CUET Physics Quiz-1 This quiz consists of 25 multiple-choice questions with 4 marks for a correct answer and -1 mark for a wrong answer. The duration of this test is 30 minutes. 1 / 25 The Fourier coefficients of the function $f(x)=\left\{\begin{matrix} 0, -\pi $a_0=1$, $a_n=0$, $b_n=\frac{1}{n \pi} [1-(-1)^n]$ $a_0=0$, $a_n=0$, $b_n=\frac{1}{n \pi} [1+(-1)^n]$ $a_0=1$, $a_n=0$, $b_n=\frac{1}{n \pi} [1+(-1)^n]$ $a_0=0$, $a_n=0$, $b_n=\frac{1}{n \pi} [1-(-1)^n]$ 2 / 25 A hemispherical shell is placed on the x-y plane centered at the origin. For a vector field $\vec{E}=\frac{-y\hat{i}+x\hat{j}}{x^2+y^2}$, the value of the integral $\int_S(\vec{\nabla}\times\vec{E})\cdot d\vec{s}$ over the hemispherical surface is $4\pi$ $2\pi$ $\pi$ $0$ 3 / 25 If $\vec{F}$ is a constant vector and $\vec{r}$ is the position vector then $\vec{\nabla}(\vec{F}\cdot\vec{r})$ would be $\vec{F}$ $(\vec{\nabla}\cdot \vec{r})\vec{F}$ $|\vec{r}|\vec{F}$ $(\vec{\nabla}cdot \vec{F})\vec{r}$ 4 / 25 Consider the differential equation $y^{\prime\prime}+2y^{\prime}+y=0$. If $y(0)=0$ and $y^{\prime}(0)=1$, then the value of $y(2)$ is $\frac{4}{e^2}$ $\frac{2}{e}$ $\frac{2}{e^2}$ $\frac{4}{e}$ 5 / 25 If the determinant of the matrix $A$ of dimension $3\times 3$ is 3, then the determinant of $2A^2$ is 72 9 96 18 6 / 25 The volume integral $\int_\nu (r^2+2) \vec{\nabla} \cdot (\frac{\hat{r}}{r^2}) d\tau$ ,where $V$ is the volume of a sphere of radius $R$Â centered at the origin, is equal to $4\pi$ $\pi$ $2\pi$ $8\pi$ 7 / 25 A unit vector perpendicular to the plane containing $\vec{A}=\hat{i}+\hat{j}-2\hat{k}$ and $\vec{B}=2\hat{i}-\hat{j}+\hat{k}$ is $\frac{1}{\sqrt{35}}(-\hat{i}+5\hat{j}-3\hat{k})$ $\frac{1}{\sqrt{19}}(-\hat{i}+3\hat{j}-3\hat{k})$ $\frac{1}{\sqrt{26}}(-\hat{i}+3\hat{j}-4\hat{k})$ $\frac{1}{\sqrt{35}}(-\hat{i}-5\hat{j}-3\hat{k})$ 8 / 25 The trace of a $2\times 2$ matrix is $4$ and its determinant is $8$ . If one of the eigenvalues is $2(1+i)$Â , the other eigenvalue is $1+2i$ $1-2i$ $2(1+i)$ $2(1-i)$ 9 / 25 The modulus and phase of the complex number $(1+i)i$Â in polar representation are $\sqrt{3}$ and $\frac{\pi}{4}$ $\sqrt{3}$ and $\frac{3\pi}{4}$ $\sqrt{2}$ and $\frac{\pi}{4}$ $\sqrt{2}$ and $\frac{3\pi}{4}$ 10 / 25 The value of the integral $\int_{-\pi/2}^{+\pi/2}\sin^2\theta\,\delta(3\theta+\pi)\,d\theta$ is $\frac{1}{3}$ $\frac{1}{2}$ $1$ $\frac{1}{4}$ 11 / 25 Three vectors $\vec{A}$, $\vec{B}$ and $\vec{C}$ are given by $\vec{A}=\alpha\hat{i}-2\hat{j}+2\hat{k}$, $\vec{B}=6\hat{i}+4\hat{j}-2\hat{k}$ and $\vec{C}=-3\hat{i}-2\hat{j}-4\hat{k}$. The value of $\alpha$ for which the vectors will be coplanar is $2$ $-1$ $1$ $-3$ 12 / 25 The value of $\sqrt{i}+\sqrt{-i}$, where $i=\sqrt{-1}$ is $-\sqrt{2}$ $\frac{1}{\sqrt{2}}$ $\sqrt{2}$ $0$ 13 / 25 The function $e^{\cos x}$ is Taylor exapnded by $x=0$. The coefficient of $x^2$ is $\frac{1}{2}$ $-\frac{1}{2}$ $\frac{e}{2}$ $-\frac{e}{2}$ 14 / 25 If $\phi(x,y,z)$ is a scalar function that satisfies the Laplace equation, then the gradient of $\phi$ is Solenoidal and irrotational Neither solenoidal nor irrotational Solenoidal but not irrotational Irrotational but not solenoidal 15 / 25 The series $\sum_{n=1}^{\infty}\frac{1}{n(\log n)^p}$ is a convergent series for p=1 divergent series for p>1 convergent series for p>1 divergent series for all values of p 16 / 25 The value of $\oint (2x\,dy-3y\,dx)$ around a square with vertices (0,2), (2,0), (-2,0) and (0,-2) is 30 20 40 10 17 / 25 What is the equation of the plane which is tangent to the surface $xyz=4$ at the point (1,2,2)? $x+4y+z=0$ $2x+y+z=6$ $x+2y+4z=12$ $4x+2y+z=12$ 18 / 25 The equation $z^2+\bar{z}^2=4$ in the complex plane (where $\bar{z}$ is the complex conjugate of $z$) represents circle of radius 2 circle of radius 4 ellipse hyperbola 19 / 25 The tangent line to the curve $x^2+xy+5=0$ aat (1,1) is represented by $x=-3y+4$ $x=3y-2$ $y=-3x+4$ $y=3x-2$ 20 / 25 Consider a vector field $\vec{F}=y\hat{i}+xz^3\hat{j}-zy\hat{k}$. Let C be the circle $x^2+y^2=4$ on the plane $z=2$, oriented counter-clockwise. The value of the contour integral $\oint \vec{F}\cdot\vec{dr}$Â is $28\pi$ $4\pi$ $-4\pi$ $-28\pi$ 21 / 25 Consider the coordinate transformation $x^\prime=\frac{x+y}{\sqrt{2}}$ and $y^\prime=\frac{x-y}{\sqrt{2}}$ . The relation between the area elements $dx^\prime\,dy^\prime$ and $dx\,dy$ is given by $dx^\prime\,dy^\prime=j\,dx\,dy$ . The value of $j$ is 2 -2 1 -1 22 / 25 What is the value of the integral $\int_{-\infty}^{\infty}\delta(x^2-\pi^2)\cos x \,dx$Â ? $-\frac{1}{\pi}$ $\pi$ $\frac{1}{\pi}$ $\frac{2}{\pi}$ 23 / 25 The value of $m$ , so that $2x-x^2+my^2$ may be harmonic is -1 2 0 1 24 / 25 A matrix is given by $M=\frac{1}{\sqrt{2}}\begin{vmatrix} i & 1\\ 1 & i \end{vmatrix}$. The eigenvalues of $M$ are real and positive real and negative purely imaginary with modulus 1 complex with modulus 1 25 / 25 The solution of the differential equation, $\sin x + y\frac{dy}{dx}=0$, where $y(0)=1$ is $y(x)=\sqrt{2\sin x-1}$ $y(x)=\sqrt{2\sin x+1}$ $y(x)=\sqrt{2\cos x-1}$ $y(x)=\sqrt{2\cos x+1}$ Your score is