List of top Mathematics Questions asked in CUET (PG)

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 ø (x, y), Ψ(x, y), \(\frac{\partial\phi}{\partial y}\) and \(\frac{\partial\psi}{\partial x}\) are continuous functions over region R bounded by simple closed curve C in xy-plane, then \(∮_C(\phi dx+\psi dy)=\int\int\limits_{R}(\frac{\partial\psi}{\partial x}-\frac{\partial\phi}{\partial y})dxdy\)
Statement II: \(\vec{F}\) be a vector point function and volume V enclosed by a closed surface, then volume integral \(=\int\int\limits_{V}\int\vec{F}.dV\)
In the light of the above statements, choose the most appropriate answer from the options given below

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:

A. vector \(\vec{V}\) is said to be solenoidal if div \(\vec{V} = 0\)
B. Vector \(\vec{F}\) is said to be irrotational if curl \(\vec{F} = \vec{0}\)
C. If \(\vec{F}\) represents the force acting on a particle along are AB, then total work done \(\int^{B}_A \vec{F}\times \vec{dr}\)
D. Volume integral \(\int \int\limits_{V} \int \vec{F}\times dV\)
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

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

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

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 directional derivative of Φ(x,y,z) = x²yz+4xz² at (1, -2, 1) in the direction of \(2\hat{i}-\hat{j}-2\hat{k}\) is

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R
Assertion A: A given family of curves is said to be 'self- orthogonal' if the family of orthogonal trajectory is the same as the given family of curves.
Reason R: For finding orthogonal trajectory, replace \(\frac{dy}{dx}by-\frac{dx}{dy}\) in \(f(x,y,\frac{dy}{dx})=0\)
In the light of the above statements, choose the correct answer from the options given below:

The general solution of the differential equation \(2x^2 \frac{d^2y}{dx^2}=x\frac{dy}{dx}-6y=0\) is :

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

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

Match List I with List II

List I Differential Equation		List II Particular Integral (P.I)
A.	(D2+6D+9)y=e3x	I.	\(\frac{x}{6}\sin3x\)
B.	(D2-6D+9)y=3	II.	\(-\frac{1}{5}\cos3x\)
C.	(D2+4)y=cos3x	III.	\(\frac{1}{3}\)
D.	(D2+9)y= cos3x	IV.	\(\frac{1}{36}e^{3x}\)

Choose the correct answer from the options given below:

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 :

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:

The solution of the differential equation \(\frac{dy}{dx}+y=3e^xy^3\) is :

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

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:

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: 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 :

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:

The integral \(\int\limits_0^1\int\limits_0^x(x^2+ y^2) dy dx\) is:

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: