List of top Questions asked in CUET (PG)

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

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: 

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

Double integral \(\int\limits_0^2\int\limits_0^{\sqrt{2x-x^2}}\frac{xdydx}{\sqrt{x^2+y^2}}\) equals:

If \(\int\int_R(x + y) dydx = A\), where R is the region bounded by x = 0, x = 2, y = x, y = x+2, then \(\frac{A}{12}\) is equal to:

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

The volume generated by the revolution of the cardioid \(r = a(1-\cosθ)\) about its axis 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 :

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: 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 volume of the cylindrical column standing on the area common to the parabolas \(y^2 = x\), \(x^2 = y\) and cut off by the surface \(z = 12+y-x^2\) 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:

The surface area of the cylinder \(x^2+z^2 = 4\) inside the cylinder \(x^2 + y^2 = 4\) is:

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

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

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:

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

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 :

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

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

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: