The energy of second orbit of hydrogen atom is $-5.45 \times 10^{-19} \mathrm{~J}$. What is the energy of first orbit of $\mathrm{Li}^{2+}$ ions (in J)?

The number of electrons with $(n+l)$ values equal to 3,4 and 5 in an element with atomic number $(Z) 24$ are respectively

( $n=$ principal quantum number and $l=$ azimuthal quantum number)

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

$$ \begin{array}{cccc} \hline & \begin{array}{c} \text { List-I } \\ \text { (Atomic number) } \end{array} & & \begin{array}{c} \text { List-II } \\ \text { (Group number and } \\ \text { period number) } \end{array} \\ \hline \text { A. } & 56 & \text { I. } & 9,4 \\ \hline \text { B. } & 50 & \text { II. } & 3,6 \\ \hline \text { C. } & 27 & \text { III. } & 14,5 \\ \hline \text { D. } & 58 & \text { IV. } & 2,6 \\ \hline \end{array} $$

The correct answer is

According to molecular orbital theory, the molecule which contains only $\pi$-bonds between the atoms is

In which of the following changes there is no change in hybridisation of the central atom?

The rate of diffusion of a gas $A$ is $\sqrt{5}$ times more than that of gas $B$. If the molar mass of $A$ is $x \mathrm{~g} \mathrm{~mol}^{-1}$, the molar mass of $B$ (in $\mathrm{g} \mathrm{mol}^{-1}$ ) is

Which one of the following has the same number of atoms as are in 6 g of $\mathrm{H}_2 \mathrm{O}$ ?

For the reaction at $25^{\circ} \mathrm{C}, X_2 \mathrm{O}_4(l) \longrightarrow 2 \mathrm{XO}_2(g), \Delta U$ and $\Delta S$ are 2.1 k cal and $20 \mathrm{cal} / \mathrm{K}$ respectively. What is $\Delta \mathrm{G}$ for the reaction at the same temperature? ( $R=2 \mathrm{cal} \mathrm{K}^{-1} \mathrm{~mol}^{-1}$ )

Which of the following does not form a buffer solution?

The hydrides of which group elements of the periodic table form electron precise hydrides?

The alkali metal with lowest $E^{\circ}{ }_{M^{+} / M}(\mathrm{~V})$ is $X$ and the alkali metal with highest $E^{\circ}{ }_{M^{+} / M}(\mathrm{~V})$ is $Y$. Then $X$ and $Y$ are respectively

Assertion (A) Be and Mg do not impart any colour to the flame.

Reason (R) The electrons in them are strongly bound to get excited by flame.

Aluminium reacts with dilute HCl and liberates a gas ' $A$ ' and with aqueous alkali liberates a gas ' $B$ '. $A$ and $B$ respectively are

Identify the incorrect statement about the oxidation states of group 14 elements.

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

The aromatic compound/species with maximum number of $\pi$-electrons is

$$ \text { In the conversion of } A \text { to } B \text {, the electrophile involved is } $$

An alkene $X$ on ozonolysis gives a mixture of propan-2-one and methanal. What is $X$ ?

The formula of a metal oxide is $M_{0.96} \mathrm{O}_1$. The fractions of metal that exists as $M^{3+}$ and $M^{2+}$ ions in that oxide are respectively

At $50^{\circ} \mathrm{C}$, the vapour pressure of pure benzene is 268 torr. The number of moles of non-volatile solute per mole of benzene required to prepare a solution having a vapour pressure of 167 torr at the same temperature is (molar mass of benzene $=78 \mathrm{~g} \mathrm{~mol}^{-1}$ )

The rate law for the decomposition of hydrogen iodide is $-\frac{d[\mathrm{HI}]}{d t}=k[\mathrm{HI}]^2$. The units of rate constant $k$ are

A current of 15.0 A is passed through a solution of $\mathrm{CrCl}_2$ for 45 minutes. The volume of $\mathrm{Cl}_2$ (in L ) obtained at the anode at 1 atm and 273 K is around (IF $=96500 \mathrm{C} \mathrm{mol}^{-1}$, atomic wt. of $\mathrm{Cl}=35.5, R=0.082 \mathrm{L}-\mathrm{atm} \mathrm{K}^{-1} \mathrm{~mol}^{-1}$ )

Which of the following can form ionic micelles in water?

The ore which is purified by leaching process is

$$ \begin{array}{r} \mathrm{P}_4+3 \mathrm{NaOH}+3 \mathrm{H}_2 \mathrm{O} \xrightarrow{\mathrm{CO}_2} 3 X+\mathrm{PH}_3 \uparrow \\ X \xrightarrow{\mathrm{HCl}(\mathrm{aq})} Y+\mathrm{NaCl} \end{array} $$

The incorrect statement about $Y$ is

The ratio of lone pair of electrons to bond pair of electrons in ozone molecule is

Consider the following statements about the oxides of halogens.

A. At room temperature, $\mathrm{OF}_2$ is thermally stable.

B. Order of stability of oxides of halogens is

$$ \mathrm{I}>\mathrm{Br}>\mathrm{Cl} . $$

C. $\mathrm{I}_2 \mathrm{O}_5$ is used in the estimation of CO .

D. $\mathrm{ClO}_2$ is used as a bleaching agent.

The correct statements are,

$$ \begin{aligned} \mathrm{XeF}_4+\mathrm{O}_2 \mathrm{~F}_2 & \longrightarrow X+\mathrm{O}_2 \\ X+\underset{(1 \text { mole })}{\mathrm{H}_2 \mathrm{O}} & \longrightarrow Y+2 \mathrm{HF} \end{aligned} $$

The shapes of molecules of $X$ and $Y$ respectively are

Among the following the incorrect statement about transition metals is

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

$$ \begin{array}{llll} \hline & \begin{array}{l} \text { List-I } \\ (\text { Complex }) \end{array} & & \begin{array}{l} \text { List-II } \\ \text { (Spin only magnetic } \\ \text { moment) } \end{array} \\ \hline \text { A. } & {\left[\mathrm{CoF}_6\right]^{3-}} & \text { I. } & 0 \\ \hline \text { B. } & {\left[\mathrm{Co}\left(\mathrm{C}_2 \mathrm{O}_4\right)_3\right]^{3-}} & \text { II. } & \sqrt{24} \\ \hline \text { C. } & {\left[\mathrm{FeF}_6\right]^{3-}} & \text { III. } & \sqrt{8} \\ \hline \text { D. } & {\left[\mathrm{Mn}(\mathrm{CN})_6\right]^{3-}} & \text { IV. } & \sqrt{35} \\ \hline & & \text { V. } & \sqrt{15} \\ \hline \end{array} $$

The correct answer is

Ziegler-Natta catalyst is used in the manufacture of

The deficiency of which vitamin causes pernicious anaemia?

$$ \text { The following molecule with the structure acts as } $$

Which of the following is least reactive towards $\mathrm{S}_{\mathrm{N}} 1$ reactions?

Consider the following sequence of reactions,

$$ \mathrm{C}_2 \mathrm{H}_4 \xrightarrow[\text { (ii) } \mathrm{Zn} / \mathrm{H}_2 \mathrm{O}]{\text { (i) } \mathrm{O}_3} P \xrightarrow[\text { (ii) } \mathrm{H}_2 \mathrm{O} / \mathrm{H}^{+}]{\text {(i) } \mathrm{CH}_3 \mathrm{MgBr} / \text { Ether }} Q \xrightarrow[\mathrm{CH}_2 \mathrm{Cl}_2]{\mathrm{PCC}} R $$

The incorrect statement about $R$ is

The common name of benzene-1,3-diol is

Identify the major product $C$ in the given sequence of reactions

Arrange the following in the correct order of their reactivity towards nucleophilic addition reactions.

$R$ is one of the monomers for the formation of a polymer called

$$ \text { What is } B \text { in the given reaction? } $$

If ${ }^n C_r$ denotes the number of combinations of $n$ distinct things taken $r$ at a time, then the domain of the function $g(x)={ }^{(16-x)} C_{(2 x-1)}$ is

Let $X=\left\{\left.\left[\begin{array}{ll}a & b \\ c & d\end{array}\right] \right\rvert\, a, b, c, d \in R\right\}$. If $f: X \rightarrow R$ is defined by $f(A)=\operatorname{det}(A) . \forall A \in X$, then $f$ is

If $f(x)$ is a function such that $f(x+y)=f(x)+f(y)$ and $f(1)=7$, then $\sum_{r=1}^n f(r)=$

If $A$ is a square matrix of order $3, \operatorname{then}\left|\operatorname{Adj}\left(\operatorname{Adj} A^2\right)\right|=$

If $A$ and $B$ are two square matrices of the same order and $(A B+B A)^T+(A B-B A)^T=2 B A$, then

If $\operatorname{adj}\left[\begin{array}{ccc}1 & 0 & 2 \\ -1 & 1 & -2 \\ 0 & 2 & 1\end{array}\right]=\left[\begin{array}{ccc}5 & m & -2 \\ 1 & 1 & 0 \\ -2 & -2 & n\end{array}\right]$, then $m+n=$

If $A=\left[\begin{array}{ll}0 & 3 \\ 0 & 0\end{array}\right]$ and $f(x)=x+x^2+x^3+\ldots \ldots+x^{2023}$, then $f(A)+I=$

If $i=\sqrt{-1}$, then $\sum_{n=0}^{\infty}\left(\frac{i}{3}\right)^n=$

If $i=\sqrt{-1}$, then $\operatorname{Arg}\left[\frac{(1+i)^{2025}}{(1-i)^{2022}}\right]=$

The locus of $z$ such that $\left|\frac{z-i}{z+i}\right|=2$, where $z=x+i y$, is

If $x_n=\cos \frac{\pi}{2^n}+i \sin \frac{\pi}{2^n}$, then $\prod_{n=1}^{\infty} x_n=$

If the roots of the equation $z^2-i=0$ are $\alpha$ and $\beta$, then $|\arg \beta-\arg \alpha|=$

If $x^2+2 p x-2 p+8>0$ for all real values of $x$, then the set of all possible values of $p$ is

If $R-(\alpha, \beta)$ is the range of $\frac{x+3}{(x-1)(x+2)}$, then the sum of the intercepts of the line $\alpha x+\beta y+1=0$ on the coordinate axes is

The quadratic equation whose roots are $\sin ^2 18^{\circ}$ and $\cos ^2 36^{\circ}$ is

The roots of the equation $x^4+x^3-4 x^2+x+1=0$ are diminished by $h$ so that, the transformed equation does not contain $x^2$ term. If the values of such $h$ are $\alpha$ and $\beta$, then $12(\alpha-\beta)^2=$

$\alpha, \beta, \gamma$ are the roots of the equation $x^3+2 x^2-x-2=0$, then $\alpha^6+\beta^6+\gamma^6=$

The number of diagonals of a polygon is 35 . If $A$ and $B$ are two distinct vertices of this polygon, then the number of all those triangles formed by joining three vertices of the polygon having $A B$ as one of its sides is

There are 10 points in a plane, of which no three points are collinear except 4. Then, the number of distinct triangles that can be formed by joining any three points of these ten points, such that at least one of the vertices of every triangle formed is from the given 4 collinear points is

A student is asked to answer 10 out of 13 questions in an examination such that he must answer atleast four questions from the first five questions. Then, the total number of possible choices available to him is

If $(-c, c)$ is the set of all values of $x$ for which the expansion of $(7-5 x)^{\frac{-2}{3}}$ is valid, then $5 c+7=$

If $n$ is a positive integer and $f(n)$ is the coefficient of $x^n$ in the expansion of $(1+x)(1-x)^n$, then $f(2023)=$

If $y=\frac{3}{4}+\frac{3 \cdot 5}{4 \cdot 8}+\frac{3 \cdot 5 \cdot 7}{4 \cdot 8 \cdot 12}+\ldots$ to $\infty$, then

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

The period of the function $f(x)=e^{\log (\sin x)}+(\tan x)^3-\operatorname{cosec}(3 x-5)$ is

If $\cos \theta=\frac{-3}{5}$ and $\pi<\theta<\frac{3 \pi}{2}$, then $\tan \frac{\theta}{2}+\sin \frac{\theta}{2}+2 \cos \frac{\theta}{2}=$

If $\sin 2 \theta$ and $\cos 2 \theta$ are solutions of $x^2+a x-c=0$, then

If $x=\log \left(y+\sqrt{y^2+1}\right)$, then $y=$

In $\triangle A B C$, if $a: b: c=4: 5: 6$, then the ratio of the circumradius to its inradius is

The perimeter of a $\triangle A B C$ is 6 times the arithmetic mean of the values of the sine of its angles. If its side $B C$ is of unit length, then $\angle A=$

If $|\mathbf{a}|=4,|\mathbf{b}|=5$ and $|\mathbf{a}-\mathbf{b}|=3$ and $\theta$ is the angle between the vectors $\mathbf{a}$ and $\mathbf{b}$, then $\cot ^2 \theta=$

If $A(1,2,3), B(3,7,-2), C(6,7,7)$ and $D(-1,0,-1)$ are points in a plane, then the vector equation of the line passing through the centroids of $\triangle A B D$ and $\triangle A C D$ is

If $\mathbf{a}+\mathbf{b}+\mathbf{c}=0,|\mathbf{a}|=3,|\mathbf{b}|=5,|\mathbf{c}|=7$, then the angle between $\mathbf{a}$ and $\mathbf{b}$ is

If $2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+3 \hat{\mathbf{k}},-12 \hat{\mathbf{i}}-\hat{\mathbf{j}}-3 \hat{\mathbf{k}},-\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-4 \hat{\mathbf{k}}$ and $\lambda \hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}$ are the position vectors of four coplanar points, then $\lambda=$

Let $\mathbf{a}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ and $\mathbf{b}=2 \hat{\mathbf{i}}-\hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ be two vectors. If the orthogonal projection vector of $\mathbf{a}$ on $\mathbf{b}$ is $\mathbf{x}$ and orthogonal projection vector of $\mathbf{b}$ on $\mathbf{a}$ is $\mathbf{y}$, then $|\mathbf{x}-\mathbf{y}|=$

The variance of 50 observations is 7 . Suppose that each observation in this data is multiplied by 6 and then 5 is subtracted from it. Then, the variance of that new data is

A bag contains four balls. Two balls are drawn randomly and found them to be white. The probability that all the balls in the bag are white is

If the coefficients $a$ and $b$ of a quadratic expression $x^2+a x+b$ are chosen from the sets $A=\{3,4,5\}$ and $B=\{1,2,3,4\}$ respectively, then the probability that the equation $x^2+a x+b=0$ has real roots is

A random variable $X$ has the following probability distribution

$$ \begin{array}{|c|l|l|l|l|l|l|l|l|} \hline \boldsymbol{X}=\boldsymbol{x} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline \boldsymbol{P}(\boldsymbol{X}=\boldsymbol{x}) & 0.15 & 0.23 & k & 0.10 & 0.20 & 0.08 & 0.07 & 0.05 \\ \hline \end{array} $$

For the events $E=\{x / x$ is a prime number $\}$ and $F=\{x / x<4\}$, then $P(E \cup F)=$

5 persons entered a lift cabin in the cellar of a 7 floor building apart from cellar. If each of them independently and with equal probability can leave the cabin at any floor out of the 7 floors beginning with the first, then the probability of all the 5 persons leaving the cabin at different floors is

If a point $P$ moves so that the distance from $(0,2)$ to $P$ is $\frac{1}{\sqrt{2}}$ times the distance of $P$ from $(-1,0)$, then the locus of the point $P$ is

Let $d$ be the distance between the parallel lines $3 x-2 y+5=0$ and $3 x-2 y+5+2 \sqrt{13}=0$.

Let $L_1 \equiv 3 x-2 y+k_1=0\left(k_1>0\right)$ and $L_2 \equiv 3 x-2 y+k_2=0\left(k_2>0\right)$ be two lines that are at the distance of $\frac{4 d}{\sqrt{13}}$ and $\frac{3 d}{\sqrt{13}}$ from the line $3 x-2 y+5=0$.

Then, the combined equation of the lines $L_1=0$ and $L_2=0$ is

If $(h, k)$ is the image of the point $(3,-4)$ with respect to the line $2 x-3 y-5=0$ and $(l, m)$ is the foot of the perpendicular from $(h, k)$ on to the line $3 x+2 y+12=0$, then $l h+m k+1=$

A straight line parallel to the line $y=\sqrt{3} x$ passes through $Q(2,3)$ and cuts the line $2 x+4 y-27=0$ at $P$. Then, the length of the line segment $P Q$ is

If a line $a x+2 y=k$ forms a triangle of area 3 sq. units with the coordinate axis and is perpendicular to the line $2 x-3 y+7=0$, then the product of all the possible values of $k$ is

The orthocenter of the triangle whose sides are given by $x+y+10=0, x-y-2=0$ and $2 x+y-7=0$ is

For $l \in R$, the equation $(2 l-3) x^2+2 l x y-y^2=0$ represents a pair of distinct lines

If the parametric equations of the circle passing through the points $(3,4),(3,2)$ and $(1,4)$ is $x=a+r \cos \theta, y=b+r \sin \theta$, then $b^a r^a=$

A tangent $P T$ is drawn to the circle $x^2+y^2=4$ at the point $P(\sqrt{3}, 1)$. If a straight line $L$ which is perpendicular to $P T$ is a tangent to the circle $(x-3)^2+y^2=1$, then a possible equation of $L$ is

If the angle between the pair of tangents drawn to the circle $x^2+y^2-2 x+4 y+3=0$ from the point $(6,-5)$ is $\theta$, then $\cot \theta=$

If the angle between the circles $x^2+y^2-4 x-6 y+k=0$ and $x^2+y^2+8 x-4 y+11=0$ is $\frac{\pi}{2}$, then the value of $k$ is

The radius of a circle touching all the four circles $(x \pm \lambda)^2+(y \pm \lambda)^2=\lambda^2$ is

If the radical centre of the given three circles $x^2+y^2=1, x^2+y^2-2 x-3=0$ and $x^2+y^2-2 y-3=0$ is $C(\alpha, \beta)$ and $r$ is the sum of the radii of the given circles, then the circle with $C(\alpha, \beta)$ as centre and $r$ as radius is

If $x-2 y+k=0$ is a tangent to the parabola $y^2-4 x-4 y+8=0$, then the value of $k$ is

If the points of intersection of the parabolas $y^2=5 x$ and $x^2=5 y$ lie on the line $L$, then the area of the triangle formed by the directrix of one parabola, latus rectum of another parabola and the line $L$ is

If the line $x \cos \alpha+y \sin \alpha=2 \sqrt{3}$ is a tangent to the ellipse $\frac{x^2}{16}+\frac{y^2}{8}=1$ and $\alpha$ is an acute angle, then $\alpha=$

If $x+\sqrt{3} y=3$ is the tangent to the ellipse $2 x^2+3 y^2=k$ at a point $P$, then the equation of the normal to this ellipse at $P$ is

If the angle between the asymptotes of a hyperbola is $30^{\circ}$, then its eccentricity is

In a $\triangle A B C$, if the mid-points of sides $A B, B C$ and $C A$ are $(3,0,0),(0,4,0)$ and $(0,0,5)$ respectively, then $A B^2+B C^2+C A^2=$

If $l, m, n$ and $a, b, c$ are direction cosines of two lines, then

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

$$ \lim _{n \rightarrow \infty} \frac{1}{n^3} \sum_{k=1}^n\left(k^2 x\right)= $$

The quadratic equation whose roots are

$$ l=\lim _{\theta \rightarrow 0}\left(\frac{3 \sin \theta-4 \sin ^3 \theta}{\theta}\right) \text { and } m=\lim _{\theta \rightarrow 0}\left(\frac{2 \tan \theta}{\theta\left(1-\tan ^2 \theta\right)}\right) \text { is } $$

On differentiation if we get $f(x, y) d y-g(x, y) d x=0$ from $2 x^2-3 x y+y^2+x+2 y-8=0$, then $\frac{g(2,2)}{f(1,1)}=$

If $f(x)=e^x, h(x)=(f \circ f)(x)$, then $\frac{h^{\prime}(x)}{h(x)}=$

If $\sin y=\sin 3 t$ and $x=\sin t$, then $\frac{d y}{d x}=$

If a line having slope 2 is a tangent to the curve $y=x^4-6 x^3+13 x^2-12 x+5$ at points $P\left(x_1, y_1\right)$ and $Q\left(x_2, y_2\right), x_1, x_2 \in N$, then $x_1 x_2-y_1 y_2=$

Let $m$ be the slope of the normal $L$ drawn at $(1,2)$ to the curve $x=t^2-7 t+7, y=t^2-4 t-10$ and $a x+b y+c=0$ be the equation of the normal $L$. If GCD of $(a, b, c)$ is 1 , then $m(a+b+c)=$

If the function $f(x)=x e^{-x}, x \in R$ attains its maximum value $\beta$ at $x=\alpha$, then $(\alpha, \beta)=$

If $\int \frac{x^{49} \tan ^{-1}\left(x^{50}\right)}{\left(1+x^{100}\right)} d x=k\left(\tan ^{-1}\left(x^{50}\right)\right)^2+C$, then $k=$

If $\int(\log x)^3 x^5 d x=\frac{x^6}{A}\left[B(\log x)^3\right. \left.+C(\log x)^2+D(\log x)-1\right]+k$ and $A, B, C, D$ are integers, then $A-(B+C+D)=$

$$ \int \frac{d x}{\left(x^2+1\right)\left(x^2+4\right)}= $$

$\int \frac{d x}{(x-1)^{\frac{3}{4}}(x+2)^{\frac{5}{4}}}=$

$$ \int_{-1}^1 x|x| d x= $$

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

The area (in sq units) of the region bounded by the curve $y=|\sin 2 x|$ and the $X$-axis in $[0,2 \pi]$ is

If $\int_0^3\left(3 x^2-4 x+2\right) d x=k$, then an integer root of $3 x^2-4 x+2=3 k / 5$ is

If the order and degree of the differential equation corresponding to the family of curves $y^2=4 a(x+a)(a$ is parameter) are $m$ and $n$ respectively, then $m+n^2=$

If the solution of the differential equation $\frac{d y}{d x}=\frac{2 x+3 y}{3 x-2 y}$ is $y=x \tan (f(x))+C$, then $f(x)=$

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

The ratio of the relative strengths of strong and weak nuclear forces is

The number of significant figures in the measurement of a length 0.079000 m is

The acceleration of a vertically projected body at its highest reaching position is

A player can throw a ball to a maximum horizontal distance of 80 m . If he throws the ball vertically with the same velocity, then the maximum height reached by the ball is

If a man of mass 50 kg is in a lift moving down with an acceleration equal to acceleration due to gravity, then the apparent weight of the man is

An engine is dragging a mass of 5000 kg with a velocity of $5 \mathrm{~ms}^{-1}$ along a smooth inclined plane of inclination 1 in 50 . Then the power of the engine is

A ball falls freely from a height $h$ on a rigid horizontal plane. If the coefficient of restitution is $e$, then the total distance travelled by the ball before hitting the plane second time is

The velocity of a particle having magnitude of $10 \mathrm{~ms}^{-1}$ in the direction of $60^{\circ}$ with positive $X$-axis is

The ratio of the radii of two solid spheres of same mass is $2: 3$. The ratio of the moments of inertia of the spheres about their diameter is

The displacement of a particle executing simple harmonic motion is given by $x=2 \cos (t)$ where $t$ is the time in seconds then the time period of the particle is

The dimensions of four wires of the same material are given below. The increase in length is maximum in the wire of

A cylindrical vessel, open at the top, contains 15 litres of water. Water drains out through a small opening at the bottom. 5 litre of water comes out in time $t_1$, the next 5 litre in further time $t_2$, and the last 5 litre in further time $t_3$, Then

A cylinder of mass $m$ and material density $\rho$ hanging from a string is lowered into a vessel of cross-sectional area $A$ containing a liquid of density $\sigma(<\rho)$ until it is fully immersed. The increase in pressure at the bottom of the vessel is

The Fahrenheit and Kelvin scales of temperature will have the same reading at a temperature of

If the ratio of densities of two substances is $5: 6$ and the ratio of their specific heat capacities is $3: 5$, then the ratio of heat energies required per unit volume so that the two substances can have same temperature rise is

In a process, the work done by the system is equal to the decrease in its internal energy. The process that the system undergoes is

N molecules each of mass $m$ of gas $A$ and 2 N molecules each of mass 2 m of gas $B$ are contained in a vessel which is maintained at a temperature $T$. The mean square velocity of the molecules of gas $B$ is denoted by $v_2^2$ and the mean square of the $x$-component velocity of the molecules of gas $A$ is denoted by $v_1^2$, then $v_1 / v_2$ is

Among the following statements, the correct statement for a wave is

A source and an observer move away from each other with same velocity of $10 \mathrm{~ms}^{-1}$ with respect to ground. If the observer finds the frequency of sound coming from the source as 1980 Hz , then actual frequency of the source is (speed of sound in air $=340 \mathrm{~ms}^{-1}$ )

Two convex lenses of focal lengths 20 cm and 30 cm are placed in contact with each other co-axially. The focal length of the combination is

A point source of light is placed at the focus of a concave mirror. Consider only paraxial rays. The shapes of the wavefronts of incident and reflected lights respectively are

If the electric potential at a point on the surface of a hollow conducting sphere of radius $R$ is $V$, then the electric potential at a point which is at distance $R / 3$ from the centre of the sphere is

The effective capacitance between points $A$ and $B$ shown in the circuit is

The Wheatstone bridge shown in the diagram is balanced. If $P_3$ is the power dissipated by $R_3$ and $P_1$ is the power dissipated by $R_1$, then the ratio $P_3 / P_1$ is

A wire of resistance $2 R$ is stretched such that its length is doubled. Then the increase in its resistance is

It is found that a non-zero current element is unable to produce any magnetic field at a particular point. Then the angle between the current element and the position vector of that point with respect to the current element is

Three long, straight, parallel wires carrying different currents are arranged as shown in the diagram. In the given arrangement, let the net force per unit length on the wire $C$ be $\mathbf{F}$. If the wire $B$ is oved without disturbing the other two wires, then the force per unit length on wire $A$ is

If $\chi$ is the susceptibility and $\mu_r$ is the relative permeability of a ferromagnetic substance, then

Metal detector works on the principle of

A copper disc of radius 0.1 m rotates about an axis passing through its centre and perpendicular to its plane with 10 revolutions per second in a uniform transverse magnetic field of 0.1 T . The emf induced across the radius of the disc is

At very high frequencies, the current ( $i$ ) in the given circuit is

Electromagnetic radiation of intensity $0.6 \mathrm{Wm}^{-2}$ is falling on a black surface. The radiation pressure on the surface is

Radiations of wavelength 400 nm incidents on a photosensitive material of work function 2.2 eV . The stopping potential is nearly

In a hypothetical Bohr hydrogen atom, if the mass of the electron is doubled then the energy of the electron in the first orbit is

The half-life period of element $X$ is same as the mean life time of element $Y$. Assume initially $X$ and $Y$ have same number of atoms. Then

Heavy water is used as moderator in nuclear reactor because

Photodiodes are mostly operated in reverse biased condition because

Which of the following statements is true about LEDs?

An amplitude modulated wave is represented by $c_m(t)=10[1+0.6 \sin (1250 t)] \sin \left(10^8 t\right)$. Then modulation index is