PPSC Paper 2015 (Lecturer in Mathematics)

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 Kaushef Salamat. We are very thankful to her for providing this paper.

  1. \(\displaystyle \int_{-4}^{0}\frac{tdt}{\sqrt{16-t62}}\)
    1. $0$
    2. divergent
    3. $-4$
    4. $4$
  2. The period of the function\(A\cos wt+B\sin wt\) is
    1. $\dfrac{\omega}{2\pi}$
    2. $2 \pi \omega$
    3. $\dfrac{\omega}{2\pi}$
    4. $\dfrac{2\pi}{\omega}$
  3. \(A=(-4x-3y+az)\underline{i}+(bx+3y+5z)\underline{j}+(4x+cy+3z)\underline{k}\) is irrational when \(a,b,c\) are
    1. \(4,-3,5\)
    2. \(4,5,-3\)
    3. \(-3,4,5\)
    4. \(2,3,5\)
  4. \(V=(-4x-6y+3z)\underline{i}+(-2x+y-5z)\underline{j}+(4x+6y+az)\underline{k}\) is is-solenoidal for \(a=------\)
    1. \(1\)
    2. \(2\)
    3. \(3\)
    4. \(4\)
  5. $\displaystyle \int_{(2,1)}^{(0,0)}(10x^4-2xy^3)dx-3x^2y^2dy$ along the path \(x^4-6xy^3=4y^2\) is ———–
    1. \(56\)
    2. \(60\)
    3. \(62\)
    4. \(64\)
  6. If \(S\) is the closed surface and \(V\) is the volume enclosed by \(S\) then \(\displaystyle \iint_{s}\underline{r}\cdot \underline{n} ds=\)————-
    1. $v$
    2. $2v$
    3. $3v$
    4. \(4v\)
  7. Centrifugal acceleration is ————
    1. $-\omega \times (\omega\times r)$
    2. $\omega \times (\omega\times r)$
    3. $\omega . (\omega\times r)$
    4. $r \times (\omega\times r)$
  8. Number of the degrees of freedom of two particles connected by a rigid road moving freely in the plane is —–
    1. $2$
    2. $3$
    3. $4$
    4. \(5\)
  9. The centroid of a uniform semicircular wire of radius \(a\) is
    1. \(\dfrac{2a}{\pi} \)
    2. \(\dfrac{4a}{\pi} \)
    3. \(\dfrac{a}{\pi} \)
    4. \(\dfrac{a}{2\pi} \)
  10. Moment of inertia of a rectangular plate with sides \(a,b\) about an axis \(\perp\) to plate and passing through vertex is ————
    1. \(\frac{1}{3}Ma^2\)
    2. \(\frac{1}{3}Mb^2\)
    3. \(\frac{1}{3}M(a^2-b^2)\)
    4. \(\frac{1}{3}M(a^2+b^2)\)
  11. Every bounded infinite set has at least one limit point is the statement of ——
    1. Hein-Borel Theorem
    2. welerstrass-Boizano theorem
    3. Cantor's intersection theorem
    4. none of these
  12. $\lim\limits_{x\longrightarrow 0}\frac{\bar{z}}{z}=$
    1. $\frac{1+i}{1-i}$
    2. $1$
    3. Does not exist
    4. \(-1\)
  13. Cauchy-Riemann equations in polar form are ——
    1. $\frac{\delta u}{\delta r}=\frac{1}{r}\frac{\delta v}{\delta \theta},\frac{\delta v}{\delta r}=-\frac{1}{r}\frac{\delta u}{\delta \theta}$
    2. $\frac{\delta u}{\delta r}=\frac{-1}{r}\frac{\delta v}{\delta \theta},\frac{\delta v}{\delta r}=\frac{1}{r}\frac{\delta u}{\delta \theta}$
    3. $\frac{\delta u}{\delta r}=\frac{-1}{r}\frac{\delta v}{\delta \theta},\frac{\delta v}{\delta r}=-\frac{1}{r}\frac{\delta u}{\delta \theta}$
    4. $\frac{\delta u}{\delta r}=\frac{1}{r}\frac{\delta v}{\delta \theta},\frac{\delta v}{\delta r}=\frac{1}{r}\frac{\delta u}{\delta \theta}$
  14. Evaluate \(\displaystyle \int_{C}\frac{z^2-z+1}{z-1}dz \), where \(C\) is the circle \(|z|=\dfrac{1}{2}\):
    1. $1$
    2. $2$
    3. $\frac{1}{2}$
    4. $0$
  15. The principal value of \((-i)^i\) is:
    1. \(e^{\frac{\pi}{2}}\)
    2. \(1\)
    3. \(e^{\frac{\pi}{2}}\)
    4. \(e^{\pi}\)
  16. The residue of \(f(z)=\dfrac{z^2-2z}{(z+1)^2(z^2+4)} \) at \(z=2i\) is ———-
    1. $\dfrac{14}{25}$
    2. $\dfrac{7+i}{25}$
    3. $\dfrac{7-i}{25}$
    4. $\dfrac{-7-i}{25}$
  17. Radius of convergence of \(\displaystyle \sum (3+4i)^nz^n\) is ———–
    1. $\dfrac{1}{5}$
    2. $5$
    3. $7$
    4. \(\infty\)
  18. \(\lim\limits_{n\to \infty}\left(1+\dfrac{x}{n}\right)^n\) is ————
    1. \(1\)
    2. \(0\)
    3. \(e^x\)
    4. \(e^n\)
  19. \(U(x,y)=e^x \cos y\) is ——–
    1. Harmonic
    2. Analytic
    3. Not harmonic
    4. None of these
  20. \(\displaystyle \int_{0}^{\infty} \frac{\sin x}{x}dx\)= ————-
    1. \(0\)
    2. \(\dfrac{-\pi}{2}\)
    3. $\dfrac{\pi}{2}$
    4. \(\pi\)
  21. \(\log(1+i)\)= ———-
    1. $\dfrac{1}{2}\ln2+\dfrac{\pi i}{4}$
    2. $\dfrac{1}{2}\ln2-\dfrac{\pi i}{4}$
    3. $\dfrac{1}{2}\ln2-\dfrac{3\pi i}{4}$
    4. $\dfrac{1}{2}\ln2+\dfrac{3\pi i}{4}$
  22. Which of the following space is complete.
    1. \(Q\)
    2. \(]0,1]\)
    3. \(Z\)
    4. \(R\)
  23. Least upper bound of \(\left\{\dfrac{1}{2},\dfrac{2}{3},\dfrac{3}{4},...\right\}\) is
    1. \(0\)
    2. \(1\)
    3. \(\infty\)
    4. \(\dfrac{n}{n+1}\)
  24. \(\lim\limits_{x \to 1}\dfrac{x^3-x}{1-x+\ln x}\) is ———–
    1. \(2\)
    2. \(-2\)
    3. \(1\)
    4. \(-1\)
  25. \(\lim\limits_{x \to 0}x^{\sin x}\) is ————
    1. \(0\)
    2. \(n\dfrac{1}{2}\)
    3. \(e\)
    4. \(\infty\)
  26. Minimum and maximum values of \(f(x)=x^{\frac{2}{3}}(x^3-8)\) in interval \(\left[-1,\dfrac{1}{2} \right]\) are
    1. \(-7,0\)
    2. \(0,6\)
    3. \(1,2\)
    4. \(-2,3\)
  27. \(\displaystyle \int_{0}^{1}\dfrac{4}{1+x^2}dx\)= ————
    1. \(0\)
    2. \(\pi\)
    3. \(\dfrac{4\pi}{3}\)
    4. \(-\pi\)
  28. \(\displaystyle \int_{0}^{\pi}cosec^2 xdx\)= ————
    1. \(0\)
    2. \(1\)
    3. \(-1\)
    4. \(\infty\)
  29. \(\lim\limits_{x\to 0}\sin \dfrac{1}{x} =\) ————-
    1. does not exist
    2. \(0\)
    3. \(-1\)
  30. $\displaystyle \int_{0}^{\frac{3\pi}{4}}|\cos x|dx=$ ————
    1. $\dfrac{1}{\sqrt{2}}$
    2. $\dfrac{-1}{\sqrt{2}}$
    3. $\infty$
    4. $2-\dfrac{1}{\sqrt{2}}$
  31. \(\sec \left(\tan^{-1}\frac{2}{3} \right)=\) ————–
    1. $\dfrac{2}{\sqrt{13}}$
    2. $\dfrac{3}{\sqrt{13}}$
    3. $\dfrac{\sqrt{13}}{3}$
    4. $\dfrac{\sqrt{13}}{2}$
  32. Which of the following is convergent series?
    1. \(\sum\dfrac{1}{n^2} \)
    2. \(\sum\dfrac{1}{\sqrt{n}} \)
    3. \(\sum\dfrac{1}{n}\)
    4. \(\sum\dfrac{1}{n^{1/3}} \)
  33. \(x-\dfrac{x^3}{3!}+\dfrac{x^5}{5!}-\dfrac{x^7}{7!}+...\) is the Maclaurin's series of ———–
    1. \(\cos x\)
    2. \(\sin x\)
    3. \(\sinh x\)
    4. \(\cosh x\)
  34. \(\displaystyle \int_{1}^{2}\frac{x}{y^2} dxdy =\)————–
    1. \(\dfrac{3}{4}\)
    2. \(\dfrac{7}{8}\)
    3. \(\dfrac{3}{2}\)
    4. \(\dfrac{1}{2}\)
  35. Domain of \(f(x)=\sqrt{1-x^2}\) is —————-
    1. $x<1$
    2. \(x>1\)
    3. \(|x|\leq 1\)
    4. \(|x|\geq1\)
  36. Domain of \(f(x)=\dfrac{1}{\sqrt{(1-x)(2-x)}}\) is —————
    1. \(\mathbb{R}\setminus [1,2]\)
    2. \(\mathbb{R}\setminus \{1,2\}\)
    3. \([1,2]\)
    4. \(]1,2[\)
  37. \(f:\mathbb{R}\to (-1,1)\) defined by \(f(x)=\)—————- is bijective.
    1. \(\dfrac{x}{1-|x|}\)
    2. \(\dfrac{x}{1+|x|}\)
    3. \(\dfrac{1}{1+|x|}\)
    4. \(\dfrac{x}{-1+|x|}\)
  38. Interval of convergence of \(\sum_{k=1}^{\infty}x^k \) is —————
    1. \( ]-1,1[\)
    2. \( [-1,1]\)
    3. \( (-\infty,+\infty)\)
    4. \( x=0\)
  39. Which of the following are in the usual metric space \( (R,d) \)?
    1. Subset of \(\mathbb{R}\)
    2. Union of open interval
    3. Intervals
    4. Singleton subsets
  40. Let \(A=(0,1]\cup (1,3]\) and \(R\) with usual metric space. Then \(A^\circ=\) ————-
    1. $A\setminus \{0\}$
    2. $A\setminus \{1\}$
    3. {$A\setminus \{3\}$}
    4. $(0,1)\cup (1,3)$
  41. Let \(A\) be finite subset of a metric space \(X\). Then \(A^d=\) ————
    1. Singleton set \(0\)
    2. \(\phi\)
    3. \(A\)
    4. \(X\setminus A\)
  42. Let \(A\) be a finite subset of \(X,d\) then \(A\) is —–
    1. Open set
    2. Open as well as closed
    3. Closed set
    4. Neither open nor closed
  43. If \(Y\) is a subset of \((X,d)\) then ————
    1. Every open set in \(Y\) is open in \(X\)
    2. Every open set in \(X\) is open in \(Y\)
    3. \(O\) is open in \(Y \Longleftrightarrow O\) is open in \(X\)
    4. \(O\) is open \( \Longleftrightarrow O=Y\cap G\) where \(G\) is open in \(X\)
  44. Let \(f(x)=1+x^3\). Then \((0,0)\) is the point of ———–
    1. Maximum value
    2. Minimum value
    3. Point of inflection
    4. None of these
  45. Number of elements in a co-finite topological space \((X,\tau)\) where \(X=\{s,t,u\}\) is —————
    1. \(2\)
    2. \(3\)
    3. \(4\)
    4. \(8\)
  46. The boundary of a subset \(B=\left\{\dfrac{1}{n}:n\in N \right\}\) of \((\mathbb{R},d)\) is —————
    1. $B$
    2. $\{0\}$
    3. $B\cup\{0\}$
    4. $\phi$
  47. The real line \(\mathbb{R}\) is a homeomorphic to —————-
    1. \((0,4)\)
    2. \(\{-1,1\}\)
    3. \(Q\)
    4. \(T_2\)-space
  48. \(\mathbb{R}\) with co-finite topology is —————
    1. \(T_0\)-space
    2. \(T_1\)-space
    3. \(T_1\)-space but not \(T_2\)-space
    4. 3
  49. Let \(X=\{a,b,c\}\), \(\tau=\{\phi,\{a\},\{b\},\{a,b\},X\}\). Then \(X\) is —–
    1. \(T_1\)-space
    2. Regular space
    3. \(T_2\)-space
    4. Normal space
  50. Which of the following is connected in \(\mathbb{R}\) with usual topology?
    1. \(\mathbb{N}\)
    2. \(\mathbb{Q}\)
    3. \((0,1]\)
    4. \(\mathbb{Z}\)
  51. Which of the following topology is not totally disconnected?
    1. $\{1\}$
    2. discrete space
    3. $\mathbb{R}$ with usual topology
    4. $\mathbb{Q}$
  52. Which of the following is nowhere dense in \(\mathbb{R}\):
    1. \(\mathbb{R}\setminus \mathbb{\mathbb{Z}}\)
    2. \(\mathbb{Z}\)
    3. \(\cup (n,n+1),n \in \mathbb{Z}\)
    4. \(\mathbb{Q}\)
  53. Which of the following is dense in \(\mathbb{R}\):
    1. \(\mathbb{N}\)
    2. \(\mathbb{Z}\)
    3. \(\mathbb{R}\setminus \mathbb{Z}\)
    4. \(\mathbb{Q}\)
  54. \(xy''+y'=0\) has a solution \(y=\ln x\) on interval ————
    1. \((0,\infty)\)
    2. \((-\infty,0)\)
    3. \((-\infty,\infty)\)
    4. \([0,\infty[\)
  55. Which of the following is not linear?
    1. $y'=(\sin x )y$
    2. $y'=(\sin y )x+e^x$
    3. $y'+xy=e^xy$
    4. $y'=5$
  56. Solution of \(y'=\dfrac{x+y}{x}\) is ————–
    1. $y=\ln |kx|$
    2. $y=\ln |x|$
    3. $y=x \ln |kx|$
    4. $y=\ln |x|+k$
  57. Which of the following differential equation is not exact?
    1. $2xydx+(1+x^2)dy=0$
    2. $ydx-xdy=0$
    3. $y'=\dfrac{2+ye^{xy}}{2y-xe^{xy}}$
    4. \(x+\sin y\)dx+\(x\cos y-2y\)dy
  58. Integrating factor for \(y'+\left(\dfrac{4}{x}\right)y=x^4\) is ———–
    1. $x^4$
    2. $\ln x^4$
    3. $4\ln |x|$
    4. $\ln |x|$
  59. The area bounded by \(y=4-x^2\) and x-axis is —————
    1. \(\dfrac{4}{3}\)
    2. \(\dfrac{8}{3}\)
    3. \(\dfrac{16}{3}\)
    4. \(\dfrac{32}{3}\)
  60. Which of the following is scalar?
    1. $(\underline{a}\cdot \underline{b})\underline{c}$
    2. $\underline{a}\cdot(\underline{b}\times\underline{c})$
    3. $\underline{a}\times(\underline{b}\times\underline{c})$
    4. $(\underline{a}\cdot \underline{b})(\underline{a}-\underline{a})$
  61. Projection of \(\underline{a}\) on \(\underline{b}\) is —————
    1. $\underline{a} \cdot \underline{b}$
    2. $\dfrac{\underline{a}}{|\underline{a}|}\cdot\underline{b}$
    3. $\underline{a}\cdot\dfrac{\underline{b}}{|\underline{b}|}$
    4. $\underline{a}\times\underline{b}$
  62. Which of the following is scalar quantity?
    1. Momentum
    2. Magnetic field intensity
    3. Special heat
    4. Moment of force
  63. A vector lying in the plane of \underline{a} and \underline{b} is ——–
    1. \((\underline{a}\times\underline{b})\times \underline{c}\)
    2. \(\underline{a}\times(\underline{b}\times\underline{c})\)
    3. \((\underline{c}\times\underline{a})\times\underline{b}\)
    4. \((\underline{c}\times\underline{b})\times\underline{a}\)
  64. Let \(\underline{t}\), \(\underline{n}\) and \(\underline{b}\) denoted respectively the tangent, principal normal and binormal vector to the cure then osculating plane to the curve at \(P\) contains ——
    1. \(\underline{t}\), \(\underline{b}\)
    2. \(\underline{n}\), \(\underline{b}\)
    3. \(\underline{t}\), \(\underline{n}\)
    4. \(\underline{t}\), \(\underline{n}\), \(\underline{b}\)
  65. Let \(\underline{t},\underline{n}\) and \(\underline{b}\) be as in the above question. Then \(\tau\underline{b}-k\underline{t}=\) ——-
    1. $\dfrac{dt}{ds}$
    2. $\dfrac{dn}{ds}$
    3. $\dfrac{db}{ds}$
    4. $\dfrac{d}{ds}\left(\dfrac{\underline{t}\times\underline{n}}{ds}\right)$
  66. Normal plane is perpendicular to ————–
    1. \(\underline{t}\)
    2. \(\underline{n}\)
    3. \(\underline{b}\)
    4. \(\underline{t}\times \underline{n}\)
  67. \(\underline{t}\times \underline{b}\)= ————–
    1. $\underline{n}$
    2. $-\underline{n}$
    3. $\underline{n}\times \underline{b}$
    4. none of these
  68. \(\{x |x \in \mathbb{C}:x^4=1\} \) is a —————
    1. Subgroup of \((\mathbb{C}\setminus\{0\},\cdot)\)
    2. Subgroup of \((\mathbb{C},+)\)
    3. None cyclic group
    4. Subgroup of \((\mathbb{Q}\setminus\{0\},\cdot)\)
  69. \(\mathbb{R}^3\) under vector product forms a ————–
    1. group
    2. semi-group
    3. groupoid
  70. An element \(x\) of group \(G\) satisfying \(x^2=x\) is called —————
    1. Involution
    2. Idempotent
    3. Transposition
    4. Cycle
  71. \(\dfrac{\mathbb{Z}}{\langle n \rangle}\) is isomorphic to ————–
    1. $n\mathbb{Z}$
    2. \(\langle n \rangle \)
    3. \(\mathbb{Z}_n\)
    4. \( \{0,\pm2n,\pm4n,....\}\)
  72. Let \( G=\langle a:a^{12}=e\rangle\). Then \(G=\) ————–
    1. $\langle a^5 \rangle$
    2. $\langle a^6 \rangle$
    3. $\langle a^2 \rangle$
    4. $\langle a^8 \rangle$
  73. Let \( G=\langle b:b^{17}=e\rangle\). Then \( G \) can be generated by —————
    1. Any element of \( G \)
    2. Any non-identity element of \(G \)
    3. \(b,b^{-1}\) are the only generators of \(G\)
    4. Identity
  74. Let \(G=\langle \alpha, \beta:\alpha^3=\beta^2=(\alpha \beta)^2=e\rangle\). Then \(N_G(\{e,\beta\})=\) ————–
    1. $\{e\}$
    2. $\{e,\beta,\alpha \beta\}$
    3. $G$
    4. $\{e,\beta\}$
  75. Let \(G=\langle \alpha, \beta:\alpha^4=\beta^2=(\alpha \beta)^2=e\rangle\). Then \(Z(G)=\) —————
    1. $\{e\}$
    2. $\{e,\alpha^2\}$
    3. $\{e,\alpha,\alpha^2,\alpha^3\}$
    4. $G$
  76. Which of the following is not true for an abelian group \(G\)?
    1. $[a,b]=e \, \forall \, a,b \in G$
    2. \(G\) is simple group of order \(60\)
    3. $G'=\{0\}$
    4. $Z(G))=G$
  77. Inner automorphism of \(Q=\{\pm1, \pm i,\pm j\}\) is ———–
    1. $\{e\}$
    2. $C_2\times C_2$
    3. $Q$
    4. $C_4$
  78. Number of conjugacy classes of a cyclic group of order \(6\) is ————
    1. \(1\)
    2. \(2\)
    3. \(3\)
    4. \(6\)
  79. Number of non-isomorphic abelian groups of order \(12\) is ———–
    1. $1$
    2. $2$
    3. $3$
    4. $4$
  80. Order of sylow-2 subgroup of \(Q_8\) is ————
    1. $1$
    2. $2$
    3. $3$
    4. $8$
  81. Which of the following is an ideal of \(\mathbb{R}\)?
    1. \(\mathbb{Z}\)
    2. \(\{0\}\)
    3. \(\mathbb{C}\)
    4. \(\mathbb{Q}\)
  82. Which of the following is not an integral domain?
    1. \(\mathbb{Z}\)
    2. \(\mathbb{Z}_7\)
    3. \(\mathbb{Q} \)
    4. Set \(M_2\) of \(2\times 2\) matrices with integer entries
  83. Which of the following is a field?
    1. $\{a+b\sqrt{2}:a,b \in \mathbb{Q}\}$
    2. \(\mathbb{Q}\setminus \{0\}\)
    3. $\mathbb{Z}$
    4. $\mathbb{Z}_6$
  84. Which of the following is not a vector space?
    1. $\mathbb{R}(\mathbb{R})$
    2. $\mathbb{R}(\mathbb{Q})$
    3. $\mathbb{R}(\mathbb{C})$
    4. $\mathbb{R}(\mathbb{Q})$
  85. Let \(\phi:\mathbb{Z} \to \mathbb{Z}_5\) be \(\phi(a)=a \pmod 5\). Then \(Ker(\phi)=\) —————
    1. $\{0\}$
    2. $\{0,\pm5,\pm10,...\}$
    3. $\mathbb{Z}_5$
    4. $\mathbb{Z}$
  86. The number of proper ideals of \(\mathbb{Z}_{17}\) is ————
    1. \(0\)
    2. \(1\)
    3. \(2\)
    4. \(3\)
  87. Which of the following is a division ring?
    1. $(\mathbb{Z},+,\cdot)$
    2. $(\mathbb{E},+,\cdot)$
    3. $(\mathbb{Q},+,\cdot)$
    4. $(\mathbb{Z}_6,\oplus_6,\odot_6)$
  88. \(\displaystyle \int_{-1}^{2}(x+|x|)dx = \)
    1. \(0\)
    2. \(4\)
    3. \(2\)
    4. \(6\)
  89. \(x=6\) in \(\mathbb{R}^3\) represents a
    1. Point
    2. Line
    3. Plane
    4. Space
  90. Kernel of \(T:\mathbb{R}^3 \to \mathbb{R}^3\), where \(T(x,y,z)=(x,y,0)\), is
    1. Point
    2. Line
    3. Plane
    4. Space
  91. Dimension of \(Hom(\mathbb{R}^3,\mathbb{R}^4)=\) —————
    1. $3$
    2. $4$
    3. $7$
    4. $12$
  92. Dimension of \(Hom(M_{2,4},P_2(t))=\) ————
    1. $4$
    2. $8$
    3. $16$
    4. $24$
  93. A dice is thrown. The probability that the dots on the top are prime numbers or odd numbers is
    1. $\dfrac{1}{3}$
    2. $\dfrac{2}{3}$
    3. $1$
    4. $\dfrac{5}{6}$
  94. A coin is tossed 4 times in succession. The probability that at least one head occurs is
    1. $\dfrac{1}{16}$
    2. $\dfrac{4}{16}$
    3. $\dfrac{12}{16}$
    4. $\dfrac{15}{16}$
  95. Number of necklaces made from 9 beads of different colours is ————–
    1. $\dfrac{8!}{2}$
    2. $8!$
    3. $7!$
    4. $9!$
  96. Period of \(3\cos\dfrac{x}{5}\) is ————–
    1. \(2\pi\)
    2. \(\dfrac{2\pi}{5}\)
    3. \(6\pi\)
    4. \(10\pi\)
  97. Range of \(\sec^{-1}x\) is ———–
    1. $[0,\pi]$
    2. $[0,\pi]\backslash \frac{\pi}{2}$
    3. $[\frac{-\pi}{2},\frac{\pi}{2}]$
    4. $[\frac{-\pi}{2},\frac{\pi}{2}]\backslash \{0\}$
  98. Solution set of \(\sin x\cos x=\dfrac{\sqrt{3}}{4}\) is ————
    1. $\{\frac{\pi}{6}+n\pi\}\cup \{\frac{\pi}{3}+n\pi\}$
    2. $\{\frac{\pi}{3}+2n\pi\}\cup \{\frac{2\pi}{3}+2n\pi\}$
    3. $\{\frac{\pi}{6}+2n\pi\}\cup \{\frac{5\pi}{6}+2n\pi\}$
    4. $\{\frac{\pi}{12}+n\pi\}\cup \{\frac{5\pi}{12}+n\pi\}$
  99. Which of the following is tautology?
    1. $p\to \sim q$
    2. $(p\to q)\cap(p\,\,q)$
    3. $p\to q \to \sim q\to \sim q$
    4. \(p \cap \sim p\)
  100. \(f(x)=\dfrac{1}{x}\) is not uniformly continuous in the region ————–
    1. $0\leq |z|\leq 1$
    2. $0 < |z|\leq 1$
    3. $0< |z|< 1$
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