====== PPSC Paper 2011 (Lecturer in Mathematics) ======
{{ :ppsc:ppsc-maths-2021.jpg|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. [[:people: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

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