Given the ratio of kinetic energy of electron in two orbitals is $16: 9$. Calculate the ratio of wavelength of electron waves?

Which of the following results is not true about photoelectric effect?

Trans uranium elements are

Arrange the following ions in the correct order with respect to their ionic radii.

What is the correct order of bond lengths in the following molecules?

I. $\mathrm{O}_2$

II. $\mathrm{O}_2^{+}$

III. $\mathrm{O}_2^{-}$

IV. $\mathrm{O}_2^{2-}$

Which one of the following compound is hypervalent?

What is the ratio of kinetic energy of 7 grams of nitrogen and 4 grams of oxygen at $T(\mathrm{~K})$ ?

Equal amount of gases are kept in two separate containers. If densities of the two gases are in $1: 2$ ratio and their temperatures are in 2:1 ratio, calculate the ratio of their respective pressures.

Given the ratio of amounts of nitrogen and oxygen in a particular gaseous mixture is $4: 1$. Calculate the ratio of number of their molecules?

Sulphuric acid reacts with sodium hydroxide as follows:

$$ \mathrm{H}_2 \mathrm{SO}_4+2 \mathrm{NaOH} \longrightarrow \mathrm{Na}_2 \mathrm{SO}_4+2 \mathrm{H}_2 \mathrm{O} $$

What will be the amount of sodium sulphate formed, when l L of 0.2 M sulphuric acid is allowed to react with l L of 0.2 M sodium hydroxide solution?

For the reactions,

$$ \begin{aligned} 2 \mathrm{Cl}(g) & \longrightarrow \mathrm{Cl}_2(g) \\ \mathrm{CO}_2(g) & \longrightarrow \mathrm{CO}(g)+\frac{1}{2} \mathrm{O}_2(g) \end{aligned} $$

What are the signs of $\Delta S$, respectively?

Which of the following statement is correct?

For a given reaction, $2 A \rightleftharpoons B+C$, the equilibrium constant is $2 \times 10^{-3}$. If at any given time the composition of the reaction mixture is $[A]=[B]=[C]=6 \times 10^{-5} \mathrm{M}$; predict in which direction the reaction will proceed and the correct value for reaction quotient.

At atmospheric pressure and very low temperature, water crystallises to

Which of the following statement(s) are correct, when alkali metals burn in the presence of oxygen?

I. Lithium forms monoxide

II. Sodium forms peroxide

III. Potassium, rubidium and cesium forms superoxide

Pick the correct statement.

I. Borax is white crystalline solid containing $\left[\mathrm{B}_4 \mathrm{O}_5(\mathrm{OH})_4\right]^{2-}$ units.

II. Aqueous solution of borax is acidic in nature.

III. Cobalt gives blue colour in borax bead test.

Which element does not show catenation property?

Acid rain is mainly caused by the emissions of which of the following gases?

I. Sulphur dioxide

II. Carbon dioxide

III. Nitrogen dioxide

IV. Methane

During the course of estimating nitrogen using Kjeldahl's method, the organic compound is heated with

An organic compound, ' $A$ ' with molecular formula $\mathrm{C}_8 \mathrm{H}_8 \mathrm{O}$ on reaction with $\mathrm{I}_2 / \mathrm{KOH}$ gives salt of carboxylic acid ' $B$ ' and a halo-compound ' $C$ '. Compound ' $C$ ' on the reaction with silver powder gives ' $D$ '. The structures of ${ }^{\prime} A^{\prime}$ and ' $D^{\prime}$ are

The order of circled $\mathrm{C}-\mathrm{H}$ bond dissociation energy in the following compound is

Intercepts of a plane in crystal is given by $a, b / 2,3 c$ in a simple cubic unit cell. The miller indices are

Relative lowering of vapour pressure of a dilute solution is 0.5 . What is the mole fraction of the non-volatile solute?

The solubility product of a sparingly soluble $A B_2$ salt is $2.56 \times 10^{-4} \mathrm{M}^3$ at $25^{\circ} \mathrm{C}$. The $K_f$ of water is 1.8 K kg mol ${ }^{-1}$. The depression in freezing point of a standard solution of $A B_2$ is

$\mathrm{Mg}^{2+}$ displaces hydrogen from acids but copper does not. A galvanic cell prepared by combining $\mathrm{Cu} / \mathrm{Cu}^{2+}$ and $\mathrm{Mg} / \mathrm{Mg}^{2+}$ has an EMF of 2.71 V at 298 K . If the potential of copper electrode is 0.34 V , what is the reduction potential of Mg electrode?

For a reversible reaction $A \rightleftharpoons B$, pre-exponential factor is same for both the forward and backward reactions and has value of $20 \mathrm{~S}^{-1}$. If the enthalpy change along the forward reaction is $-41.5 \mathrm{~kJ} / \mathrm{mol}$, the value of equilibrium constant at 500 K is

Which of the following is true for spontaneous adsorption?

Which element does not exist in elemental state in the earth's crust?

Phosphine is prepared by the reaction of $\mathrm{P}_4$ with which of the following?

The most stable form of sulphur allotrope is

When $\left[\mathrm{Ti}\left(\mathrm{H}_2 \mathrm{O}\right)_6\right] \mathrm{Cl}_3$ is heated at $250^{\circ} \mathrm{C}$, the change in colour is from

Which compound is zero valent metal complex?

$$ \text { The structure given below is an example of } $$

I. A condensation and biodegradable polymer.

II. A biodegradable and thermoplastic polymer.

III. A biodegradable and thermosetting polymer.

IV. A polyester and thermoplastic polymer.

154. Which of the following statements support the cyclic form for glucose?

(i) It doesn't give Schiff's test.

(ii) It is found to exist in two different crystalline forms.

(iii) It oxidises with nitric acid to give saccharic acid.

(iv) Pentaacetate of glucose doesn't react with hydroxylamine.

$$ \text { The structure of paracetamol is } $$

Arrange the following in the correct order of reactivity towards nucleophilic substitution reaction.

I. 1-chloro-2-nitrobenzene

II. Chlorobenzene

III. 1-chloro-3-nitrobenzene

Reactions that produce $n$-butanol in the following are :

I. $\mathrm{CH}_3 \mathrm{CH}_2 \mathrm{CH}=\mathrm{CH}_2 \xrightarrow[\text { (ii) } \mathrm{NaOH} / \mathrm{H}_2 \mathrm{O}_2]{\text { (i) } \mathrm{B}_2 \mathrm{H}_6}$

II. $\mathrm{CH}_3 \mathrm{CH}_2 \mathrm{CHO} \xrightarrow[\text { (ii) } \mathrm{H}_2 \mathrm{O}]{\text { (i) } \mathrm{CH}_3 \mathrm{Mgl}}$

III. $\mathrm{CH}_3 \mathrm{CH}_2 \mathrm{CH}_2 \mathrm{CN} \xrightarrow[\substack{\text { (ii) } \mathrm{H}_3 \mathrm{O} \\ \text { (iii) } \mathrm{NaBH}_4}]{\text { (i) } \mathrm{SnCl}_2, \mathrm{HCl}}$

IV. $\mathrm{CH}_3 \mathrm{CH}_2 \mathrm{CH}_2 \mathrm{CO}_2 \mathrm{H} \xrightarrow{\mathrm{B}_2 \mathrm{H}_6}$

$$ \text { What are } X \text { and } Y \text { in the following reaction sequence? } $$

$$ \text { Predict } A \text { and } B \text { in the following reaction sequence : } $$

$$ \mathrm{CH}_3-\mathrm{CH}_2-\mathrm{CH}_2-\mathrm{CH}_2-\mathrm{COOH} $$

$$ \xrightarrow[\text { Red phosphorus }]{\mathrm{Br}_2} A \xrightarrow{\mathrm{KOH} \text { (alc.) }} B$$

The major product formed by the reaction of benzylamine with nitrous acid is

If $f: Z \rightarrow N$ is defined by

$$ f(n)=\left\{\begin{array}{cll} 2 n, & \text { if } & n>0 \\ 1, & \text { if } & n=0, \text { then } f \text { is } \\ -2 n-1, & \text { if } & n<0 \end{array}\right. $$

Domain of $\cos ^{-1}\left[\log _5\left(x^2+7 x+15\right)\right]$ is

Let $f(n)=A(-2)^n+B(-3)^n \forall A, B \in \mathbf{R}$ and $n \in \mathbf{N}-\{1,2\}$. If $f(n)+a f(n-1)+b f(n-2)=0$, then $(a+b)(b-a)=$

If $a$ and $b$ are any two real numbers, then

$$ \left|\begin{array}{ccc} 2 a-2 b-4 & 4 a & 4 a \\ 4 & 2-b-a & 4 \\ 2 b & 2 b & b-a-2 \end{array}\right|= $$

Let $A=\left[\begin{array}{ccc}2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & x\end{array}\right]$ and $A^2=A$. If $r$ is the rank of $A$, then $r+x=$

Let $a, b, c, d \in \mathbf{R}$ be such that $a d-b c \neq 0$ and $e$ be a positive number other than 1 .

If $x^a y^b=e^m, x^c y^d=e^n, \Delta_1=\left|\begin{array}{ll}m & b \\ n & d\end{array}\right|, \Delta_2=\left|\begin{array}{cc}a & m \\ c & n\end{array}\right|$ and $\Delta_3=\left|\begin{array}{ll}a & b \\ c & d\end{array}\right|$, then the values of $x$ and $y$ are respectively.

Let $z$ be a complex number such that $|z|-z=2+i$, where $i=\sqrt{-1}$. Then, $|z|=$

If the amplitude of $z-2-3 i$ is $\pi / 4$, then the locus of $z=x+i y$ is

If $1+\frac{\cos \theta}{2}+\frac{\cos 2 \theta}{4}+\frac{\cos 3 \theta}{8}+\ldots \ldots=\frac{a-2 \cos \theta}{5+b \cos \theta}$ for some $a, b \in \mathbf{R}$, then $(a-b)^2=$

For $n>1$ and $n \in \mathbf{N}$, if $z_1, z_2, \ldots, z_n$ are the roots of the equation $(z+1)^n=z^n$, then $\sum_{i=1}^n \frac{\cot ^{-1}\left(2\left|\operatorname{Im} z_i\right|\right)-1}{2 \operatorname{Re} z_i}=$

If $x$ is real, then the maximum and minimum values of $\frac{x^2+14 x+9}{x^2+2 x+3}$ are respectively

When $\mathbf{R}$ is the set of all real numbers,

$$ \left\{x \in \mathbf{R}: \frac{\sqrt{12-x-x^2}}{x+10} \leq \frac{\sqrt{12-x-x^2}}{2 x+9}\right\}= $$

If $\alpha$ and $\beta$ are two complex roots of the equation $6 x^6-25 x^5+31 x^4-31 x^2+25 x-6=0$, then $\alpha+\beta=$

If $\alpha$ is a root of multiplicity 3 of the equation $x^5-8 x^4+25 x^3-38 x^2+28 x-8=0$, then $\alpha^2-5 \alpha+6=$

Consider the following statements:

I. The number of positive integral solutions of $x_1+x_2+x_3+x_4=10$ is 286 .

II. If $25!=10^n \times k,(k \in \mathbf{N})$, then $n=6$

Which one of the following options is true?

A student is allowed to select at least $(n+1)$ books but not all books from a collection of ( $2 n+1$ ) books. If the total number of ways in which he can select these books is 255 , then the number of books in that collection is

If $x$ is so small that all terms containing $x^2$ and higher powers of $x$ can be neglected, then the approximate value of $\frac{\left(1+\frac{2 x}{3}\right)^{-4}(4+5 x)^{1 / 2}}{(9+x)^{3 / 2}}$, when $x=\frac{6}{371}$, is

The sum of the coefficients of $x^{-3 / 2}$ and $x^3$ in the expansion of $\sqrt{3+x}+\sqrt{5+x}$ when $3 < x< 5$, is

If $\frac{x^5-5}{x^3+x^2}=f(x)+\frac{A}{x}+\frac{B}{x^2}+\frac{C}{x+1}$, then the larger value of $K$ for which $f(K)+A+B+C=1$, is

If $\cos x-\sin x=\sqrt{a} \sin x$, then $a \sin x+\cos x-\sin x=$

$$ \text { Match the items of List-I to the items of List-II } $$

$$ \text { The correct match is } $$

If $\cot \left(\frac{A}{2}\right)=\sqrt{\frac{1+a}{1-a}} \cdot \cot \left(\frac{\theta}{2}\right)$, then $\cos \theta=$

If $\sum\limits_{n=1}^k \tan ^{-1}\left(\frac{1}{n^2+3 n+3}\right)=\tan ^{-1} \alpha$, then $\alpha=$

The set of values of $x$ such that $\tan ^{-1}\left(\frac{x}{x-2}\right)-\tan ^{-1}\left(\frac{x}{2 x-1}\right)=\tan ^{-1}\left(\frac{2}{3}\right)$ is

If $\sin \theta \cosh \alpha=\tan x, \cos \theta \sinh \alpha=\sec x$, then $\cos 2 \theta \cosh 2 \alpha=$

If the sides of a triangle are three consecutive natural numbers and its largest angle is twice the smallest one, then the area (in sq. units) of that triangle is

In $\triangle A B C, A D$ and $B E$ are medians drawn from $A$ and $B$. If $A D=\frac{7}{2}, \angle D A B=\frac{\pi}{8}$ and $\angle A B E=\frac{\pi}{4}$, then the area (in sq. units) of $\triangle A B C$ is

If the radius of the incircle of a triangle with sides $5 k, 6 k$ and $5 k$ is 6 , then the largest angle of that triangle is

If $\mathbf{a , b , c}$ are three independent vectors and there exists a non zero scalar traid $(l, m, n)$ such that $l(3 \mathbf{a}+2 \mathbf{b}+\mathbf{c})+m(2 \mathbf{a}+2 \mathbf{b}+3 \mathbf{c})+n(\mathbf{a}+2 \mathbf{b}+5 \mathbf{c})=\mathbf{0}$, then

If $\mathbf{a}$ and $\mathbf{b}$ represent two non collinear vectors, the equation $\mathbf{r}=t \mathbf{a}+(1-t) \mathbf{b}$ represents

Let $\mathbf{a , b , c}$ be three vectors such that the magnitude of $\mathbf{b}$ is twice that of $\mathbf{a}$ and magnitude of $\mathbf{c}$ is three times that of $\mathbf{a}$. If the angle between each pair of vectors is $\frac{\pi}{3}$ and $|\mathbf{a}+\mathbf{b}+\mathbf{c}|=5$, then $|\mathbf{c}|+|\mathbf{a}|+|\mathbf{b}|=$

If $\mathbf{a , b , c}$ are three mutually perpendicular vectors such that the magnitudes of $\mathbf{b}$ and $\mathbf{c}$ are $1 / 2$ times and $\sqrt{3} / 2$ times that of $\mathbf{a}$, respectively, then the angle between the vectors $\mathbf{a}+\mathbf{b}+\mathbf{c}$ and $\mathbf{b}$ is

The locus of the point $P(\mathbf{r})$ which encloses a triangle $A B P$ of area 1 sq. unit with the fixed points $A(\hat{\mathbf{i}})$ and $B(\hat{\mathbf{j}})$ is

The shortest distance between the skew-lines $\mathbf{r}=(-\hat{\mathbf{i}}+3 \hat{\mathbf{k}})+t(2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+6 \hat{\mathbf{k}})$ and $\mathbf{r}=(3 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}})+s(2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}})$ is

In a discrete data $\frac{1 \text { th }}{4}$ of the observations are equal to $a$, another $\frac{1 \text { th }}{4}$ of the observations are equal to $-a$. Out of the remaining, half of them are equal to $b$ and the rest are equal to $-b$. If the variance of all the observations is $(a b)$, then

For the following distribution, the mean deviation about the median is

$$ \begin{array}{cccccccc} \hline x_i & 6 & 12 & 18 & 24 & 30 & 36 & 42 \\ \hline f_i & 4 & 7 & 9 & 18 & 15 & 10 & 5 \\ \hline \end{array} $$

If a man throws a die until he gets a number bigger than 3 , then the probability that he gets a 5 in his last throw is

A diagnostic test has the probability 0.95 of giving a positive result when applied to a person suffering from a certain disease and a probability 0.10 of giving a positive result when given to a non-sufferer. It is estimated that $0.5 \%$ of the population are suffering from the disease. If this test is now administered to a person from this population about whom there is no information relating to the incidence of this disease and the test gives a positive result, then the probability that he is a sufferer, is

Consider the following statements

Assertion (A) If $P_1, P_2, P_3$ are probability of happening of three independent events, then probability of happening of atleast one of them is $1-\left[\left(1-P_1\right)\left(1-P_2\right)\left(1-P_3\right)\right]$

Reason (R) For any three independent events $A, B$ and $C$

$$ \begin{array}{r} P(A \cup B \cup C)=P(A)+P(B)+P(C)-P(A) P(B)-P(A) P(C) -P(B) P(C)+P(A) P(B) P(C) \end{array} $$

The correct option among the following is

If probability function of a discrete random variable $X$ is $P(X=r)=r / k, r=1,2,3,4,5$, then $P\left(X=2\right.$ or $\left.X=\frac{k}{3}\right)$, is

If the probability that an individual will suffer a reaction from an injection of a drug is 0.001 , then the probability that out of 2000 individuals having that injection, more than 2 individuals will suffer a reaction, is

Let $A=(2,3), B=(3,-5)$ be two vertices of $\triangle A B C$ such that $C$ is a point on the line $L \equiv 3 x+4 y-5=0$. Then the locus of the centroid of $\triangle A B C$ is a line parallel to

If $a \alpha^2+b \beta^2+c \alpha \beta+d=0$ is the transformed equation of $4 x^2+\sqrt{3} x y+5 y^2-4=0$ obtained by using $\alpha=\frac{\sqrt{3}}{2} x+\frac{y}{2}$ and $\beta=-\frac{x}{2}+\frac{\sqrt{3}}{2} y$, then $c(a+b+d)=$

If the normal form of the equation of a straight line $4 x+3 y+2=0$ is $x \cos \alpha+y \sin \alpha=p$ and its intercept form is $\frac{x}{a}+\frac{y}{b}=1$, then $\frac{p \sec \alpha}{a b}=$

For an integer $K$, if the point $P\left(K^2, K+1\right)$ and the origin $O(0,0)$ lie in the same region between the lines $x+2 y-5=0$ and $3 x-y+1=0$, then the possible number of such points $P$ is

The area (in square units) of the quadrilateral formed by the point of intersection of the lines $x+y-1=0$, $x-y+1=0$, the point $(1,1)$ and the feet of the perpendiculars from this point on to the lines is

The condition that the lines joining the origin to the points of intersection of the two curves $x^2+y^2+g x+c=0, x^2+y^2+2 f y-c=0$ are at right angles, is

If $\alpha$ represent the square of the distance between the origin and the point of intersection of the lines $x^2-y^2-x+3 y-2=0$ and $\beta$ represent the product of the perpendicular distances from the origin on the pair of lines, then $\alpha \beta=$

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 $b^a r^a=$

From a point $P$ on the circle $x^2+y^2-4 x-6 y+9=0$, a pair of tangents $P Q$ and $P R$ are drawn touching the circle $x^2+y^2-4 x-6 y+12=0$ at $Q$ and $R$. If $C$ is the centre of the concentric circles, then the area of the $\triangle C Q R$ (in sq. units) is

The equations of the tangents drawn from the origin to the circle $x^2+y^2+2 g x+2 f y+g^2=0$ are

If $2 x+y=0$ is the equation of a chord of the circle $x^2+y^2-2 x-6 y+3=0$, then the circle with this chord as diameter passes through the point

If the radical axis of the circles $x^2+y^2+2 \alpha x+2 \beta y+c=0$ and $x^2+y^2+\frac{3}{2} x+4 y+c=0$ touches the circle $x^2+y^2+2 x+2 y+1=0$, then $4 \alpha \beta-8 \alpha-3 \beta+10=$

If $P Q$ is a focal chord of the parabola $y^2=4 x$ with focus $S$ and $P=(4,4)$, then $S Q=$

If the parabola $x^2=4 a y,(a>0)$ makes an intercept of length $\sqrt{40}$ units on the line $y=1+2 x$ then $4 a=$

If tangents are drawn to the ellipse $x^2+2 y^2=2$, then the locus of the mid-points of the intercepts made by those tangents between the coordinate axes is

The area (in sq. units) of the quadrilateral formed by the tangents drawn at the end points of the latus rectum to the ellipse $S \equiv \frac{x^2}{16}+\frac{y^2}{12}=1$ is

If $p, q$ are the eccentricities of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and its conjugate hyperbola respectively, then the area of the square (in sq. units) formed by the points of intersection of the ellipse $\frac{x^2}{p^2}+\frac{y^2}{q^2}=1$ and the pair of lines $x^2-y^2=0$ is

$\Pi_1, \Pi_2, \Pi_3$ are three planes which are respectively parallel to the $Y Z, Z X$ and $X Y$ planes at distances $a, b$ and $c$ forming a rectangular parallelopiped. $d_1$ is a diagonal of the face of $X Y$-plane not passing through the origin and $d_2$ is a diagonal of the plane $\Pi_2$ coterminous with $d_1$. If none of the coordinates of the vertices of the parallelopiped are negative, then the angle between $d_1$ and $d_2$ is

The obtuse angle between the lines whose direction ratios are determined by the equations $a+b+c=0$, $2 a b+2 a c-b c=0$ is

A plane meets the coordinate axes at $A, B, C$ respectively such that the centroid of the $\triangle A B C$ is $(2,3,5)$. Then, the equation of that plane is

Let $[x]$ denote the greatest integer less than or equal to $x$ and $k \geq 2$ be an integer. Then

$$ \mathop {Lt}\limits_{x \to k} \frac{\sin \left(2 \pi\left([x]-\left[\frac{x}{k}\right]\right)-x\right)+\sin k}{x-k}= $$

Define $f(x)=\left\{\begin{array}{ll}1+x, & 0 \leq x \leq 2 \\ 3-x, & 2 < x \leq 3\end{array}\right.$.

If $f \circ f(x)$ is discontinuous at $a$ and $b$ in $[0,3]$ and $a

$$ \frac{d}{d x}\left[\operatorname{cosech}^{-1}(\tan 2 x)\right]= $$

If $f(x)=\frac{1}{x^3} \int_5^x\left(2 u^2-u f^{\prime}(u) d u\right.$, then $f^{\prime}(5)=$

Let $f: R \rightarrow R$ be defined by $f\left(\frac{x+y}{2}\right)=\frac{f(x)+f(y)}{2}$ for all $x$ and $y$. If $f^{\prime}(0)$ exists and equals -1 and $f(0)=1$, then $f(2)=$

The angle $A$ of $\triangle A B C$ is found by measurement to be $67 \frac{1^{\circ}}{2}$ and the area of $\triangle A B C$ is calculated from the measurements of $b, c, A$. In measuring $A$, an error of 9 min is made then the percentage error in the area of the triangle is

Let $f: R \rightarrow R$ be a bijection. A curve represented by $y=f(x)$ is such that $f^{\prime}(x)>0 \forall x \in \mathbf{R}$. The tangent and normal drawn at $P(\alpha, 1)$ on the curve cuts the $X$-axis at $A, B$ respectively and $C$ is the foot of the perpendicular from $P$ onto the $X$-axis. If $P(\alpha, 1)$ is such a point that $A C+C B$ is minimum, then the tangent at $P$ is parallel to the line

The $x$-coordinate changes on the curve $y=3 x^5+15 x-8$ at the rate of $\frac{1}{5}$ units/sec. $A\left(x_1, y_1\right), B\left(x_2, y_2\right)$ are the points on the curve at which the $y$-coordinate changes at the rate of 6 units/sec, then the slope of $A B=$

In $\triangle A B C, \angle B=90^{\circ}$ and $(b+a)$ is always a constant. In order that $\triangle A B C$ encloses the maximum area, $\angle C=$

If $\int \frac{(x-1) d x}{(x+1) \sqrt{x^3+x^2+x}}=A \cdot \tan ^{-1} \sqrt{f(x)}+$ constant, then the ordered pair $(A, f(-1))=$

If $f\left(\frac{2 x+3}{3 x+5}\right)=x+4, x \neq \frac{-5}{3}, \frac{2}{3}$ and $\int f(x) d x=A x+B \ln |3 x-2|+C$, then $3 B-A=$

If $\int e^x\left(\frac{x^2-8 x+19}{(x-1)^5}\right) d x=\frac{e^x(l x+m)}{(x-1)^4}+C$, then $4 l+m=$

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

Assertion (A) $\int\limits_{-a}^a f(x) d x=\int_0^a(f(x)+f(-x)) d x$

Reason (R) $\int\limits_a^b f(x) d x=\int_{g(a)}^{g(b)} f(g(u)) g^{\prime}(u) d u$

The correct option among the following is

If $\cos x+\cos 2 x+\ldots+\cos n x=\frac{A(x)}{2 \sin x / 2}$, then $\int\limits_0^\pi A(x) d x=$

The area (in sq. units) bounded by the parabola $y=x^2+3$, the tangent to the parabola at $(3,12)$ and the coordinate axes and lying in the first quadrant is

The order and degree of the differential equation $\frac{d^2 y}{d x^2}+y+\left(\frac{d y}{d x}-\frac{d^3 y}{d x^3}\right)^{3 / 2}=0$, are respectively.

The general solution of the differential equation $\frac{d y}{d x}=\frac{2 x-3 y+4}{3 x+2 y-7}$ is

The general solution of $\frac{d y}{d x}=\frac{x+y+1}{y-x+1}$ is

Identify the incorrect statement.

In an experiment the angles are required to be measured using an instrument in which 29 divisions of the main scale exactly coincide with the 30 divisions of the vernier scale. If the smallest division of the main scale is half a degree $\left(=0.5^{\circ}\right)$, then the least count of the instrument is

A ball is thrown straight upward from ground with a speed of $20 \mathrm{~m} / \mathrm{s}$. The ball was caught on its way down at a point 5 m above the ground. The time taken by the ball during entire trip is (assume, $g=10 \mathrm{~m} / \mathrm{s}^2$ )

A motor-bike starts from rest, attains a velocity of 10 $\mathrm{m} / \mathrm{s}$ with an acceleration of $0.5 \mathrm{~m} / \mathrm{s}^2$, travels 10 km with this uniform velocity and then comes to halt with a uniform deceleration of $0.2 \mathrm{~m} / \mathrm{s}^2$. The total time of travel is

If $\mathbf{r}_1=2 \hat{\mathbf{x}}, \mathbf{r}_2=2 \hat{\mathbf{y}}$, where $\hat{\mathbf{x}}$ and $\hat{\mathbf{y}}$ are unit vectors along the $X$-axis and $Y$-axis respectively, then the magnitude of $\mathbf{r}_1+\mathbf{r}_2$ is

Let $\mathbf{A}_1+\mathbf{A}_2=5 \mathbf{A}_3, \mathbf{A}_1-\mathbf{A}_2=3 \mathbf{A}_3, \mathbf{A}_3=2 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}$, then $\frac{\left|\mathbf{A}_1\right|}{\left|\mathbf{A}_2\right|}$ is

The velocity of an object of mass 2 kg is given by $\mathbf{v}=\left(8 t \hat{\mathbf{i}}+3 t^2 \hat{\mathbf{j}}\right) \mathrm{m} / \mathrm{s}$, where $t$ is time in seconds. What will be the direction of net force on the object relative to the positive direction of $X$-axis, at the instant when its magnitude is 20 N ?

A box of mass $m$ is in equilibrium under the application of three forces as shown below. If the magnitude of $\mathbf{F}_1$ is 10 N , what is the magnitude of $\mathbf{F}_3$ ?

A moving body with a mass $m_1$ and velocity $u$ strikes a stationary body of mass $m_2$. The masses $m_1$ and $m_2$ should be in the ratio $\frac{m_1}{m_2}$, so as to decrease the velocity of the first body to $\frac{2 u}{3}$ and giving a velocity of $v$ to $m_2$ assuming a perfectly elastic impact. Then, the ratio $\frac{m_1}{m_2}$ is

A force acts on a 30 g particle in such a way that the position of the particle as a function of time is given by $x=\alpha t^2$, where $x$ is in metre, $t$ is in seconds and $\alpha=1 \mathrm{m} / \mathrm{s}^2$. The work done during the first 4 s is

A straight rod of length $L$ is made of a material having mass per unit length $m(x)=\lambda|x|$, where $x$ is measured from the centre of rod. The moment of inertia about an axis perpendicular to the rod and passing through one end of the rod will be $L=1 \mathrm{~m}$ and $\lambda=16 \mathrm{~kg} / \mathrm{m}^2$.

Consider a uniform horizontal solid cylinder of mass 10 kg such that its length is 9 times its radius. Let the radius be 40 cm . Calculate the moment of inertia of the cylinder about a line passing through its edge and perpendicular to its axis.

A particle is executing simple harmonic motion in one-dimension. If the amplitude of oscillations is 0.2 cm and if its velocity at the mean position is $5 \mathrm{~m} / \mathrm{s}$, then the angular frequency of the oscillation is

A mass $M$ is split into two parts $m_0$ and $M-m_0$. These two masses are then separated by a distance $D$. If the gravitational force between the parts is maximum, then the ratio $\frac{m_0}{M}$ is

Two metal wires $A$ and $B$ have length $L$ and $3 L$ respectively. The radius of cross-sectional circular area of wire $A$ and $B$ are $R$ and $2 R$, respectively. These wires are joined end to end along their axis. When one end of the combined system is fixed and other end is pulled with a constant force $F$, the elongation in both the wires is equal. If $Y_A$ and $Y_B$ are Young's modulus of wire $A$ and $B$, then the $Y_B / Y_A$ is

A hydraulic lift as shown in the figure is used to lift a mass of 1000 kg , which is placed on a piston $\left(P_1\right)$ of area $1 \mathrm{~m}^2$. If the cross-section area of the piston $\left(P_2\right)$ at the other end is $0.01 \mathrm{~m}^2$, then how much mass needs to be put on it to lift the 1000 kg ?

If $\alpha_V$ and $T$ are the coefficient of volume expansion and temperature for an ideal gas respectively, then

If $\lambda$ denotes the wavelength at which the radiative emission from a black body at a temperature $T$ is maximum, then

A Carnot engine $C_1$ operates between temperature $T_1$ and $T_2\left(T_1>T_2\right)$. A second Carnot engine $C_2$ uses all the heat rejected by the engine $C_1$ and operates between temperature $T_2$ and $T_3$ (where $T_2>T_3$ ). The efficiency of this combined ( $C_1$ and $C_2$ together) engine is

All gases deviate from gas laws at

A body is oscillating in simple harmonic motion according to the equation $x=6 \cos \left(2 \pi t+\frac{\pi}{3}\right) \mathrm{m}$. The magnitude of the acceleration (in $\mathrm{m} / \mathrm{s}^2$ ) of the body at $t=\mathrm{ls}$

A short straight object of length $l$ lies along the central axis of a spherical concave mirror, at a distance $X$ from the mirror. The focal length of the mirror is $F$. If the length of the image in the mirror is $l^{\prime}$, then ratio $\left(\frac{l^{\prime}}{l}\right)$ is (assume, $l \ll X$ and $l \ll F$ )

Consider a glass prism immersed in a liquid as shown below. The refractive index of glass and liquid is 1.5 and 1.2, respectively. A ray of light enters the prism perpendicular to the face $A B$. The largest value of angle $\theta$ is, if the ray is totally reflected at the face $A C$, then

If in a Young's double slit experiment the slit separation is doubled and the distance of the screen from the slits is reduced to half, then the fringe widths become how many times their original value?

In a uniformly charged sphere of total charge $Q$ and radius $R$, the electric field $E$ is plotted as function of distance from the centre of the sphere. The graph which would correspond to the above description will be

Electric charges $+q$ and $-q$ are placed at points $A$ and $B$ respectively which are at a distance of $2 L$ apart. If $C$ is the midpoint between $A$ and $B$, then the work done in moving a charge $+Q$ along the semi-circle $C R D$ is

Find the equivalent resistance between point $A$ and $B$ in the following circuit. (The resistance of each resistor is $R$ )

In a meter bridge the balancing length from the left end is found to be 25 cm . The value of the unknown resistance is (assume, standard resistance of $1 \Omega$ is in the right gap)

Two infinite wires carrying opposite electrical currents $I$ and $i$ are placed a distance $x$ apart. A point $P$ at a distance $y$ away from the wire carrying current $i$ is shown in the figure. If the magnetic field is zero at point $P$, then the magnitude of $i$ is

A solenoid of length 2 m carries a current of 20 A . The diameter of the solenoid is 3 cm . If the magnetic field inside the solenoid is 20 mT , then the length of wire forming the solenoid is (assume, $\mu_0=4 \pi \times 10^{-7} \mathrm{H} / \mathrm{m}$ )

Which of the following is desirable for making permanent magnets?

Consider two solenoids $X$ and $Y$ such that the area and length of $Y$ are twice that of $X$ respectively and the magnetic energy stored in both the solenoids is same, then the ratio of magnitude of magnetic fields of the two solenoids $\frac{\left|\mathbf{B}_X\right|}{\left|\mathbf{B}_Y\right|}$ is

Which one of the following curves represents the variation of impedance ( $Z$ ) with frequency $f$ in a series $L-C-R$ circuit, when connected to an AC source?

The radiation energy emitted per second by a point source is 100 W . If the efficiency of the source is $4 \%$, then the rms value of the electric field at distance of 2 m is [use $\frac{1}{4 \pi \varepsilon_0}=9 \times 10^9$ in SI unit]

A light of wavelength 310 nm is used in a photoelectric experiment. The metal electrode of work function of 2.5 eV is used in the experiment. The stopping potential for the photoelectrons will be (assume, $h c=1240 \mathrm{eV}-\mathrm{nm}$ )

If the first line in the Lyman series has wavelength $\lambda$, then the first line in Balmer series has the wavelength

The half-life of a radiocative isotope is 30 h . How long will it take to get reduced to $12.5 \%$ of its initial amount?

Which of the following circuits satisfies the logic condition $A=1, B=1$ and $D=1$ ?

In an $n-p-n$ transistor, $95 \%$ of emitted electrons reach the collector. If the base current is 2 mA , then collector current is

A message signal of frequency $50 / \pi \mathrm{kHz}$ and peak voltage of 5 V is used to modulate a carrier of frequency 1 MHz and peak voltage 20 V . The modulation index is