- $4$

- \(\underline{a}\cdot (\underline{b}\times\underline{c})\) is not equal to

- \(\underline{a}\cdot(\underline{c}\times\underline{b})\)
- \((\underline{a}\times\underline{b})\cdot\underline{c}\)
- \(\underline{b}\cdot(\underline{c}\times\underline{a})\)
- \(\underline{a}\cdot(\underline{a}\times\underline{b})\)

- Let \(G\) be a group. Then the derived group \(G^{'}\) is subgroup of \(G\)

- cyclic
- abelian
- normal
- none of these

- Let \(G\) be a group. Then the factor group \(G/G\) is ——-

- abelian
- cyclic
- normal
- none of these

- Finite simple abelian group are of order

- $4$
- prime power
- power of \(2\)
- prime number

- Set of integers \(Z\) is

- Field
- group under multiplication
- integral domain
- division ring

- Set of integers \(\mathbb{Z}\) is ——- of the set \(\mathbb{Q} \) of rationals.

- prime ideal
- sub ring
- maximal ideal
- none of these

- Solution set of the equation \(1+\cos x=0\) is

- \( \{\pi+n\pi:n\in \mathbb{Z}\}\)
- \( \left\{2n\pi:n\in \mathbb{Z}\right\}\)
- \( \{\dfrac{\pi}{2}+n\pi:n\in \mathbb{Z}\}\)
- \( \{\pi+2n\pi:n\in \mathbb{Z}\}\)

- Non-zero elements of a field from a group under

- addition
- multiplication
- subtraction
- division

- Let \(\mathbb{Q}\) be the set of rational numbers. Then \(\mathbb{Q}(\sqrt{3})=\{a+b\sqrt{3}:a,b \in \mathbb{Q}\}\) is a vector space over \(g\) with dimension

- $1$
- $2$
- $3$
- $4$

- Let \(W\) be a subspace of the space \(\mathbb{R}^3\). if \(\dim W=0\) then \(W\) is a

- line through the origin \(0\)
- plane through the origin \(0\)
- entire space \(R^3\)
- a point

- Let \(P_n(t)\) be a vector space of all polynomials of degree \(\leq n\). Then

- \(\dim P_n(t)=n-1\)
- \(\dim P_n(t)=n\)
- \( \dim P_n(t)=n+1\)
- \(2\)

- A one to one linear transformation preserves —————

- basis but not dimension
- basis and dimension
- dimension but not basis
- none of these

- In a group \((\mathbb{Z},\circ)\) of all integers where \(a \circ b=a+b+1\) for \(a,b\in \mathbb{Z}\), the inverse of \(-3\) is

- \(-3\)
- \(0\)
- \(3\)
- \(1\)

- The set \(\mathbb{Z} \) of all integers is not a vector space over the field \(\mathbb{R} \) of real numbers under ordinary addition `$+$', multiplication `\(\times\)' of real numbers, because

- \((\mathbb{Z},+)\) is a ring
- \((\mathbb{Z},+,\times)\) is not a field
- \((\mathbb{R},\times)\) is not a group
- ordinary multiplication of real numbers does not define a scalar multiplication of \(\mathbb{Z}\) by \(\mathbb{R}\).

- Let \(G\) be an abelian group. Then \(\varphi:G\longrightarrow G\) given by ———– is an automorphism.

- $\varphi(x)=x^3$
- $\varphi(x)=e$
- $\varphi(x)=x^2$
- $\varphi(x)=x^{-1}$

- Let \(G\) be a group in which \(g^2=1\) for all \(g\) is $G$. Then \(G\) is ———-

- Abelian
- cyclic
- abelian but not cyclic
- non abelian

- Let \(G=\langle a,b:b^2=1=a^2,ab=ba^{-1} \rangle\). Then the number of distinct left cosets of $H=\langle b \rangle$ in $G$ is ————

- 1
- 2
- 4
- 3

- A linear transformation \(T:U \to V \) is one-to-one if and only if kernel of $T$ is equal to

- U
- V
- \( \{ 0\}\)
- $\Im (T)$

- For a scalar point function $\varphi(x,y,z)$, $text{div grad} \varphi $ is

- scalar point function
- vector point function
- guage function
- neither

- A particle moves along a curve \(F=(e^{-1},2\cos 3t,2\sin 3t)\), where \(t\) is time. The velocity at \(t=0\) is

- $(-1,0,6)$
- $(-1,-6,0)$
- $(1,2,0)$
- $(-1,2,2)$

- The coordinates surface for the cylindrical coordinates \(x=r\cos \varphi\),\(y=r\sin \varphi,z=z\) are given by

- \(r=c, \varphi=c\)
- \(r=c_1, \varphi=c,z=c_3\)
- \(r=c_1, z=c_3\)
- \(\varphi=c_2,z=c_3\)

- The metric coefficients in cylindrical coordinates are

- \((1,1,1)\)
- \((1,0,1)\)
- \((1,r,1)\)
- neither

- The value of the quantity \(\delta_ix_ix_j\) is

- \(x_i\)
- zero
- \(x_{ij}\)
- \(x_ix_j\)

- A tensor of rank \(5\) in a space of \(4\) dimensions has components

- $5$
- $4$
- $625$
- $1024$

- A vector is said to be irrational if

- $\bigtriangledown \bar{F}=1$
- $\bigtriangledown \bar{F}=0$
- $\bigtriangledown \times\bar{F}=0$
- none

- The moment of inertia of a rigid hemisphere of mass \(M\) and radius \(a\) about a diameter of a base is

- $Ma^2/5$
- $Ma^2/2$
- $2Ma^2/5$
- more information needed

- Radius of gyration of a rigid body of mass \(4gm\) having moment of inertia \(32gm(cm)^2\) is:

- $8(cm)^2$
- $2\sqrt{2}cm$
- $\sqrt{2}$
- $2\sqrt{2}gm$

- Equation for the ellipsoid of inertia for a rigid body having moments and products of inerti \(1_{xx}=18\)units, \(1_{yy}=18\)units, \(1_{zz}=36\)units, \(1_{xy}=-13.5\)units, \(1_{xz}=0\), \(1_{yz}=0\)

- \(18(x^2+y^2+z^2)-27xy=1\)
- \(18(x^2+y^2+2z^2)-27xy=1\)
- \(18(x^2+y^2)+2z^2-27xy=1\)
- more information needed

- The neighbourhood of \(0,\) under the usual topology for the real line \(r\), is

- $]\frac{-1}{2},\frac{1}{2}]$
- \(]-1,0]\)
- \(]0,1]\)
- $[0,\frac{1}{2}[$

- Let\(A=[0,1]\) be a subset of \(R\) with Euclidean metric Then interior of \(A\) is

- $[0,1[$
- $]0,1[$
- $[0,1]$
- $]0,1]$

- Number of non-isomorphic groups of order \(8\) is

- \(5\)
- \(2\)
- \(3\)
- \(4\)

- Suppose \(a\) and \(c\) are real numbers, \(c>0\), and \(f\) is defined on \([-1,1]\) by \[f(x)=\left\{ \begin{array}{c} x^a\sin(x^{-c}) \\ 0 \end{array} \right.\begin{array}{l} (if\,\, x\neq 0), \\ (if\,\, x=0). \end{array}\] \(f\) is bounded if and only if

- \(a>1+c\)
- \(b>2+c\)
- \(a\geq 1+c\)
- \(a\geq 2+c\)

- Let \(M_{2,3}\) be a vector space of all \(2\times 3\) matrices over \(R\). Then dimension of \(Hom(M_{2,3},\mathbb{R}^4)\)

- \(12\)
- \(6\)
- \(8\)
- \(24\)

- Let \(X=\{a,b,c,d,e\}\). Which one of the following classes of subsets of \(X\) is a topology on \(X\).

- $T_1=\{X,\phi,\{a\},\{a,b\},\{a,c\}\}$
- $T_2=\{X,\phi,\{a,b,c\},\{a,b,d\},\{a,b,c,d\}\}$
- $T_3=\{X,\phi,\{a\},\{a,b\},\{a,c,d\},\{a,b,c,d\}\}$
- $T_4=\{\phi,\{a\},\{a,b\},\{a,c\}\}$

- Let $T=\{X,\phi,\{a\},\{a,b\},\{a,c,d\},\{a,b,c,d\},\{a,b,e\}\}$ be a topology on \(X=\{a,b,c,d,e\}\) and \(A=\{a,b,c\}\) be the subset of \(X\). Then interior of \(A\) is

- \(\{a,b,c\}\)
- \(\{a,b\}\)
- \(\{a\}\)
- \(\{b,c\}\)

- The value of \(\sin(\cos^{-1}\dfrac{\sqrt{3}}{2})\) is

- $\dfrac{\sqrt{3}}{2}$
- $\dfrac{1}{\sqrt{2}}$
- $\dfrac{1}{2}$
- $1$

- The smallest field containing set of integers \(\mathbb{Z}\) is

- \(\mathbb{Q}\sqrt{2}\)
- \(\mathbb{Q}\)
- \(\mathbb{Q}\sqrt{6}\)
- \(\mathbb{Q}\sqrt{3}\)

- Let $\mathbb{R}$ be the usual metric space. Then which of the following set is not closed.

- Set of integers
- Set of rational numbers
- \([0,1]\)
- \(\displaystyle \left\{1,\frac{1}{2},\frac{1}{3},...\right\} \)

- Let $\mathbb{R}$ be the usual metric space and \(\mathbb{Z}\) be the set of integers, then clouser of $\mathbb{Z}$ is

- \(\mathbb{Z}\)
- set of rational number \(\mathbb{Q}\)
- set of real number \(\mathbb{R}\)
- set of natural number \(\mathbb{N}\).

- A subspace \(A\) of a complete metric space \(X\) is complete if and only if \(A\) is

- $X$
- open
- closed
- empty set

- A subset \(A\) of a topological space \(x\) is open if and only if \(A\) is

- $A$ is neighbourhood of each of its point
- $A$ is neighbourhood of some of its point
- $A$ contain all of its limits points
- $A$ contain all of its boundary points

- Non-zero elements of a finite filed form ———— group.

- non-cyclic
- An abelian group but not cyclic
- Non-abelian
- a cyclic

- Let \(R\) be the co-finite topology. Then \(R\) is a

- $T_0$ but not \(T_1\)
- $T_1$ but not \(T_2\)
- $T_2$ but not \(T_3\)
- $T_3$ but not \(T_1\)

- Let $f(x)=\dfrac{x+5}{(x-1)(x-2)}$ then range of \(f\) is

- set of all real numbers $R$
- $R-\{1,2\}$
- $R^+$
- $R^-$

- The value of \(\displaystyle \int_{0}^{1}xe^ydx\) is

- $-1$
- $1$
- $c$
- $2c$

- The solution of the congruence \(4x\equiv5 \pmod{9}\) is

- $x\equiv6\pmod{9}$
- $x\equiv 7 \pmod{9}$
- $x\equiv8\pmod{9}$
- $x\equiv2\pmod{9}$

- The series \(x-\dfrac{x^2}{2}+\dfrac{x^3}{3}-\dfrac{x^4}{4}+...\) is convergent for

- \(|x|<1\) only
- \(|x|\leq 1\)
- \(-1<x\leq1\)
- all real values of \(x\)

- The general solution of the differential equation \((x^2+y^2)dx-2xdy=0\) is

- $x^2-cx-y^2=0$, \(c\) is an arbitrary constant.
- $(x-y)^2=cx$, \(c\) is an arbitrary constant.
- $x+y+2xy=c$, \(c\) is an arbitrary constant.
- $y=x^2-2x+c$, \(c\) is an arbitrary constant.

- Let \(f\) be defined on \(\mathbb{R}\) by setting \(f(x)=x\), if \(x\) is rational and \(f(x)=1-x\) if \(x\) is irrational. Then

- $f$ is continuous on \(\mathbb{R}\)
- $f$ is continuous only at \(x=\dfrac{1}{2}\)
- $f$ is continuous everywhere except at \(x=\dfrac{1}{2}\)
- \(f\) is discontinuous everywhere.

- The differential equation $ydx-2xdy=0$ represents

- a family of straight lines
- a family of parabola
- a family of hyperbola
- a family of circles

- A particular integral of the differential equation \((D^2+4)y=x\) is

- \(xc^{-2x}\)
- \(x\cos2x\)
- \(x\sin2x \)
- \(\dfrac{x}{4}\)

- The area of the cardioid \(r=a(1+\cos\theta)\) is equal to

- $4\pi a^2$
- $8\pi a$
- $\dfrac{3\pi a^2}{4}$
- $2\pi a^2$

- The value of \(\sqrt{3}\sin x+\cos x\) will be greatest when \(x\) is equal to

- $\dfrac{\pi}{2}$
- $\dfrac{\pi}{4}$
- $\dfrac{\pi}{6}$
- $\dfrac{\pi}{8}$

- If a particle in equilibrium is subjected to four forces \(F_1=2\hat i-5\hat j+6\hat k\), \(F_2=\hat i+3\hat j-7\hat k\), \(F_3=2\hat i-2\hat j-3\hat k\) and \(F_4\) then \(F_4\) is equal to

- $-5\hat i+4\hat j+4\hat k$
- $5\hat i-4\hat j-4\hat k$
- $3\hat i-2\hat j-\hat k$
- $3\hat i+\hat j-10\hat k$

- The function \(f(x)=|x|+|x-1|\) is

- Continuous and differentiable for \(x=0,x=1\)

- Continuous but not differentiable for \(x=0,x=1\)

- Discontinuous but differentiable for \(x=0,x=1\)

- Neither continuous nor differentiable for \(x=0,x=1\)

- Evaluate \(\displaystyle \lim\limits_{x \to 0}\left(\dfrac{\tan x}{x}\right)^{\frac{3}{x^3}}\)

- $0$
- $e$
- $e^{\frac{1}{3}}$
- $e^3$

- If \(z=x2\tan^{-1}(\frac{x}{y})-y^2tan^{-1}(\frac{x}{y})\), then \(\dfrac{d^2z}{dxdy}\) is

- \(\dfrac{x^2}{y^2}\)
- \(\dfrac{x^2+y^2}{x^2-y^2}\)
- \(\dfrac{x^2-y^2}{x^2+y^2}\)
- None of these

- The radius of curvature is

- Double the measure of curvature
- Square of curvature
- Reciprocal of curvature
- None of these

- Suppose \(a\) and \(c\) are real numbers, \(c>0\), and \(f\) is defined on \([-1,1]\) by \[f(x)=\left\{ \begin{array}{c} x^a\sin(x^{-e}) \\ 0 \end{array} \right.\begin{array}{c} (\text{if } x\neq 0), \\ (\text{if } x=0). \end{array}\] \(f\) is continuous if and only if

- $a\geq 1$
- $a>1$
- $a\geq 0$
- $a>0$

- The value of $\displaystyle\int_{0}^{\infty}\frac{dx}{1+x^2}$ is

- $\dfrac{\pi}{2}$
- $\dfrac{\pi}{4}$
- $0$
- $\infty$

- Which of the following function is a bijection from \(\mathbb{R}\) to \(\mathbb{R}\).

- $f(x)=x^2+1$
- $f(x)=x^3$
- $f(x)=\dfrac{(x^2+1)}{(x^2+2)}$
- $f(x)=x^2$

- $f(z)=\dfrac{1}{z}$ is not uniformly continuous in the region

- $0\leq|z|\leq1$
- $0\leq|z|<1$
- $0<|z|<1$
- $0<|z|\leq1$

- $f(z)=z^3+3i$ is ……………..

- analytic everywhere except $z=3i$
- analytic everywhere except $z=0$
- analytic everywhere except $z=-3i$
- analytic everywhere

- If \(C\) is the circle \(|z|=3\), then \(\displaystyle \int_{c}\frac{dz}{1+z^2}\) is equal to

- $3$
- $2$
- $0$
- $1$

- The series \(\displaystyle \sum_{n=0}^{\infty}\dfrac{n^1}{(2i)^n}\) is

- convergent
- absolutely convergent
- divergent
- none of these

- The radius of convergence of \(\sinh z\) is

- $R=\infty$
- $R=0$
- $R=1$
- $R=2$

- Four married couples have bought \(8\) seats in a concert. In how many different ways can they be seated if each couple is to sit together?

- $24$
- $96$
- $384$
- None of these

- A coin is biased so that a head is twice as likely to occur as a tail. if the coin is tossed \(3\) times, then the probability of getting \(2\) tails and \(1\) head is

- $\dfrac{1}{9}$
- $\dfrac{2}{9}$
- $\dfrac{4}{9}$
- none of these

- If \(X\) represents the outcome when a die is tossed. Then the expected value of \(X\) is

- $\dfrac{1}{2}$
- $\dfrac{5}{2}$
- $\dfrac{7}{2}$
- $\dfrac{3}{2}$

### Answers

1-a, 2-b, 3-d, 4-c, 5-a, 6-a, 7-b, 8-b, 9-a, 10-c, 11-a, 12-c, 13-b, 14-c, 15-b, 16-d, 17-b, 18-a, 19-d, 20-c, 21-a, 22-a, 23-b, 24-a, 25-b, 26-d, 27-d, 28-d, 29-b, 30-d, 31-c, 32-a, 33-c, 34-a, 35-d, 36-b, 37-c, 38-d, 39-b, 40-b, 41-d, 42-c, 43-b, 44-c, 45-d, 46-b, 47-c, 48-c, 49-c, 50-c, 51-a, 52-a, 53-d, 54-d, 55-d, 56-c, 57-c, 58-b, 59-b, 60-a, 61-b, 62-a, 63-a, 64-d, 65-c, 66-b, 67-c, 68-b, 69-c, 70-a, 71-a, 72-a, 73-b, 74-b, 75-a, 76-b, 77-c, 78-a, 79-a, 80-a, 81-b, 82-d, 83-c, 84-b, 85-a, 86-a, 87-a, 88-c, 89-c, 90-d, 91-a, 92-b, 93-a, 94-d, 95-c, 96-c, 97-a, 98-c, 99-d, 100-d