In the following sequence of reaction compound 'M' is

Identify the ion having $$4 f^6$$ electronic configuration.

Metallic conductors and semiconductors are heated separately. What are the changes with respect to conductivity?

The equivalent weight of $$\mathrm{Na}_2 \mathrm{S}_2 \mathrm{O}_3(\mathrm{Gram}$$ molecular weight $$=\mathrm{M})$$ in the given reaction is

$$\mathrm{I}_2+2 \mathrm{Na}_2 \mathrm{~S}_2 \mathrm{O}_3=2 \mathrm{NaI}+\mathrm{Na}_2 \mathrm{~S}_4 \mathrm{O}_6$$

The reactivity order of the following molecules towards $$\mathrm{S}_{\mathrm{N}} 1$$ reaction is

$$\begin{array}{ccc} \text { Allyl chloride } & \text { Chlorobenzene } & \text { Ethyl chloride } \\ \text { (I) } & \text { (II) } & \text { (III) } \end{array}$$

Toluene reacts with mixed acid at $$25^{\circ} \mathrm{C}$$ to produce

The product 'P' in the above reaction is

The decreasing order of reactivity of the following alkenes towards $$\mathrm{HBr}$$ addition is

Ozonolysis of $$\underline{o}$$-xylene produces

The compounds A and B are respectively

The compound that does not give positive test for nitrogen in Lassaigne's test is

The correct acidity order of phenol (I), 4-hydroxybenzaldehyde (II) and 3-hydroxybenzaldehyde (III) is

The major product of the following reaction is :

Which of the following statements is correct for a spontaneous polymerization reaction ?

At 25$$^\circ$$C, the ionic product of water is 10$$^{-14}$$. The free energy change for the self-ionization of water in kCal mol$$^{-1}$$ is close to

Consider an electron moving in the first Bohr orbit of a $$\mathrm{He}^{+}$$ ion with a velocity $$v_1$$. If it is allowed to move in the third Bohr orbit with a velocity $$v_3$$, then indicate the correct $$v_3: v_1$$ ratio.

The compressibility factor for a van der Waal gas at high pressure is

For a spontaneous process, the incorrect statement is

Identify the incorrect statement among the following :

Which of the following statements is true about equilibrium constant and rate constant of a single step chemical reaction?

After the emission of a $$\beta$$-particle followed by an $$\alpha$$-particle from $${ }_{83}^{214} \mathrm{Bi}$$, the number of neutrons in the atom is -

Which hydrogen like species will have the same radius as that of $$1^{\text {st }}$$ Bohr orbit of hydrogen atom?

For a first order reaction with rate constant $$\mathrm{k}$$, the slope of the plot of $$\log$$ (reactant concentration) against time is

Equal volumes of aqueous solution of $$0.1(\mathrm{M}) \mathrm{HCl}$$ and $$0.2(\mathrm{M}) \mathrm{H}_2 \mathrm{SO}_4$$ are mixed. The concentration of $$\mathrm{H}^{+}$$ ions in the resulting solution is

The correct order of boiling point of the given aqueous solutions is

Correct solubility order of $$\mathrm{AgF}, \mathrm{AgCl}, \mathrm{AgBr}, \mathrm{AgI}$$ in water is

What will be the change in acidity if

(i) $$\mathrm{CuSO}_4$$ is added in saturated $$(\mathrm{NH}_4)_2 \mathrm{SO}_4$$ solution

(ii) $$\mathrm{SbF}_5$$ is added in anhydrous $$\mathrm{HF}$$

Which of the following contains maximum number of lone pairs on the central atom?

Number of moles of ions produced by complete dissociation of one mole of Mohr's salt in water is

Which of the following species exhibits both LMCT and paramagnetism?

How many $$\mathrm{P}-\mathrm{O}-\mathrm{P}$$ linkages are there in $$\mathrm{P}_4 \mathrm{O}_{10}$$

$$\mathrm{Q}$$ and $$\mathrm{R}$$ in the above reaction sequences are respectively

$$\mathrm{pH}$$ of $$10^{-8}(\mathrm{M}) \mathrm{~HCl}$$ solution is

The specific conductance $$(\mathrm{k})$$ of $$0.02(\mathrm{M})$$ aqueous acetic acid solution at $$298 \mathrm{~K}$$ is $$1.65 \times 10^{-4} \mathrm{~S} \mathrm{~cm}^{-1}$$. The degree of dissociation of acetic acid is

$$[\lambda_{\mathrm{O}^{+}}+=349.1 \mathrm{~S} \mathrm{~cm}^2 \mathrm{~mol}^{-1} \text { and } \lambda_{{ }^{\circ} \mathrm{CH}_3 \mathrm{COO}^{-}}=40.9 \mathrm{~S} \mathrm{~cm}^2 \mathrm{~mol}^{-1}]$$

The number(s) of $$-\mathrm{OH}$$ group(s) present in $$\mathrm{H}_3 \mathrm{PO}_3$$ and $$\mathrm{H}_3 \mathrm{PO}_4$$ is/are

Which of the following statements about the $$\mathrm{S}_{\mathrm{N}} 2$$ reaction mechanism is/are true?

Which of the following represent(s) the enantiomer of Y ?

Identify the correct statement(s) :

Which of the following ion/ions is/are diamagnetic ?

Which of the following statement/statements is/are correct ?

All values of a for which the inequality $$\frac{1}{\sqrt{a}} \int_\limits1^a\left(\frac{3}{2} \sqrt{x}+1-\frac{1}{\sqrt{x}}\right) \mathrm{d} x<4$$ is satisfied, lie in the interval

For any integer $$\mathrm{n}, \int_\limits0^\pi \mathrm{e}^{\cos ^2 x} \cdot \cos ^3(2 n+1) x \mathrm{~d} x$$ has the value :

Let $$\mathrm{f}$$ be a differential function with $$\lim _\limits{x \rightarrow \infty} \mathrm{f}(x)=0$$. If $$\mathrm{y}^{\prime}+\mathrm{yf}^{\prime}(x)-\mathrm{f}(x) \mathrm{f}^{\prime}(x)=0$$, $$\lim _\limits{x \rightarrow \infty} y(x)=0$$ then

If $$x y^{\prime}+y-e^x=0, y(a)=b$$, then $$\lim _\limits{x \rightarrow 1} y(x)$$ is

The area bounded by the curves $$x=4-y^2$$ and the Y-axis is

$$f(x)=\cos x-1+\frac{x^2}{2!}, x \in \mathbb{R}$$ Then $$\mathrm{f}(x)$$ is

Let $$\mathrm{y}=\mathrm{f}(x)$$ be any curve on the $$\mathrm{X}-\mathrm{Y}$$ plane & $$\mathrm{P}$$ be a point on the curve. Let $$\mathrm{C}$$ be a fixed point not on the curve. The length $$\mathrm{PC}$$ is either a maximum or a minimum, then

If a particle moves in a straight line according to the law $$x=a \sin (\sqrt{\lambda} t+b)$$, then the particle will come to rest at two points whose distance is [symbols have their usual meaning]

A unit vector in XY-plane making an angle $$45^{\circ}$$ with $$\hat{i}+\hat{j}$$ and an angle $$60^{\circ}$$ with $$3 \hat{i}-4 \hat{j}$$ is

Let $$\mathrm{f}: \mathbb{R} \rightarrow \mathbb{R}$$ be given by $$\mathrm{f}(x)=\left|x^2-1\right|$$, then

Given an A.P. and a G.P. with positive terms, with the first and second terms of the progressions being equal. If $$a_n$$ and $$b_n$$ be the $$n^{\text {th }}$$ term of A.P. and G.P. respectively then

If for the series $$a_1, a_2, a_3$$, ...... etc, $$\mathrm{a}_{\mathrm{r}}-\mathrm{a}_{\mathrm{r}+\mathrm{i}}$$ bears a constant ratio with $$\mathrm{a}_{\mathrm{r}} \cdot \mathrm{a}_{\mathrm{r}+1}$$; then $$\mathrm{a}_1, \mathrm{a}_2, \mathrm{a}_3 \ldots .$$. are in

If $$z_1$$ and $$z_2$$ be two roots of the equation $$z^2+a z+b=0, a^2<4 b$$, then the origin, $$\mathrm{z}_1$$ and $$\mathrm{z}_2$$ form an equilateral triangle if

If $$\cos \theta+i \sin \theta, \theta \in \mathbb{R}$$, is a root of the equation

$$a_0 x^n+a_1 x^{n-1}+\ldots .+a_{n-1} x+a_n=0, a_0, a_1, \ldots . a_n \in \mathbb{R}, a_0 \neq 0,$$

then the value of $$a_1 \sin \theta+a_2 \sin 2 \theta+\ldots .+a_n \sin n \theta$$ is

If $$\left(x^2 \log _x 27\right) \cdot \log _9 x=x+4$$ then the value of $$x$$ is

If $$\mathrm{P}(x)=\mathrm{a} x^2+\mathrm{b} x+\mathrm{c}$$ and $$\mathrm{Q}(x)=-\mathrm{a} x^2+\mathrm{d} x+\mathrm{c}$$ where $$\mathrm{ac} \neq 0$$, then $$\mathrm{P}(x) \cdot \mathrm{Q}(x)=0$$ has (a, b, c, d are real)

Let $$\mathrm{N}$$ be the number of quadratic equations with coefficients from $$\{0,1,2, \ldots, 9\}$$ such that 0 is a solution of each equation. Then the value of $$\mathrm{N}$$ is

If $$a, b, c$$ are distinct odd natural numbers, then the number of rational roots of the equation $$a x^2+b x+c=0$$

The numbers $$1,2,3, \ldots \ldots, \mathrm{m}$$ are arranged in random order. The number of ways this can be done, so that the numbers $$1,2, \ldots \ldots ., \mathrm{r}(\mathrm{r}<\mathrm{m})$$ appears as neighbours is

If $$A=\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right)$$ and $$\theta=\frac{2 \pi}{7}$$, then $$A^{100}=A \times A \times \ldots .(100$$ times) is equal to

If $$\left(1+x+x^2+x^3\right)^5=\sum_\limits{k=0}^{15} a_k x^k$$ then $$\sum_\limits{k=0}^7(-1)^{\mathbf{k}} \cdot a_{2 k}$$ is equal to

The coefficient of $$a^{10} b^7 c^3$$ in the expansion of $$(b c+c a+a b)^{10}$$ is

$$ \text { If }\left|\begin{array}{lll} x^k & x^{k+2} & x^{k+3} \\ y^k & y^{k+2} & y^{k+3} \\ z^k & z^{k+2} & z^{k+3} \end{array}\right|=(x-y)(y-z)(z-x)\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right) \text {, then } $$

If $$\left[\begin{array}{ll}2 & 1 \\ 3 & 2\end{array}\right] \cdot A \cdot\left[\begin{array}{cc}-3 & 2 \\ 5 & -3\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$$, then $$A=$$

$$ \text { Let } f(x)=\left|\begin{array}{ccc} \cos x & x & 1 \\ 2 \sin x & x^3 & 2 x \\ \tan x & x & 1 \end{array}\right| \text {, then } \lim _\limits{x \rightarrow 0} \frac{f(x)}{x^2}= $$

In R, a relation p is defined as follows: $$\forall a, b \in \mathbb{R}, a p$$ holds iff $$a^2-4 a b+3 b^2=0$$. Then

Let $$\mathrm{f}: \mathbb{R} \rightarrow \mathbb{R}$$ be a function defined by $$\mathrm{f}(x)=\frac{\mathrm{e}^{|x|}-\mathrm{e}^{-x}}{\mathrm{e}^x+\mathrm{e}^{-x}}$$, then

Let A be the set of even natural numbers that are < 8 & B be the set of prime integers that are $$<7$$ The number of relations from A to B are

Two smallest squares are chosen one by one on a chess board. The probability that they have a side in common is

Two integers $$\mathrm{r}$$ and $$\mathrm{s}$$ are drawn one at a time without replacement from the set $$\{1,2, \ldots, \mathrm{n}\}$$. Then $$\mathrm{P}(\mathrm{r} \leq \mathrm{k} / \mathrm{s} \leq \mathrm{k})=$$

(k is an integer < n)

A biased coin with probability $$\mathrm{p}(0<\mathrm{p}<1)$$ of getting head is tossed until a head appears for the first time. If the probability that the number of tosses required is even is $$\frac{2}{5}$$, then $$\mathrm{p}=$$

The expression $$\cos ^2 \phi+\cos ^2(\theta+\phi)-2 \cos \theta \cos \phi \cos (\theta+\phi)$$ is

If $$0< \theta<\frac{\pi}{2}$$ and $$\tan 3 \theta \neq 0$$, then $$\tan \theta+\tan 2 \theta+\tan 3 \theta=0$$ if $$\tan \theta \cdot \tan 2 \theta=\mathrm{k}$$ where $$\mathrm{k}=$$

The equation $$\mathrm{r} \cos \theta=2 \mathrm{a} \sin ^2 \theta$$ represents the curve

If $$(1,5)$$ be the midpoint of the segment of a line between the line $$5 x-y-4=0$$ and $$3 x+4 y-4=0$$, then the equation of the line will be

In $$\triangle \mathrm{ABC}$$, co-ordinates of $$\mathrm{A}$$ are $$(1,2)$$ and the equation of the medians through $$\mathrm{B}$$ and C are $$x+\mathrm{y}=5$$ and $$x=4$$ respectively. Then the midpoint of $$\mathrm{BC}$$ is

A line of fixed length $$\mathrm{a}+\mathrm{b} . \mathrm{a} \neq \mathrm{b}$$ moves so that its ends are always on two fixed perpendicular straight lines. The locus of a point which divides the line into two parts of length a and b is

With origin as a focus and $$x=4$$ as corresponding directrix, a family of ellipse are drawn. Then the locus of an end of minor axis is

Chords $$\mathrm{AB}$$ & $$\mathrm{CD}$$ of a circle intersect at right angle at the point $$\mathrm{P}$$. If the length of AP, PB, CP, PD are 2, 6, 3, 4 units respectively, then the radius of the circle is

The plane $$2 x-y+3 z+5=0$$ is rotated through $$90^{\circ}$$ about its line of intersection with the plane $$x+y+z=1$$. The equation of the plane in new position is

If the relation between the direction ratios of two lines in $$\mathbb{R}^3$$ are given by

$$l+\mathrm{m}+\mathrm{n}=0,2 l \mathrm{~m}+2 \mathrm{mn}-l \mathrm{n}=0$$

then the angle between the lines is ($$l, \mathrm{~m}, \mathrm{n}$$ have their usual meaning)

$$\triangle \mathrm{OAB}$$ is an equilateral triangle inscribed in the parabola $$\mathrm{y}^2=4 \mathrm{a} x, \mathrm{a}>0$$ with O as the vertex, then the length of the side of $$\triangle \mathrm{O A B}$$ is

For every real number $$x \neq-1$$, let $$\mathrm{f}(x)=\frac{x}{x+1}$$. Write $$\mathrm{f}_1(x)=\mathrm{f}(x)$$ & for $$\mathrm{n} \geq 2, \mathrm{f}_{\mathrm{n}}(x)=\mathrm{f}\left(\mathrm{f}_{\mathrm{n}-1}(x)\right)$$. Then $$\mathrm{f}_1(-2) \cdot \mathrm{f}_2(-2) \ldots . . \mathrm{f}_{\mathrm{n}}(-2)$$ must be

If $$\mathrm{U}_{\mathrm{n}}(\mathrm{n}=1,2)$$ denotes the $$\mathrm{n}^{\text {th }}$$ derivative $$(\mathrm{n}=1,2)$$ of $$\mathrm{U}(x)=\frac{\mathrm{L} x+\mathrm{M}}{x^2-2 \mathrm{~B} x+\mathrm{C}}$$ (L, M, B, C are constants), then $$\mathrm{PU}_2+\mathrm{QU}_1+\mathrm{RU}=0$$, holds for

The equation $$2^x+5^x=3^x+4^x$$ has

Consider the function $$\mathrm{f}(x)=(x-2) \log _{\mathrm{e}} x$$. Then the equation $$x \log _{\mathrm{e}} x=2-x$$

If $$\alpha, \beta$$ are the roots of the equation $$a x^2+b x+c=0$$ then $$\lim _\limits{x \rightarrow \beta} \frac{1-\cos \left(a x^2+b x+c\right)}{(x-\beta)^2}$$ is

If $$\mathrm{f}(x)=\frac{\mathrm{e}^x}{1+\mathrm{e}^x}, \mathrm{I}_1=\int_\limits{\mathrm{f}(-\mathrm{a})}^{\mathrm{f}(\mathrm{a})} x \mathrm{~g}(x(1-x)) \mathrm{d} x$$ and $$\mathrm{I}_2=\int_\limits{\mathrm{f}(-\mathrm{a})}^{\mathrm{f}(\mathrm{a})} \mathrm{g}(x(1-x)) \mathrm{d} x$$, then the value of $$\frac{I_2}{I_1}$$ is

Let $$f: \mathbb{R} \rightarrow \mathbb{R}$$ be a differentiable function and $$f(1)=4$$. Then the value of $$\lim _\limits{x \rightarrow 1} \int_\limits4^{f(x)} \frac{2 t}{x-1} d t$$, if $$f^{\prime}(1)=2$$ is

$$ \text { If } \int \frac{\log _e\left(x+\sqrt{1+x^2}\right)}{\sqrt{1+x^2}} \mathrm{~d} x=\mathrm{f}(\mathrm{g}(x))+\mathrm{c} \text { then } $$

Let $$\mathrm{I}(\mathrm{R})=\int_\limits0^{\mathrm{R}} \mathrm{e}^{-\mathrm{R} \sin x} \mathrm{~d} x, \mathrm{R}>0$$. then,

Consider the function $$\mathrm{f}(x)=x(x-1)(x-2) \ldots(x-100)$$. Which one of the following is correct?

In a plane $$\vec{a}$$ and $$\vec{b}$$ are the position vectors of two points A and B respectively. A point $P$ with position vector $$\overrightarrow{\mathrm{r}}$$ moves on that plane in such a way that $$|\overrightarrow{\vec{r}}-\vec{a}| \sim|\vec{r}-\vec{b}|=c$$ (real constant). The locus of P is a conic section whose eccentricity is

Five balls of different colours are to be placed in three boxes of different sizes. The number of ways in which we can place the balls in the boxes so that no box remains empty is

Let $$A=\left(\begin{array}{ccc}1 & -1 & 0 \\ 0 & 1 & -1 \\ 1 & 1 & 1\end{array}\right), B=\left(\begin{array}{l}2 \\ 1 \\ 7\end{array}\right)$$

Then for the validity of the result $$\mathrm{AX}=\mathrm{B}, \mathrm{X}$$ is

If $$\alpha_1, \alpha_2, \ldots, \alpha_n$$ are in A.P. with common difference $$\theta$$, then the sum of the series $$ \sec \alpha_1 \sec \alpha_2+\sec \alpha_2 \sec \alpha_3+\ldots .+\sec \alpha_{n-1} \sec \alpha_n=k\left(\tan \alpha_n-\tan \alpha_1\right)$$ where $$\mathrm{k}=$$

For the real numbers $$x$$ & $$y$$, we write $$x$$ p y iff $$x-y+\sqrt{2}$$ is an irrational number. Then relation p is

Let $$A=\left[\begin{array}{ccc}0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0\end{array}\right]$$, then

$$ \text { If } 1000!=3^n \times m \text { where } m \text { is an integer not divisible by } 3 \text {, then } n= $$

If $$A$$ and $$B$$ are acute angles such that $$\sin A=\sin ^2 B$$ and $$2 \cos ^2 A=3 \cos ^2 B$$, then $$(A, B)=$$

If two circles which pass through the points $$(0, a)$$ and $$(0,-a)$$ and touch the line $$\mathrm{y}=\mathrm{m} x+\mathrm{c}$$, cut orthogonally then

The locus of the midpoint of the system of parallel chords parallel to the line $$y=2 x$$ to the hyperbola $$9 x^2-4 y^2=36$$ is

Angle between two diagonals of a cube will be

$$ \text { If } y=\tan ^{-1}\left[\frac{\log _e\left(\frac{e}{x^2}\right)}{\log _e\left(e x^2\right)}\right]+\tan ^{-1}\left[\frac{3+2 \log _e x}{1-6 \cdot \log _e x}\right] \text {, then } \frac{d^2 y}{d x^2}= $$

$$\lim _\limits{n \rightarrow \infty} \frac{1}{n^{k+1}}[2^k+4^k+6^k+\ldots .+(2 n)^k]=$$

The acceleration f $$\mathrm{ft} / \mathrm{sec}^2$$ of a particle after a time $$\mathrm{t}$$ sec starting from rest is given by $$\mathrm{f}=6-\sqrt{1.2 \mathrm{t}}$$. Then the maximum velocity $$\mathrm{v}$$ and time $$\mathrm{T}$$ to attend this velocity are

Let $$\Gamma$$ be the curve $$\mathrm{y}=\mathrm{be}^{-x / a}$$ & $$\mathrm{L}$$ be the straight line $$\frac{x}{\mathrm{a}}+\frac{\mathrm{y}}{\mathrm{b}}=1$$ where $$\mathrm{a}, \mathrm{b} \in \mathbb{R}$$. Then

If $$n$$ is a positive integer, the value of $$(2 n+1){ }^n C_0+(2 n-1){ }^n C_1+(2 n-3){ }^n C_2 +\ldots .+1 \cdot{ }^n C_n$$ is

If the quadratic equation $$a x^2+b x+c=0(a>0)$$ has two roots $$\alpha$$ and $$\beta$$ such that $$\alpha<-2$$ and $$\beta>2$$, then

If $$\mathrm{a}_{\mathrm{i}}, \mathrm{b}_{\mathrm{i}}, \mathrm{c}_{\mathrm{i}} \in \mathbb{R}(\mathrm{i}=1,2,3)$$ and $$x \in \mathbb{R}$$ and $$\left|\begin{array}{lll}\mathrm{a}_1+b_1 x & a_1 x+b_1 & c_1 \\ \mathrm{a}_2+b_2 x & a_2 x+b_2 & c_2 \\ \mathrm{a}_3+b_3 x & a_3 x+b_3 & c_3\end{array}\right|=0$$, then

The function $$\mathrm{f}: \mathbb{R} \rightarrow \mathbb{R}$$ defined by $$\mathrm{f}(x)=\mathrm{e}^x+\mathrm{e}^{-x}$$ is :

A square with each side equal to '$$a$$' above the $$x$$-axis and has one vertex at the origin. One of the sides passing through the origin makes an angle $$\alpha$$ $$\left(0<\alpha< \frac{\pi}{4}\right)$$ with the positive direction of the axis. Equation of the diagonals of the square

If $$\mathrm{ABC}$$ is an isosceles triangle and the coordinates of the base points are $$B(1,3)$$ and $$C(-2,7)$$. The coordinates of $$A$$ can be

$$ \text { The points of extremum of } \int_\limits0^{x^2} \frac{t^2-5 t+4}{2+e^t} d t \text { are } $$

Choose the correct statement :

Let $$\theta$$ be the angle between two vectors $$\vec{A}$$ and $$\vec{B}$$. If $$\hat{a}_{\perp}$$ is the unit vector perpendicular to $$\vec{A}$$, then the direction of $$ \overrightarrow{\mathrm{B}}-\mathrm{B} \sin \theta \hat{\mathrm{a}}_{\perp} \text { is }$$

The Power $$(\mathrm{P})$$ radiated from an accelerated charged particle is given by $$\mathrm{P} \propto \frac{(q \mathrm{a})^{\mathrm{m}}}{\mathrm{c}^{\mathrm{n}}}$$ where $$\mathrm{q}$$ is the charge, $$\mathrm{a}$$ is the acceleration of the particle and $$\mathrm{c}$$ is speed of light in vacuum. From dimensional analysis, the value of $$m$$ and $$n$$ respectively, are

Two convex lens $$(\mathrm{L}_1$$ and $$\mathrm{L}_2)$$ of equal focal length $$\mathrm{f}$$ are placed at a distance $$\frac{\mathrm{f}}{2}$$ apart. An object is placed at a distance $$4 \mathrm{f}$$ in the left of $$\mathrm{L_1}$$ as shown in figure. The final image is at

Which of the following quantity has the dimension of length ?

(h is Planck's constant, m is the mass of electron and c is the velocity of light)

The speed distribution for a sample of $$\mathrm{N}$$ gas particles is shown below. $$\mathrm{P}(\mathrm{v})=0$$ for $$\mathrm{v}>2 \mathrm{v}_0$$. How many particles have speeds between $$1.2 \mathrm{v}_0$$ and $$1.8 \mathrm{v}_0$$ ?

The internal energy of a thermodynamic system is given by $$U=a s^{4 / 3} V^\alpha$$ where $$\mathrm{s}$$ is entropy, $$\mathrm{V}$$ is volume and '$$\mathrm{a}$$' and '$$\alpha$$' are constants. The value of $$\alpha$$ is

A particle of mass '$$m$$' moves in one dimension under the action of a conservative force whose potential energy has the form of $$U(x)=-\frac{\alpha x}{x^2+\beta^2}$$ where $$\alpha$$ and $$\beta$$ are dimensional parameters. The angular frequency of the oscillation is proportional to

Longitudinal waves cannot

A $$2 \mathrm{~V}$$ cell is connected across the points $$\mathrm{A}$$ and $$\mathrm{B}$$ as shown in the figure. Assume that the resistance of each diode is zero in forward bias and infinity in reverse bias. The current supplied by the cell is

A charge Q is placed at the centre of a cube of sides a. The total flux of electric field through the six surfaces of the cube is

The elastic potential energy of a strained body is

Which of the following statement(s) is/are truc in respect of nuclear binding energy ?

(i) The mass energy of a nucleus is larger than the total mass energy of its individual protons and neutrons.

(ii) If a nucleus could be separated into its nucleons, an energy equal to the binding energy would have to be transferred to the particles during the separating process.

(iii) The binding energy is a measure of how well the nucleons in a nucleus are held together.

(iv) The nuclear fission is somehow related to acquiring higher binding energy.

A satellite of mass $$\mathrm{m}$$ rotates round the earth in a circular orbit of radius R. If the angular momentum of the satellite is J, then its kinetic energy $$(\mathrm{K})$$ and the total energy (E) of the satellite are

What force $$\mathrm{F}$$ is required to start moving this $$10 \mathrm{~kg}$$ block shown in the figure if it acts at an angle of $$60^{\circ}$$ as shown? $$(\mu_s=0.6)$$

Light of wavelength $$6000 \mathop A\limits^o$$ is incident on a thin glass plate of r.i. 1.5 such that the angle of refraction into the plate is $$60^{\circ}$$. Calculate the smallest thickness of the plate which will make dark fringe by reflected beam interference.

Consider a circuit where a cell of emf $$E_0$$ and internal resistance $$\mathrm{r}$$ is connected across the terminal $$\mathrm{A}$$ and $$\mathrm{B}$$ as shown in figure. The value of $$\mathrm{R}$$ for which the power generated in the circuit is maximum, is given by

The equivalent capacitance of a combination of connected capacitors shown in the figure between the points $$\mathrm{P}$$ and $$\mathrm{N}$$ is

In a single-slit diffraction experiment, the slit is illuminated by light of two wavelengths $$\lambda_1$$ and $$\lambda_2$$. It is observed that the $$2^{\text {nd }}$$ order diffraction minimum for $$\lambda_1$$ coincides with the $$3^{\text {rd }}$$ diffraction minimum for $$\lambda_2$$. Then

The acceleration-time graph of a particle moving in a straight line is shown in the figure. If the initial velocity of the particle is zero then the velocity-time graph of the particle will be

The position vector of a particle of mass $$\mathrm{m}$$ moving with a constant velocity $$\vec{v}$$ is given by $$\vec{r}=x(t) \hat{i}+b \hat{j}$$, where $$\mathrm{b}$$ is a constant. At an instant, $$\vec{r}$$ makes an angle $$\theta$$ with the $$x$$-axis as shown in the figure. The variation of the angular momentum of the particle about the origin with $$\theta$$ will be

The position of the centre of mass of the uniform plate as shown in the figure is

In a series LCR circuit, the rms voltage across the resistor and the capacitor are $$30 \mathrm{~V}$$ and $$90 \mathrm{~V}$$ respectively. If the applied voltage is $$50 \sqrt{2} \sin \omega t$$, then the peak voltage across the inductor is

A small ball of mass m is suspended from the ceiling of a floor by a string of length $$\mathrm{L}$$. The ball moves along a horizontal circle with constant angular velocity $$\omega$$, as shown in the figure. The torque about the centre (O) of the horizontal circle is

If $$\hat{n}_1, \hat{n}_2$$ and $$\hat{\mathrm{t}}$$ represent, unit vectors along the incident ray, reflected ray and normal to the surface respectively, then

A beam of light of wavelength $$\lambda$$ falls on a metal having work function $$\phi$$ placed in a magnetic field B. The most energetic electrons, perpendicular to the field are bent in circular arcs of radius R. If the experiment is performed for different values of $$\lambda$$, then $$\mathrm{B}^2$$ vs. $$\frac{1}{\lambda}$$ graph will look like (keeping all other quantities constant)

A charged particle moving with a velocity $$\vec{v}=v_1 \hat{i}+v_2 \hat{j}$$ in a magnetic field $$\vec{B}$$ experiences a force $$\vec{F}=F_1 \hat{i}+F_2 \hat{j}$$. Here $$v_1, v_2, F_1, F_2$$ all are constants. Then $$\overrightarrow{\mathrm{B}}$$ can be

Two straight conducting plates form an angle $$\theta$$ where their ends are joined. A conducting bar in contact with the plates and forming an isosceles triangle with them starts at the vertex at time $$t=0$$ and moves with constant velocity $$\vec{v}$$ to the right as shown in figure. A magnetic field $$\vec{B}$$ points out of the page. The magnitude of emf induced at $$t=1$$ second will be

Three point charges $$\mathrm{q},-2 \mathrm{q}$$ and $$\mathrm{q}$$ are placed along $$x$$ axis at $$x=-{a}, 0$$ and $a$ respectively. As $$\mathrm{a} \rightarrow 0$$ and $$\mathrm{q} \rightarrow \infty$$ while $$\mathrm{q} \mathrm{a}^2=\mathrm{Q}$$ remains finite, the electric field at a point P, at a distance $$x(x \gg a)$$ from $$x=0$$ is $$\overrightarrow{\mathrm{E}}=\frac{\alpha \mathrm{Q}}{4 \pi \epsilon_0 x^\beta} \hat{i}$$. Then

A body floats with $$\frac{1}{n}$$ of its volume keeping outside of water. If the body has been taken to height $$\mathrm{h}$$ inside water and released, it will come to the surface after time t. Then

A small sphere of mass m and radius r slides down the smooth surface of a large hemispherical bowl of radius R. If the sphere starts sliding from rest, the total kinetic energy of the sphere at the lowest point $$\mathrm{A}$$ of the bowl will be [given, moment of inertia of sphere $$=\frac{2}{5} \mathrm{mr}^2$$]

When a convex lens is placed above an empty tank, the image of a mark at the bottom of the tank, which is 45 cm from the lens is formed 36 cm above the lens. When a liquid is poured in the tank to a depth of 40 cm, the distance of the image of the mark above the lens is 48 cm. The refractive index of the liquid is

In the given network of AND and OR gates, output Q can be written as (assuming n even)

Water is filled in a cylindrical vessel of height $$\mathrm{H}$$. A hole is made at height $$\mathrm{z}$$ from the bottom, as shown in the figure. The value of z for which the range (R) of the emerging water through the hole will be maximum for

A metal plate of area $$10^{-2} \mathrm{~m}^2$$ rests on a layer of castor oil, $$2 \times 10^{-3} \mathrm{~m}$$ thick, whose coefficient of viscosity is $$1.55 \mathrm{~Ns} \mathrm{~m}^{-2}$$. The approximate horizontal force required to move the plate with a uniform speed of $$3 \times 10^{-2} \mathrm{~ms}^{-1}$$ is

The following figure shows the variation of potential energy $$V(x)$$ of a particle with distance $$x$$. The particle has

Consider the integral form of the Gauss' law in electrostatics

$$\oint {\overrightarrow E .d\overrightarrow S } = {Q \over {{\varepsilon _0}}}$$

Which of the following statements are correct?

A uniform rod $$\mathrm{AB}$$ of length $$1 \mathrm{~m}$$ and mass $$4 \mathrm{~kg}$$ is sliding along two mutually perpendicular frictionless walls OX and OY. The velocity of the two ends of the $$\operatorname{rod} \mathrm{A}$$ and $$\mathrm{B}$$ are $$3 \mathrm{~m} / \mathrm{s}$$ and $$4 \mathrm{~m} / \mathrm{s}$$ respectively, as shown in the figure. Then which of the following statement(s) is/are correct?

The variation of impedance $$\mathrm{Z}$$ of a series $$\mathrm{L C R}$$ circuit with frequency of the source is shown in the figure. Which of the following statement(s) is/are true ?

The electric field of a plane electromagnetic wave in a medium is given by

$$ \overrightarrow{\mathrm{E}}(x, y, z, t)=\mathrm{E}_0 \hat{\mathrm{n}} \mathrm{e}^{i k_o[(x+y+z)-c t]} $$

where $$\mathrm{c}$$ is the speed of light in free space. $$\overrightarrow{\mathrm{E}}$$ field is polarized in the $$x-\mathrm{z}$$ plane. The speed of wave is $$v$$ in the medium. Then