Which of the following represents the wavelength of spectral line of Balmer series of $\mathrm{He}^{+}$ion?

$(R=$ Rydberg constant, $n>2)$

The work functions (in eV ) of $\mathrm{Mg}, \mathrm{Cu}, \mathrm{Ag}, \mathrm{Na}$ respectively are $3.7,4.8,4.3,2.3$. From how many metals, the electrons will be ejected if their surfaces are irradiated with an electromagnetic radiation of wavelength 300 nm ?

$\left(h=6.6 \times 10^{-34} \mathrm{Js}, 1 \mathrm{eV}=1.6 \times 10^{-19} \mathrm{~J}\right)$

The order of negative electron gain enthalpy of $\mathrm{Li}, \mathrm{Na}$, $\mathrm{S}, \mathrm{Cl}$ is

The number of molecules having lone pair of electrons on central atom in the following is

$\mathrm{BF}_3, \mathrm{SF}_4, \mathrm{SiCl}_4, \mathrm{XeF}_4, \mathrm{NCl}_3, \mathrm{XeF}_6, \mathrm{PCl}_5, \mathrm{HgCl}_2, \mathrm{SnCl}_2$

Observe the following substances.

Ethanol, acetic acid, ethylamine, trimethylamine, salicylic acid. ethanal.

In the above list, the number of substances with H -bonding is

126. Consider the following

Statement I If thermal energy is stronger than intermolecular forces, the substance prefers to be in gaseous state.

Statement II At constant temperature, the density of an ideal gas is proportional to its pressure.

The correct answer is

At $27^{\circ} \mathrm{C}, 1 \mathrm{~L}$ of $\mathrm{H}_2$ with a pressure of 1 bar is mixed with 2 L of $\mathrm{O}_2$ with a pressure of 2 bar in a 10 L flask. What is the pressure exerted by gaseous mixture in bar? (Assume $\mathrm{H}_2$ and $\mathrm{O}_2$ as ideal gases)

Two acids $A$ and $B$ are titrated separately, 25 mL of $0.5 \mathrm{M} \mathrm{Na}_2 \mathrm{CO}_3$ solution requires 10 mL of $A$ and 40 mL of $B$ for complete neutralisation. The volume (in L ) of $A$ and $B$ required to produce 1 L of 1 N acid solution respectively are

If $\Delta_r H^{\ominus}$ and $\Delta_r S^{\ominus}$ are standard enthalpy change and standard entropy change respectively for a reaction, the incorrect option is

The $\mathrm{C}_p$ of $\mathrm{H}_2 \mathrm{O}(l)$ is $75.3 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}$. What is the energy (in J ) required to raise 180 g of liquid water from $10^{\circ} \mathrm{C}$ to $15^{\circ} \mathrm{C}$ ? $\left(\mathrm{H}_2 \mathrm{O}=18 \mathrm{u}\right)$

At $T(\mathrm{~K})$, consider the following gaseous reaction, which is in equilibrium.

$$ \mathrm{N}_2 \mathrm{O}_5 \rightleftharpoons 2 \mathrm{NO}_2+\frac{1}{2} \mathrm{O}_2 $$

What is the fraction of $\mathrm{N}_2 \mathrm{O}_5$ decomposed at constant volume and temperature, if the initial pressure is 300 mm Hg and pressure at equilibrium is 480 mm Hg ? (Assume all gases as ideal)

Observe the following molecules/ions $\mathrm{NH}_4^{+}, \mathrm{NH}_3, \mathrm{BF}_3, \mathrm{OH}^{-}, \mathrm{CH}_3^{+}, \mathrm{H}^{+}, \mathrm{CO}, \mathrm{C}_2 \mathrm{H}_4$.

The number of Lewis bases in the above list is

Observe the following reactions

I. $\mathrm{N}_2(g)+3 \mathrm{H}_2(g) \xrightarrow[773 \mathrm{~K}, 200 \mathrm{~atm}]{x} 2 \mathrm{NH}_3(g)$

II. $\mathrm{CO}(\mathrm{g})+\mathrm{H}_2 \mathrm{O}(\mathrm{g}) \xrightarrow[673 \mathrm{~K}]{Y} \mathrm{CO}_2(\mathrm{~g})+\mathrm{H}_2(\mathrm{~g})$

III. $\mathrm{CH}_4(g)+\mathrm{H}_2 \mathrm{O}(g) \xrightarrow[1270 \mathrm{~K}]{\mathrm{z}} \mathrm{CO}(g)+3 \mathrm{H}_2(g)$

Catalysts $X, Y, Z$ respectively are

Consider the following

Statement - I : Both $\mathrm{BeSO}_4$ and $\mathrm{MgSO}_4$ are readily soluble in water.

Statement - II : Among the nitrates of alkaline earth metals, only $\mathrm{Be}\left(\mathrm{NO}_3\right)_2$ on strong heating gives its oxide, $\mathrm{NO}_2$ and $\mathrm{O}_2$.

The correct answer is

Which of the following is not associated with water molecules ?

Identify the incorrect statement about silica.

Which one of the following statements related to photochemical smog is not correct?

In compound $(X)$, hyperconjugation is present and in $(Y)$, resonance effect is present. What are $X$ and $Y$, respectively?

An alcohol $X\left(\mathrm{C}_4 \mathrm{H}_{10} \mathrm{O}\right)$ on dehydration gave alkene $\left(\mathrm{C}_4 \mathrm{H}_8\right)$ as major product, which on bromination followed by treatment with $Y$ gave alkyne $\mathrm{C}_4 \mathrm{H}_6$. Alkyne $\mathrm{C}_4 \mathrm{H}_6$, does not react with sodium metal. What are $X$ and $Y$ ?

An element occurs in the body centred cubic structure with edge length of 288 pm . The density of the element is $7.2 \mathrm{~g} \mathrm{~cm}^{-3}$. The number of atoms present in 208 g of the element is nearly

An aqueous solution containing 0.2 g of a non volatile solute ' $A$ ' in 21.5 g of water freezes at 272.814 K . If the freezing point of water is 273.16 K , the molar mass (in $\mathrm{g} \mathrm{mol}^{-1}$ ) of solute $A$ is $\left[\mathrm{K}_f\left(\mathrm{H}_2 \mathrm{O}\right)=1.86 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}\right]$

At $T(\mathrm{~K})$, the vapour pressure of $x$ molal aqueous solution containing a non-volatile solute is $12.078 \mathrm{kPa}_{\mathrm{d}}$, The vapour pressure of pure water at $T(\mathrm{~K})$ is 12.3 kPa . What is the value of $x$ ?

Consider the following cell reaction

$$ 2 \mathrm{Fe}^{3+}(a q)+2 \mathrm{I}^{-}(a q) \rightleftharpoons 2 \mathrm{Fe}^{2+}(a q)+\mathrm{I}_2(s) $$

At 298 K , the cell emf is 0.237 V . The equilibrium constant for the reaction is $10^x$. The value of $x$ is $\left(F=96500 \mathrm{C} \mathrm{mol}^{-1} ; R=8.3 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}\right)$.

For a first order reaction, the ratio between the time taken to complete $\frac{3}{4}$ th of the reaction and time taken to complete half of the reaction is

What is the indicator used in argentometric titrations?

In a Freundlich adsorption isotherm, if the slope is unity and $k$ is 0.1 , the extent of adsorption at 2 atm is ( $\log 2=0.30$ )

$$ \text { Match the following } $$

$$ \begin{array}{llll} \hline & \text { List-I (Process) } & & \text { List-II (Metal) } \\ \hline \text { (A) } & \text { Hall-Heroult process } & \text { (I) } & \mathrm{Ti} \\ \hline \text { (B) } & \text { Mond process } & \text { (II) } & \mathrm{In} \\ \hline \text { (C) } & \text { van-Arkel process } & \text { (III) } & \mathrm{Al} \\ \hline \text { (D) } & \text { Zone refining process } & \text { (IV) } & \mathrm{Ni} \\ \hline \end{array} $$

The correct answer is

The number of $\mathrm{P}=\mathrm{O}, \mathrm{P}-\mathrm{P}$ bonds present in oxoacid of phosphorus, prepared by treating red $\mathrm{P}_4$ with alkali are respectively

Which one of the following statements is not correct?

The coordination number of chromium in $\mathrm{K}\left[\mathrm{Cr}\left(\mathrm{H}_2 \mathrm{O}\right)_2\left(\mathrm{C}_2 \mathrm{O}_4\right)_2\right]$ is

Consider the following

Statement I Nylon 6 is a condensation copolymer.

Statement II Nylon 6, 6 is a condensation polymer of adipic acid and tetra-methylene diamine.

The correct answer is

$$ \text { Match the following } $$

The list given below contains essential amino acids that are basic $(X)$ and also non essential amino acids that are neutral $(Y) . X$ and $Y$, respectively are

I. Lysine

II. Alanine

III. Serine

IV. Arginine

V. Tyrosine

The artificial sweetener $X$ contains glycosidic linkage and $Y$ contains amide, ester linkages. $X$ and $Y$ respectively are

Which one of the following halogen compounds is least reactive towards hydrolysis by $\mathrm{S}_{\mathrm{N}} 1$ mechanism?

$p$-chlorotoluene is the major product in which of the following reactions?

Arrange the following in decreasing order of electrophilicity of carbonyl carbon.

$$ \mathrm{CH}_3 \mathrm{CH}_2 \mathrm{CHO} $$

What is the ratio of $s p^3$ carbons to $s p^2$ carbons in the product ' $P$ ' of the given sequence of reactions?

$$ \text { The final product }(C) \text { in the given reaction sequence is } $$

$$ \mathrm{C}_6 \mathrm{H}_5 \mathrm{COOH} \xrightarrow{\mathrm{SOCl}_2}(A) \xrightarrow[\text { anhy } \cdot \mathrm{AlCl}_3]{\mathrm{C}_6 \mathrm{H}_6}(B) \xrightarrow[\text { (ii) } \mathrm{KOH}_4 /\left(\mathrm{CH}_2 \mathrm{OH}\right)_2]{\text { (i) } \mathrm{NH}_2-\mathrm{NH}_2}(C) $$

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

$$ \mathrm{C}_6 \mathrm{H}_5 \mathrm{~N}_2^{+} \mathrm{Cl}^{-} \xrightarrow{X} \mathrm{C}_6 \mathrm{H}_5 \mathrm{CN} \xrightarrow[\text { (ii) } \mathrm{H}_2 \mathrm{O}]{\text { (i) } \mathrm{CH}_3 \mathrm{MgBr}} Y $$

The domain of the function, $f(x)=\sqrt{\log _e\left(\frac{1}{x^2-4 x+4}\right)}+\sin ^{-1}\left(x^2-2\right)$ is

If $a$ is the determinant of the adjoint of the matrix $\left[\begin{array}{lll}1 & 1 & 2 \\ 1 & 2 & 3 \\ 2 & 3 & 3\end{array}\right]$ and $b$ is the determinant of the inverse of the matrix $\left[\begin{array}{ccc}1 & 2 & 3 \\ 4 & -3 & -1 \\ 2 & 1 & -4\end{array}\right]$, then $\frac{b+1}{18 b}=$

Consider two systems of 3 linear equations in 3 unknowns $A X=B$ and $C X=D$. If $A X=B$ has unique solution $D$ and $C X=D$ has unique solution $B$, then the solution of $\left(A-C^{-1}\right) X=0$ is

$f(x)$ is an $n$th degree polynomial satisfying $f(x)=\frac{1}{2}\left|\begin{array}{cc}f(x) & f\left(\frac{1}{x}\right)-f(x) \\ 1 & f\left(\frac{1}{x}\right)\end{array}\right|$. If $f(2)=33$, then the value of $f(3)$ is

If the point $P$ denotes the complex number $z=x+i y$ in the argand plane and $\frac{z-(2-i)}{z+(1+2 i)}$ is purely imaginary number, then the locus of $P$ is

If $(\sqrt{3}-i)^n=2^n, n \in N$, then the least possible value of $n$ is

$$ (1+\sqrt{5}+i \sqrt{10-2 \sqrt{5}})^5= $$

The number of solutions of the equation $\sqrt{3 x^2+x+5}=x-3$ is

The set of all real values of $x$ for which $\frac{x^2-1}{(x-4)(x-3)} \geq 1$ is

Two roots of the equation, $a x^4+b x^3+c x^2+d x+e=0$ are positive and equal. If the product of the other two real roots is 1 , then

The number of integers between 10 and 10,000 such that in every integer every digit is greater than its immediate preceeding digit, is

The number of ways in which a cricket team of 11 members can be formed out of 6 batsmen, 6 bowlers, 4 all-rounders and 4 wicket keepers by selecting atleast 4 batsmen, atleast 3 bowlers, atleast 2 all-rounders and only one wicket keeper is

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

Sum of the coefficients of $x^4$ and $x^6$ in the expansion of $\left(1+x-x^2\right)^6$ is

19. If $\frac{3 x^3-7 x+1}{(x-2)^5}=\frac{A}{x-2}+\frac{B}{(x-2)^2}+\frac{C}{(x-2)^3}+\frac{D}{(x-2)^4}+\frac{E}{(x-2)^5}, \text { then } A(B+C+D+E)= $

$$ \tan \frac{2 \pi}{7} \cdot \tan \frac{4 \pi}{7}+\tan \frac{4 \pi}{7} \cdot \tan \frac{\pi}{7}+\tan \frac{\pi}{7} \cdot \tan \frac{2 \pi}{7}= $$

$$ \cos 13^{\circ} \sin 17^{\circ} \sin 21^{\circ} \cos 47^{\circ}= $$

If $\cot \left(\cos ^{-1} x\right)=\sec \left\{\tan ^{-1}\left(\frac{a}{\sqrt{b^2-a^2}}\right)\right\}, b>a$ then $x=$

If $\sinh ^{-1} x=\log 3$ and $\cosh ^{-1} y=\log \frac{3}{2}$, then $\tanh ^{-1}(x-y)=$

In a $\triangle A B C$, if $a, b, c$ are in arithmetic progression and the angle $A$ is twice the angle $C$, then $\cos A: \cos B: \cos C=$

In a $\triangle A B C, A, B$ and $C$ are in arithmetic progression, $r r_3=r_1 r_2$ and $c=10$, then $a^2+b^2+c^2=$

In a $\triangle A B C, \frac{2\left(r_1+r_3\right)}{a c(1+\cos B)}=$

In a right angled triangle, if the position vector of the vertex having the right angle is $-3 \hat{\mathbf{i}}+5 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ and the position vector of the mid-point of its hypotenuse is $6 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}$, then the position vector of its centroid is

If the position vectors of the vertices $A, B, C$ of a triangle are $3 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}-\hat{\mathbf{k}}, \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+\hat{\mathbf{k}}, 5(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}})$ respectively, then the magnitude of the altitude drawn from $A$ on to the side $B C$ is

If the vectors $2 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}-3 \hat{\mathbf{k}},-\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ and $p \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{\mathbf{k}}$ are coplanar, then the unit vector in the direction of the vector $9 p \hat{\mathbf{i}}-4 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}$ is

Assertion (A) For the lines $\mathbf{r}=\mathbf{a}+t \mathbf{b}$ and $\mathbf{r}=\mathbf{p}+s \mathbf{q}$, if $(\mathbf{a}-\mathbf{p}) \cdot(\mathbf{b} \times \mathbf{q}) \neq 0$, then the two lines are coplanar.

Reason $(\mathrm{R})|(\mathbf{a}-\mathbf{p}) \cdot(\mathbf{b} \times \mathbf{q})|$ is $|\mathbf{b} \times \mathbf{q}|$ times the shortest distance between the lines $\mathbf{r}=\mathbf{a}+t \mathbf{b}$ and $\mathbf{r}=\mathbf{p}+s \mathbf{q}$.

Let $\mathbf{a}=4 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}$ and $\mathbf{b}$ be two perpendicular vectors in the $X O Y$-plane. A vector $\mathbf{c}$ in the same plane and having projections 1 and 2 respectively on $\mathbf{a}$ and $\mathbf{b}$ is

The mean deviation about the mean for the following data is

$$ \begin{array}{llllll} \hline \text { Class Interval } & 0-2 & 2-4 & 4-6 & 6-8 & 8-10 \\ \hline \text { Frequency } & 1 & 3 & 4 & 1 & 2 \\ \hline \end{array} $$

A basket contains 5 apples and 7 oranges and another basket contains 4 apples and 8 oranges. If one fruit is picked out at random from each basket, then the probability of getting one apple and one orange is

Two cards are drawn from a pack of 52 playing cards one after the other without replacement. If the first card drawn is a queen, then the probability of getting a face card from a black suit in the second draw is

An item is tested on a device for its defectiveness. The probability that such an item is defective is 0.3 . The device gives accurate result in 8 out of 10 such tests.

If the device reports that an item tested is not defective, then the probability that it is actually defective is

In a school there are 3 sections $A, B$ and $C$. Section $A$ contains 20 girls and 30 boys, section $B$ contains 40 girls and 20 boys and section $C$ contains 10 girls and 30 boys. The probabilities of selecting the section $A, B$ and $C$ are $0.2,0.3$ and 0.5 respectively. If a student selected at random from the school is a girl, then the probability that she belongs to section $A$ is

If the probability distribution of a random variable $X$ is as follows, then the mean of $X$ is

$$ \begin{array}{ccccc} \hline \boldsymbol{X}=\boldsymbol{x}_{\boldsymbol{i}} & -1 & 0 & 1 & 2 \\ \hline \boldsymbol{P}\left(\boldsymbol{X}=\boldsymbol{x}_{\boldsymbol{i}}\right) & \boldsymbol{k}^3 & 2 \boldsymbol{k}^3+\boldsymbol{k} & 4 \boldsymbol{k}-10 \boldsymbol{k}^2 & 4 \boldsymbol{k}-1 \\ \hline \end{array} $$

If $X$ is a binomial variate with mean $\frac{16}{5}$ and variance $\frac{48}{25}$, then $P(X \leq 2)=$

$A(a, 0)$ is a fixed point and $\theta$ is a parameter such that $0<\theta<2 \pi$. If $P(a \cos \theta, a \sin \theta)$ is a point on the circle $x^2+y^2=a^2$ and $Q(b \sin \theta,-b \cos \theta)$ is a point on the circle $x^2+y^2=b^2$, then the locus of the centroid of the $\triangle A P Q$ is

The point $P(4,1)$ undergoes the following transformations in succession :

(i) origin is shifted to the point $(1,6)$ by translation of axes.

(ii) translation through a distance of 2 units along the positive direction of $X$-axis.

(iii) rotation of axes through an angle of $90^{\circ}$ in the positive direction.

Then, the coordinates of the point $P$ in its final position are

$L_1 \equiv a x-3 y+5=0$ and $L_2 \equiv 4 x-6 y+8=0$ are two parallel lines. If $p, q$ are the intercepts made by $L_1=0$ and $m, n$ are the intercepts made by $L_2=0$ on the $X$, $Y$-coordinate axes respectively, then the equation of the line passing through the points $(p, q)$ and $(m, n)$ is

If $(h, k)$ is the image of the point $(2,-3)$ with respect to the line $5 x-3 y=2$, then $h+k=$

If the pair of lines $a x^2-7 x y-3 y^2=0$ and $2 x^2+x y-6 y^2=0$ have exactly one line in common and ' $a$ ' is an integer, then the equation of the pair of bisectors of the angles between the lines $a x^2-7 x y-3 y^2=0$ is

If the angle between the pair of lines $2 x^2+2 h x y+2 y^2-x+y-1=0$ is $\tan ^{-1}\left(\frac{3}{4}\right)$ and $h$ is a positive rational number, then the point of intersection of these two lines is

If the equation of the circle passing through the point $(8,8)$ and having the lines $x+2 y-2=0$ and $2 x+3 y-1=0$ as its diameters is $x^2+y^2+p x+q y+r=0$, then $p^2+q^2+r=$

If $2 x-3 y+1=0$ is the equation of the polar of a point $P\left(x_1, y_1\right)$ with respect to the circle $x^2+y^2-2 x+4 y+3=0$, then $3 x_1-y_1=$

If a unit circle $S \equiv x^2+y^2+2 g x+2 f y+c=0$ touches the circle $S^{\prime} \equiv x^2+y^2-6 x+6 y+2=0$ externally at the point $(-1,-3)$, then $g+f+c=$

$3 x+4 y-43=0$ is a tangent to the circle $S \equiv x^2+y^2-6 x+8 y+k=0$ at a point $P$. If $C$ is the centre of the circle and $Q$ is a point which divides $C P$ in the ratio $-1: 2$, then the power of the point $Q$ with respect to the circle $S=0$ is

If the radical axis of the circles $x^2+y^2+2 g x+2 f y+c=0$ and $2 x^2+2 y^2+3 x+8 y+2 c=0$ touches the circle $x^2+y^2+2 x+2 y+1=0$, then

Tangents are drawn at three points $P\left(t_1\right), Q\left(t_2\right), R\left(t_3\right)$ on the parabola $y^2=x$. Let these tangents intersect each other at the points $L, M, N$. If $t_1=2, t_2=-4, t_3=6$, then the area of the $\triangle L M N$ is

The area (in sq. units) of the triangle formed by the tangent and normal to the ellipse $9 x^2+4 y^2=72$ at the point $(2,3)$ with the $X$-axis is

If $3 \sqrt{2} x-4 y=12$ is a tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and $\frac{5}{4}$ is its eccentricity, then $a^2-b^2=$

If the normal drawn to the hyperbola $x y=16$ at $(8,2)$ meets the hyperbola again at a point $(\alpha, \beta)$, then $|\beta|+\frac{1}{|\alpha|}=$

The locus of a point at which the line joining the points $(-3,1,2),(1,-2,4)$ subtends a right angle, is

If $A(1,2,3), B(2,3,-1), C(3,-1,-2)$ are the vertices of a $\triangle A B C$, then the direction ratios of the bisector of $\angle A B C$ are

Let $A=(2,0,-1), B=(1,-2,0), C=(1,2,-1)$ and $D=(0,-1,-2)$ be four points.

If $\theta$ is the acute angle between the plane determined by $A, B, C$ and the plane determined by $A, C, D$, then $\tan \theta=$

$[x]$ represents the greatest integer function. If $\mathop {\lim }\limits_{x \to 0 + } \frac{\cos [x]-\cos (k x-[x])}{x^2}=5$, then $k=$

$$ \mathop {\lim }\limits_{x \to 0} \frac{x \tan 2 x-2 x \tan x}{(1-\cos 2 x)^2}= $$

If $f(x)=\left\{\begin{array}{cl}\frac{\left(e^{a x}-1\right) \log (1+x)}{\sin ^2 x}, & \text { if } x>0 \\ 2, & \text { if } x=0 \\ \frac{\cos 4 x-\cos b x}{\tan ^2 x}, & \text { if } x<0\end{array}\right.$ is continuous at $x=0$, then $\sqrt{b^2-a^2}=$

If $y=\tan ^{-1}\left(\frac{3 x-x^3}{1-3 x^2}\right)+\tan ^{-1}\left(\frac{7 x}{1-12 x^2}\right)$, then at $x=0, \frac{d y}{d x}=$

If $y=\sqrt{\frac{x^4 \sqrt{3 x-5}}{\left(x^2-3\right)(2 x-3)}}$, then $\left(\frac{d y}{d x}\right)_{x=2}=$

If $x^2+y^2+\sin y=4$, then the value of $\frac{d^2 y}{d x^2}$ at $x=-2$ is

If the surface area of a spherical bubble is increasing at the rate of $4 \mathrm{sq} . \mathrm{cm} / \mathrm{sec}$, then the rate of change in its volume (in cubic $\mathrm{cm} / \mathrm{sec}$ ) when its radius is 8 cms is

The number of turning points of the curve $f(x)=2 \cos x-\sin 2 x$ in the interval $[-\pi, \pi]$ is

The radius and the height of a right circular solid cone are measured as 7 feet each. If there is an error of 0.002 ft for every feet in measuring them, then the error in the total surface area of the cone (in sq. ft ) is

$$ \int(\sqrt{\tan x}+\sqrt{\cot x}) d x= $$

$\int \frac{\sqrt{x-2}}{2 x+4} d x=$

If $\int x^{49}\left[\tan ^{-1} x^{50}+\frac{x^{50}}{1+x^{100}}\right] d x=\frac{x^n}{k} f(x)+c$, then

$$ f(x)-f\left(\sqrt[k]{x^n}\right)= $$

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

For $0 < x < 1, \int\left[\tan ^{-1}\left(1-x+x^2\right)+\tan ^{-1}(1-x)\right] d x=$

$$ \int_{-2 \pi}^{2 \pi} \sin ^4(2 x) \cos ^6(2 x) d x= $$

If $f(t)=\int_0^t \tan ^{(2 n-1)} x d x, n \in N$, then $f(t+\pi)=$

$$ \int_0^2 x^8\left(\frac{4}{x^2}-1\right)^{\frac{5}{2}} d x= $$

The area (in sq. units) of the region bounded by the curves $y=x^2$ and $y=8-x^2$ is

The solution of the differential equation $x^2(y+1) \frac{d y}{d x}+y^2(x+1)^2=0$, when $y(1)=2$, is

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

If $x \log x \frac{d y}{d x}+y=\log x^2$ and $y(e)=0$, then $y\left(e^2\right)=$

If the error in the measurement of the surface area of a sphere is $1.2 \%$, then the error in the determination of the volume of the sphere is

A body starts from rest with uniform acceleration and its velocity at a time of ' $n$ ' seconds is ' $v$ '. The total displacement of the body in the $n$th and $(n-1)$ th seconds of its motion is

If the range of a body projected with a velocity of $60 \mathrm{~ms}^{-1}$ is $180 \sqrt{3} \mathrm{~m}$, then the angle of projection of the body is

(Acceleration due to gravity $=10 \mathrm{~ms}^{-2}$ )

If the height of a projectile at a time of 2 s from the beginning of motion is 60 m , then the time of flight of the projectile is

(Acceleration due to gravity $=10 \mathrm{~ms}^{-2}$ )

A disc of mass 0.2 kg is kept floating in air without falling by vertically firing bullets each of mass 0.05 kg on the disc at the rate of 10 bullets per every second. If the bullets rebound with the same speed, then the speed of each bullet is

(Acceleration due to gravity $=10 \mathrm{~ms}^{-2}$ )

Two bodies $A$ and $B$ of masses 1.5 kg and 3 kg are moving with velocities $20 \mathrm{~ms}^{-1}$ and $15 \mathrm{~ms}^{-1}$ respectively. If the same retarding force is applied on the two bodies, then the ratio of the distances travelled by the bodies $A$ and $B$ before they come to rest is

If a force $\mathbf{F}=(3 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}) \mathrm{N}$ acting on a body displaces it from point $(1 \mathrm{~m}, 2 \mathrm{~m})$ to point $(2 \mathrm{~m}, 0 \mathrm{~m})$, then work done by the force is

A body moving along a straight line collides another body of same mass moving in the same direction with half of the velocity of the first body. If the coefficient of restitution between the two bodies is 0.5 , then the ratio of the velocities of the two bodies after collision is (Treat the collision as one dimensional)

If a solid sphere is rolling without slipping on a horizontal plane, then the ratio of its rotational and total kinetic energies is

As shown in the figure, two thin coplanar circular discs $A$ and $B$ each of mass $M^{\prime}$ and radius ' $r$ ' are attached to form a rigid body. The moment of inertia of this system about an axis perpendicular to the plane of disc $B$ and passing through its centre is

The time period of a simple pendulum on the surface of the Earth is $T$. If the pendulum is taken to a height equal to half of the radius of the Earth, then its time period is

A particle is executing simple harmonic motion starting from its mean position. If the time period of the particle is 1.5 s , then the minimum time at which the ratio of the kinetic and total energies of the particle becomes 3:4 is

If the escape velocity of a body from the surface of the Earth is $11.2 \mathrm{~km} \mathrm{~s}^{-1}$, then the orbital velocity of a satellite in an orbit which is at a height equal to the radius of the Earth is

In a water tank, an air bubble rises from the bottom to the top surface of the water. If the depth of the water in the tank is 7.28 m and atmospheric pressure is 10 m of water, then the ratio of the radii of the bubble at the bottom of the tank and at the top surface of the water is

(Temperature of the water in the tank is constant)

A wire of length 0.5 m and area of cross-section $4 \times 10^{-6} \mathrm{~m}^2$ at a temperature of $100^{\circ} \mathrm{C}$ is suspended vertically by fixing its upper end to the ceiling. The wire is then cooled to $0^{\circ} \mathrm{C}$, but is prevented from contracting by attaching a mass at the lower end. If the mass of the wire is negligible, then the value of the mass attached to the wire is

[Young's modulus of material of the wire $=10^{11} \mathrm{Nm}^{-2}$, coefficient of linear expansion of the material of the wire $=10^{-5} \mathrm{~K}^{-1}$ and acceleration due to gravity $=10 \mathrm{~ms}^{-2}$ ]

A wire is stretched 1 mm by a force $F$. If a second wire of same material, same length and 4 times the diameter of the first wire is stretched by the same force $F$, then the elongation of the second wire is

When 80 J of heat is supplied to a gas at constant pressure, if the work done by the gas is 20 J , then the ratio of the specific heat capacities of the gas is

A refrigerator of coefficient of performance 5 that extracts heat from the cooling compartment at the rate of 250 J per cycle is placed in a room. The heat released per cycle to the room by the refrigerator is

In a container of volume $16.62 \mathrm{~m}^3$ at $0{ }^{\circ} \mathrm{C}$ temperature, 2 moles of oxygen 5 moles of nitrogen and 3 moles of hydrogen are present, then the pressure in the container is

(Universal gas constant $=8.31 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}$ )

If a travelling wave is given by $y(x, t)=0.5 \sin (70.1 x-10 \pi t)$, where $x$ and $y$ are in metre the time $t$ is in second, then the frequency of the wave is

The ratio of the focal lengths of a convex lens when kept in air and when it is immersed in a liquid is $1: 2$. If the refractive index of the material of the lens is 1.5 , then the refractive index of the liquid is

The path difference between two waves given by the equations

$y_1=a_1 \sin \left(\omega t-\frac{2 \pi x}{\lambda}\right)$ and $y_2=a_2 \sin \left(\omega t-\frac{2 \pi x}{\lambda}+\phi\right)$ is

The sum of two point positive charges separated by a distance of 1.5 m in air is $25 \mu \mathrm{C}$. If the electrostatic force between the two charges is 0.6 N , then the difference between the two charges is

The energy stored in a capacitor of capacitance $10 \mu \mathrm{~F}$ when charged to a potential of 6 kV is

A parallel plate capacitor has plates of area $0.4 \pi \mathrm{~m}^2$ and spacing of 0.5 mm . If a slab of thickness 0.5 mm and dielectric constant 4.5 is introduced in between the plates of the capacitor, then the capacitance of the capacitor is

In the given circuit, the potential difference across the plates of the capacitor $C$ in steady state is

The potential difference across a conducting wire of length 20 cm is 30 V . If the electron mobility is $2 \times 10^{-6} \mathrm{~m}^2 \mathrm{~V}^{-1} \mathrm{~s}^{-1}$, then the drift velocity of the electrons is

A maximum current of 0.5 mA can pass through a galvanometer of resistance $15 \Omega$. The resistance to be connected in series to the galvanometer to convert it into a voltmeter of range $0-10 \mathrm{~V}$ is

Two charged particles of specific charges in the ratio 2:1 and masses in the ratio $1: 4$ moving with same kinetic energy enter a uniform magnetic field at right angles to the direction of the field. The ratio of the radii of the circular paths in which the particles move under the influence of the magnetic field is

A sample of paramagnetic salt contains $2 \times 10^{24}$ atomic dipoles each of dipole moment $15 \times 10^{-23} \mathrm{JT}^{-1}$. The sample is placed under homogeneous magnetic field of 0.6 T and cooled to a temperature 4.2 K . The degree of magnetic saturation achived is $20 \%$. Then total dipole moment of the sample for a magnetic field of 0.9 T and a temperature of 2.8 K is

A coil of resistance $200 \Omega$ is placed in a magnetic field. If the magnetic flux $\phi$ (in weber) linked with the coil varies with time ' $t$ ' (in second) as per the equation $\phi=50 t^2+4$, then the current induced in the coil at a time $t=2 \mathrm{~s}$ is

If the voltage and current in an AC circuit are respectively $50 \sin (50 t) \mathrm{V}$ and $50 \sin \left(50 t+\frac{\pi}{4}\right) \mathrm{mA}$, then the power dissipated in the circuit is nearly

The oscillating electric and magnetic field vectors of an electromagnetic wave are along

A laser produces a beam of light of frequency $5 \times 10^{14}$ Hz with an output power of 33 mW . The average number of photons emitted by the laser per second is (Planck's constant $=6.6 \times 10^{-34} \mathrm{Js}$ )

The ratio of energies of photons produced due to transition of an electron in hydrogen atom from second energy level to first energy level and fifth energy level to second energy level is

The half life of a radioactive substance is 10 minutes. If $n_1$ and $n_2$ are the number of atoms decayed in 20 and 30 minutes respectively, then $n_1: n_2=$

If $X, Y$ and $Z$ are the sizes of the emitter, base and collector of a transistor respectively, then

The logic gate equivalent to the circuit given in the figure is

If the ratio of the maximum and minimum amplitudes of an amplitude modulated wave is $7: 3$, then the modulation index is