The energy associated with electron in first orbit of hydrogen atom is $-2.18 \times 10^{-18} \mathrm{~J}$. The frequency of the light required (in Hz ) to excite the electron to fifth orbit is ( $h=6.6 \times 10^{-34} \mathrm{Js}$ )

In $\mathrm{Sr}(Z=38)$, the number of electrons with $l=0$ is $x$, number of electrons with $l=2$ is $y \cdot(x-y)$ is equal to ( $l=$ Azimuthal quantum number)

$$ \text { Match the following. } $$

The correct answer is

Observe the following data ( $\Delta_t H_1, \Delta_t H_2$ and $\Delta_{\mathrm{eg}} H$ represent the first, second ionisation enthalpies and electron gain enthalpy respectively)

Using the data identify the most reactive metal.

The sum of bond order of $\mathrm{O}_2^{+}, \mathrm{O}_2^{-}, \mathrm{O}_2$ and $\mathrm{O}_2^{2+}$ is equal to

Observe the following statements

Statement-I Hybridisation is not same in both $\mathrm{SF}_6$ and $\mathrm{BrF}_5$.

Statement-II $\mathrm{BrF}_5$ is square pyramidal while $\mathrm{SF}_6$ is octahedral in shape.

The correct answer is

At $T(\mathrm{~K})$ root mean square (rms) velocity of argon (molar mass $40 \mathrm{~g} \mathrm{~mol}^{-1}$ ) is $20 \mathrm{~ms}^{-1}$. The average kinetic energy of the same gas at $T(\mathrm{~K})$ (in $\mathrm{J} \mathrm{mol}^{-1}$ ) is

4.0 g of a mixture containing $\mathrm{Na}_2 \mathrm{CO}_3$ and $\mathrm{NaHCO}_3$ is heated to 673 K . Loss in mass of the mixture is found to be 0.62 g . The percentage of sodium carbonate in the mixture is

At 298 K , if the standard Gibbs energy change $\Delta_r G^{\circ}$ of a reaction is -115 kJ , the value of $\log _{10} K_p$ will be $\left(R=8.314 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}\right)$

200 mL of an aqueous solution of $\mathrm{HCl}(\mathrm{pH}=2)$ is mixed with 300 mL of aqueous solution of NaOH $(\mathrm{pH}=12)$ and is diluted to 1.0 L . The pH of the resulting solution is ( $\mathrm{pH}=2$ )

Identify the electron rich hydrides from the following

The incorrect statement about Castner-kellner cell process is

The incorrect statement about Castner-kellner cell process is

Which of the following is an incorrect statement about the compounds of group 13 elements?

The incorrect statement about the oxidation states of group 14 elements is

In drinking water, if the maximum prescribed concentration of copper is $x \mathrm{mgdm}^{-3}$, the maximum prescribed concentration of zinc will be

The empirical formula of the compound ' $D$ ' formed in the given reaction sequence is

$$ \mathrm{C}_2 \mathrm{H}_4 \xrightarrow{\mathrm{Br}_2 / \mathrm{CCl}_4} A \xrightarrow[\text { (ii) } \mathrm{NaNH}_2]{\text { (i) } \mathrm{Alc} \cdot \mathrm{KOH}} B \xrightarrow[\text { polymerisation }]{\text { Cyclic }} C \xrightarrow[\text { DryAlCla }{ }_3 \text {, dark.cold }]{\mathrm{Cl}_2 \text { (excess) }} $$

Which one of the following mixtures can be separated by steam distillation technique?

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

An alkyne has the molecular formula $\mathrm{C}_6 \mathrm{H}_{10}$. The number of 1 -alkyne isomers (excluding stereoisomers) possible for it is

A metal crystallises in two cubic phases, fcc and bcc with edge lengths $3.5 \mathop {\rm{A}}\limits^{\rm{o}}$ and $3 \mathop {\rm{A}}\limits^{\rm{o}}$ respectively. The ratio of densities of fcc and bcc is approximately

Observe the following data given in the table ( $K_H=$ Henry's law constant)

$$ \begin{aligned} &\begin{array}{ccccc} \hline \text { Gas } & \mathrm{CO}_2 & \mathrm{Ar} & \mathrm{HCHO} & \mathrm{CH}_4 \\ \hline\left(\boldsymbol{K}_{\mathrm{H}} \text { bar at } \mathbf{2 9 8 ~ K}\right) & 1.67 & 40.3 & 1.83 \times 10^{-5} & 0.413 \\ \hline \end{array}\\ &\text { The correct order of their solubility in water is } \end{aligned} $$

The Gibbs energy change of the reaction (in $\mathrm{kJ} \mathrm{mol}^{-1}$ ) corresponding to the following cell

$\mathrm{Cr}\left|\mathrm{Cr}^{3+}(0.1 \mathrm{M}) \| \mathrm{Fe}^{2+}(0.001 \mathrm{M})\right| \mathrm{Fe}$

(Given $E_{\mathrm{Cr}^{3+} \mid \mathrm{Cr}}^{\circ}=-0.75 \mathrm{~V} ; E_{\mathrm{Fe}^{2+} \mid \mathrm{Fe}}^{\circ}=-0.45 \mathrm{~V}$,

$\left.\mathrm{IF}=96,500 \mathrm{C} \mathrm{mol}^{-1}\right)$

For a first order decomposition of a certain reaction, rate constant is given by the equation. $\log k\left(s^{-1}\right)=7.14-\frac{1 \times 10^4 \mathrm{~K}}{T}$. The activation energy of the reaction ( in $\mathrm{kJ} \mathrm{mol}^{-1}$ ) is

$$ \left(R=8.3 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}\right) $$

The source of an enzyme is malt and that enzyme converts $X$ into $Y . X$ and $Y$ respectively are

In the extraction of iron using blast furnace to remove the impurity $(X)$, chemical $(Y)$ is added to the ore. $X$ and $Y$ are respectively

Thionyl chloride on reaction with white phosphorus gives a compound of phosphorus ' $C$ ' which on hydrolysis gives an oxo acid ' $O$ '. The correct statement about $C$ and $O$ are

I. Shape of ' $C$ ' is pyramidal

II. ' $O$ ' is a dibasic acid

III. ' $O$ ' is a monobasic acid

IV. ' $C$ ' on reaction with acetic acid given ' $O$ '

Which one of the following statements is not correct?

Consider the following.

Assertion (A) Phosphorus can form both phosphorus (III) and phosphorus (V) chlorides but nitrogen cannot form nitrogen (V) chloride.

Reason (R) The electronegativity of nitrogen is more than that of phosphorus.

The correct answer is

$E_{\mathrm{M}^{3}| \mathrm{M}^{2+}}^{\circ}($ in V$)$ is highest for

Arrange the following complexes in the increasing order of their spin only magnetic moment (in B.M)

I. $\left[\mathrm{Fe}(\mathrm{CN})_6\right]^{4-}$

II. $\left[\mathrm{MnCl}_4\right]^{2-}$

III. $\left[\mathrm{Mn}(\mathrm{CN})_6\right]^{4-}$

IV. $\left.\left[\mathrm{Cr}(\mathrm{NH})_3\right)_6\right]^{3+}$

Neoprene is the polymer of a monomer $X$. IUPAC name of $X$ is

On prolonged heating with HI , glucose gives a compound ' $C$ ' which can be obtained by Wurtz reaction using sodium metal and compound ' $D$ ' . Identify ' $D$ '

$$ \text { Match the following } $$

What is the product ' $Z$ ' in the following reaction sequence?

$$ \mathrm{C}_6 \mathrm{H}_5 \mathrm{~N}_2 \mathrm{Cl} \xrightarrow[\mathrm{HCl}]{\mathrm{Cu}_2 \mathrm{Cl}_2} X \xrightarrow[\text { Na/dry ether }]{\mathrm{CH}_3 \mathrm{Cl}} Y \xrightarrow[\text { Dark }]{\mathrm{Cl}_2 / \mathrm{Fe}} Z $$

Identify the compounds $A$ and $B$ involved in the formation of given aldol

In which of the following, intramolecular hydrogen bonding is present?

The products $C$ and $D$ are

The reagent which is used to distinguish primary, secondary and tertiary amines from the mixture is

If $D \subseteq R$ and $f: D \rightarrow R$ defined by $f(x)=\frac{x^2+x+a}{x^2-x+a}$ is a surjection, then ' $a$ ' lies in the interval.

If the domain of the real valued function $f(x)=\frac{1}{\sqrt{\log _{\frac{1}{3}}\left(\frac{x-1}{2-x}\right)}}$ is $(a, b)$, then $2 b=$

If $\frac{1}{2 \cdot 7}+\frac{1}{7 \cdot 12}+\frac{1}{12 \cdot 17}+\frac{1}{17 \cdot 22}+\ldots$ to 10 terms $=k$, then $k=$

If the system of simultaneous linear equations $x+\lambda y-2 z=1, x-y+\lambda z=2$ and $x-2 y+3 z=3$ is inconsistent for $\lambda=\lambda_1$ and $\lambda_2$, then $\lambda_1+\lambda_2=$

The system of linear equation $(\sin \theta) x+y-2 z=0$, $2 x-y+(\cos \theta) z=0$ and $-3 x+(\sec \theta) y+3 z=0$, where $\theta \neq(2 n+1) \frac{\pi}{2}$, has non-trivial solution for

If $A=\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right]$, then $\operatorname{adj}(\operatorname{adj}(\operatorname{adj} A))$

The sum of all the roots of the equation

$\left|\begin{array}{ccc}x & -3 & 2 \\ -1 & -2 & (x-1) \\ 1 & (x-2) & 3\end{array}\right|=0$ is

One of the values of $\sqrt{24-70 i}+\sqrt{-24+70 i}$ is

The set of all values of $\theta$ such that $\frac{1-i \cos \theta}{1+2 i \sin \theta}$ is purely imaginary is

If $\cos \alpha+\cos \beta+\cos \gamma=0=\sin \alpha+\sin \beta+\sin \gamma$, then $\sin 2 \alpha+\sin 2 \beta+\sin 2 \gamma=$

If $\alpha$ is a root of the equation $x^2-x+1=0$, then

$\left(\alpha+\frac{1}{\alpha}\right)^3+\left(\alpha^2+\frac{1}{\alpha^2}\right)^3+\left(\alpha^3+\frac{1}{\alpha^3}\right)^3+\left(\alpha^4+\frac{1}{\alpha^4}\right)^3+\ldots$ to 12 terms $=$

If the equations $x^2+p x+2=0$ and $x^2+x+2 p=0$ have a common root, then the sum of the roots of the equation $x^2+2 p x+8=0$ is

If both roots of the equation $x^2-5 a x+6 a=0$ exceed 1 , then the range of ' $a$ ' is

If $\alpha, \beta, \gamma$ and $\delta$ are the roots of the equation $x^4-4 x^3+3 x^2+2 x-2=0$ such that $\alpha$ and $\beta$ are integers and $\gamma, \delta$ are irrational numbers, then $\alpha+2 \beta+\gamma^2+\delta^2=$

The equation having the multiple root of the equation $x^4+4 x^3-16 x-16=0$ as its roots is

There are 15 stations on a train route and the train has to be stopped at exactly 5 stations among these 15 stations. If it stops at atleast two consecutive stations, then the number of ways in which the train can be stopped is

Number of all possible ways of distributing eight identical apples among three persons is

Number of all possible words (with or without meaning) that can be formed using all the letters of the word CABINET in which neither the word CAB nor the word NET appear is

Numerically greatest term in the expansion of $(2 x-3 y)^n$ when $x=\frac{7}{2}, y=\frac{3}{7}$ and $n=13$ is

If $C_0, C_1, C_2, \ldots, C_8$ are the binomial coefficients in the expansion of $(1+x)^8$, then $\sum\limits_{r = 1}^8 {} r^3 \frac{C_r}{C_{r-1}}=$

If $\frac{x+3}{(x+1)\left(x^2+2\right)}=\frac{a}{x+1}+\frac{b x+c}{x^2+2}$, then $a-b+c=$

If $3 \sin \theta+4 \cos \theta=3$ and $\theta \neq(2 n+1) \frac{\pi}{2}$, then $\sin 2 \theta=$

$$ \frac{\cos 15^{\circ} \cos ^2 22 \frac{1^{\circ}}{2}-\sin 75^{\circ} \sin ^2 \cdot 52 \frac{1^{\circ}}{2}}{\cos ^2 15^{\circ}-\cos ^2 75^{\circ}} $$

$16 \sin 12^{\circ} \cos 18^{\circ} \sin 48^{\circ}=$

Number of solutions of the equation $\sin ^2 \theta+2 \cos ^2 \theta-\sqrt{3} \sin \theta \cos \theta=2$ lying in the interval ( $-\pi, \pi$ ) is

If $0 \leq x<\frac{3}{4}$, then the number of values of $x$ satisfying the equation $\tan ^{-1}(2 x-1)+\tan ^{-1} 2 x= \tan ^{-1} 4 x-\tan ^{-1}(2 x+1)$ is

If $\sinh ^{-1} x=\cosh ^{-1} y=\log (1+\sqrt{2})$, then $\tan ^{-1}(x+y)$

In a $\triangle A B C$, if $c^2-a^2=b(\sqrt{3} c-b)$ and $b^2-a^2=c(c-a)$ then, $\angle A B C$

Let $A B C$ be a triangle right angled at $B$. If $a=13$ and $c=84$, then $r+R=$

If $\mathbf{a}=(x+2 y-3) \hat{\mathbf{i}}+(2 x-y+3) \hat{\mathbf{j}}$ and $\mathbf{b}=(3 x-2 y) \hat{\mathbf{i}} +(x-y+1) \hat{\mathbf{j}}$ are two vectors such that $\mathbf{a}=2 \mathbf{b}$, then $y-5 x=$

$7 \hat{\mathbf{i}}-4 \hat{\mathbf{j}}+7 \hat{\mathbf{k}}, \hat{\mathbf{i}}-6 \hat{\mathbf{j}}+10 \hat{\mathbf{k}},-\hat{\mathbf{i}}-3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}, 5 \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}$ are the position vectors of the points $A, B, C$ and $D$ respectively. If $p \hat{\mathbf{i}}+q \hat{\mathbf{j}}+r \hat{\mathbf{k}}$ is the position vector of the point of intersection of the diagonals of the quadrilateral $A B C D$, then $p+q+r=$

If $\mathbf{a}=\hat{\mathbf{i}}+\sqrt{11} \hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ and $\mathbf{b}=\hat{\mathbf{i}}+\sqrt{11} \hat{\mathbf{j}}-10 \hat{\mathbf{k}}$ are two vectors, then the component of $\mathbf{b}$ perpendicular to $\mathbf{a}$ is

Let $\mathbf{a}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ and $\mathbf{b}=2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+p \hat{\mathbf{k}}$ be two vectors.

If $(\mathbf{a}, \mathbf{b})=60^{\circ}$, then $p=$

Let $\pi_1$ be the plane determined by the vectors $\hat{\mathbf{i}}+\hat{\mathbf{j}}$. $\hat{\mathbf{i}}+\hat{\mathbf{k}}$ and $\pi_2$ be the plane determined by the vectors $\hat{\mathbf{j}}-\hat{\mathbf{k}}, \hat{\mathbf{k}}-\hat{\mathbf{i}}$. Let $\mathbf{a}$ be a non-zero vector parallel to the line of intersection of the planes $\pi_1$ and $\pi_2$. If $\mathbf{b}=\hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}$, then the angle between the vectors $\mathbf{a}$ and $\mathbf{b}$ is

The variance of the discrete data $3,4,5,6,7,8,10,13$ is

If a number $x$ is drawn randomly from the set of numbers $\{1,2,3, \ldots ., 50\}$, then the probability that number $x$ that is drawn satisfies the inequation $x+\frac{10}{x} \leq 11$ is

If a coin is tossed seven times, then the probability of getting exactly three heads such that number two heads occur consecutively is

Two cards are drawn randomly from a pack of 52 playing cards one after the other with replacement. If $A$ is the event of drawing a face card in first draw and $B$ is the event of drawing a clubs card in second draw, then $P\left(\frac{\bar{B}}{A}\right)=$

If $X$ is a random variable with probability distribution $P(X=k)=\frac{(2 k+3) c}{3^k}, k=0,1,2, \ldots .$. to $\infty$, then $P(X=3)=$

If a possion variate $X$ satisfies the relation $P(X=3)=P(X=5)$, then $P(X=4)=$

The equation of the locus of a point, which is at a distance of 5 units from a fixed point $(1,4)$ and also from a fixed line $2 x+3 y-1=0$ is

If $2 x^2+x y-6 y^2+k=0$ is the transformed equation of $2 x^2+x y-6 y^2-13 x+9 y+15=0$ when the origin is shifted to the point $(a, b)$ by translation of axes, then $k=$

The line $L \equiv 6 x+3 y+k=0$ divides the line segment joining the points $(3,5)$ and $(4,6)$ in the ratio $-5: 4$. If the point of intersection of the lines $L=0$ and $x-y+1=0$ is $P(g, h)$, then $h=$

A straight line through the point $P(1,2)$ makes an angle $\theta$ with positive X -axis in anticlockwise direction and meets the line $x+\sqrt{3 y}-2 \sqrt{3}=0$ at $Q$. If $P Q=\frac{1}{2}$, then $\theta=$

The lines $x-2 y+1=0,2 x-3 y-1=0$ and $3 x-y+k=0$ are concurrent. The angle between the lines $3 x-y+k=0$ and $m x-3 y+6=0$ is $45^{\circ}$. If $m$ is an integer, then $m-k=$

If $\tan ^{-1}(2 \sqrt{10})$ is the angle between the lines $a x^2+4 x y-2 y^2=0$ and $a \in Z$, then the product of the slopes of given lines is

If the equation of the circumcircle of the triangle formed by the lines $L_1 \equiv x+y=0$,

$L_2 \equiv 2 x+y-1=0, L_3 \equiv x-3 y+2=0$ is $\lambda_1 L_1 L_2+\lambda_2 L_2 L_3+\lambda_3 L_3 L_1=0$, then $\frac{7 \lambda_1}{\lambda_2}+\frac{\lambda_3}{\lambda_1}=$

A circle $C$ touches $X$-axis and makes an intercept of length 2 units on $Y$-axis. If the centre of this circle lies on the line $y=x+1$, then a circle passing through the centre of the circle $C$ is

If $m_1, m_2$ are the slopes of the tangents drawn through the point $(-1,-2)$ to the circle $(x-3)^2+(y-4)^2=4$, then $\sqrt{3}\left|m_1-m_2\right|=$

A line meets the circle $x^2+y^2-4 x-4 y-8=0$ in two points $A$ and $B$. If $P(2,-2)$ is a point on the circle such that $P A=P B=2$, then the equation of the line $A B$ is

If the centre $(\alpha, \beta)$ of a circle cutting the circles $x^2+y^2-2 y-3=0$ and $x^2+y^2+4 x+3=0$ orthogonally lies on the line $2 x-3 y+4=0$, then $2 \alpha+\beta=$

The radius of a circle $C_1$ is thrice the radius of another circle $C_2$ and the centres of $C_1$ and $C_2$ are $(1,2)$ and $(3,-2)$ respectively. If they cut each other orthogonally and the radius of the circle $C_1$ is $3 r$, then the equation of the circle with $r$ as radius and $(1,-2)$ as centre is

If the normals drawn at the points $P\left(\frac{3}{4}, \frac{3}{2}\right)$ and $Q(3,3)$ on the parabola $y^2=3 x$ intersect again on $y^2=3 x$ at $R$, then $R=$

If $\theta$ is the acute angle between the tangents drawn from the point $(1,5)$ to the parabola $y^2=9 x$, then

Let $P$ be a point on the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ and let the perpendicular drawn through $P$ to the major axis meet its auxiliary circle at $Q$. If the normals drawn at $P$ and $Q$ to the ellipse and the auxiliary circle respectively meet in $R$, then the equation of the locus of $R$ is

The mid-point of the chord of the ellipse $x^2+\frac{y^2}{4}=1$ formed on the line $y=x+1$ is

If the tangent drawn at the point $P(3 \sqrt{2}, 4)$ on the hyperbola $\frac{x^2}{9}-\frac{y^2}{16}=1$ meets its directrix at $Q(\alpha, \beta)$ in fourth quadrant, then $\beta=$

If $m: n$ is the ratio in which the point $\left(\frac{8}{5},-\frac{1}{5}, \frac{8}{5}\right)$ divides the segment joining the points $(2, p, 2)$ and $(p,-2, p)$, where $p$ is an integer than $\frac{3 m+n}{3 n}=$

If $(\alpha, \beta \gamma)$ is the foot of the perpendicular drawn from a point $(-1,2,-1)$ to the line joining the points $(2,-1,1)$ and ( $1,1-2$ ), then $\alpha+\beta+\gamma=$

If $A(2,1,-1), B(6,-3,2), C(-3,12,4)$ are the vertices of a $\triangle A B C$ and the equation of the plane containing the $\triangle A B C$ is $53 x+b y+c z+d=0$, then $\frac{d}{b+c}=$

If $\{x\}=x-[x]$, where $[x]$ is the greatest integer $\leq x$ and $\mathop {\lim }\limits_{x \to {0^ - }} \frac{\cos ^{-1}\left(1-\{x\}^2\right) \sin ^{-1}(1-\{x\})}{\{x\}-\{x\}^4}=\theta$, then $\tan \theta$

For $a \neq 0$ and $b \neq 0$, if the real valued function $f(x)=\frac{\sqrt[5]{a(625+x)}-5}{\sqrt[4]{625+b x}-5}$ is continuous at $x=0$, then $f(0)=$

If $3^x y^x=x^{3 y}$, then the value of $\frac{d y}{d x}$ at $x=1$ is

The value of $x$ at which the real valued function $f(x)=7|2 x+1|-19|3 x-5|$ is not differentiable is

If $y=\left(1-x^2\right) \tanh ^{-1} x$, then $\frac{d^2 y}{d x^2}=$

If $f(x)=\log _{\left(x^2-2 x+1\right)}\left(x^2-3 x+2\right), x \in R-[1,2]$ and $x \neq 0$, then $f^{\prime}(3)=$

If the normal drawn at the point $P$ on the curve $y^2=x^3-x+1$ makes equal intercepts on the coordinate axes, then the equation of the tangent drawn to the curve at $P$ is

If a balloon lying at an altitude of 30 m from an observed at a particular instant is moving horizontally. At the rate of $1 \mathrm{~m} / \mathrm{s}$ away from him, then the rate at which the balloon is moving away directly from the observer at the 40 th second is (in m/s) .

The approximate value of $\sqrt{6560}$ is

A real valued function $f:[4, \infty) \rightarrow R$ is defined as $f(x)=\left(x^2+x+1\right)^{\left(x^2-3 x-4\right)}$, then $f$ is

If a normal is drawn at a variable point $P(x, y)$ on the curve $9 x^2+16 y^2-144=0$, then the maximum distance from the centre of the curve to the normal is

$$ \int e^{-x}\left(x^3-2 x^2+3 x-4\right) d x= $$

$$ \int\left(1+\tan ^2 x\right)(1+2 x \tan x) d x= $$

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

$$ \int \frac{\log x}{(1+x)^3} d x= $$

$$ \int_0^{\pi / 4} \frac{1}{5 \cos ^2 x+16 \sin ^2 x+8 \sin x \cos x} d x= $$

$$ \int_8^{18} \frac{1}{(x+2) \sqrt{x-3}} d x= $$

If [.] denotes the greatest integer function, then $\int_1^2\left[x^2\right] d x=$

The differential equation of a family of hyperbolas whose axes are parallel to coordinate axes, centres lie on the line $y=2 x$ and eccentricity is $\sqrt{3}$ is

The general solution of the differential equation $\left(x^3-y^3\right) d x=\left(x^2 y-x y^2\right) d y$ is

The phenomenon of physics that deals with the constitution and structure of matter at the minute scales of atoms and nuclei is

If the length of a rod is measured as 830600 mm , then the number of significant figures in the measurement is

A particle initially at rest is moving along a straight line with an acceleration of $2 \mathrm{~ms}^{-2}$. At a time of 3 s after the beginning of motion, the direction of acceleration is reversed. The time from the beginning of the motion in which the particle returns to its initial position is

If a body projected with a velocity of $19.6 \mathrm{~ms}^{-1}$ reaches a maximum height of 9.8 m , then the range of the projectile is

(Neglect air resistance)

A force separately produces accelerations of $18 \mathrm{~ms}^{-2}$, $9 \mathrm{~ms}^{-2}$ and $6 \mathrm{~ms}^{-2}$ in three bodies of masses $P, Q$ and $R$ respectively. If the same force is applied on a body of mass $P+Q+R$, then the acceleration of that body is

A body of mass 500 g is falling from rest from a height of 3.2 m from the ground. If the body reaches the ground with a velocity of $6 \mathrm{~ms}^{-1}$, then the energy lost by the body due to air resistance is (Acceleration due to gravity $=10 \mathrm{~ms}^{-2}$ )

A body of mass ' $m$ ' moving with a velocity of ' $v$ ' collides head on with another body of mass ' 2 m ' at rest. If the coefficient of restitution between the two bodies is ' $~ e$ ', then the ratio of the velocities of the two bodies after collision is

A thin uniform circular disc of mass $\frac{10}{\pi^2} \mathrm{~kg}$ and radius 2 m is rotating about an axis passing through its centre and perpendicular to its plane. The work done to increase the angular speed of the disc from $90 \mathrm{rev} / \mathrm{min}$ to $120 \mathrm{rev} / \mathrm{min}$ is

A solid cylinder of mass 2 kg , length 40 cm and radius 10 cm is placed in contact with a solid sphere of mass 0.5 kg and radius 10 cm such that the centres of the two bodies lie along the geometrical axis of the cylinder. The distance of the centre of mass of the system of two bodies from the centre of the sphere is

If the amplitude of a damped harmonic oscillator becomes half of its initial amplitude in a time of 10 s , then the time taken for the mechanical energy of the oscillator to become half of its initial mechanical energy is

A body is projected from the Earth's surface with a speed $\sqrt{5}$ times the escape speed $\left(V_e\right)$. The speed of the body when it escapes from the gravitational influence of the Earth is

A metal rod of area of cross-section $3 \mathrm{~cm}^2$ is stretched along its length by applying a force of $9 \times 10^4 \mathrm{~N}$. If the Young's modulus of the material of the rod is $2 \times 10^{11} \mathrm{Nm}^{-2}$, the energy stored per unit volume in the stretched rod is

An air bubble rises from the bottom to the top of a water tank in which the temperature of the water is uniform. The surface area of the bubble at the top of the tank is $125 \%$ more than its surface area at the bottom of the tank. If the atmospheric pressure is equal to the pressure of 10 m water column, then the depth of water in the tank is

If $W_1$ is the work done in increasing the radius of a soap bubble from ' $r$ ' to ' $2 r$ ' and $W_2$ is the work done in increasing the radius of the soap bubble from ' $2 r$ ' to ' $3 r$ ', then $W_1: W_2=$

To increase the length of a metal rod by $0.4 \%$ the temperature of the rod is to be increased by (Coefficient of linear expansion of the metal $\left.=20 \times 10^{-6}{ }^{\circ} \mathrm{C}^{-1}\right)$

The power of a refrigerator that can make 15 kg of ice at $0^{\circ} \mathrm{C}$ from water at $30^{\circ} \mathrm{C}$ in one hour is

Three moles of an ideal gas undergoes a cyclic process $A B C A$ as shown in the figure. The pressure, volume and absolute temperature at points $A, B$ and $C$ are respectively $\left(p_1, V_1, T_1\right),\left(p_2, 3 V_1, T_1\right)$ and $\left(p_2, V_1, T_2\right)$. Then, the total work done in the cycle $A B C A$ is ( $R=$ Universal gas constant).

The pressure of a mixture of 64 g of oxygen, 28 g of nitrogen and 132 g of carbon dioxide gases in a closed vessel is $p$. Under isothermal conditions if entire oxygen is removed from the vessel, the pressure of the mixture of remaining two gases is

A sound wave of frequency 210 Hz travels with a speed of $330 \mathrm{~ms}^{-1}$ along the positive $X$-axis. Each particle of the wave moves a distance of 10 cm between the two extreme points. The equation of the displacement function ( s ) of this wave is ( $x$ in metre, $t$ in second)

A string vibrates in its fundamental mode when a tension $T_1$ is applied to it. If the length of the string is decreased by $25 \%$ and the tension applied is changed to $T_2$, the fundamental frequency of the string increases by $100 \%$, then $\frac{T_2}{T_1}=$

(Linear density of the string is constant)

An object of height 3.6 cm is placed normally on the principal axis of a concave mirror of radius of curvature 30 cm . If the object is at a distance of 10 cm from the principal focus of the mirror, then the height of the real image formed due to the mirror is

Monochromatic light of wavelength $6000 \mathop {\rm{A}}\limits^{\rm{o}} $ incidents on a small angled prism. If the angle of the prism is $6^{\circ}$, the refractive indices of the material of the prism for violet and red lights are respectively 1.52 and 1.48 , then the angle of dispersion produced for this incident light is

In Young's double slit experiment, if the distance between 5th bright and 7th dark fringes is 3 mm , then the distance between 5th dark and 7th bright fringes is

Four electric charges $2 \mu \mathrm{C}, Q, 4 \mu \mathrm{C}$ and $12 \mu \mathrm{C}$ are placed on $X$-axis at distance $x=0,1 \mathrm{~cm}, 2 \mathrm{~cm}$ and 4 cm respectively. If the net force acting on the charge at origin is zero, then $Q=$

If a particle of mass 10 mg and charge $2 \mu \mathrm{C}$ at rest is subjected to a uniform electric field of potential difference 160 V , then the velocity acquired by the particle is

The potential difference between points $C$ and $D$ of the electrical circuit shown in the figure is

The length of a potentiometer wire is 2.5 m and its resistance is $8 \Omega$. A cell of negligible internal resistance and emf of 2.5 V is connected in series with a resistance of $242 \Omega$ in the primary circuit. The potential difference between two points separated by a distance of 20 cm on the potentiometer wire is

The magnetic field due to a current carrying circular coil on its axis at a distance of $\sqrt{2} \mathrm{~d}$ from the centre of the coil is $B$. If $d$ is the diameter of the coil, then the magnetic field at the centre of the coil is

A square coil of side 10 cm having 200 turns is placed in a uniform magnetic field of 2 T such that the plane of the coil is in the direction of magnetic field. If the current through the coil is 3 mA , then the torque acting on the coil is

The magnetic field at a point $P$ on the axis of a short bar magnet of magnetic moment $M$ is $B$. If another short bar magnet of magnetic moment 2 M is placed on the first magnet such that their axes are perpendicular and their centres coincide. The resultant magnetic field at the point $P$ due to both the magnets is

A circular coil of area $3 \times 10^{-2} \mathrm{~m}^2, 900$ turns and a resistance of $1.8 \Omega$ is placed with its plane perpendicular to a uniform magnetic field of $3.5 \times 10^{-5} \mathrm{~T}$. The current induced in the coil when it is rotated through $180^{\circ}$ in half a second is

An electric bulb, an open coil inductor, an AC source and a key are all connected in series to form a closed circuit. They key is closed and after some time an iron rod is inserted into the interior of the inductor, then

If the rate of change in electric flux between the plates of a capacitor is $9 \pi \times 10^3 \mathrm{Vms}^{-1}$, then the displacement current inside the capacitor is

20 kV electrons can produce X- rays with a minimum wavelength of

The ratio of wavelengths of second line in Balmer series and the first line in Lyman series of hydrogen atom is

A radioactive material of half-life 2.5 hours emits radiation that is 32 times the safe maximum level. The time (in hours) after which the material can be handled safely is

If the number of uranium nuclei required per hour to produce a power of 64 kW is $7.2 \times 10^{18}$, then the energy released per fission is

According to a graph drawn between the input and output voltages of a transistor connected in common emitter configuration, the region in which transistor acts as a switch is

If the energy gap of a semiconductor used for the fabrication of an LED is nearly 1.9 eV , then the color of the light emitted by the LED is

When the receiving antenna is on the ground, the range of a transmitting antenna of height 980 m is (Radius of the Earth $=6400 \mathrm{~km}$ )