On this page, we have given question from old (past) paper of Lecturer in Mathematics conducted in year 2021. 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.
How many women candidates won National Assembly seats directly in General Election \(2018\)?
$4$
\(6\)
$8$
$10$
Let \(X\) and \(Y\) be Banach spaces. Then the product space \(X\times Y\), with the norm defined by: \(\parallel (x,y) \parallel=\parallel x\parallel+\parallel y\parallel, \,\forall \, (x,y)\in X \times Y\)
Banach space
Normed space
Linear space
All of these
Let \(f(x)=[x],\) greatest integer \(\leq x;\) be integrable function on \([0,4]\), then \(\displaystyle \int_{0}^{4}[x]dx \) is equal to:
\(3\)
\(7\)
\(6\)
\(4\)
Solution of \((\bigtriangleup ^2=2\bigtriangleup +1) u_x=3x+2\) is:
\(u_x=3x+4\)
\(u_x=4x+3\)
\(u_x=3x-4\)
\(u_x=4x-3\)
The sequence \(\left\{\dfrac{2ni}{n+i}-\dfrac{(9-12i)n+2}{3n+1+7i}\right\}\) converges to :
\(3+6i\)
\(-3-6i\)
\(-3+6i\)
\(3-6i\)
Which of those is not an analytical method to solve partial differential equation?
Change of variable
Superposition principle
Finite element method
Integral transform
Order of convergence of Newton's method is :
Quadratic
Cubic
$4^{th}$
Undefined
After discretizing the partial differential equations take which if these forms?
Exponential equations
Trigonometric equations
Logarithmic equations
Algebraic equations
For a function \(f\), if \(f_{xx}=f_{xy}=f_{yy}=0\), the point \((x,y)\) will be multiple point of order:
Lower than two
Two
Higher
Higher than the two
Suppose that \(X\) and \(Y\) are closed subspaces of a Hilbert space \(H\) such that \(X \perp Y\), then \(X+Y\) is :
A closed subspace
\(X^\perp +Y^\perp\)
\(X^\perp + Y\)
Normed subspace
The function \(f(x)=x^{(-1)}\) is not:
Uniform continuous on \((0,1)\)
Continuous on \((0,1)\)
Differentiable on \([0,1)\)
Both \(A\) and \(B\)
${\rm Ln} z$ is discontinuous on axis:
Positive real
Nonpositive real
Negative
Nonnegative
Normal component of an acceleration is :
$v/p$
$v^2/p$
$p^2/v$
$p/v$
Every metric space is a :
Hausdroff space
$T_2$ space
Both \(A\) and \(B\)
$T_3$
\(\mathbb{N}'\), derived set of \(\mathbb{N}\), is:
\(\phi \)
\(\mathbb{R}\)
\(\mathbb{Q}\)
Not exist
Given \( \left(a_n\right)_{n\in \mathbb{N}}\), where \(a_n>0 \,\forall \, n\). If \(\lim\limits_{n \to \infty} a_n=l>0\), then \(\lim\limits_{n \to \infty}(a_n\ldots a_1)^{\frac{1}{n}}\) is
\(l^n\)
\(1/l\)
\(l\)
\(1/l^n\)
If \(\vec{F}\) is a continuously differentiable vector point function and \(V\) is the volume bounded by a closed surface \(S\), then \(\displaystyle \iint_{c}\vec{F}\times \vec{n}dS=\iiint_{c}{\rm div} \vec{F} dV\) is called
Gauss' divergence theorem
Surface integral
Volume integral
None of these
\(c_{ijk}c_{ijk}\) is equal :
$4$
$3$
$6$
\(2\)
\(\displaystyle \sum_{d|n} 1/d=\sigma(n)/n\) for each integer:
\(n\geq 1\)
\(n\geq 3\)
\(n\geq 7\)
None of these
Let \(A\) and \(B\) be two non-square matrices such that \(AB=A\) and \(BA=B\), then \(A\) and \(B\) are matrices:
Idempotent
Involuntary
Nilpotent
No conclusion
The Cauchy-Riemann equations can be satisfied at a point $z$, but the function $f(z)=u(x,y)+iv(x,y)$ can be at $z$:
Differentiable
Non-differentiable
Continuous
None of these
Let $G$ and $G'$ be two groups. Then a homomorphism $f:G\to G'$ is one-one iff:
\(\text{Ker } f=\{e\} \)
\(\text{Im } f=\{e\}\)
$\text{Ker } f \neq \{e\}$
\(\text{Im } f=\{e\} \)
The sequence $\left(1 / n^2 \right)_{n\in \mathbb{N}}$ is:
convergent
Cauchy
Bounded
Both A \& B
Kronecker delta is a tensor of rank :
\(3\)
\(2\)
\(1\)
any
Let \(G\) be a group and \(a\in G\) is of finite order \(n\) such that \(a^m=0\), then
\(m |n\)
\(n|m\)
\((m,n)=1\)
\(n\)
\(4\times \) volume of a tetrahedron is equal to the volume of:
Parallelepiped
Cube
Cuboid
None
Let \(A\) be the real matrix with the rows from an orthonormal set, then \(A\) is :
Normal
Orthogonal
Column of \(A\) from an orthonormal set
Both \(A\) and \(C\)
The solution \(\sin2x\) and \(\cos2x\) of the differential equation \(y''+4y=0\) are
Independent
Dependent
Wronskian of both is zero
Both \(A\) and \(C\)
Let \(V\) be an inner product space and \(u,v\in V\), Then \(|\textless u,v\textgreater|=||u||\,||v||\) iff:
\(u\) nd \( v\) are linearly independent
\(u\) and \( v\) are linearly dependent
\(u \) and \( v\) are scalar multiple of each other
Both \(B\) and \(C\)
Let \(U \) \& \(V\) be two vectors spaces such that \(T:V\to U\), a linear transformation. Then :
\(\dfrac{V}{{\rm Ker} T}\cong U\)
\( \dfrac{V}{{\rm Ker} T}\cong V\)
\(\dfrac{U}{{\rm Ker} T}\cong V\)
\(\dfrac{V}{{\rm Ker} T}\cong T(V)\)
Let $x,p\in \mathbb{R}$, $x+1>0$, $p\neq 0,1$ be such that \((1+x)^p<1+px\), then:
$0\leq p\leq 1$
$0\leq p< 1$
$0< p\leq 1$
$0< p< 1$
The Diophantine equation \(15x+51y=14\) has solution:
Unique
More than one
No
Arbitrary
Kepler stated the first law of planetary motion in:
\(1709 \)
\(1609 \)
\(1507\)
\(1607 \)
Rings of integers has characteristic:
\(1\)
\(0\)
\(\infty\)
\(-1\)
Every invertible diagonal matrix is a matrix:
Scalar
Lower triangular
Upper triangular
Both \(B\) \& \(C\)
\(\tanh^{-1}z\) is not defined for \(z\) equal to:
$1$
\(-1\)
\(\pm 1\)
Complex plane
Every homogenous system of linear equations has solution:
Trivial
Non-trivial
Parametric
None of these
Let \(H\) be a subgroup of a group \(G\) such that \(Ha\neq Hb\), then:
\(aH=bH\)
\(aH \subseteq bH\)
\(bH \subseteq aH\)
\(aH\neq bH\)
If \(S_1,S_2\) are subsets of \(V(F)\) and \(L(S_1)\) is the linear space of \(S_1\), then:
\( L(L(S_1))=L(S_1)\)
\( L(S_1\cup S_2)=L(S_1)+L(S_2)\)
Both \(A\) and \(B\)
\( L(S_1)\subseteq L(S_2)\)
The summation index is also called:
Dummy index
Free index
Convention
Both \(A\) and \(B\)
Reduced echelon form of a matrix is:
Unique
Not unique
Pivot element are 1
Both A and C
The number of asymptotes of an algebraic curve of the $n$th degree:
Exceed \(n\)
Cannot exceed \(n\)
Exactly \(n\)
Both $A$ and \(C\)
Let $H$, $K$ be subgroups of a group \(G\), Then \(HK\) is a subgroup of \(G\) iff:
\(HK = KH\)
\(HK \neq KH\)
\(H^{-1}=K^{-1}\)
\((HK)^{-1}=K^{-1}H^{-1}\)
If \(A\) and \(B\) are two ideals of a ring \(R\), Then \(A+B\) is an ideal of \(R\) containing:
\(A\)
\(B\)
Both $A$ \& \(C\)
None of these
A normed space \(X\) is finite dimension iff \(X\) is:
Compact
Connected
Locally compact
Homeomorphic
The symbol $A_{ijk}$, $\{i,j,k=1,2,3\}$ denotes numbers:
\(27\)
\(9\)
\(8\)
\(4\)
The function \(f(z)\) is analytical in a domain \(D\) and \(f(z)=c+iv(x,y)\), where \(c\) is a real constant. Then \(f\) in \(D\) is a:
Constant
Nonconstant
Continuous
None of these
The centre of curvature at any point \(P\) of a curve is the point which on the positive direction of the normal at \(P\) and is at a distance
\(x\,(keps)\)
\(\ell(rho)\)
\(\dfrac{1}{x}\)
\(\dfrac{1}{\ell}\)
Time of flight of the projectile is:
\(\dfrac{2v_0 \sin\alpha}{-g}\)
\(\dfrac{2v_0 \cos\alpha}{g}\)
\(\dfrac{v_0 \sin2\alpha \sec \alpha}{g}\)
\(\dfrac{v_0 \sin2\alpha}{-g\cos\alpha}\)
For a function \(f(x,y)\) in a region \(D\) in \(xy\) plane, the condition \(|f(x,y_2)-f(x,y_1)|\leq K|y_2-y_1|\) is called Lipschitz, provided that:
\(K=0\)
\(K>0\)
\(K<0\)
\(K\in \mathbb{R}\)
If \(k\) integers \(a_1,a_2,...,a_k\) form a complete residue system modulo \(m\), then:
\(m<k\)
\(m\geq k\)
\(m\leq k\)
Both \(B\) \& \(C\)
the function \(f(x)=x|x|\) at the origin is only
Differentiable
Continuous
Both A and B
None of these
Every subgroup of an abelian group is :
Normal
Cyclic
Abelian
None of these
A point where a curve cross on a tangent is known as:
Maximum point
Minimum point
Point of inflection
None of these
A vector space \(V(F)\), if possible, have:
A unique basis
Two bases
At least one basis
None of these
Method of factorization is also called :
Method of factorization
Method of triangularization
Indirect method
Both A and B
Every infinite dimensional normed space has a subspace which is:
Closed
Not closed
Connected
Both B and C
For any real matrix \(A\) such that \(AA^t=A^tA\), we have \(A\):
Orthogonal
Normal
Both A and B
None of these
The locus of the centres of curvature of a curve is called its evolute and a curve is said to be an:
Evolute of its involute
Involute of its evolute
Both A and B
None of these
Suppose that \(u_1,u_2\) are non-zero orthogonal vectors in \(\mathbb{R}^n\), then for \(v\in \mathbb{R}^n\) we have:
\(v=\dfrac{v\cdot u_1}{u_1\cdot u_1}u_1+\dfrac{v\cdot u_2}{u_2\cdot u_2}u_2\)
\(v=\dfrac{v\cdot u_1}{u_1\cdot u_1}u_2+\dfrac{v\cdot u_2}{u_2\cdot u_2}u_1\)
\(v=\dfrac{v\cdot u_1}{u_1\cdot u_2}u_1-\dfrac{v\cdot u_2}{u_2\cdot u_2}u_2\)
None of these
The zeros of the function \(f(z)=\sin\dfrac{\pi(1-z)}{z}\) are:
$1$
$-1$
$\pi$
$-\pi$
The interval \([3,5)\) with respect to usual topology is:
Open
Closed
Semi open
None of these
Which one is more reliable, Simpson's rule or Trapezoidal rule?
Trapezoidal rule
Simpson's rule
Both A and B
None of these
Let \(X\) be a normed space such that norm obeys the parallelogram law, \(X\) can be made:
Inner product
Hilbert space
A linear space
No conclusion
The greatest and the least values of a function \(f\) in an interval \([a,b]\) are \(f(a)\) or \(f(b)\) or are given by the values of \(x\) for which?
\(f'(x)>0\)
\(f'(x)<0\)
Both A and B
\(f'(x=0)\)
The mapping \(w=z^2+1\) is conformable at:
$z=-1$
$z=1$
$z=\pm1$
None of these
The order of the continuity equation of steady two-dimensional flow is:
\(1\)
\(0\)
\(2\)
\(3\)
Let \(a\) and \(m>0\) be integers with \(a^{\phi(m)}\equiv1 \pmod m\) provided that:
$a>m$
$m<a$
$(a,m)=1$
(a,m)$\neq 1$
Let \(\vec{F}(\vec{r})\) be a continuous vector point function defined on smooth curve \(C\) given by $\vec{r}=\vec{f}(t)$, then $\displaystyle \int_{C} \vec{F} \times d\vec{r}$ is called
Line integral
Surface integral
Volume integral
None of these
The frequency of a simple harmonic motion is:
\(\dfrac{\sqrt{\lambda}}{2pi}\)
\(\dfrac{\lambda}{\pi}\)
\(\dfrac{\lambda}{\sqrt{2\pi}}\)
\(\sqrt{\dfrac{\lambda}{2\pi}}\)
Let \(u_1,u_2,...,u_n\) be a linear dependent set of functions on \(x\in [a,b]\) and let each function be \((n-1)\) times differentiable in \((a,b)\). Then the Wronskian of the set of the functions is:
Zero
Positive
Negative
An integer
\(\dfrac{\tan x}{x}>\dfrac{x}{\sin x}\) is true for:
$0\leq x \leq \dfrac{\pi}{2}$
$0< x< \dfrac{\pi}{2}$
$0\leq x< \dfrac{\pi}{2}$
$0< x\leq \dfrac{\pi}{2}$
While solving a partial differential equation using a variable separable method, we equate the ratio to a constant?
Can be positive or negative integer or zero
Can be positive or negative number or zero
Must be a positive integer
Must be negative integer
A particle of mass \(m\) moves in a circle of radius \(r\) with constant speed \(v\) and \(F\) an acting force, then:
\(F \propto \frac{mv^2}{r}\)
\(F \propto \frac{mv}{r}\)
\(F \propto\frac{(mv^2)}{\sqrt{r}}\)
None of these
Let \(W(F)\) be a subspace of a finite dimensional vector space \(V(F)\), then \(\dim(V/W)\) is :
$\dim V-\dim W$
$\dim V+\dim W$
$\dim V+\dim W-\dim (V \cap W)$
$\dim V-\dim W+\dim (V \cap W)$
If $L$,$M$ and $N$ are three subspaces of a vector space $V$ such that \(M\subseteq L\), then:
$L \cap (M+N)=(L\cap M)+(L\cap N)$
$L \cap (M+N)=M+(L\cap N)$
$L \cup (M+N)=(L\cup M)+(L\cup N)$
Both A and B
Changes in sign but not in magnitude when the cyclic order is changed is possible in:
Vector triple product
Scalar triple product
Mixed product
Both B and C
Let \(A\) be a subspace of a topological space \(X\); let \(\bar{A}\) be its closure, then \(\bar{A}\) is equal: (provided that $A^\circ$, $b(A)$ and $A'$ are respectively interior, boundary and set of accumulation points of \(A\) respectively)
$A^\circ \cup b(A)$
$A \cup A'$
Both A and B
$A\cap A'$
Let \(\vec{f}(x,y,z)\) be continuously differentiable vector point function then CurlCurl\(\vec{f}+\nabla^2 f\) is:
grad div\(\vec{f}\)
div grad\(\vec{f}\)
div Curl\(\vec{f}\)
Curl div\(\vec{f}\)
A square unitary matrix with real entries is:
Orthogonal
Normal
None
Leslie
Every triangular matrix is:
Diagonal
Lower triangular
Invertible diagonal
Both A and B
Printing press was invented by
Mary Anderson
Johannes Gutenbery
George Antheil
Victor Adler
Which of the following is used for the purification of water?
Oxygen
Ammonia
Chlorine
Carbon Dioxide
Faiz Ahmed Faiz was imprisoned for his alleged involvement in ———– conspiracy.
Agartala
Lahore
Attock
Rawalpindi
Turkey connects which two continents?
Asia and Europe
Asia and Africa
South America and North America
Asia and Australia
Aljazeera TV channel belong to:
Qatar
Kuwait
Egypt
Bahrain
In the period of which pious caliph Quranic verses were collected in one place?
Hazrat Umar (RA)
Hazrat Abu Bakar (RA)
Hazrat Ali (RA)
Hazrat Usman (RA)
Given the name of the Sahabi who was given the title of Ateeq
Hazrat Abu Bakar (RA)
Hazrat Umar (RA)
Hazrat Ali (RA)
Hazrat Zaid Bin Sabit (RA)
Choose the correct pronoun;- Can you please return the calculator —– you borrow yesterday?
Who
Whom
That
Whose
Fill in the suitable preposition;- “The shop is open from \(7 \)am —– \(5\)pm.
At
Until
Above
On
In computing, WAN stands for:
World Area Network
Wide Area Network
World Access Network
Wireless Access Network
What is the shortcut key to hide entire column in MS Excel sheet?
CTRL+O
CTRL+A
CTRL+H
CTRL+I
Which one is the capital city of Oman?
Adam
Muscat
As Sib
Bahia
Tarbela is ———– dam.
Rock fill
Earth fill
Concrete
None of these
The famous pre-historic monument Stonehenge is found in —–
Greece
China
England
None of these
When India stopped supply of water to Pakistan from every canal flowing from India to Pakistan for first time after creation
April 1st, 1947
April 1st, 1948
April 1st, 1949
April 26, 1948
The Pakistan International Airlines came into being in the year
1952
1953
1954
1955
Sepoys in the British army raised in revolt from the city of —–
Meerut
Delhi
Lucknow
Calcutta
کے درست مطلب کا انتخاب کریں ``By Leaps and Bounds”
کتے کی طرح لپکنا
جنگجو ہونا
رفتہ رفتہ آگے بڑھنا
تیزی سے ترقی کرنا
اردو میں منقوط حروف کی تعدادہے
پندرہ
سترہ
انیس
اکیس