The major products U and V in the following reaction are

Among $\mathrm{N}_2 \mathrm{O}, \mathrm{ClF}_2^{-}, \mathrm{SO}_2$ and $\mathrm{I}_3^{+}$, the species having the linear structures are

In the following sequence of reactions, what is the end product ' $Z$ '?

A compound $(\mathrm{X})$ when treated with $\mathrm{CuSO}_4$ solution yields a brown precipitate．On adding hypo solution the precipitate turns white．The compound $(\mathrm{X})$ is

Three engines $\mathrm{A}, \mathrm{B}$ and C take steam at $130^{\circ} \mathrm{C}$ and reject it at $20^{\circ} \mathrm{C}, 40^{\circ} \mathrm{C}$ and $50^{\circ} \mathrm{C}$ respectively．The most efficient engine will be

In a conductance experiment, aqueous $\mathrm{AgNO}_3$ solution is added to aqueous KCl solution gradually and simultaneously the molar conductivity ( $\lambda_{\mathrm{m}}$ ) is measured. The correct plot of $\lambda_{\mathrm{m}}$ versus volume of $\mathrm{AgNO}_3$ solution is

Indicate the major product of the following reaction:

The van't Hoff Factor (i) for a dilute aqueous solution of $\mathrm{Na}_2 \mathrm{SO}_4$ is

Which of the following is the structure of pyrosulphuric acid?

Peroxide ion is

How many isomers can a compound with molecular formula $\mathrm{C}_3 \mathrm{H}_5 \mathrm{Br}$ have？

Which one of the following cations gives a chocolate brown precipitate upon addition of aqueous solution of $\mathrm{K}_4\left[\mathrm{Fe}(\mathrm{CN})_6\right]$ ？

A compound contains two types of atoms $A$ and $B$ ．Its crystal structure is a cubic lattice with＇$A$＇ atoms at the corner of the unit cells and＇$B$＇atoms at the body centres．The simplest formula of the compound will be

The plot of radial probability density（ $4 \pi r^2 R^2$ ）against $r$ for an electron in $n p$ orbital of a many electron atom is given below．

The value of $n$ is

A buffer solution contains 100 ml of $0.01(\mathrm{M}) \mathrm{CH}_3 \mathrm{COOH}$ and 200 ml of $0.02(\mathrm{M}) \mathrm{CH}_3 \mathrm{COONa}_a .700 \mathrm{ml}$ of water is added subsequently to the buffer solution．The pH before and after dilution are ［given，$p K_a=4.74 ; \log 2=0.301$ ］

The correct order of conductivity of $0.001(\mathrm{M})$ separate aqueous solutions of $\left[\mathrm{Pt}\left(\mathrm{NH}_3\right)_6\right] \mathrm{Cl}_4(\mathrm{i})$; $\left[\mathrm{Cr}\left(\mathrm{NH}_3\right)_6\right] \mathrm{Cl}_3$ (ii); $\left[\mathrm{Co}\left(\mathrm{NH}_3\right)_4 \mathrm{Cl}_2\right] \mathrm{Cl}$ (iii) and $\mathrm{K}_2 \mathrm{PtCl}_6$ (iv) each containing octahedral complex species is

Borazole is prepared by heating the product isolated by reacting

The increasing order of basicity of the following compounds is

The products $\mathbf{X}$ and $\mathbf{Y}$ in the following reaction sequence are

The van der Waal's equation : $\left(P+\frac{a}{4 V^2}\right)\left(V-\frac{b}{2}\right)=\frac{R T}{2}$ is valid for

Which one of the following does not lose water even in conc. $\mathrm{H}_2 \mathrm{SO}_4$ ?

The major product in the following reaction is

Rank the following anions in order of decreasing nucleophilicity in a polar protic solvent (most → least nucleophilic).

In which of the following species, $\mathrm{sp}^3 \mathrm{~d}^2$ hybridisation is not associated?

For the reaction $A \rightarrow B$ ，variation of concentration is plotted against time as shown below．

Which of the following statements is true？

In a first order reaction，the concentration of reactant decreases from 400 moles lit $^{-1}$ to 50 moles lit ${ }^{-1}$ in $7 \cdot 5 \times 10^3 \mathrm{~s}$ ．The rate constant of the reaction is（approximately）

In the following reaction sequence, the product Y is

$$ \mathrm{Br}\left(\mathrm{CH}_2\right)_{12}-\mathrm{C} \equiv \mathrm{CH} \xrightarrow{\mathrm{NaNH}_2} \mathrm{X} \xrightarrow[\text { catalyst }]{\mathrm{H}_2, \text { Lindler }} \mathrm{Y} $$

The mass of an electron is $9 \cdot 1 \times 10^{-31} \mathrm{~kg}$ ．If its K ．E．is $3 \cdot 0 \times 10^{-25} \mathrm{~J}$ ，its wavelength is（approximately）

The calculated magnetic moment for low spin $[\operatorname{Ru}(E D T A)]^{-}$is

Glucose is added to 1 litre of water to such an extent that $\Delta T_f / K_f$ equals to $\frac{1}{1000}$ ．The weight of glucose added is

A $5.0 \mathrm{~cm}^3$ solution of $\mathrm{H}_2 \mathrm{O}_2$ liberates 1.27 g of iodine from an acidified KI solution. The percentage strength of $\mathrm{H}_2 \mathrm{O}_2$ is close to

An organic compound undergoes first order decomposition．The time taken for its decomposition to $\frac{1}{8}$ th and $\frac{1}{10}$ th of its initial concentration are $t_{1 / 8}$ and $t_{1 / 10}$ respectively．The value of $\left[\frac{t_{1 / 8}}{t_{1 / 10}}\right]$ is ［Given $\log _{10} 2=0 \cdot 3$ ］

For the metal complex $\left[\mathrm{Co}\left(\mathrm{NH}_3\right)_5 \mathrm{SO}_4\right] \mathrm{Br}$ ，coordination number，oxidation number，number of $d$－electrons and number of unpaired $d$－electrons are respectively

$$ \mathrm{RO}-\mathrm{CH}_2-\mathrm{C} \equiv \mathrm{CH} \xrightarrow{\mathrm{X}} \xrightarrow{\mathrm{Y}} \mathrm{RO}^{-} \mathrm{CH}_2-\mathrm{C} \equiv \mathrm{C}-\mathrm{CH}_2 \mathrm{CH}_2-\mathrm{Br} $$

To carry out the above conversion X and Y are respectively

Which of the following statement(s) is/are correct about the given compound?

1 mole of an ideal gas undergoes the following processes：

Process $A \rightarrow$ Isothermal expansion at 400 K from volume $\mathrm{V}_1$ to volume $\mathrm{V}_2$ ，such that $\mathrm{V}_2=4 \mathrm{~V}_1$

Process $B \rightarrow$ Adiabatic expansion from volume $\mathrm{V}_1$ to volume $\mathrm{V}_2$ ，such that $\mathrm{V}_2=4 \mathrm{~V}_1$

Consider the following statements and select the correct one／s．

Which of the following statement（s）is／are correct？

Which of the following have tetrahedral structures？

Which of the following plot(s) is/are correct representation(s) of Boyle's Law?

Given $P(x)=x^4+a x^3+b x^2+c x+d$ such that $x=0$ is the only real root of $P^{\prime}(x)=0$. If $P(-1) < P(1)$, then in the interval $[-1,1]$.

If $\alpha, \beta$ are the roots of the equation $x^2-p x+q=0$ and $\alpha>0, \beta>0$, then $\alpha^{\frac{1}{4}}+\beta^{\frac{1}{4}}=\left(p+6 \sqrt{p}+4 q^{\frac{1}{4}} \sqrt{p+2 \sqrt{q}}\right)^k$, where $K$ is

If $\sum\limits_{r=1}^{\infty} \tan ^{-1}\left(\frac{1}{2 r^2}\right)=a$, then $\tan a$ is equal to

Consider a function $f(x)$ which has exactly two roots at $x=a$. If $\mathop {\lim }\limits_{x \to a}\left(\frac{\lambda f^{\prime}(x)}{f(x)}-\frac{1}{x-a}\right)=m(\neq 0)$, then the value of $\lambda$ ix

A vector given by $\vec{P}=f(t) \hat{i}+g(t)+\hat{k}$ moves in such a way that it is always parallel to the vector $\vec{Q}=-f^{\prime \prime}(t) \hat{i}+f^{\prime}(t) \hat{j}+\hat{k}$.

The expression $\sum_{k=1}^{32}(3 K+2)\left\{\sum_{r=1}^{10}\left(\sin \frac{2 r \pi}{11}-i \cos \frac{2 r \pi}{11}\right)\right\}^k$ represents

$\theta$ elimination from the equation $x^2+y^2=\frac{x \cos 3 \theta+y \sin 3 \theta}{\cos ^3 \theta}=\frac{y \cos 3 \theta-x \sin 3 \theta}{\sin ^3 \theta}$ will be

If $t_n$ denotes the $n^{\text {th }}$ term of an A.P. and $t_p=\frac{1}{q}, t_q=\frac{1}{p}$, then which one of the following options is a root of the equation $(p+2 q-3 r) x^2+(q+2 x-3 p) x+(r+2 p-3 q)=0 ?$

Consider the sequence of numbers $(1,2,3, \ldots \ldots, 13)$. A person choose three numbers at random from the sequence. The probability that the chosen three number form an A.P. is

If $f(x)=\frac{1+x}{1-x}$ and $A$ is a matrix such that $A^3=0$, then $f(A)=$

Which of the following statements is always true?

If $0<\alpha<\beta<\gamma<\frac{\pi}{2}$, then the equation $\frac{1}{x-\sin \alpha}+\frac{1}{x-\sin \beta}+\frac{1}{x-\sin \gamma}=0$ has

On the set $\mathbb{R}$ of real numbers the relation $\rho$, defined by $\mathrm{x} \rho \mathrm{y}(\mathrm{x}, \mathrm{y} \in \mathbb{R})$ iff

If $\int \frac{\operatorname{cosec}^2 x-2010}{\cos ^{2010} x} d x=-\frac{f(x)}{(g(x))^{2010}}+c$, where $f\left(\frac{\pi}{4}\right)=1$; then the number of solutions of the equation $\frac{f(x)}{g(x)}=\{x\}$ in $[0,2 \pi]$ is/are (where $\{\cdot\}$ represents fractional part function)

If the locus of mid point of any normal chord of the parabola $y^2=4 x$ is $x-\lambda=\frac{\mu}{y^2}+\frac{y^2}{v}$, where $\lambda, \mu, v \in N$, then ( $\lambda+\mu+v$ ) equals to

The true set of values of ' $K$ ' for which $\sin ^{-1}\left(\frac{1}{1+\sin ^2 x}\right)=\frac{K \pi}{6}$ may have a solution is

A mapping is selected at random from all mappings $f: A \rightarrow A$, where set $A=\{1,2,3 \ldots, n\}$. If the probability that the mapping is injective is $\frac{3}{32}$, then the value of $n$ is

Let $A=[a, \infty)$ denotes the domain, then $f:(a, \infty) \rightarrow B$, which is defined by $f(x)=2 x^3-3 x^2+6$ will have an inverse for the smallest real value of ' $a$ ' if

If $a=\mathop {\lim }\limits_{n \to \infty } \cos ^{2 n} x,(x=n \pi)$ and $b=\mathop {\lim }\limits_{n \to \infty } \cos ^{2 n} x,(x \neq n \pi)$, then numerical value of the area of the triangle whose vertices are (a, b), (-2, 1) and (2, 1) is

The position vectors of two adjacent sides $\overrightarrow{O A}$ and $\overrightarrow{O B}$ of a rectangle $O A C B$ are $\vec{a}$ and $\vec{b}$ respectively, where $O$ is the origin. If $16|\vec{a} \times \vec{b}|=3(|\vec{a}|+|\vec{b}|)^2$ and $\theta$ be the acute angle between the diagonals $O C$ and $A B$, then the value of $\tan \left(\frac{\theta}{2}\right)$ is

The point of intersection of $\vec{r} \times \vec{a}=\vec{b} \times \vec{a}$ and $\vec{r} \times \vec{b}=\vec{a} \times \vec{b}$, where $\vec{a}=\hat{i}+\hat{j}$ and $\vec{b}=2 \hat{i}-\hat{k}$ is

Let $a_1, a_2, a_3 \ldots$ are in G.P. such that $n>m, a_n>a_m$ and $a_1+a_n=66, a_2 \cdot a_{n-1}=128$. If $\sum_{r=1}^n a_r=126$, then $n$ is

The minimum length of intercept on any tangent to the ellipse $\frac{x^2}{4}+\frac{y^2}{9}=1$ cut by the circle $x^2+y^2=25$ is

Intercepts of the plane $\vec{r} \cdot \vec{n}=d(\neq 0)$ on the coordinate axes respectively are

The general solution of the equation $\sin ^{100} \mathrm{x}-\cos ^{100} \mathrm{x}=1$ is

If $\vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{b}=\hat{i}-\hat{j}+\hat{k}, \vec{c}=\hat{i}+2 \hat{j}-\hat{k}$, then the value of $\left|\begin{array}{lll}\vec{a} \cdot \vec{a} & \vec{a} \cdot \vec{b} & \vec{a} \cdot \vec{c} \\ \vec{b} \cdot \vec{a} & \vec{b} \cdot \vec{b} & \vec{b} \cdot \vec{c} \\ \vec{c} \cdot \vec{a} & \vec{c} \cdot \vec{b} & \vec{c} \cdot \vec{c}\end{array}\right|$ is equal to

Number of elements in the range set of $f(x)=\left[\frac{x}{15}\right]\left[-\frac{15}{x}\right]$, for all $x \in(0,90$ ); (where [.] denotes the greatest integer function) is

Let 10 Bags $B_1, B_2, \ldots, B_{10}$ which contains $21,22, \ldots, 30$ different articles respectively. Then the total number of ways to bring out 10 articles from a Bag is

Let domain and range of $f(x)$ and $g(x)$ is $[0, \infty)$. If $f(x)$ is an increasing function, $g(x)$ is a decreasing function, $h(x)= f\{g(x)\}, h(0)=0$ and $p(x)=h\left(x^3-2 x^2+2 x\right)-h(4)$, then for all $x \in(0,2)$

Consider the following ellipse :

$\frac{x^2}{f\left(K^2+2 K+5\right)}+\frac{y^2}{f(K+11)}=1$, where $f(x)$ is a positive decreasing function. Then the value (values) of $K$ for which the major axis coincides with $x$-axis is

The solution of the differential equation $2 x^2 y \frac{d y}{d x}=\tan \left(x^2 y^2\right)-2 x y^2$, given $y(1)=\sqrt{\frac{\pi}{2}}$ is

$$ \int \frac{\left(\sqrt[3]{x+\sqrt{2-x^2}}\right)\left(\sqrt[6]{1-x \sqrt{2-x^2}}\right)}{\sqrt[3]{1-x^2}} d x ;(x \in(0,1))= $$

Consider the function $y=f(x)$ defined implicitly by the equation $y^3-3 y+x=0$ on the interval $(-\infty,-2) \cup(2, \infty)$. The area of the region bounded by the curve $y=f(x)$, the $x$-axis and the lines $x=a, x=b$, where $-\infty< a< b< -2$ is

The total number of polynomials of the form $x^3+a x^2+b x+c$ which is divisible by $x^2+1$, where $a, b, c \in\{1,2,3, \ldots ., 10\}$ is

The term independent of $x$ in the expansion of $\left(\frac{x+1}{x^{\frac{2}{3}}-x^{\frac{1}{3}}+1}-\frac{x-1}{x-x^{\frac{1}{2}}}\right)^{15}$ is equal to

For a real number $y$, consider $(y)$ denotes the greatest integer less than or equal to $y$. If $f(x)=\frac{\tan (\pi[x-\pi])}{1+[x]^2}$, then

If $\int_0^1\left(\sum_{r=1}^{2013} \frac{x}{x^2+r^2}\right)\left(\prod_{r=1}^{2013}\left(x^2+r^2\right)\right) d x=\frac{1}{2}\left[\left(\prod_{r=1}^{2013}\left(1+r^2\right)-K^2\right]\right.$, then $K$ is

The least positive value of ' $a$ ' for which the equation $\int_0^x\left(t^2-8 t+13\right) d t=x \sin \frac{a}{x}$ has a solution is

Let all the points on the curve $x^2+y^2-10 x=0$ are reflected about the line $y=x+3$. If the locus of the reflected points is in the form $x^2+y^2+g x+f y+c=0$, then the value of $(g+f+c)$ is

The equation $|x+1|^{\log _{(x+1)}\left(3+2 x-x^2\right)}=(x-3)|x|$ has

If the domain of $f(x)$ is $(0,1)$, then the domain of $y=f\left(e^x\right)+f(\ln |x|)$ is

The number of 3-digit numbers we of the form $x y z$ with $x< y, z< y$ and $x \neq 0$ is

Suppose $A$ is denoted the set of all numbers between 1 and 700 which are divisible by 3 and let $B$ is denoted the set of all numbers between 1 and 300 which are divisible by 7 . If $C=\{(a, b) \mid a \in A, b \in B, a \neq b$ and $a+b=$ even number $\}$, then order of C is

Let us define the power of a matrix $A$ as the maximum $m \in Z^{+}$such that $A^m=I$. For two matrices $A$ and $B$ if $A^5=I$ and $A B A^{-1}=B^2$, then the power of the matrix $B$ is between

If for two real numbers $\mathrm{a}, \mathrm{b}$ with $|\mathrm{a}| \leq 1$ and $|\mathrm{b}| \leq 1$,

$\frac{1}{3}+\frac{\sin ^{-1} a+\sin ^{-1} b}{4}+\frac{\left(\sin ^{-1} a+\sin ^{-1} b\right)^2}{16}+\frac{\left(\sin ^{-1} a+\sin ^{-1} b\right)^3}{64}+\cdots=\frac{2(8-3 \pi)}{3(16+3 \pi)}, \quad$ then the value of $\sin ^{-1}\left(a \sqrt{1-b^2}+b \sqrt{1-a^2}\right)$ is

Let $\operatorname{det} A=\left|\begin{array}{ccc}\mathrm{l} & \mathrm{m} & \mathrm{n} \\ \mathrm{p} & \mathrm{q} & \mathrm{r} \\ \mathrm{l} & \mathrm{l} & \mathrm{l}\end{array}\right|$ If $(I-m)^2+(p-q)^2=9,(m-n)^2+(q-r)^2=16,(n-I)^2+(r-p)^2=25$, then the value of $(\operatorname{det} A)^2$ is

Let $f:(0,1) \rightarrow(0,1)$ be a differentiable function such that $f^{\prime}(x) \neq 0 \forall x \in(0,1)$ and $f\left(\frac{1}{2}\right)=\frac{\sqrt{3}}{2}$. Suppose for all $x$, $\mathop {\lim }\limits_{t \to x} \frac{\int_0^t \sqrt{1-(f(s))^2} d s-\int_0^x \sqrt{1-(f(s))^2} d s}{f(t)-f(x)}=f(x)$. Then the value of $f\left(\frac{1}{4}\right)$ belongs to

If ' $a$ ' is an integer lying in $[-5,30]$, then the probability that the graph of $y=x^2+2(a+4) x-5 a+64$ lies above the $x-$ axis is

Consider a square $A B C D$ of diagonal length 2a. The square is folded along the diagonal $A C$ so that the plane of $\triangle A B C$ is perpendicular to the plane of $\triangle A D C$. In this case the shortest distance between $A B$ and $C D$ is

If $\int \frac{\left(1-x^2\right)}{\sqrt{x} \sqrt{\left(1+x^2\right)^3}}=\alpha \frac{x^\beta}{\left(1+x^2\right)^\gamma}+C ; \alpha, \beta, \gamma \in \mathbb{R}$ and $C$ is constant of integration, then $\alpha: \beta: \gamma$ will be

Let $\vec{a}=(x, y, z)$ be the vector with $|\vec{a}|=2 \sqrt{3}$, which makes equal angles with the vector $\vec{b}=(y,-2 z, 3 x)$ and $\vec{c}=(2 z, 3 x,-y)$ and is perpendicular to the vector $\vec{d}=(1,-1,2)$. If the angle between $\vec{a}$ and the unit vector $\hat{j}$ is obtuse, then $\vec{a}$ is

Let $A_1, A_2, \ldots, A_6$ are six sets, each with four elements and $B_1, B_2, \ldots ., B_n$ are $n$ sets, each with two elements. Let $S=A_1 \cup A_2 \cup \ldots \cup A_6=B_1 \cup B_2 \cup \ldots \cup B_n$.

Given that each element of $S$ belongs to exactly four of the A's and to exactly three of the B's. Then $n$ is

A figure is bounded by the curves $y=x^2+1, y=0, x=0$ and $x=1$. The point at which a tangent should be drawn to the curve $y=x^2+1$ for it to cut off trapezium of the greatest area from the figure is

The ends $A$, $B$ of a straight line segment of constant length $c$ slide upon the fixed rectangular axes $O X, O Y$ respectively. If the rectangle $O A P B$ completed, then the locus of the foot of perpendicular drawn from $P$ to $A B$ is

Let 1 lies between the roots of the equation $y^2-m y+1=0$ and $[x]$ denotes the greatest integer function. Then the value of $\left[\left(\frac{4|x|}{x^2+16}\right)^m\right]$ is

Let $f(x)$ be a twice differentiable function in $[1,3]$ and $f(1)=f(3)$. Further if $\left|f^{\prime \prime}(x)\right| \leq 2$, then for all $x$ in $[1,3]$

The quantities $a_1, a_2, a_3, \ldots$ form an infinite decreasing G.P. If $a_1=1$, then the common ratio of the progression for which the expression $6 a_5-16 a_4-3 a_3+12 a_2$ is at a maximum is

If $f$ be a real valued function defined for all real numbers $x$ such that for some fixed $a>0$, it satisfies $f(x+a)=\frac{1}{2}+\sqrt{f(x)-(f(x))^2} \forall x$, then $f(x)$ is periodic with period

Four natural numbers selected at random are multiplied together, then the probability that the digit in the unit's place in the product be $1,3,7$ or 9 is

Let $f(x)$ be a real valued $f$ unction which is monotonic and differentiable. Then for any reals a and $b, \int_{f(a)}^{f(b)} 2 x\left\{b-f^{-1}(x)\right\} d x=$

Tangent at a point $P_1$ (other than $(0,0)$ ) on the curve $y=x^3$ meets the curve again at $P_2$. The tangent at $P_2$ meets the curve at $\mathrm{P}_3$ and so on. Then the abscissae of $\mathrm{P}_1, \mathrm{P}_2, \mathrm{P}_3, \ldots, \mathrm{P}_{\mathrm{n}}$ form

The equation $x^3+5 x^2+p x+q=0$ and $x^3+7 x^2+p x+r=0$ have two roots in common. If the third root of each equation is represented by $x_1$ and $x_2$ respectively, the GCD of $x_1, x_2$ will be

Let $a, b, c$ be non-zero real numbers, such that $\int_0^r\left(1+\cos ^8 x\right)\left(a x^2+b x+c\right) d x=\int_0^{2^{\prime}}\left(1+\cos ^8 x\right)\left(a x^2+b x+c\right) d x$, then $a x^2+b x+c=0$ has

Let $Z_1, Z_2$ be the roots of the equation $Z^2+p Z+q=0$, where the coefficients $p$ and $q$ may be complex numbers and also let $A, B$ represent $Z_1, Z_2$ respectively in the complex plane. If $\angle A O B=\alpha \neq 0$ and $O A=O B$, where $O$ is the origin, then the value of $\frac{p^2}{q} \sec ^2 \frac{\alpha}{2}$ will be

Let $g(x)=a x+b$, where $a<0$ and $g$ is defined from $[1,3]$ onto $[0,2]$. Then the value of $\cot \left(\cos ^{-1}(|\sin x|+|\cos x|)+\right. \left.\sin ^{-1}(-|\cos x|-|\sin x|)\right)$ is equal to

If $\sum_{r=0}^{2 n} a_r(x-2)^r=\sum_{r=0}^{2 n} b_r(x-3)^r$ and $a_k=1 \forall k \geq 1$, then the value of $\frac{b_n}{{ }^{2 n+1} C_{n+1}}$ is

If $f(x)$ is differentiable for all $x \in \mathbb{R}$ and satisfies the relation

$x=\mathop {\lim }\limits_{n \to \infty }\frac{\left[1^2(f(x))^x\right]+\left[2^2(f(x))^x\right]+\ldots+\left[n^2(f(x))^x\right]}{n^3}$ where [.] denotes the greatest integer function, then $f^{\prime}(x)=$

If a differentiable function satisfies $(x-y) f(x+y)-(x+y) f(x-y)=2\left(x^2 y-y^3\right) \forall x, y \in \mathbb{R}$ and $f(I)=2$, then

Let $f(x)>0$ for all $x \in \mathbb{R}$ and $f(x)$ is bounded. If $\mathop {\lim }\limits_{n \to \infty } \sum_{r-1}^n a^{r-1} \int_{(r-1) a}^{r a} \frac{f(x) d x}{f(x)+f(2 r a-a-x)}=\frac{3}{5}$ where $0< a< 1$, then the value(s) of a is are

Consider the curve $x=1-3 t^2, y=t-3 t^3$. The tangent to the curve at the point $t$ is inclined at an angle $\phi$ to OX and the tangent at $\mathrm{P}(-2,2)$ meets the curve again at Q . Then

If $f(x)=x\left(1331 x^2-3630 x+3300\right)$, then for $a=\cos ^2\left(\tan ^{-1}\left(\sin \left(\cot ^{-1} 3\right)\right)\right)$

Let $\vec{r}=\sin x(\vec{a} \times \vec{b})+\cos y(\vec{b} \times \vec{c})+2(\vec{c} \times \vec{a})$ ，where $\vec{a}, \vec{b}$ and $\vec{c}$ are three non－coplanar vectors． It is given that $\vec{r}$ is perpendicular to $(\vec{a}+\vec{b}+\vec{c})$ ．Then the possible value（s）of $\left(x^2+y^2\right)$ is／are

The parabola $y=4-x^2$ has vertex P. It intersects $x$-axis at A and B. If the parabola is translated from its initial position to a new position by moving its vertex along the line $y=x+4$, so that it intersects $x$-axis at B and C , then the abscissa of C will be

If $A_1, A_2, A_3, \ldots, A_{1006}$ be independent events such that $P\left(A_l\right)=\frac{1}{2 i},(i=1,2, \ldots, 1006)$ and the probability that none of the events occurs be $\frac{\alpha!}{2^a(\beta!)^2}$ ，then

If $\left(4^{\sec ^2 \alpha}\right) x^2+2 x+\left(\beta^2-\beta+\frac{1}{2}\right)=0$ has real roots，then the value／values of $\left(\cos \alpha+\cos ^{-1} \beta\right)$ is／are

A body of density＇$\rho$＇is dropped slowly on the surface of a lake of depth $d$ ．If the density of the lake water be＇$\rho^{\prime \prime}\left(\rho^{\prime}<\rho\right)$ then the time taken by the body to reach the bottom of the lake is ‘

Beyond what distance，the ray optics is sufficiently valid when the aperture is 6 mm wide and the wavelength is $6000 \mathop {\rm{A}}\limits^{\rm{o}}$ ？

A plano－convex lens fits exactly into a plano－concave lens．Their plane surfaces are parallel to each other．If lenses are made of different materials of refractive indices $\mu_1$ and $\mu_2$ and $R$ is the radius of curvature of the curved surface of the lenses，then the focal length of the combination is

Consider a fuse wire of length $l$ and radius $r$ ．The time of heating $(t)$ for passing the maximum current will depend on

Density and volume of a body are given as $(20 \pm 4) \mathrm{gm} / \mathrm{cm}^3$ and $(10 \pm 1) \mathrm{cm}^3$ respectively. The absolute error in measurement of mass is

If a vector $\vec{v}=3 \hat{i}$ is rotated in the $x-z$ plane by an angle $\theta$ with respect to $x$－axis in the clockwise direction，then for an observer at $+y$ axis the vector will be

A circular coil, carrying current, has radius $R$. The distance from the centre of the coil on the axis where the magnetic induction will be $\frac{1}{27}$ th of its value at the centre of the coil is

A square of side $L$ lies in the $x-y$ plane，where the magnetic field is given by $B=B_0(2 \hat{i}+3 \hat{j}+4 \hat{k})$ where $B_0$ is constant．The magnetic flux passing through the square is $L$

A resistor of resistance＇$R$＇draws power＇$P$＇when connected to an AC source．If an inductance is now placed in series with $R$ ，such that the impedance of the circuit becomes＇$Z$＇，the power drawn will be

A simple pendulum of length $l$ has a bob of mass $m$ ，with a charge $q$ ．On it a vertical sheet of charge， with surface charge density＇$\sigma$＇passes through the point of suspension．At equilibrium，if the string makes an angle $\theta$ with the vertical，then

The equation of a transverse wave is $y=y_0 \sin 2 \pi\left(f t-\frac{x}{\lambda}\right)$ ．If the maximum particle velocity be four times that of wave velocity then

A uniform but time varying magnetic field is present in a circular region of radius ' $R$ '. The magnetic field is perpendicular and into the plane of loop and the magnitude of field is increasing at a constant rate $\alpha$. There is a straight conducting rod of length 2 R placed as shown in figure. The magnitude of induced emf across the rod is

From a tower of height $H$ ，a particle is thrown vertically upwards with a speed $u$ ．The time taken by the particle to hit the ground is $n$ times that taken by it to reach the highest point of its path．The relation between $H, u$ and $n$ is

There is a ring of radius $r$ having linear charge density $\lambda$ and rotating with a uniform angular velocity $\omega$. The magnitude of the magnetic field produced by this ring at its own centre would be ( $\mu_0=$ permeability of air)

Three vectors $\vec{a}, \vec{b}$ and $\vec{c}$ are such that $|\vec{a}|=1,|\vec{b}|=2$ and $|\vec{c}|=4$ along with $\vec{a}+\vec{b}+\vec{c}=0$ ．Then，the value of $4 \vec{a} \cdot \vec{b}+3 \vec{b} \cdot \vec{c}+3 \vec{c} \cdot \vec{a}$ will be

Two identical metal bars are heated in two different temperatures and allowed to cool in the same surroundings. Which one of the following figures correctly shows their cooling curves?

The inputs to a digital circuit are as shown below. The output Y is

The velocity $v$ of a particle at time $t$ is given by $v=a t+\frac{b}{t+c}$, where $a, b$ and $c$ are constants. The dimension of $a, b$ and $c$ are, respectively

A body initially at rest and sliding along a frictionless track from a height＇$h$＇（as shown in figure） just completes a vertical circle of diameter $\mathrm{AB}=d$ ．The height＇$h$＇is equal to

The I-V characteristics graph shown below is exhibited by

A person has a minimum distance of distinct vision of 50 cm ．The power of lenses required to read a book at a distance of 25 cm is

Three blocks of masses $m_1=2 \mathrm{~kg}, m_2=3 \mathrm{~kg}$ and $m_3=5 \mathrm{~kg}$ are placed on a horizontal frictionless surface and a force of 30 N pulls the system as shown below．The value of tension $T$ will be

Which one of the following graphs represents the velocity-time $(v-t)$ graph of a small spherical body falling in a viscous liquid?

A ray of light travelling in air is incident on one face of a parallel glass slab of thickness $t$ and refractive index $\mu$ at an angle of incidence $i$ ．Total time spent by the ray inside the slab is

Radiation of wavelength $\lambda$ is incident on a photocell．The fastest emitted electron has speed $v$ . If the wavelength is changed to $\frac{3 \lambda}{4}$ ，then the speed of the fastest emitted electron will be

Two spherical soap bubbles of radii $r_1$ and $r_2$ in vaccum coalesce under isothermal condition．The newly formed bubble has a radius（ $r$ ）given by

A radioactive element ${ }_{92}^{242} \mathrm{X}$ emits two $\alpha$－particles，one electron and two positrons．The transformed nucleus is represented by ${ }_P^{234} Y$ ．The value of $P$ is

The magnetic moment of an iron bar is $M$. It is now bent in such a way that it forms an arc section of a circle subtending an angle of $60^{\circ}$ at the centre. The magnetic moment of the arc section is

A uniform $\operatorname{rod} A B$ is suspended from a point $P$ ，at a variable distance $x$ ，from $A$ ，as shown in figure． To make the rod horizontal，a mass＇$m$＇is suspended from its end $A$ ．Which set of variables will give a straight line when they are plotted？

A pipe $A$ is connected with other pipes $B$ and $C$ as shown in the figure．The areas of cross－section of $A, B$ and $C$ are respectively $\alpha, \frac{\alpha}{2}$ and $\frac{\alpha}{4}$ ．If the velocities of flow of water through $A$ and $B$ are $10 \mathrm{~m} / \mathrm{sec}$ and $6 \mathrm{~m} / \mathrm{sec}$ ，respectively，then velocity of flow，$V_c$ along $C$ is

A particle of mass $m$ is suspended from a point O by a string of length $R$ ．It is given a velocity $u=3 \sqrt{g R}$ at the bottom．The difference in tension at point $B$ and at the point $C$ is $m$

The de-Broglie wavelength of an electron in 4th orbit is (where $r=$ radius of the 1st orbit)

An electromagnetic wave，whose wave normal makes an angle of $45^{\circ}$ with the vertical，is travelling in air and strikes a horizontal liquid surface．While travelling through the liquid，it gets deviated by $15^{\circ}$ ．If the speed of electromagnetic wave in air is $3 \times 10^8 \mathrm{~m} / \mathrm{s}$ ，then the speed of electromagnetic wave in the liquid will be

The circuit has two oppositely connected ideal diodes in parallel as shown in the figure．What is the current flowing in the circuit？

2 moles of an ideal gas with $\frac{C_p}{C_v}=\frac{5}{3}$ are mixed with 3 moles of another ideal gas with $\frac{C_p}{C_v}=\frac{4}{3}$ ．The value of $\frac{C_p}{C_v}$ for the mixture is

Which of the velocity-time $(v-t)$ graph(s) can possibly represent one-dimensional motion of a particle?

The moment of inertia of a thin disc about axes $a, b, c, d$ are $\mathbf{I}_1, \mathbf{I}_2, \mathbf{I}_3$ and $\mathbf{I}_4$ respectively，as shown in figure．If the moment of inertia about an axis passing through the centre and perpendicular to the plane of the disc is I then，

The displacement current flows through a capacitor when the voltage across its plates

Two points of monochromatic and coherent sources of light of wavelength $\lambda$ each, are placed as shown in figure. The initial phase difference between the sources is zero, $(D \gg d)$. Mark the correct statement(s).

For Boolean variables $A$ and $B, A \oplus B=A \bar{B}+\bar{A} B$ ．Then，which of the following statements is / are correct？