The wavenumber of the first line $\left(n_2=3\right)$ in the Balmer series of hydrogen is overline $\bar{v}_1 \mathrm{~cm}^{-1}$. What is the wavenumber (in $\mathrm{cm}^{-1}$ ) of the second line ( $n_2=4$ ) in the Balmer series of $\mathrm{He}^{+}$?

Which of the following sets of quantum numbers is not possible for the electron?

The correct order of atomic radii of $\mathrm{C}, \mathrm{Al}$ and S is

How many of the following molecules / ions have trigonal planar structure?

$$ \mathrm{BO}_3^{3-}, \mathrm{NH}_3, \mathrm{PCl}_3, \mathrm{BCl}_3, \mathrm{ClF}_3, \mathrm{XeO}_3 $$

Consider the following

Assertion (A) Dipole moment of $\mathrm{NF}_3$ is lesser than $\mathrm{NH}_3$.

Reason (R) In $\mathrm{NF}_3$, the orbital dipole due to lone pair of electrons is in the opposite direction to the resultant dipole moment of the three $\mathrm{N}-\mathrm{F}$ bonds.

The correct answer is

At $\mathrm{T}(\mathrm{K})$, a gaseous mixture contains $\mathrm{H}_2$ and $\mathrm{O}_2$. The total pressure of the mixture is 2 bar. The partial pressure of $\mathrm{H}_2$ is 1.778 bar. What is the weight $(w / w)$ percentage of $\mathrm{H}_2$ in the mixture?

1.84 g of a mixture of $\mathrm{CaCO}_3$ and $\mathrm{MgCO}_3$ is strongly heated to get a residue of 0.96 g . The percentage of $\mathrm{CaCO}_3$ in the mixture is

Identify the correct statements from the following.

I. Work is a path function.

II. Enthalpy is an extensive property.

III. Lattice enthalpy of ionic compounds can be obtained from Born-Haber cycle.

Which of the following processes entropy change $(\Delta S)$ is negative?

I. Sublimation of dry ice

II. Freezing of water

III. Crystallisation of the dissolved substance

IV. Burning of rocket fuel

At $25^{\circ} \mathrm{C}$, the percentage of ionisation of ' $x$ ' M acetic acid is 4.242 . What is the value of $x$ ? $\left(K_{\mathrm{a}}=1.8 \times 10^{-5}\right)$

At $T(\mathrm{~K}), K_C$ value for

$\mathrm{AO}_2(\mathrm{~g})+\mathrm{BO}_2(\mathrm{~g}) \rightleftharpoons \mathrm{AO}_3(\mathrm{~g})+\mathrm{BO}(\mathrm{g})$ is 16 . In a closed 1 L flask, one mole each of $A \mathrm{O}_2, B \mathrm{O}_2, A \mathrm{O}_3$ and $B \mathrm{O}$ are taken and heated to $T(\mathrm{~K})$.

What is the concentration (in $\mathrm{mol} \mathrm{L}^{-1}$ ) of $\mathrm{AO}_3$ at equilibrium?

The incorrect statement in the following is

Which of the following statements are correct regarding lithium and magnesium?

I. They react slowly with water.

II. Their bicarbonates are solids.

III. Their chlorides are not soluble in ethanol.

IV. Their nitrates decompose easily on heating.

The correct option is

The incorrect statement from the following is

$$ \text { Match the following } $$

The correct answer is

Consider the following

Statement I Kolbe's electrolysis of sodium propionate gives $n$-hexane as product.

Statement II In Kolbe's process $\mathrm{CO}_2$ is liberated at anode and $\mathrm{H}_2$ is liberated at cathode.

Correct answer is

The correct decreasing order of priority for the functional group of organic compounds in the IUPAC method of nomenclature is

A compound is formed by two elements $A$ and $B$. Atoms of the element $B$ (as anion) make ccp lattice and those of element $A$ (as cation) occupy all tetrahedral voids. The formula of the compound is

The mole fractions of glucose and water in aqueous glucose solution are 0.0244 and 0.9756 respectively. What is the weight percentage $(w / w)$ of glucose in this solution?

At $T(\mathrm{~K})$, the vapour pressure of an aqueous solution of a non-volatile solute, whose mole fraction is 0.02 is found to be 34.65 mm Hg . What is the vapour pressure (in mm Hg ) of pure water at the same temperature?

If $E_{\mathrm{Fe}^{2+} / \mathrm{Fe}}^{\circ}=-0.441 \mathrm{~V}$ and $E_{\mathrm{Fe}^{3+} / \mathrm{Fe}^{2+}}^{\circ}=0.771 \mathrm{~V}$, the standard emf of the cell reaction $\mathrm{Fe}(s)+2 \mathrm{Fe}^{3+}(a q) \longrightarrow 3 \mathrm{Fe}^{2+}(a q)$ is

At $T(\mathrm{~K})$ the following equation is obtained for a first order reaction $\log \frac{k}{A}=-\frac{x}{T}$. The activation energy for this reaction is equal to ( $R=$ gas constant)

Which one of the following is not the correct characteristic property of physical adsorption?

In each of four separate beakers (I, II, III, IV), $X \mathrm{~mL}$ of $y \mathrm{M} \mathrm{Fe}_2 \mathrm{O}_3 x \mathrm{H}_2 \mathrm{O}$ colloidal solution is present. Equal volume and equal concentration of KCl , $\mathrm{K}_4\left[\mathrm{Fe}(\mathrm{CN})_6\right], \mathrm{K}_3 \mathrm{PO}_4$ and $\mathrm{K}_2 \mathrm{SO}_4$ was added into I, II, III and IV respectively.

The efficiency of precipitations in these beakers follows the order

In the extraction of iron from haematite, the impurity $(x)$ of the ore is removed in the form of ' $y$ ', what are $x$ and $y$ respectively?

Which of the following is not correct?

How many of the following lanthanide elements exhibit +4 oxidation state?

Ce, Pr, Nd, Pm, Sm, Eu, Gd, Tb, Dy

In which of the following, complex ions are not in correct order with respect to their magnitude of crystal field splitting?

Novolac is formed by the polymerisation of monomer ' $x$ ' in the presence of $\mathrm{OH}^{-}$ions. What is ' $x$ '?

Which of the following contain $\alpha$-D-glucose units?

I. Cane sugar

II. Milk sugar

III. Cellulose

IV. Amylose

Identify the set containing purine and pyrimidine base of DNA respectively.

Bithionol is added to soaps to impart antiseptic properties. The number of -OH and -Cl groups in its structure are respectively.

Which of the following is the product of Fittig reaction?

$$ \text { Match the following } $$

Which of the following represents Etard reaction?

The correct order of boiling points of the compounds given below is

(A) methoxy ethane

(B) propan-1-ol

(C) propanal

(D) propanone

The correct statement about the product of the following reaction is

$$ \mathrm{CH}_3 \mathrm{CHO} \xrightarrow[\text { (ii) } \mathrm{H}_2 \mathrm{O}]{\text { (i) } \mathrm{C}_2 \mathrm{H}_5 \mathrm{MgBr}} \text { product } $$

How many amines with molecular formula $\mathrm{C}_3 \mathrm{H}_9 \mathrm{~N}$ can react with benzenE sulphonyl chloride?

Let $f: N \rightarrow N$ be a function such that $f(x+y)=f(x)+f(y)+x y$ for every $x, y \in N$. If $f(\mathbb{l})=2$, then $\sum_{k=0}^{10} f(k)=$

If a real valued function $f:[-1,2] \rightarrow B$ defined by

$$ f(x)= \begin{cases}1-x, & \text { when }-1 \leq x \leq 1 \\ x-1, & \text { when } 1 < x \leq 2\end{cases} $$

is a surjection, then $B=$

For all $n \in N, \frac{3^n-1}{2} \geq$

The value of $p$ and $q$ is that system of equations $2 x+p y+6 z=8, x+2 y+q z=5$ and $x+y+3 z=4$ may have no solution are

$A$ is the set of all matrices of order 3 with entries 0 or 1 only. $B$ is the subset of $A$ consisting of all matrices with determinant value 1 . If $C$ is the subset of $A$ consisting of all matrices with determinant value -1 , then

Consider the matrices $A=\left[\begin{array}{ccc}x & y & 0 \\ -3 & 1 & 2 \\ 1 & -2 & z\end{array}\right]$ and $B=\left[\begin{array}{ccc}1 & -2 & -2 \\ 2 & 0 & 1 \\ 2 & 1 & 0\end{array}\right]$

If the cofactors of the elements $z, 1$ in 3rd row and $x$ of $A$ are $9,4,3$, respectively then $A B=$

If $z=x+i y$ and if the point $P$ in the argand diagram represents $z$, then the locus of the point $P$ satisfying the equation $2|z-2-3 i|=3|z+i-2|$ is a circle with centre

If $z$ is a non-real root of $x^7=1$, then $1+3 z+5 z^2+7 z^3+9 z^4+11 z^5+13 z^6=$

If $(2 k-1) x^2-2(3 k-2) x+4 k>0$ for every $x \in R$, then the sum of all possible integral values of $k$ is

The sum of the least positive integer and the greatest negative integer in the range of the function $f(x)=\frac{x^2-5 x+7}{x^2-5 x-7}$ is

If $\alpha$ is a repeated root of multiplicity 2 of the equation $18 x^3-33 x^2+20 x-4=0$, then

The equation $6 x^4-5 x^3+13 x^2-5 x+6=0$ will have

All the letters of the word LETTER are arranged in all possible ways and the words (with or without meaning) thus formed are arranged in dictionary order.

Then, the rank of the word TETLER is

5-digit numbers are formed by using the digits $0,1,2$, $3,5,7$ without repetetion and all of them are arranged in ascending order. Then, the rank of the number 70513 is

The number of divisors of 7 ! is

If $k$ is a positive integer and $10^k$ is a divisor of the number $9^{11}+11^9$, then the greatest value of $k$ is

If $\frac{x+1}{(x-1)^2\left(x^2+1\right)}=\frac{A}{x-1}+\frac{B}{(x-1)^2}+\frac{C x+D}{x^2+1}$, then $\sqrt{3 A^2+4 D^2+5 C^2+B^2}=$

If $A+B=\frac{\pi}{4}$, then $\frac{\cos B-\sin B}{\cos B+\sin B}=$

If $7 \cos \theta-\sin \theta=5$ and $\tan \theta>0$, then $\tan \theta=$

$$ \sin ^3 10^{\circ}+\sin ^3 50^{\circ}-\sin ^3 70^{\circ}= $$

The number of solutions of $\sin 2 x+\cos 4 x=2$ in the interval $[-\pi, \pi]$ is

The range of the real valued function $f(x)=\cos ^{-1}(-x)+\sin ^{-1}(-x)+\operatorname{cosec}^{-1}(x)$ is

If $\cosh 2 x=199$, then $\cot h x=$

The horizontal distance between a tower and a building is $10 \sqrt{3}$ units. If the angle of depression of the foot of the building from the top of the tower is $60^{\circ}$ and the angle of elevation of the top of the building from the foot of the tower is $30^{\circ}$, then the sum of the heights of the tower and the building is

In a $\triangle A B C, A-B=120^{\circ}, R=8 r$, then $\frac{1+\cos C}{1-\cos C}=$

In $\triangle A B C, \sqrt{\frac{r \cdot r_2}{r_3 r_1}}=$

$\hat{\mathbf{i}}-2 \hat{\mathbf{j}}$ is a point on the line parallel to the vector $2 \hat{\mathbf{i}}+\hat{\mathbf{k}}$. If $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}$ is a point on the plane parallel to the vectors $2 \hat{\mathbf{j}}-\hat{\mathbf{k}}$ and $\hat{\mathbf{i}}+2 \hat{\mathbf{k}}$, then the point of intersection of the line and the plane is

Points $P$ and $Q$ are given by $\mathbf{O P}=\hat{\mathbf{i}}-\hat{\mathbf{j}}-\hat{\mathbf{k}}$ and $\mathbf{O Q}=-\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$. A line along the vector $\mathbf{a}=\hat{\mathbf{i}}+\hat{\mathbf{j}}$ passes through the point $P$ and another line along the vector $\mathbf{b}=\hat{\mathbf{j}}-\hat{\mathbf{k}}$ passes through the point $Q$. If a line along the vector $\mathbf{c}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}$ intersects both the lines along the vectors $\mathbf{a}$ and $\mathbf{b}$ at $L$ and $M$, respectively, then $\mathbf{P M}=$

Angle between a diagonal of a cube and a diagonal of its face which are coterminus is

For $a \in R$, if the vectors $\mathbf{p}=(a+1) \hat{\mathbf{i}}+a \hat{\mathbf{j}}+a \hat{\mathbf{k}}$, $\mathbf{q}=a \hat{\mathbf{i}}+(a+1) \hat{\mathbf{j}}+a \hat{\mathbf{k}}$ and $\mathbf{r}=a \hat{\mathbf{i}}+a \hat{\mathbf{j}}+(a+1) \hat{\mathbf{k}}$ are coplanar and $3(\mathbf{p} \cdot \mathbf{q})^2-\lambda|\mathbf{r} \times \mathbf{q}|^2=0$, then the value of $\lambda$ is

If $\mathbf{a}=\hat{\mathbf{i}}+4 \hat{\mathbf{j}}-4 \hat{\mathbf{k}}, \mathbf{b}=-2 \hat{\mathbf{i}}+5 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ and $\mathbf{c}=3 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}-4 \hat{\mathbf{k}}$ are three vectors such that $(\mathbf{b} \times \mathbf{c}) \times \mathbf{a}=x \hat{\mathbf{i}}+y \hat{\mathbf{j}}+z \hat{\mathbf{k}}$, then $x+y-z=$

The following data represents the frequency distribution of 20 observations

$$ \begin{array}{ccccccc} \hline x_i & 3 & 4 & 5 & 8 & 10 & 11 \\ \hline f_i & \alpha+2 & (\alpha-1)^2 & 4 & \alpha-1 & 2 & \alpha \\ \hline \end{array} $$

Then, its mean deviation about the mean is

If $l, m$ represent any two elements (identical or different) of the set $\{1,2,3,4,5,6,7\}$, then the probability that $l x^2+m x+1>0 \forall x \in R$ is

$A$ and $B$ are playing chess game with each other. The probability that $A$ wins the game is 0.6 . the probability that he loses is 0.3 and the probability its draw is 0.1 . If they played three games, then the probability that $A$ wins atleast two games is

$U_1, U_2, U_3$ are three urns. $U_1$ contains 5 red, 3 white, 2 back balls: $U_2$ contains 4 red 4 white, 2 black balls and $U_3$ contains 3 red. 4 white, 3 black balls. If a ball is chosen at random from an urn chosen at random, then the probability of not getting a black ball is

If the probability distribution of a random variable $X$ is as follows, then $P(X \leq 2)=$

$$ \begin{array}{cccccc}\hline x_i & 0 & 1 & 2 & 3 & 4 \\ \hline P\left(X=x_i\right) & 3 k & 5 k & 3 k^2 & 4 k^2+k & 3 k^2 \\ \hline \end{array} $$

If $X$ follows poisson distribution with variance 2 , then $P(X \geq 3)=$

A straight line passing through a fixed point $(2,3)$ intersects the coordinate axes at points $P$ and $Q$. If $O$ is the origin and $R$ is a variable point such that $O P R Q$ is a rectangle, then the locus of $R$ is

By rotating the axes about the origin in anti-clockwise direction with certain angle, if the equation $x^2+4 x y+y^2=1$ is transformed to $\frac{x^2}{a^2}-\frac{y^2}{b^2}=l$, then $\sqrt{\frac{a^2+b^2}{a^2}}=$

If the lines $x+2 a y+a=0, x+3 b y+b=0$, $x+4 c y+c=0$ are concurrent, then $a, b, c$ are in

If $M$ is the foot of the perpendicular drawn from the origin to the line $x-2 y+3=0$ which meets the $X$ and $Y$-axes at $A$ and $B$, respectively, then $A M=$

One line of the pair of lines $x^2+x y-2 y^2=0$ is perpendicular to one line of the pair of lines $3 y^2-5 x y-2 x^2=0$ If the combined equation of the two lines other than those two perpendicular lines is $a x^2+2 h x y+b y^2=0$, then $a+2 h+b=$

If the angle between the lines joining the origin to the points of intersection of $x+2 y+\lambda=0$ and $2 x^2-2 x y+3 y^2+2 x-y-1=0$ is $\frac{\pi}{2}$, then a value of $\lambda$. is

If $Q$ is the inverse point of $P(-1,1)$ with respect to the circle $x^2+y^2-2 x+2 y=0$, then the line containing $Q$ is

If the circle passing through $(3,5),(5,5)$ and $(3,-3)$ cuts the circle $x^2+y^2+2 x+2 f y=0$ orthogonally, then $f=$

Length of the common chord of two circles of same radius is $2 \sqrt{17}$. If one of the two circles is $x^2+y^2+6 x+4 y-12=0$, then acute angle between the two circles is

A circle $S \equiv x^2+y^2-16=0$ intersects another circle $S^{\prime}=0$ of radius 5 units such that their common chord is of maximum length. If the slope of that chord is $\frac{3}{4}$, then the centre of such a circle $S^{\prime}=0$ is

Let $\theta$ be the angle between the circles $S \equiv x^2+y^2+2 x-2 y+c=0$ and $S^{\prime} \equiv x^2+y^2-6 x-8 y+9=0$. If $c$ is an integer and $\cos \theta=\frac{5}{16}$, then the radius of the circle $S=0$ is

$P Q$ is a focal chord of the parabola $y^2=4 x$ with focus $S$. If $P=(4,4)$, then $S Q=$

The angle between the tangents drawn from a point $(-3,2)$ to the ellipse $4 x^2+9 y^2-36=0$ is

If a tangent to the hyperbola $x y=-1$ is also a tangent to the parabola $y^2=8 x$, then the equation of that tangent is

The distance between the tangents of the hyperbola $2 x^2-3 y^2=6$ which are perpendicular to the line $x-2 y+5=0$ is

If $A(0,0,0) B(3,4,0)$ and $C(0,12,5)$ are the vertices of a $\triangle A B C$, then the $x$-coordinate of its incentre is

If $A=(0,4,-3), B=(5,0,12)$ and $C=(7,24,0)$, then $\sqrt{B A C}=$

A plane $\pi$ is passing through the points $A(1,-2,3)$ and $B(6,4,5)$. If the plane $\pi$ is perpendicular the plane $3 x-y+z=2$, then the perpendicular distance from $(0,0,0)$ to the plane $\pi$ is

$$ \mathop {\lim }\limits_{y \to 0} \frac{\sqrt{1+\sqrt{1+y^4}}-\sqrt{2}}{y^4}= $$

If $\mathop {\lim }\limits_{x \to 0} \frac{\cos 2 x-\cos 4 x}{1-\cos 2 x}=k$, then $\lim\limits_{x \rightarrow k} \frac{x^k-27}{x^{k+1}-81}=$

If the function $f(x)=\left\{\begin{array}{l}1+\cos x, x \leq 0 \\ a-x, 02\end{array}\right.$ everywhere, then $a^2+b^2=$

If $x=2 \cos ^3 \theta$ and $y=3 \sin ^2 \theta$, then $\frac{d y}{d x}=$

Assertion (A) If $y=f(x)=(|x|-|x-1|)^2$, then $\left(\frac{d y}{d x}\right)_{x=1}=1$

Reason (R) $\mathop {\lim }\limits_{x \to a} \frac{f(x)-f(a)}{x-a}$ exist, then it is called derivative of $f(x)$ at $x=a$.

If $y=|\cos x-\sin x|+|\tan x-\cot x|$, then

$$ \left(\frac{d y}{d x}\right)_{x=\frac{\pi}{3}}+\left(\frac{d y}{d x}\right)_{x=\frac{\pi}{6}}= $$

If the tangent drawn at the point $(\alpha, \beta)$ on the curve $x^{\frac{2}{3}}+y^{\frac{2}{3}}=4$ is parallel to the line $\sqrt{3 x}+y=1$, then $\alpha^2+\beta^2=$

The displacement $S$ of a particle measured from a fixed point $O$ on a line is given by $S=t^3-16 t^2+64 t-16$. Then, the time at which displacement of the particle is maximum is

If the extreme value of the function $f(x)=\frac{4}{\sin x}+\frac{1}{1-\sin x}$ in $\left[0, \frac{\pi}{2}\right]$ is $m$ and it exists at $x=k$, then $\cos k=$

The interval in which the curve represented by $f(x)=2 x+\log \left(\frac{x}{2+x}\right)$ is

$$ \int \frac{1}{9 \cos ^2 x-24 \sin x \cos x+16 \sin ^2 x} d x= $$

If $\int \frac{1}{\cot \frac{x}{2} \cot \frac{x}{3} \cot \frac{x}{6}} d x=A \log \left|\cos \frac{x}{2}\right| +B \log \left|\cos \frac{x}{3}\right|+C \log \left|\cos \frac{x}{6}\right|+k$, then $A+B+C=$

$$ \int \frac{\sin x+\cos x}{\sin x-\cos x} d x= $$

$$ \int \frac{x^4-1}{x^2 \sqrt{x^4+x^2+1}} d x= $$

$$ \int \frac{(3 x-2) \tan \left(\sqrt{9 x^2-12 x+1}\right)}{\sqrt{9 x^2-12 x+1}} d x= $$

$\int_{\frac{-\pi}{4}}^{\frac{\pi}{3}}\left|\tan \left(x-\frac{\pi}{6}\right)\right| d x=$

$$ \int_0^\pi \frac{x \sin x}{\sin ^2 x+2 \cos ^2 x} d x= $$

The area of the region lying between the curves $y=\sqrt{4-x^2}, y^2=3 x$ and the $Y$-axis is

$$ \mathop {\lim }\limits_{n \to \infty }\left(\frac{1}{1^2+n^2}+\frac{2}{2^2+n^2}+\frac{3}{3^2+n^2}+\ldots+\frac{n}{n^2+n^2}\right)= $$

The general solution of the differential equation $\frac{d y}{d x}=\frac{2 x^2-x y-y^2}{x^2-y^2}$ is

If the degree of the differential equation corresponding to the family of curves $y=a x+\frac{1}{a}$ (where $a \neq 0$ is an arbitary constant) is $r$ and it's order is $m$. Then, the solution of $\frac{d y}{d x}=\frac{y}{2 x}, y(\mathrm{l})=\sqrt{r+m}$ is

The general solution of the differential equation $y+\cos x\left(\frac{d y}{d x}\right)-\cos ^2 x=0$ is

The dimensional formula of Planck's constant is

The ratio of the displacements of a freely falling body during second and fifth seconds of its motion is

The magnitudes of two vectors are $A$ and $B(A>B)$. If the maximum resultant magnitude of the two vectors is ' $n$ ' times their minimum resultant magnitude, then $\frac{A}{B}=$

A particle crossing the origin at time $t=0$ moves in the $X Y$-plane with a constant acceleration ' $a$ ' in $y$-direction. If the equation of motion of the particle is $y=b x^2$ (where $b$ is a constant), then its velocity component in the $x$-direction is

A train of mass $10^6 \mathrm{~kg}$ is moving at a constant speed of $108 \mathrm{~km} / \mathrm{h}$. If the frictional force acting on it is 0.5 N per 100 kg , then the power of the train is

Two balls each of mass 250 g moving in opposite directions each with a speed $16 \mathrm{~ms}^{-1}$ collide and rebound with the same speeds. The impulse imparted to one ball due to the other is

A body is moving along a straight line under the influence of a constant power source. If the relation between the displacement $(s)$ of the body and time $(t)$ is $s \propto t^x$, then $x=$

A body is projected at an angle of $60^{\circ}$ with the horizontal. If the initial kinetic energy of the body is $X$, then its kinetic energy at the highest point is

A thin uniform circular disc rolls with a constant velocity without slipping on a horizontal surface. Its total kinetic energy is

Three thin uniform rods each of mass $M$ and length $L$ are placed along the three axes of a cartesian co-ordinate system with one end of all the rods at origin. The moment of inertia of the system of the rods about $Z$-axis is

For a particle executing simple harmonic motion, the ratio of kinetic and potential energies at a point where displacement is one half of the amplitude is

When the mass attached to a spring is increased from 4 kg to 9 kg , the time period of oscillation increases by $0.2 \pi \mathrm{~s}$. Then, the spring constant of the spring is

Two solid spheres each of radius ' $R$ ' made of same material are placed in contact with each other. If the gravitational force acting between them is $F$, then

The force required to stretch a steel wire of area of cross-section $1 \mathrm{~mm}^2$ to double its length is

(Young's modulus of steel $=2 \times 10^{11} \mathrm{~N}-\mathrm{m}^{-2}$ )

In a hydraulic lift, if the radius of the smaller piston is 5 cm and the radius of the larger piston is 50 cm , then the weight that the larger piston can support when a force of 250 N is applied to the smaller piston is

If the values of the temperature of a body in Fahrenheit and Celsius scales are in the ratio of $13: 5$, then the temperature of the body is

A Carnot heat engine absorbs 600 J of heat from a source at a temperature of $127^{\circ} \mathrm{C}$ and rejects 400 J of heat to a sink in each cycle. The temperature of the sink is

During adiabatic expansion, if the temperature of 3 moles of a diatomic gas decreases by $50^{\circ} \mathrm{C}$, then the work done by the gas is

( $R=$ Universal gas constant)

The fundamental limitation to the coefficient of performance of a refrigerator is given by

If the ratio of specific heats of a gas at constant pressure and at constant volume is $\gamma$, then the number of degrees of freedom of the rigid molecules of the gas is

A steel wire of length 81 cm has a mass of $5 \times 10^{-3} \mathrm{~kg}$.

If the wire is under a tension of 50 N , then the speed of transverse waves on the wire is

A light ray incidents on an equilateral prism made of material of refractive index $\sqrt{3}$. Inside the prism, if the light ray moves parallel to the base of the prism, then the angle of incidence of the light ray is

An unpolarised beam of light incidents on a group of three polarising sheets arranged such that the angle between the axes of any two adjascent sheets is $30^{\circ}$. The ratio of the intensities of polarised light emerging from the second and third sheets is

In a region, the electric field is given by $\mathbf{E}=(3 \hat{\mathbf{i}}+5 \hat{\mathbf{j}}+7 \hat{\mathbf{k}}) \mathrm{NC}^{-1}$. The electric flux through a surface of area $3 \mathrm{~m}^2$ in $y z$-plane is (in SI units)

The energy stored in a capacitor is $W$. To double the charge on the plates of the capacitor, the additional work to be done is

The velocity acquired by an electron at rest when subjected to a uniform electric field of potential difference 180 V is

(Mass of electron $=9 \times 10^{-31} \mathrm{~kg}$ and charge of electron $=1.6 \times 10^{-19} \mathrm{C}$ )

Charge ' $Q$ ' (in coulomb) flowing through a conductor in terms of time ' $t$ ' (in second) is given by the equation $Q=3 t^2+t$. The current in the conductor at time $t=3 \mathrm{~s}$ is

In a metal, the charge carrier density is $9.1 \times 10^{28} \mathrm{~m}^{-3}$ and its electrical conductivity is $6.4 \times 10^7 \mathrm{~S} \mathrm{~m}^{-1}$. When an electric field of $10 \mathrm{NC}^{-1}$ is applied to the metal, then the average time between two successive collisions of electrons in the metal is

(Mass of electron $=9.1 \times 10^{-31} \mathrm{~kg}$, charge of electron $=1.6 \times 10^{-19} \mathrm{C}$ )

The force per unit length on a straight wire carrying current of 8 A making an angle of $30^{\circ}$ with a uniform magnetic field of 0.15 T is

A wire of length 10 m carrying current of 1 A is bent in to a circular loop. If a magnetic field of $2 \pi \times 10^{-4} \mathrm{~T}$ is applied on the loop, then the maximum torque acting on it is

A short bar magnet has a magnetic moment of $0.48 \mathrm{JT}^{-1}$. The magnitude of magnetic field at a point at 10 cm distance from the centre of the magnet on its axis is

A coil of 45 turns and radius 4 cm is placed in a uniform magnetic field such that its plane is perpendicular to the direction of the field. If the magnetic field increases from 0 to 0.70 T at a constant rate in a time interval of 220 s , then the induced emf in the coil is

For better tuning of a series $L C R$ circuit in a communication system, the preferred combination is

The magnitude of the electric field of a plane electromagnetic wave travelling in free space is $E$. If $\mu_0$ and $\varepsilon_0$ are respectively permeability and permittivity of the free space, then the magnitude of magnetic field of the wave is

An alpha particle moves along a circular path of radius 0.5 mm in a magnetic field of $2 \times 10^{-2} \mathrm{~T}$. The de-Broglie wavelength associated with the alpha particle is nearly (Planck's constant $=6.63 \times 10^{-34} \mathrm{~J} \mathrm{~s}$ )

The difference between the frequencies of second and first Paschen lines of hydrogen atom is ( $R=$ Rydberg constant and $c=$ speed of light in vacuum)

If the time taken for a radioactive substance to decay $8 \%$ to $77 \%$ is 12 minutes, then the half life of the substance in minutes is

A camera is fabricated using a semiconducting material having a band gap of 3 eV . The wavelength of light if can detect is nearly

If in an amplitude modulated wave, the maximum amplitude is 14 V and the modulation index is 0.4 , then the amplitude of the carrier wave is