The uncertainty in the velocities of two particles $A$ and $B$ are 0.03 and $0.01 \mathrm{~ms}^{-1}$ respectively. The mass of $B$ is four times to the mass of $A$. The ratio of uncertainties in their positions is

The total maximum number of electrons possible in $3 d$. $6 d, 5 s$ and $4 f$ orbitals with $m_l$ (magnetic quantum number) value -2 is

The period and group numbers of the element having maximum electronegativity in the long form of periodic table respectively, are

Identify the set containing isoelectronic species

Choose the incorrect statement from the following.

An ideal gas mixture of $\mathrm{C}_2 \mathrm{H}_6$ and $\mathrm{C}_2 \mathrm{H}_4$ occupies a volume of 28 L at 1 atm and 273 K . This mixture reacts completely with 128 g of $\mathrm{O}_2$ to produce $\mathrm{CO}_2$ and $\mathrm{H}_2 \mathrm{O}(l)$. What is the mole fraction of $\mathrm{C}_2 \mathrm{H}_4$ in the mixture ?

Complete combustion of ethane gives only gaseous products. In a closed vessel, 15 g of ethane and 112 g of $\mathrm{O}_2$ were allowed to completely react. What is the total number of moles of gaseous substances present in the vessel at the end of the reaction?

Identify the incorrect statements from the following.

I. For adiabatic process, $\Delta U=w_{\text {ad }}$

II. Enthalpy is an intensive property

III. For the process, $\mathrm{H}_2 \mathrm{O}(l) \rightarrow \mathrm{H}_2 \mathrm{O}(s)$, the entropy increases

The correct answer is

Enthalpy of formation of $\mathrm{CO}_2(\mathrm{~g}), \mathrm{H}_2 \mathrm{O}(\mathrm{l})$ and $\mathrm{C}_6 \mathrm{H}_{12} \mathrm{O}_6(\mathrm{~s})$ are $-393,-286$ and $-1170 \mathrm{~kJ} \mathrm{~mol}^{-1}$ respectively. The quantity of heat liberated when 18 g of $\mathrm{C}_6 \mathrm{H}_{12} \mathrm{O}_6(s)$ is burnt completely in oxygen is

The percentage of ionisation of 1 L of $x \mathrm{M}$ acetic acid is 4.242 and is called solution " $A$ ". The percentage of ionisation of 1 L of $y \mathrm{M}$ acetic acid is 3 and is called solution " $B$ ". Solution " $A$ " is mixed with solution " $B$ ". What is the concentration of acetic acid in the resultant solution? ( $K_{\mathrm{a}}$ of acetic acid $=1.8 \times 10^{-5}$ )

At 298 K , the value of $K_p$ for $\mathrm{N}_2 \mathrm{O}_4(g) \rightleftharpoons 2 \mathrm{NO}_2(g)$ is 0.113 atm . The partial pressure of $\mathrm{N}_2 \mathrm{O}_4$ at equilibrium is 0.2 atm . What is the partial pressure (in atm) of $\mathrm{NO}_2$ equilibrium?

$\mathrm{H}_2 \mathrm{O}_2$ reduces $\mathrm{KMnO}_4$ in acidic medium to ' $x$ ' and in basic medium to ' $y$ '. What are $x$ and $y$ ?

Identify the incorrect statement about the group 13 elements.

Which of the following statements are correct?

(I) $\mathrm{SnF}_4$ is ionic in nature.

(II) Stability of dihalides of group 14 elements increases down the group.

(III) $\mathrm{GeCl}_2$ is more stable than $\mathrm{GeCl}_4$.

Which of the following when present in excess in drinking water causes the disease methemoglobinemia?

$$ \text { IUPAC name of the following compound is } $$

The empirical formula weight of ' $Z$ ' in the given reaction sequence is

$$ n \text {-propyl bromide } \xrightarrow[\text { Dry ether }]{\mathrm{Na}} X \xrightarrow[\substack{773 \mathrm{~K} \\ 20 \mathrm{~atm}}]{\mathrm{v}_2 \mathrm{O}_5} Y \xrightarrow[\text { UV } 500 \mathrm{~K}]{\mathrm{Cl}_2} Z $$

If AgCl is doped with $1 \times 10^{-4}$ mole percent of $\mathrm{CdCl}_2$ the number of cation vacancies (in $\mathrm{mol}^{-1}$ ) is

In aqueous glucose solution, the mole fraction of wate is 40 times to mole fraction of glucose. What is the weight percentage $(w / w)$ of glucose in the solution?

Benzoic acid molecules undergo dimerisation in benzene. 2.44 g of benzoic acid when dissolved in 30 g of benzene caused depression in freezing point of 2 K . What is the percentage of association of it?

(Given $K_f\left(\mathrm{C}_6 \mathrm{H}_6\right)=5 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}$; molar mass of benzoic acid $=122 \mathrm{~g} \mathrm{~mol}^{-1}$ )

When the lead storage battery is in use (during discharge) the reaction that occurs at the anode is

The following equation is obtained for a first order reaction at 300 K

$$ \log _{10} \frac{k}{A}=0.00174 $$

What is the activation energy (in $\mathrm{J} \mathrm{mol}^{-1}$ ) of the reaction?

$$ \left(R=8314 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}\right) $$

$$ \text { Match the following } $$

The correct answer

Adsorption of a gas on solids follows Freundlich adsorption isotherm. The graph drawn between $\log \frac{x}{m}$ (on $y$-axis) and $\log p$ (on $x$-axis) is a straight line with slope equal to 3 and intercept equal to 0.30 . What is the value of $\frac{x}{m}$ at a pressure of 2 atm ?

(Given: $\log 2=0.3$ )

Which of the following reactions is an example of roasting?

Nature of two oxides of nitrogen $X$ and $Y$ formed in the reaction of sodium nitrite with hydrochloric acid is

$$ \text { Match the following } $$

$$ \begin{array}{cccc} \hline & \begin{array}{c} \text { List-I } \\ \text { (Transition metal, M) } \end{array} & & \begin{array}{c} \text { List-II } \\ \left(E_{M^{2+} / M}^{\ominus}\right) \end{array} \\ \hline \text { (A) } & \mathrm{Ni} & \text { (I) } & -1.18 \\ \hline \text { (B) } & \mathrm{Mn} & \text { (II) } & -0.91 \\ \hline \text { (C) } & \mathrm{Fe} & \text { (III) } & -0.25 \\ \hline \text { (D) } & \mathrm{Cr} & \text { (IV) } & -0.44 \\ \hline \end{array} $$

The correct answer is

Identify the complex ion with spin only magnetic moment of 4.90 BM .

What are $X$ and $Y$ in the following reaction?

$$ n \mathrm{ClCH}=\mathrm{CH}_2 \xrightarrow{x} Y $$

Consider the following

Statement-I : Cane sugar is a disaccharide of $\alpha$-D-glucose and $\beta$-D-fructose.

Statement-II : Milk sugar is a disaccharide of $\alpha$-D-glucose and $\beta$-D-galactose.

The correct answer is

The deficiency of vitamin $(X)$ causes convulsions. Source of $X$ is $Y$. What are $X$ and $Y$ ?

Which of the following is not an example of synthetic detergent?

The most reactive compound towards nucleophilic substitution with an aqueous NaOH is

An alkyl bromide $X\left(\mathrm{C}_5 \mathrm{H}_{11} \mathrm{Br}\right)$ undergoes hydrolysis in a two step mechanism $X$ is converted to Grignard reagent and then reacted with $\mathrm{CO}_2$ in dry ether followed by acidification gave $Y$. What is $Y$ ?

Consider the following sequence of reactions

$$ \mathrm{C}_6 \mathrm{H}_5 \mathrm{COONa} \xrightarrow[\Delta]{\mathrm{NaOH} / \mathrm{CaO}} X \xrightarrow[\text { Anhy } \cdot \mathrm{AlCl}_3]{\text { CO+ } \mathrm{HCl}} Y \xrightarrow[\text { NaOH }]{\text { Conc. }} A+B $$

If $A$ is the reduction product of $Y$, what is $B$ ?

$$ \text { What is } A \text { in the following reaction? } $$

The correct statement regarding $X$ and $Y$ formed in the following reaction is

$$ \left(\mathrm{CH}_3\right)_3 \mathrm{COC}_2 \mathrm{H}_5 \xrightarrow[\Delta]{\mathrm{HI}} \text { halide }(X)+\text { alcohol }(Y) $$

Consider the following

Statement-I In the nitration of aniline, more amount of $m$-nitroaniline is formed than expected.

Statement-II In the presence of a strongly acidic medium, aniline is protonated to form anilinium ion, which is meta-directing.

The correct answer is

If $f(x)=(x+1)^2-1, x \geq-1$, then $\left\{x \mid f(x)=f^{-1}(x)\right\}$ is

If $11^{12}-11^2=k\left(5 \times 10^9+6 \times 10^9+33 \times 10^8\right. \left.+110 \times 10^7+\ldots+33\right)$, then $k=$

If $P=\left[\begin{array}{lll}1 & \alpha & 3 \\ 1 & 3 & 3 \\ 2 & 4 & 4\end{array}\right]$ is the adjoint of a matrix $A$ and det $A=4$, then the value of $\alpha$ is

If $\alpha$ is a real root of the equation $x^3+6 x^2+5 x-42=0$, then the determinant of the matrix

$\left[\begin{array}{lll}\alpha-1 & \alpha+1 & \alpha+2 \\ \alpha-2 & \alpha+3 & \alpha-3 \\ \alpha+4 & \alpha-4 & \alpha+5\end{array}\right]$ is

The rank of the matrix $\left[\begin{array}{cccc}2 & -3 & 4 & 0 \\ 5 & -4 & 2 & 1 \\ 1 & -3 & 5 & -4\end{array}\right]$ is

If the least positive integer $n$ satisfying the equation $\left(\frac{\sqrt{3}+i}{\sqrt{3}-i}\right)^n=-1$ is $p$ and the least positive integer $m$ satisfying the equation $\left(\frac{1-\sqrt{3 i}}{1+\sqrt{3} i}\right)^m=\operatorname{cis} \frac{2 \pi}{3}$ is $q$, then $\sqrt{p^2+q^2}=$

Sum of the squares of the imaginary roots of the equation $z^8-20 z^4+64=0$ is

If the roots of the equation $x^2+2 a x+b=0$ are real, distinct and differ atmost by 2 m , then $b$ lies in the interval

The cubic equation whose roots are the squares of the roots of the equation $x^3-2 x^2+3 x-4=0$ is

If all possible 4 -digit numbers are formed by choosing 4 different digits from the given digits $1,2,3,5,8$ then the sum of all such 4 -digit numbers is

The number of ways in which a committee of 7 members can be formed from 6 teachers, 5 fathers and 4 students in such a way that at least one from each group is included and teachers form the majority among them, is

If $C_0, C_2, \ldots, C_n$ are the binomial coefficients in the expansion of $(1+x)^n$, then

$$ \left(C_0+C_1\right)-\left(C_2+C_3\right)+\left(C_4+C_5\right)-\left(C_6+C_7\right)+\ldots= $$

$$ 1+\frac{4}{15}+\frac{4 \cdot 10}{15 \cdot 30}+\frac{4 \cdot 10 \cdot 16}{15 \cdot 30 \cdot 45}+\ldots . .+\infty= $$

If $\frac{3 x+1}{(x-1)\left(x^2+2\right)}=\frac{A}{x-1}+\frac{B x+C}{x^2+2}$, then $5(A-B)=$

$\operatorname{cosec} 48^{\circ}+\operatorname{cosec} 96^{\circ}+\operatorname{cosec} 192^{\circ}+\operatorname{cosec} 384^{\circ}=$

If $\sqrt{3} \cos \theta+\sin \theta>0$, then

If $\cos \theta=\frac{-3}{5}$ and $\theta$ does not lie in second quadrant, then $\tan \frac{\theta}{2}=$

The general solution satisfying both the equations $\sin x=-\frac{3}{5}$ and $\cos x=-\frac{4}{5}$ is

The number of solution of $\tan ^{-1} 1+\frac{1}{2} \cos ^{-1} x^2-\tan ^{-1} \left(\frac{\sqrt{1+x^2}+\sqrt{1-x^2}}{\sqrt{1+x^2}-\sqrt{1-x^2}}\right)=0$ is

$$ \tanh ^{-1}(\sin \theta)= $$

In $\triangle A B C$, if $a=8, b=10, c=12$, then $\frac{r}{R}=$

In $\triangle A B C$, if $a=13, b=8, c=7$, then $\cos (B+C)=$

In a $\triangle A B C$, if $\left(r_1-r_3\right)\left(r_1-r_2\right)-2 r_2 r_3=0$, then $a^2-b^2=$

If the median $A D$ of the $\triangle A B C$ is bisected at $E$ and $B E$ meets $A C$ in $E$, then $A F: A C=$

If $\mathbf{a}=2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}$ and $\mathbf{b}=-\hat{\mathbf{i}}+3 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ are two vectors, then the vector of magnitude 28 units in the direction of the vector $\mathbf{a}-\mathbf{b}$ is

If $\bar{a}$ is a unit vector, then

$$ |\mathbf{a} \times \hat{\mathbf{i}}|^2+|\mathbf{a} \times \hat{\mathbf{j}}|^2+|\mathbf{a} \times \hat{\mathbf{k}}|^2= $$

If $\mathbf{a}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}, \mathbf{b}=-2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}, \mathbf{c}=5 \hat{\mathbf{i}}-4 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ and $\mathbf{d}=3 \hat{\mathbf{i}}+\hat{\mathbf{j}}+5 \hat{\mathbf{k}}$ are four vectors, then $(\mathbf{a} \times \mathbf{b}) \times(\mathbf{c} \times \mathbf{d})=$

$3 \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, 2 \hat{\mathbf{i}}+\hat{\mathbf{k}}, \hat{\mathbf{i}}+5 \hat{\mathbf{j}}$ are the position vectors of three non-collinear points $A, B, C$ respectively. If the perpendicular drawn from $C$ onto $\mathbf{A B}$ meets $\mathbf{A B}$ at the point $a \hat{\mathbf{i}}+b \hat{\mathbf{j}}+c \hat{\mathbf{k}}$, then $a+b+c=$

Let $x_1, x_2, \ldots, x_{11}$ be the observations satisfying $\sum\limits_{i=1}^{11}\left(x_i-4\right)=22$ and $\sum\limits_{i=1}^{11}\left(x_i-4\right)^2=154$. If the mean and variance of the observations are $\alpha$ and $\beta$, then the quadratic equation having the roots $\frac{\alpha}{\beta}$ and $\frac{\beta}{\alpha}$ is

A die is thrown twice. Let A be the event of getting a prime number when the die is thrown first time and $B$ be the event of getting an even number when the die is thrown second time. Then, $P(A / \bar{B})=$

A bag contains 5 balls of unknown colours. There are equal chances that out of these five balls, there may be 0 or 12 or or 3 or 4 or 5 red balls, A ball is taken out from the bag at random and is found to be red. The probability that it is the only red ball in the bag is

If $X \sim B(9, p)$ is a binomial variate satisfying the equation $P(X=3)=P(X=6)$, then $P(X<3)=$

The mean and variance of a binomial distribution are $x$ and 5 respectively. If $x$ is an integer, then the possible values for $x$ are

After the coordinate axes are rotated through an angle $\frac{\pi}{4}$ in the anti-clockwise direction without shifting the origin, if the equation $x^2+y^2-2 x-4 y-20=0$ transforms to $a x^2+2 h x y+b y^2+2 g x+2 f y+c=0$ in the new coordinate system, then

$$ \left|\begin{array}{lll} a & h & g \\ h & b & f \\ g & f & c \end{array}\right|= $$

If $\alpha$ is the angle made by the perpendicular drawn from origin to the line $12 x-5 y+13=0$ with the positive $X$-axis in anti-clockwise direction, then $\alpha=$

If the equation of the pair of lines passing through $(1,1)$ and perpendicular to the pair of line $2 x^2+x y-y^2-x+2 y-1=0$ is $a x^2+2 h x y+b y^2+2 g x+3 y=0$, then $\frac{b}{a}=$

If the combined equation of the lines joining the origin to the point of intersection of the curve $x^2+y^2-2 x-4 y+2=0$ and the line $x+y-2=0$ is $\left(l_1 x+m_1 y\right)\left(l_2 x+m_2 y\right)=0$, then $l_1+l_2+m_1+m_2=$

If the circles $x^2+y^2+5 k x+2 y+k=0$ and $2 x^2+2 y^2+2 k x+3 y-1=0, k \in R$ intersect at points $P$ and $Q$ then the line $4 x+5 y-k=0$ passes through $P$ and $Q$ for

The slope of one of the direct common tangents drawn to the circles $x^2+y^2-2 x+4 y+1=0$ and $x^2+y^2-4 x-2 y+4=0$ is

If the pole of the line $x+2 b y-5=0$ with respect to the circle $S \equiv x^2+y^2-4 x-6 y+4=0$ lies on the line $x+b y+1=0$, then the polar of the point $(b,-b)$ with respect to the circle $S=0$ is

If $P(\alpha, \beta)$ is the radical centre of the circles $S \equiv x^2+y^2+4 x+7=0, S^{\prime}=2 x^2+2 y^2+3 x+5 y+9=0$ and $S^{\prime \prime} \equiv x^2+y^2+y=0$, then the length of the tangent drawn from $P$ to $S^{\prime}=0$ is

If the tangents of the parabola $y^2=8 x$ passing through the point $P(1,3)$ touches the parabola at $A$ and $B$, then the area (in sq. units) of $\triangle P A B$ is

The equation of the normal drawn at the point $(\sqrt{2}+1,-1)$ to the ellipse $x^2+2 y^2-2 x+8 y+5=0$ is

If $3 x+2 \sqrt{2} y+k=0$ is a normal to the hyperbola $4 x^2-9 y^2-36=0$ making positive intercepts on both the axes, then $k=$

If a hyperbola has asymptotes $3 x-4 y-1=0$ and $4 x-3 y-6=0$, then the transverse and conjugate axes of that hyperbola are

If $A(0,1,2), B(2,-1,3)$ and $C(1,-3,1)$ are the vertices of a triangle, then the distance between its circumcentre and orthocentre is

If the direction cosines of two lines satisfy the equations $l-2 m+n=0, l m+10 m n-2 n l=0$ and $\theta$ is the angle between the lines, then $\cos \theta=$

If $(2,-1,3)$ is the foot of the perpendicular drawn from the origin $(0,0,0)$ to a plane, then the equation of that plane is

$$\mathop {\lim }\limits_{x \to 0} \frac{x^2 \sin ^2(3 x)+\sin ^4(6 x)}{(1-\cos 3 x)^2}= $$

If a real valued function

$$ f(x)=\left\{\begin{array}{cc} (1+\sin x)^{\cos x}, & -\pi / 2 < x < 0 \\ a, & x=0 \\ \frac{e^{2 / x}+e^{3 / x}}{a e^{2 / x}+b e^{3 / x}}, & 0 < x < \pi / 2 \end{array}\right. $$

is continuous at $x=0$, then $a b=$

$$ \mathop {\lim }\limits_{x \to 0} \frac{(\operatorname{cosec} x-\cot x)\left(e^x-e^{-x}\right)}{\sqrt{3}-\sqrt{2+\cos x}}= $$

If $y=\sqrt{\cosh x+\sqrt{\cosh x}}$, then $\frac{d y}{d x}=$

If $y=(\log x)^{1 / x}+x^{\log x}$, at $x=e, \frac{d y}{d x}=$

The interval in which the function $f(x)=\tan ^{-1}(\sin x+\cos x)$ is an increasing function is

The function $f(x)=x e^{-x} \forall x \in R$ attains a maximum value at $x=k$, then $k=$

If $m$ and $M$ are the absolute minimum and absolute maximum values of the function $f(x)=2 \sqrt{2} \sin x-\tan x$ in the interval $[0, \pi / 3]$, then $m+M=$

$$ \int \frac{\sec ^2 x}{\sin ^7 x} d x-\int \frac{7}{\sin ^7 x} d x= $$

$$ \int \frac{d x}{(x+1) \sqrt{x^2+1}}= $$

If $\int \frac{d x}{2 \cos x+3 \sin x+4}=\frac{2}{\sqrt{3}} f(x)+C$, then $f\left(\frac{2 \pi}{3}\right)=$

If $\int \frac{1}{\left((x+4)^3(x+1)^5\right)^{1 / 4}} d x=A \cdot\left(\frac{x+4}{x+1}\right)^n+C$

$$ \int_{-\pi / 2}^{\pi / 2} \sin ^2 x \cos ^2 x(\sin x+\cos x) d x= $$

$$ \int_{1 / 5}^{1 / 2} \frac{\sqrt{x-x^2}}{x^3} d x= $$

$$ \int_0^{400 \pi} \sqrt{1-\cos 2 x} d x= $$

Area of the region (in sq. units) bounded by the curve $y=x^2-5 x+4, x=0, x=2$ and the $X$-axis is

If the order and degree of the differential equation $x \frac{d^2 y}{d x^2}=\left(1+\left(\frac{d^2 y}{d x^2}\right)^2\right)^{-1 / 2}$ are $k$ and $l$ respectively, then $k, l$ are the roots of

The equation of the curve passing through the point $(0, \pi)$ and satisfying the differential equation $y d x=\left(x+y^3 \cos y\right) d y$ is

The general solution of the differential equation $(x-(x+y) \log (x+y)) d x+x d y=0$ is

If the equation for the velocity of a particle at time ' $t$ ' is $v=$ at $+\frac{b}{t+c}$, then the dimensions of $a, b, c$ are respectively

If $\alpha, \beta$ and $\gamma$ are the angles made by a vector with $x, y$ and $z$ axes respectively, then $\sin ^2 \alpha+\sin ^2 \beta=$

A particle moving along a straight line covers the first half of the distance with a speed of $3 \mathrm{~ms}^{-1}$, the other half of the distance is covered in two equal time intervals with speeds of $4.5 \mathrm{~ms}^{-1}$ and $7.5 \mathrm{~ms}^{-1}$ respectively, then the average speed of particle during the motion is

Water flowing through a pipe of area of cross-section $2 \times 10^{-3} \mathrm{~m}^2$ hits a vertical wall horizontally with a velocity of $12 \mathrm{~ms}^{-1}$. If the water does not rebound after hitting the wall, then the force acting on the wall due to water is

Two blocks $A$ and $B$ of masses 2 kg and 4 kg respectively are kept on a rough horizontal surface. If same force of 20 N is applied on each block, then the ratio of the accelerations of the blocks $A$ and $B$ is (Coefficient of kinetic friction between the surface and the blocks is 0.3 and acceleration due to gravity $=10 \mathrm{~ms}^{-2}$ )

If a force of $\left(6 x^2-4 x\right) \mathrm{N}$ acts on a body of mass 10 kg , then work to be done by the force in displacing the body from $x=2 \mathrm{~m}$ to $x=4 \mathrm{~m}$ is

A circular well of diameter 2 m has water upto the ground level. If the bottom of the well is at a depth of 14 m , the time taken in seconds to empty the well using a 1.4 kW motor is

(Acceleration due to gravity $=10 \mathrm{~ms}^{-2}$ )

The co-ordinates of the centre of mass of a uniform $L$ shaped plate of mass 3 kg shown in the figure is

A circular disc of mass 20 kg and radius 1 m is rotating about an axis passing through its centre and perpendicular to its plane with an angular velocity of $2 \mathrm{rad} \mathrm{s}^{-1}$. Then, the rotational kinetic energy of the disc is

The equations for the displacements of two particles in simple harmonic motion are $y_1=0.1 \sin \left(100 \pi t+\frac{\pi}{3}\right)$ and $y_2=0.1 \cos \pi t$ respectively. The phase difference between the velocities of the two particles at a time $t=0$ is

A spring is stretched by 0.2 m when a mass of 0.5 kg is suspended to it. The time period of the spring when 0.5 kg mass is replaced with a mass of 0.25 kg is suspended to it is

(Acceleration due to gravity $=10 \mathrm{~ms}^{-2}$ )

An artificial satellite is revolving around a planet of radius $R$ in a circular orbit of radius ' $a$ '. If the time period of revolution of the satellite. $T \propto a^{3 / 2} g^x R^y$, then the values of $x$ and $y$ are respectively

[ $g=$ acceleration due to gravity]

If the longitudinal strain of a stretched wire is $0.2 \%$ and the Poisson's ratio of the material of the wire is 0.3 , then the volume strain of the wire is

If two soap bubbles $A$ and $B$ of radii $r_1$ and $r_2$ respectively are kept in vacuum at constant temperature, then the ratio of masses of air inside the bubbles $A$ and $B$ is

A small quantity of water of mass ' $m$ ' at temperature $\theta^{\circ} \mathrm{C}$ is mixed with a large mass ' $M$ ' of ice which is at its melting point. If ' $s$ ' is specific heat capacity of water and ' $L$ ' is the latent heat of fusion of ice, then the mass of ice melted is

In a Carnot engine, if the absolute temperature of the source is $25 \%$ more than the absolute temperature of the sink, then the efficiency of the engine is

The work done by 6 moles of helium gas when its temperature increases by $20^{\circ} \mathrm{C}$ at constant pressure is (Universal gas constant $=8.31 \mathrm{Jmol}^{-1} \mathrm{~K}^{-1}$ )

If a heat engine and a refrigerator are working between the same two temperatures $T_1$ and $T_2\left(T_1>T_2\right)$, then the ratio of efficiency of heat engine to coefficient of performance of refrigerator is

If the internal energy of 3 moles of a gas at a temperature of $27^{\circ} \mathrm{C}$ is 2250 R , then the number of degrees of freedom of the gas is

( $R=$ Universal gas constant)

If two progressive sound waves represented by $y_1=3 \sin 250 \pi t$ and $y_2=2 \sin 260 \pi t$ (where displacement is in metre and time is in second) superimpose, then the time interval between two successive maximum intensities is

If the least distance of distinct vision for a boy is 35 cm , then the lens to be used by the boy for correcting the defect of his eye is

In Young's double slit experiment, if the distance between the slits is increased to 3 times initial distance, then the ratio of initial and final fringe widths is

A solid of mass 1 kg has $6 \times 10^{24}$ atoms. If one electron is removed from every one atom of $0.005 \%$ of the atoms, then the charge gained by the solid is

One of the two identical capacitors having the same capacitance $C$, is charged to a potential $V_1$ and the other is charged to a potential $V_2$. If they are connected with their like plates together, then the decrease in the electrostatic potential energy of the combined system is

If the energy stored in a spherical conductor having a charge of $12 \mu \mathrm{C}$ is 6 J , then the radius of the spherical conductor is

A part of a circuit is shown in the figure. The ratio of the potential differences between the points $A$ and $C$ and the points $D$ and $E$ is

A solenoid of one metre length and 3.55 cm inner diameter carries a current of 5 A . If the solenoid consists of five closely packed layers each with 700 turns along its length, then the magnetic field at its centre is

The work done in rotating a bar magnet which is initially in the direction of a uniform magnetic field through $45^{\circ}$ is $W$. The additional work to be done to rotate the magnet further through $15^{\circ}$ is

When a current of 4 mA passes through an inductor, if the flux linked with it is $32 \times 10^{-6} \mathrm{Tm}^2$, then the energy stored in the inductor is

In a series resonant LCR circuit, for the power dissipated to become half of the maximum power dissipated, the current amplitude is

The density (in $\mathrm{kg} \mathrm{m}^{-3}$ ) of nuclear matter is of the order of

In common emitter amplifier of a transistor, if the ratio of the voltage gain and current amplification factor is 4 , then the ratio of the collector and base resistances is

If three logic gates are connected as shown in the figure, then the correct truth table of the circuit is

Ionosphere acts as a reflector for the frequency range of