List of top Mathematics Questions

If \(\vec{F} = (x+2y+az)\hat{i} + (bx −3y-z)\hat{j}+(4x+cy+2z)\hat{k}\) is irrotational, where a, b and c are constant, then a² + b² + c² is equal to :

If \(\int \int\limits_{R} \int xyz\ dxdydz=\frac{m}{n}\) where, m,n, are coprime and R:0≤x≤1,1≤ y ≤2, 2 ≤ z ≤3 , then m.n is equal to:

Match List I with List II

Choose the correct answer from the options given below:

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R
Assertion A: The integral \(\int \int \int (x^2+y^2+z^2)dxdydz\) taken over the volume enclosed by the sphere x²+ y²+z² = 1 is \(\frac{4\pi}{5}\)
Reason R: \(\int^{1}_{0}\int^{1}_{0}x\ dxdy=\frac{1}{2}\)
In the light of the above statements, choose the most appropriate answer from the options given below:

If \(\vec{r}=x\hat{i}+y\hat{j}+z\hat{k}\) and \(r=\sqrt{x^2+y^2+z^2}\), then grad \((\frac{1}{r})\) is equal to :

If\(ƒ(x + iy) = x^3 − 3.xy^2 + ¡\Psi(x,y) \space where \space i= \sqrt{-1}\) and ƒ (x+iy) is an analytic function, then \(\Psi (x, y)\) is: 

Let \(Z^3 = \bar Z\) where Z is a complex number on the unit circle then Z is a solution of _____:

Given below are two statements
Statement I: If A = \(\begin{bmatrix}2 &2\\ 1& 3\end{bmatrix}\) then sum of eigenvalues of A is 3.
Statement II: If \(λ\) is an eigenvalue of \(T\), where \(T\) is invertible linear operator, then \(λ^{-1}\) is an eigenvalue of \(T^{-1}\)
In the light of the above statements, choose the correct answer from the options given below

If a right circular cylinder a height 14cm is increased in a sphere of radius 8cm then volume of the cylinder (in cm³)- (use π = \(\frac{22}{7}\)).

The orthogonal trajectory of the cardioid r = a(1+cos θ), a being the parameter is:

Consider the following linear equations:-
3x+7y+z=0
5x+9y-z=0
9x+13y+kz=0
For what values of k the above system of equations has an infinite number of solutions -

If there is no feasible region in LPP, then the problem has:

The area bounded by the curves y = x² and y = 4 - x² is

Given below are two statements
Statement I: In cylindrical co-ordinates, \(Volume = \int \int\limits_{V} \int rdrdødz \)
Statement II: In spherical polar Co-ordinates, \(Volume = \int \int\limits_{V} \int r^2\ \cos\theta\ drd\theta d\phi\)
In the light of the above statements, choose the correct answer from the options given below :

The natural domain of definition of the function f(z) = \(\frac{1}{1-|z|^2}\) is ________.

A. \(∮_{C}\vec{F}.d\bar{r}=\int\int\limits_{s}\text{Curl}\ \vec{F}.\hat{n}ds\)
B. \(\int\limits_{C}Mdx+Ndy=\int\int\limits_{R}(\frac{∂N}{∂x}+\frac{∂M}{∂y})dxdy\)
C. grad is a vector normal to the surface ∅ = C
D. ∇∅ is the maximum rate of change of ∅
Choose the correct answer from the options given below

Given below are two statements:
Statement I : If x²y" - 2xy' - 4y = x⁴, then \(C.F.=\frac{C_1}{x}+C_2x^4\)
Statement II: If (D²-8D+15) y = 0, then auxiliary equation has equal roots.
In the light of the above statements, choose the correct answer from the options given below

The region represented by the inequation system x,y≥o, y ≥5,x+y≥3 is

Match List I with List II

List I		List II
A.	Φ=y2,∇Φ at (1, 1, 1)	I.	\(\hat{i}\)
B.	Φ=x,∇Φ at (1, -1, 2)	II.	\(-6\hat{k}\)
C.	Φ=2x3,∇Φ at (0, 1, 2)	III.	\(2\hat{j}\)
D.	Φ=3z2,∇Φ at (1, 2, -1)	IV.	\(\vec{0}\)

Choose the correct answer from the options given below:

The rank of matrix A = \(\begin{bmatrix} 1&3&1&-2&-3\\1&4&3&-1&-4\\2&3&-4&-7&-3\\3&8&1&-7&-8 \end{bmatrix}\)

Let A =\(\begin{bmatrix}2&3\\4&-1\end{bmatrix}\) then the matrix B that represents the linear operator A relative to the basis
S = {\(u_1,u_2\)}=\({[1, 3]^T, [2, 5]^T}\), is:

Which one of the following is a cyclic group?

A scalar potential \(\Psi\) has the gradient defined as  \(\nabla\Psi=yz\hat{i}+xz\hat{j}+xy\hat{k}\). The value of the integral \(\int_c\nabla\Psi.d\overrightarrow{r}\) on the curve \(\overrightarrow{r}=x\hat{i}+y\hat{j}+z\hat{k}\), where curve C: x=t, y = t², z = 3t² (1 ≤ t ≤ 3) is:

Given below are two statements
Statement I: If f(z) = u + iv is an analytic function, then u and v are both harmonic function.
Statement II: If f (z) is analytic within and on a closed curve C, and if a is any point within C, then \(f(a)=\frac{1}{2πi}\int_c\frac{f(z)}{z-a}dz\)
In the light of the above statements, choose the most appropriate answer from the options given below

$ \begin{vmatrix} 1+a & 1 & 1 & 1 \\ 1 & 1+b & 1 & 1 \\ 1 & 1 & 1+c & 1 \\ 1 & 1 & 1 & 1=d  \end{vmatrix} $ is equal to