List of top Mathematics Questions

Given below are two statements :
Statement I: Mdx+Ndy = 0 is said to be an exact differential equation if it satisfies the following condition \(\frac{∂M}{∂x}=\frac{∂N}{∂y}\)
Statement II: If Mdx + Ndy = 0 is not an exact differential equation and \(\frac{1}{N}(\frac{∂M}{∂y}-\frac{∂N}{∂x})=f(x)\), then \(I.F.=e^{\int f(x)dx}\)
In the light of the above statements, choose the correct answer from the options given below :

The value of \(\lim\limits_{n \to \infty} \frac{1}{n} [1+2^{\frac{1}{2}}+3^{\frac{1}{3}}+...n^{\frac{1}{n}}]\) is

If particular Integral (P.I) of \((D^2-4D+4)y=x^3e^{2x}\) is \(e^{mx}\frac{x^n}{20}\), then m²+n² is equal to:

Which of the following set is convex?

If \(u = x^2 - y^2\) is real part of an analytic function f(z), then f(z) 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 solution of the Linear Programming Problem
maximize Z = 107x + y
subject to constraints x + y ≤2
-3x + y ≥ 3
x, y ≥ 0 is

Integrating factors of the equation y (2xy + e^x) dx - e^x dy = 0 is :

Given below are two statements
Statement I: Draw back in Lagrange's method of undetermined multipliers is that nature of stationary point cannot be determined
Statement II: \(\displaystyle\sum_{n=1}^{∞} (-1)^{n-1}\frac{1}{n\sqrt n}\)convergent
In the light of the above statements, 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 given vector \(\vec{F}=(y^2-z^2+3yz-2x)\hat{i} +(3xz+2xy)\hat{j}+(3xy-2xz+2z)\hat{k}\) is solenoidal
Reason R: A vector \(\vec{F}\) is said to be solenoidal if div \(\vec{F}\) = 0
In the light of the above statements, choose the correct answer from the options given below :

If the curl of vector \(\vec{A} = (2xy-3yz)\hat{i} +(x^2+axz −4z^2)\hat{j}-(3xy+byz)\hat{k}\) is zero, then a + b is equal to :

For what value(s) of k the set of vectors {(1, k, 5), (1, -3, 2), (2, -1, 1)} form a basis in R³ ?

The work done by the force \(\overrightarrow F = (x^2-y^2)\hat{i} + (x+y)\hat{j}\) in moving a particle along the closed path C containing the curves x + y = 0, x²+ y² = 16 and y = x in the first and fourth quadrant is

If \(f(x,y)=x^2+y^2+6x+12\), then minimum value of f 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

The general solution of the differential equation y"+y = 6sin x is:

Match List I with List II

List I Differential Equation		List II I.F.
A.	y'+y=sinx	I.	x
B.	y'-y=x2	II.	\(\frac{1}{x}\)
C.	\(y'+\frac{1}{x}y=e^x\)	III.	ex
D.	\(y'-\frac{1}{x}y=1\)	IV.	e-x

Choose the correct answer from the options given below:

The value of \(∫_c \frac{3z^2+7z+1}{z+1} dz\), where C is the circle |z|=\(\frac{1}{2}\) is:

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 :

Evaluate the integral\(\oint\limits_C\frac{dz}{(z^2+4)^2},C:|z-i|=2\)

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

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:

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

The value of double integal \(\int\limits_0^∞\int\limits_0^xe^{-xy} ydydx\) is equal to:

Given below are two statements
Statement I: If \(x=\frac{1}{3}(2u + v)\) and \(y =\frac{1}{3}(u − v)\), then \(dxdy=\frac{-1}{3}\ dudv\)
Statement II: Area in Polar Co-ordinater \(\int\limits^{\theta_1}_{\theta_1} \int\limits^{r_2}_{r_1} rd\theta dr\)
In the light of the above statements, choose the correct answer from the options given below: