The radius of second orbit of hydrogen atom is same as that of orbit ( $n$ ) of an ion ( $x$ ), $n$ and $x$ are respectively.

An electromagnetic radiation of wavelength 331.5 nm is made to strike the surface of a metal. Electrons are emitted with a kinetic energy of $12 \times 10^5 \mathrm{~J} \mathrm{~mol}^{-1}$. The work function (in eV ) of the metal is $\left(h=6.63 \times 10^{-34} \mathrm{Js}, N_A=6 \times 10^{23} \mathrm{~mol}^{-1}\right)$

Match the following

The correct answer is

In long form of periodic table an element ' $E$ ' has atomic number 78 . The period and group number of the element are $x$ and $y$ respectively. $(x+y)$ is equal to

In which of the following options, the molecules are correctly arranged in the increasing order of their bond angles?

In which of the following the compounds are correctly arranged in the decreasing order of boiling points?

The force $(F)$ required to maintain the flow of layers of a liquid is equal to

( $A=$ area of contact of layers

$d z=$ distance between the layers

$d u=$ change in velocity

$\eta=$ coefficient of viscosity)

$$ \begin{aligned} &\text { Consider the following redox reaction in basic medium. }\\ &\begin{aligned} x \mathrm{Cr}(\mathrm{OH})_3+y\left(\mathrm{IO}_3\right)^{-} & +\mathrm{z}(\mathrm{OH})^{-} \rightarrow a\left(\mathrm{CrO}_4\right)^{2-}+b(\mathrm{I})^{-}+\mathrm{c}\left(\mathrm{H}_2 \mathrm{O}\right) \end{aligned} \end{aligned} $$

The incorrect option about it is

The entropy and enthalpy changes for the reaction $\mathrm{CO}(\mathrm{g})+\mathrm{H}_2 \mathrm{O}(\mathrm{g}) \rightleftharpoons \mathrm{CO}_2(\mathrm{~g})+\mathrm{H}_2(\mathrm{~g})$ at 300 K and 1 atm are respectively $-42.4 \mathrm{JK}^{-1}$ and -41.2 kJ . The temperature at which the reaction will go in the reverse direction is

The volume of water required to dissolve $0.1 \mathrm{~g} \mathrm{PbCl}_2$ to get a saturated solution (in mL ) is (Given $K_{s p}\left(\mathrm{PbCl}_2\right)=3.2 \times 10^{-8}$; Atomic mass of $\mathrm{Pb}=207 \mathrm{u}$ )

1 mL of " $x$ volume" $\mathrm{H}_2 \mathrm{O}_2$ solution on heating gives 20 mL of oxygen gas at STP. The $(w / v) \%$ corresponding to " $x$ volume" of $\mathrm{H}_2 \mathrm{O}_2$ is

Identify the correct statement from the following

I. LiF is less soluble in water than NaF

II. Both LiCl and $\mathrm{MgCl}_2$ are insoluble in ethanol

III. Both Li and Mg form nitrides

IV. $\mathrm{Na}_2 \mathrm{CO}_3$ gives $\mathrm{CO}_2$ on heating

The major ingredient $(51 \%)$ in Portland cement is

Boron trifluoride on reaction with lithium aluminium hydride in ether gives $\mathrm{LiF}^{\circ} \mathrm{AlF}_3$ and $X . X$ on reaction with $\mathrm{NH}_3$ gives $Y$. $Y$ on further heating gives a compound $Z$. The number of $\sigma$-bonds and $\pi$-bonds in $Z$ are $x$ and $y$ respectively. $(x+y)$ is equal to

Consider the following sequence of reaction

$$ 2 \mathrm{CH}_3 \mathrm{Cl}+\mathrm{Si} \xrightarrow[573 \mathrm{~K}]{\mathrm{Cu}} X \xrightarrow{\mathrm{H}_2 \mathrm{O}} Y \xrightarrow{\text { Polymerisation }} Z $$

The repeating structural unit in Z is

Which of the following is not the common component of photochemical smog?

Identify the compound $(Z)$ in the following reaction sequence

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

Which purification method is generally used for a high boiling organic liquid compound which decompose below its boiling point?

$$ \text { Match the following } $$

The correct answer is

Sodium metal crystallises in a body centred cubic lattice with edge length of $x \mathop {\rm{A}}\limits^{\rm{o}}$. If the radius of sodium atom is $1.86 \mathop {\rm{A}}\limits^{\rm{o}}$ the value of $x$ is

In a mixture of liquids $A$ and $B$, if the mole fractions of component $A$ in vapour phase and liquid mixture are $x_1$ and $x_2$ respectively, then the total vapour pressure of liquid mixture is

(Where $p_{\mathrm{A}}^0$ and $p_{\mathrm{B}}^0$ are the vapour pressure of pure $A$ and $B$ )

A current of 0.5 ampere is passed through molten $\mathrm{AlCl}_3$ for 96.5 seconds. The mass of aluminium deposited at cathode is $x \mathrm{mg}$ and volume of chlorine liberated (at STP) at anode is $y \mathrm{~mL} . x$ and $y$ are respectively.

$R \rightarrow P$ is a first order reaction. For this reaction a graph of $\ln [R]$ (on $y$-axis) and time (on x -axis) gave a straight line with negative slope. The intercept on $y$-axis is equal to ( $k=$ rate constant)

The correct statements about the properties of colloidal solutions are

A. Tyndall effect is used to distinguish between a colloidal solution and a true solution.

B. Zeta potential is related to movement of colloidal particles.

C. Brownian motion in colloidal solution is faster if the viscosity of the solution is very high.

D. Brownian motion stabilises the sols.

The ore of which metal is concentrated by leaching?

Arrange the following molecules in the correct order of their bond angles

What are the products formed when ammonium dichromate is thermally decomposed?

Sulphur dioxide on reaction with chlorine in the presence of charcoal gives compound $(A)$. This on reaction with white phosphorus gives $\mathrm{SO}_2$ and compound $(B)$. The correct statement about ' $B$ ' is

In which of the following transition metal ion (aquated) is not correctly matched with its colour?

Which one of the following complex ions is diamagnetic in nature?

Polymer $X$ is an example of polyester and $Y$ is an example of polyamide $X$ and $Y$ are respectively

The general structure of alpha amino acid can be represented as

Which amino acid is not correctly matched with $R$ given?

Consider the following.

Assertion (A) Aspirin is useful in the prevention of heart attacks.

Reason (R) Aspirin acts as anti-blood clotting agent The correct answer is

Chlorobenzene when subjected to fittig reaction gives a compound ' $X$ '. The sum of $\sigma$ and $\pi$ - bonds in $X$ is

Cumene on oxidation in air gives a compound, $X$. This on reaction with dilute acid gives $Y$ and $Z . Y$ reacts with sodium metal and not $Z$. What is $Z$ ?

The reaction of benzene with CO and HCl in the presence of anhydrous $\mathrm{AlCl}_3$ gives a compound $X \cdot X$ can also be obtained from which of the following reaction?

What is the product ' $Z$ ' in the given sequence of reactions?

$$ \text { The ratio of } \sigma \text { bonds to } \pi \text { bonds in } Q \text { is } $$

What is the major product ' $Z$ ' in the given reaction sequence?

If $f(x)=x^2+b x+c$ and $f(1+k)=f(1-k) \forall k \in R$, for two real numbers $b$ and $c$ then

The domain of the real valued function $f(x)=\log _{\sqrt{2}}\left(\sqrt{x^2+x}+\sqrt{x^2-x}\right)$ is

$t_1, t_2, t_3, \ldots, t_n$ are positive integers, $S_n=t_1+t_2+t_3+\ldots+t_n$, $S_1=1^2, S_2=3^2, S_3=6^2, S_4=10^2, S_5=15^2$ and similarly other terms are there. Following this pattern, if $S_{10}=k^2$ then $k=$

If $x=\alpha, y=\beta, z=\gamma$ is the solution of the system of equations $2 x+3 y+z=-1,3 x+y+z=4$, $x-3 y-2 z=1$, then the value of $\beta$ is

The positive value of ' $a$ ' for which the system of linear homogeneous equations $x+a y+z=0, a x+2 y-z=0$, $2 x+3 y+z=0$ has non-trivial solution is

If $A=\left[\begin{array}{lll}1 & 2 & 2 \\ 2 & 1 & 1 \\ 1 & 2 & 1\end{array}\right]$ then $|\operatorname{adj}|\left(A^2\right) \mid=$

$K=\left|\begin{array}{cc}3 & 4 \\ 5 & 4\end{array}\right|+\left|\begin{array}{cc}1 & -1 \\ 5 & 4\end{array}\right|+\left|\begin{array}{cc}\frac{1}{3} & \frac{1}{4} \\ 5 & 4\end{array}\right|+\left|\begin{array}{cc}\frac{1}{9} & -\frac{1}{16} \\ 5 & 4\end{array}\right|+\ldots$ to $\infty$, then $K=$

$$ \left(\frac{1+i}{1-i}\right)^{228}= $$

Let $z=x+i y$ represent a point of $P(x, y)$ in the argand plane. If $z$ satisfies the condition that amplitude of $\frac{z-3}{z-2 i}=-\frac{\pi}{2}$ then the locus of $P$ is

$$ (1-i \sqrt{3})^{2025}= $$

One of the roots of the equation $(x+1)^4+81=0$ is

If $\alpha, \beta$ are the roots of the equation $x^2+3 x+k=0$ and $\alpha+\frac{1}{\alpha}, \beta+\frac{1}{\beta}$ are the roots of the equation $4 x^2+p x+18=0$, then $k$ satisfies the equation

If $f(x)$ is a second degree polynomial such that $f(x) \geq 0 \forall x \in R, f(-3)=0$ and $f(0)=18$, then $f(3)=$

If one of the roots of the equation $6 x^3-25 x^2+2 x+8=0$ is an integer and $\alpha>0, \beta<0$ are the other two roots, then $\frac{4}{\alpha}+\frac{1}{\beta}=$

If $\alpha, \beta, \gamma, \delta$ and $\varepsilon$ are the roots of the equation $x^5+x^4-13 x^3-13 x^2+36 x+36=0$ and $\alpha<\beta<\gamma<\delta<\varepsilon$ then $\frac{\varepsilon}{\alpha}+\frac{\delta}{\beta}+\frac{1}{\gamma}=$

5 boys and 5 girls have to sit around a table. The number of ways in which all of them can sit so that no two boys and no two girls are together is

All possible words (with or without meaning) the contain the word 'GENTLE' are formed using all the letters of the word 'INTELLIGENCE'. Then, the number of words in which the word 'GENTLE' appears among the first nine positions only is

$$ { }^{20} P_5-{ }^{19} P_5= $$

If $C_0, C_1, C_2, \ldots, C_{10}$ represent the binomial coefficients in the expansion of $(1+x)^{10}$, then

$$ C_0 C_6+C_1 C_7+C_2 C_8+C_3 C_9+C_4 C_{10}= $$

When $|x|<\frac{1}{2}$ the coefficient of $x^6$ in the expansion of $\left(\frac{2-x}{1+2 x}\right)^2$ is

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

If $\cos \theta+\sin \theta=\sqrt{2} \cos \theta$ and $0<\theta<\frac{\pi}{2}$, then $\sec 2 \theta+\tan 2 \theta=$

If $0 \leq A, B \leq \frac{\pi}{4}$ and $\cot A+\cot B+\tan A+ \tan B=\cot A \cot B-\tan A \tan B$, then $\sin (A+B)=$

If the extreme values of the function $f(x)=(2 \sqrt{6}+1) \cos x+(2 \sqrt{2}-\sqrt{3}) \sin x-6$ are $m$ and $M$ then $\sqrt{\left|M^2-m^2\right|}=$

Number of solutions of the equation $\tan ^2 x+3 \cot ^2 x=2 \sec ^2 x$ lying in the interval $[0,2 \pi]$ is

$$ \sin ^{-1}(-\cos 2)+\cos ^{-1}(\sin 3)+\tan ^{-1}(\cot 5)= $$

If $x=\log _e 3$, then $\tanh 2 x+\operatorname{sech} 2 x=$

If $a=3, b=5, c=7$ are the sides of a $\triangle A B C$, then $\cot A+\cot B+\cot C=$

Let $p_1, p_2$ and $p_3$ be the altitudes of a $\triangle A B C$ drawn through the vertices $A, B$ and $C$ respectively. If $r_1=4$, $r_2=6, r_3=12$ are the ex-radii of $\triangle A B C$, then $\frac{1}{p_1^2}+\frac{1}{p_2^2}+\frac{1}{p_3^2}=$

$A B C D$ is a tetrahedron, $\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}},-2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+3 \hat{\mathbf{k}}$, $3 \bar{i}+2 \bar{j}-\bar{k}$ are the the position vectors of the points $A, B$ and $C$ respectively. $-\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}$ is the position vector of the centroid of the triangular face $B C D$. If G is the centroid of the tetrahedron, then $G D=$

If $\mathbf{a}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}, \mathbf{b}=6 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}, \mathbf{c}=-4 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+12 \hat{\mathbf{k}}$ are three vectors, then $\sqrt{(|\mathbf{a}|+|\mathbf{b}|+|\mathbf{c}|)+|\mathbf{a}+\mathbf{b}+\mathbf{c}|}=$

Let $\mathbf{a}$ and $\mathbf{b}$ be two vectors such that $|\mathbf{a}|=|\mathbf{b}|$ and $|\mathbf{a}+2 \mathbf{b}|=|2 \mathbf{a}-\mathbf{b}|$. If $\mathbf{c}$ is a vector parallel to $\mathbf{a}$, then the angle between $\mathbf{b}$ and $\mathbf{c}$ is

If $\mathbf{a}$ and $\mathbf{b}$ are two vectors such that $|\mathbf{a}|=|\mathbf{b}|=\sqrt{6}$ and $\mathbf{a} \cdot \mathbf{b}=-1$, then $|\mathbf{a} \times \mathbf{b}| \sin (\mathbf{a}, \mathbf{b})=$

If the volume of a tetrahedron having $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}, 2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-3 \hat{\mathbf{k}}$ and $3 \hat{\mathbf{i}}-\hat{\mathbf{j}}+p \hat{\mathbf{k}}$ as its coterminous edges is 2 , then the values of $\mathbf{p}$ are the roots of the equation

$$ \text { The coefficient of variation for the following data is } $$

$$ \begin{array}{llllll} \hline \text { Class interval } & 0-2 & 2-4 & 4-6 & 6-8 & 8-10 \\ \hline \text { Frequency } & 2 & 3 & 5 & 3 & 2 \\ \hline \end{array} $$

If two smallest squares are chosen at random on a chess board, then the probability of getting these squares such that they do not have a side in common is

Let $A$ and $B$ be two events in a random experiment . If $P(A \cap \bar{B})=0.1, P(\bar{A} \cap B)=0.2$ and $P(B)=0.5$, then $P(\bar{A} \cap \bar{B})=$

An urn contains 7 red, 5 white and 3 black balls. Three balls are drawn randomly one after the other without replacement. If it is known that first ball drawn is red and the second ball drawn is white, then the probability that the third ball drawn is not red is

The range of a discrete random variable $X$ is $\{1,2,3\}$ and the probabilities of its elements are given by $P(X=1)=3 k^3, P(X=2)=2 k^2$ and $P(X=3)=7-19 \mathrm{k}$. Then, $P(X=3)=$

Among every 8 units of a product, one is likely to be defective. If a consumer has order 5 units of that product, then the probability that atmost one unit is defective among them is

If $A=(0,1), B=(1,2), C=(-2,1)$, then the equation of the locus of a point $P$ such that area of $\triangle P A B=$ area of $\triangle P A C$ is

$(a, b)$ are the new coordinates of the point $(2,3)$ after shifting the origin to the point $(3,2)$ by translation of axes. If $(c, d)$ are the new coordinates of the point $(a, b)$ after rotating the axes through an angle $\frac{\pi}{4}$ about the origin in the anti-clockwise direction, then $d-c=$

The lines $x+y+4=0, x-2 y-4=0$ and $3 x+4 y-2=0$

The area of the triangle formed by the line $L$ with the coordinate axes is 12 sq. units. If $L$ passes through the point $(12,4)$ and the product $P$ of $X$ - intercept of $L$ and square of the $Y$-intercept of $L$ is negative, then $P=$

The area of the quadrilateral formed by the lines $x+2 y+3=0,2 x+4 y+9=0, x-2 y+3=0$ and $3 x-6 y+11=0$

If $(-1,-1)$ is the point of intersection of the pair of lines $2 x^2+5 x y-3 y^2+2 g x+2 f y+c=0$. Then $g+f$

If the length of the chord $2 x+3 y+k=0$ of the circle $x^2+y^2-2 x+4 y-11=0$ is $2 \sqrt{3}$, then the sum of all possible values of $k$ is

The power of a point $(2,-1)$ with respect to a circle $C$ of radius 4 is 9 . The centre of the circle $C$ lies on the lines $x+y=0$ and in the 2nd quadrant. If ( $\alpha, \beta$ ) is the centre of the circle $C$ then $\beta-\alpha=$

The angle between the tangents drawn from the point $P(k, 6 k)$ to the circle $x^2+y^2+6 x-6 y+2=0$ is $2 \tan ^{-1}\left(\frac{4}{3}\right)$. If the coordinates of $P$ are integers, then $k=$

The tangents drawn from a point $(2,-1)$ touch the circle $x^2+y^2+4 x-2 y+1=0$ at the points $A$ and $B$. If $C$ is the centre of the circle, then the area (in sq. units) of the $\triangle A B C$ is

If $\theta$ is the angle between the circles $x^2+y^2-4 x+2 y-4=0$ and $x^2+y^2-2 x+4 y-11=0$ then $\sin \theta=$

If the line $x+y=2$ cuts the circle $x^2+y^2+2 x-4 y+4=0$ at two points $A$ and $B$, then the radius of the circle passing through $A, B$ and orthogonal to $x^2+y^2-2 x-4 y-4=0$ is

A normal chord $P Q$ drawn at a point $P$ on the parabola $y^2=5 x$ subtends a right angle at the vertex. If $P$ lies in the first quadrant, then the other end $Q$ of the normal chord is

If $L(p, q), q>3$ is one end of the latus rectum of the parabola $(y-2)^2=3(x-1)$, then the equation of the tangent at $L$ to this parabola is

If $P$ is any point on the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$ and $S, S^{\prime}$ are its foci, then the maximum area (in sq. units) of $\triangle S P S^{\prime}=$

Let $e$ be the eccentricity of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$.

If $a=5, b=4$ and the equation of the normal drawn at one end of the latus rectum that lies in the first quadrant is $l x+m y=27$ then $l+m=$

If the latus rectum through one of the foci of a hyperbola $\frac{x^2}{9}-\frac{y^2}{b^2}=1$ subtends a right angle at the farther vertex of the hyperbola, then $b^2=$

The equation of the locus of a point whose distance from $X Y$-plane is twice its distance from $Z$-axis is

If $\alpha$ is the angle between any two diagonals of a cube and $\beta$ is the angle between a diagonal of a cube and a diagonal of its face, which intersects this diagonal of the cube, then $\cos \alpha+\cos ^2 \beta=$

If the angle between the planes $a x-y+3 z=2 a$ and $3 x+a y+z=3 a$ is $\frac{\pi}{3}$, then the direction ratio of the line perpendicular to the plane $(a+2) x+(a-4) y+2 a z=a$ are

If $\mathop {\lim }\limits_{x \to 0} \frac{3^{x^3}-\left(1-x^3\right)^{\frac{2}{3}}}{x^2 \sin x}=p+\log q$, then $p q=$

If $[x]$ is the greatest integer function and

$$ f(x)=\left\{\begin{array}{cc} 2[x]-\frac{x}{|x|}, & x \neq 0 \\ 1, & x=0 \end{array}\right. $$

is a real valued function, then $f$ is

If $x=2 \sqrt{2} \sqrt{\cos 2 \theta}$ and $y=2 \sqrt{2} \sqrt{\sin 2 \theta}, 0<\theta<\frac{\pi}{4}$, then the value of $\frac{d y}{d x}$ at $\theta=22 \frac{1}{2}^{\circ}$ is

The domain of the derivative of the function $f(x)=\cos ^{-1}(2 x-5)-\sin ^{-1}(x-2)$ is

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

If $y=x^{\log x}+(\log x)^x, x>1$, then $\left(\frac{d y}{d x}\right)_{x=e}=$

If the curves $y^2=12 x-3$ and $y^2=12-k x$ cut each other orthogonally, then the length of the sub-tangent at $(1, b)$ on the curve $y^2=12-k x$ is

A rod of length 41 m with an end $A$ on the floor and another end $B$ on the wall perpendicular to the floor is sliding away horizontally from the wall at the rate of $3 \mathrm{fit} / \mathrm{min}$. When the end $B$ is at the height of 9 ft from the floor, then the rate at which the area of the triangle formed by the rod with wall and floor changes at that instant is (in $\mathrm{ft} / \mathrm{min}$ )

There is a possible error of 0.02 cm in measuring the base diameter of a right circular cone as 14 cm . If the semi-vertical angle of the cone is $45^{\circ}$, then the approximate error in its volume is (in $\mathrm{cu} . \mathrm{cm}$ )

The real valued function $f(x)=\frac{x^2}{2}-\log \left(x^2+x+1\right)$ is

If $x$ and $y$ are two positive real numbers such that $x y=4$, then the minimum value of $\left(\sqrt{x}+\frac{y^2}{2}\right)$ is

If $\int x^3 \sin 3 x d x=\frac{1}{27}[f(x) \cos 3 x+g(x) \sin 3 x]+C$, then $f(\mathrm{l})+g(\mathrm{l})=$

If $I_1=\int \sin ^6 x d x$ and $I_2=\int \cos ^6 x d x$, then $I_1+I_2=$

$$ \int \frac{x+\cos x}{1-\sin x} d x= $$

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

$$ \int_{-4}^5 \frac{1}{\sqrt{20+x-x^2}} d x= $$

$$ \int_0^{\frac{\pi}{2}} \frac{d x}{\cos x-\sqrt{3} \sin x}= $$

$$ \int_0^{\frac{\pi}{2}} \sqrt{\tan x d x}= $$

If $y=f(x)$ is the solution of the differential equation $\left(1+\cos ^2 x\right) f^{\prime}(x)-4 \sin 2 x-f(x) \sin 2 x=0$ when $f(0)=0$, then $f\left(\frac{\pi}{3}\right)=$

The differential equation corresponding to the family of ellipses $\frac{x^2}{a^2}+\frac{y^2}{4}=1$, where ' $a$ ' is an arbitrary constant is

Match the "Technology" given in List-I with the "Principle of physics" given in List-II.

$$ \begin{array}{l|l|l|l} \hline & \text { List-I (Technology) } & & \text { List-II (Principle of physics) } \\ \hline \text { (A) } & \text { Steam engine } & \text { I } & \begin{array}{l} \text { Magnetic confinement of } \\ \text { plasma } \end{array} \\ \hline \text { (B) } & \text { Electron microscope } & \text { II } & \text { Laws of thermodynamics } \\ \hline \text { (C) } & \text { Non-reflecting coatings } & \text { III } & \text { Wave nature of electrons } \\ \hline \text { (D) } & \text { Tokamak } & \text { IV } & \text { Interference of light } \\ \hline \end{array} $$

The relation between the displacement ' $x$ ' (in metre) and the time ' $t$ ' (in second) of a particle is $t=2 x^2+3 x$. If the displacement of the particle is 25 cm from the origin $(x=0)$, then the acceleration of the particle is

A body projected at certain angle $\left(\neq 90^{\circ}\right)$ from the ground crosses a point in its path at a time of 2.3 s and from there it reaches the ground after a time of 5.7 s . The maximum heigh reached by the body is (Acceleration due to gravity $=10 \mathrm{~ms}^{-2}$ )

A circular path of radius 75 m is banked at an angle of $\tan ^{-1}(0.2)$. If the coefficient of static friction between the tyres of the car and the circular path is 0.1 , then the maximum permissible speed of the car to avoid slipping is

A horizontal force of 10 N is applied on a block of mass 1.5 kg which is initially at rest on a rough horizontal surface. The work done by the applied force in a time of 6 s from the beginning of the motion is (Acceleration due to gravity $=10 \mathrm{~ms}^{-2}$; the coefficient of kinetic friction between the block and the surface is 0.2)

A ball is allowed to fall freely from a height of 42 m form the ground. If the coefficient of restitution between the ball and the ground is 0.4 , then the total distance travelled by the ball before it comes to rest is

A thin uniform wire of mass ' $m$ ' and linear density ' $\rho$ ' is bent in the form of a circular ring. The moment of inertia of the ring about a tangent parallel to its diameter is ' $m$ '

A solid sphere and a thin uniform circular disc of same radius are rolling down an inclined plane without slipping. If the acceleration of the sphere is $3 \mathrm{~ms}^{-2}$, then the acceleration of the disc is

If the amplitudes of a damped harmonic oscillator at times $t=0, t_1$ and $t_2$ are $A_0, A_1$ and $A_2$ respectively, then the amplitude of the oscillator at a time of $\left(t_1+t_2\right)$ is

A meteor of mass ' $m$ ' having a speed ' $V$ ' at infinity reaches the surface of the Earth with a speed of ( $v_c$ is escape speed from the Earth's surface)

The work to be done to produce a strain of $10^{-3}$ in a steel wire of mass 2.96 kg and density $7.4 \mathrm{~g} \mathrm{~cm}^{-3}$ is

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

A wooden block of outer volume 1 litre and specific gravity $\frac{3}{4}$ having a cavity floats with half of its volume immersed in water. Then, the volume of the cavity is

When ' $n$ ' identical mercury drops combine to form a single big drop

The temperature of a body shown by a faulty Celsius thermometer is $49^{\circ} \mathrm{C}$ and by a correct Fahrenheit thermometer is $122^{\circ} \mathrm{F}$. The correction to be applied to the faulty thermometer is

If the radiation emitted by a perfect radiator has maximum intensity at a wavelength of $2900 \mathop {\rm{A}}\limits^{\rm{o}}$, the intensity of radiation emitted by it is

(Stefan-Boltzmann's constant $=5.67 \times 10^{-8} \mathrm{Wm}^{-2} \mathrm{~K}^{-4}$ and Wein's constant $=2.9 \times 10^{-3} \mathrm{mK}$ )

The ratio of the work done, change in internal energy and heat absorbed when a diatomic gas expands at constant pressure is

If the temperature of a gas is increased from $127^{\circ} \mathrm{C}$ to $527^{\circ} \mathrm{C}$, then the rms speed of the gas molecules

An air column in a tube of length 50 cm , closed at one end is vibrating in its fifth harmonic. The phase difference between a particle at the open end and a particle at 42 cm from the open end is

A metal rod of length 125 cm is clamped at its midpoint. If the speed of the sound in the metal is $5000 \mathrm{~ms}^{-1}$, then the fundamental frequency of the longitudinal vibrations of the rod is

If the distances of the object and its real image from the principal focus of a concave mirror are 16 cm and 9 cm respectively, then the focal length of the mirror is

If the angle of minimum deviation produced by an equilateral prism is equal to the angle of the prism, then the refractive index of the material of the prism is nearly

When two light waves of equal intensity superimpose, the maximum intensity obtained is $I$. If the intensity of one of the waves is quadrupled, then the maximum intensity obtained is

The electric field due to an infinitely long thin straight wire with uniform linear charge density of $2.5 \times 10^{-7} \mathrm{~cm}^{-1}$ at a radial distance of $x$ from the wire is $7.5 \times 10^4 \mathrm{NC}^{-1}$. Then, $x=$

A parallel plate capacitor of capacitance $10 \mu \mathrm{~F}$ is charged by a 220 V supply. The capacitor is then disconnected from the supply and is connected to another uncharged parallel plate capacitor of capacitance $12 \mu \mathrm{~F}$. The loss of electrostatic energy in this process is

The lengths of two wires made of the same material are in the ratio $2: 3$ and their radii are in the ratio $1: 2$. If the two wires are connected in parallel to a battery, then the ratio of the drift velocities of free electrons in the two wires is

In a potentiometer experiment for the determination of the internal resistance of a cell, when an external resistance of $R$ is connected parallel to the cell, the balancing length decreases by $10 \%$. The internal resistance of the cell is

The number of turns of two circular coils $A$ and $B$ are 300 and 200 respectively. The magnetic moments of the two coils $A$ and $B$ are in the ratio $1: 2$. If the two coils carry equal currents, then the ratio of radii of coils $A$ and $B$ is

Two long straight parallel wires carry currents of 8 A and 10 A in opposite directions. If the distance of separation between the wires is 9 cm , then the net magnetic field at a point between the two wires, which is at a perpendicular distance of 4 cm from the wire carrying 8 A current is

A short bar magnet of magnetic moment $2.5 \mathrm{Am}^2$ is kept in a uniform magnetic field of $4 \times 10^{-5} \mathrm{~T}$. The work done in moving the magnet from its most stable position to most unstable position is

The radius of a coil of $N$ turns is $R$. If the plane of the coil is placed parallel to a uniform magnetic field $B$, then the flux linked with the coil is

The inductance $L$, capacitance $C$ and resistance $R$ are the values of the components connected in series to an AC source of angular frequency $\omega$. The inductive and capacitive reactances are $X_L$ and $X_C$ respectively. If the circuit is purely resistive, then

If the rate of change of electric field across the plates of a parallel plate capacitor is $E$ and the displacement current is $I$, then the area of one plate of the capacitor is ( $\varepsilon_0$ is permittivity of free space)

The work done to accelerate an electron from rest so that it can have a de-Broglie wavelength of $6600 \mathop {\rm{A}}\limits^{\rm{o}}$ is nearly

(Planck's constant $=6.6 \times 10^{-34} \mathrm{Js}$ and mass of electron $=9 \times 10^{-31} \mathrm{~kg}$ )

If the total energy of an electron in an orbit is positive, then

If $87.5 \%$ of atoms of a radioactive element decay in 6 days, then the fraction of atoms of the element that decay in 8 days is

If the ratio of the mass numbers of two nuclei is $27: 125$, then the ratio of their surface areas is

At absolute zero temperature, a semiconductor behaves like

Three logic gates are connected as shown in the figure. If the inputs are $A=1, B=0$ and $C=1$, then the values of $y_1, y_2$ and $y_3$ respectively are

The radio horizon of a transmitting antenna of height 39.2 m is (Radius of the Earth $=6400 \mathrm{~km}$ )