The wavelength of a particular electron transition for $\mathrm{He}^{+}$is 100 nm . The wavelength (in $\AA$ ) of H atom for the same transition is

The energy of second Bohr orbit of hydrogen atom is -3.4 eV . The energy of the fourth Bohr orbit of the $\mathrm{He}^{+}$ ion will be

$$ \text { Observe the following data : } $$

$$ \begin{array}{lllll} \hline \text { Ion } & Q^{a+} & X^{b+} & Y^{c+} & Z^{\alpha+} \\ \hline \text { Radius (pm) } & 53 & 66 & 40 & 100 \\ \hline \end{array} $$

$Q^{a+}, X^{b+}, Y^{c+}, Z^{d+}$ are respectively

$$ \text { Which of the following sets are correctly matched? } $$

$$ \begin{array}{llll} \hline & \text { Molecule } & \text { Hybridisation } & \text { Geometry } \\ \hline \text { I. } & \mathrm{BrF}_5 & s p^3 d^2 & \text { Square pyramidal } \\ \hline \text { II. } & \mathrm{XeF}_6 & s p^3 d^3 & \text { Distorted octahedral } \\ \hline \text { III. } & \mathrm{SF}_4 & d s p^2 & \text { Square planar } \\ \hline \text { IV. } & \mathrm{PbCl}_2 & s p & \text { Linear } \\ \hline \end{array} $$

The order of dipole moments of $\mathrm{H}_2 \mathrm{O}(A), \mathrm{CHCl}_3(B)$ and $\mathrm{NH}_3(C)$ is

Identify the correct graph for an ideal gas $(y$-axis $=$ compressibility factor, $Z: x$-axis $=$ pressure, $p)$

Identify the correct statements from the following:

I. Glass is an extremely viscous liquid.

II. Increase in temperature decreases the surface tension of liquids.

III. Compressibility factor for an ideal gas is zero.

Identify the correct statements about the following stoichiometric equation.

$$ a \mathrm{P}_4+b^{-} \mathrm{OH}+c \mathrm{H}_2 \mathrm{O} \longrightarrow d \mathrm{PH}_3+e \mathrm{H}_2 \mathrm{PO}_2^{-} $$

I. $a+b+c=5$

II. $b+c-e=3$

III. The oxidation state of P in $\mathrm{H}_2 \mathrm{PO}_2^{-}$is +l .

5 moles of a gas is allowed to pass through a series of changes as shown in the graph, in a cyclic process. The processes $C \rightarrow A, B \rightarrow C$ and $A \rightarrow B$ respectively are

1 mole of an ideal gas is allowed to expand isothermally and reversibly from $\mathrm{1L}$ to 5 L at 300 K . The change in enthalpy (in kJ ) is $\left(R=8.3 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}\right)$

Consider the following equilibrium reaction in gaseous state at $T(\mathrm{~K})$.

$$ A+2 B \rightleftharpoons 2 C+D $$

The initial concentration of $B$ is 1.5 times that of $A$. At equilibrium, the concentrations of $A$ and $B$ are equal. The equilibrium constant for the reaction is

At $T(\mathrm{~K}) K_{\mathrm{sp}}$ of two ionic salts $M X_2$ and $M X$ is $5 \times 10^{-13}$ and $1.6 \times 10^{-11}$ respectively. The ratio of molar solubility of $M X_2$ and $M X$ is

Consider the following.

Statement $\mathrm{IH}_2 \mathrm{O}_2$ acts as an oxidising as well as reducing agent in both acidic and basic mediim.

Statement $\mathrm{II} 10 \mathrm{~V} \mathrm{H}_2 \mathrm{O}_2$ sample means it contains $6 \% \left(\frac{w}{v}\right) \mathrm{H}_2 \mathrm{O}_2$.

The correct answer is

Identify the correct statements from the following.

I. All alkaline earth metals give hydrides on heating with hydrogen.

II. Calcium hydroxide is used to purify sugar.

Select the correct statements from the following

(A) Aluminium liberates $\mathrm{H}_2$ gas with dil. HCl but not with aqueous NaOH .

(B) Formula of sodium metaborate is $\mathrm{Na}_3 \mathrm{BO}_3$.

(C) Boric acid is a weak monobasic acid.

(D) For thallium, +1 state is more stable than +3 state.

The number of amphoteric oxides from the following is $\mathrm{CO}_2, \mathrm{GeO}_2, \mathrm{SnO}_2, \mathrm{PbO}_2$, $\mathrm{CO}, \mathrm{GeO}, \mathrm{SnO}, \mathrm{PbO}$

Which of the following statements is not correct?

Consider the sets I, II and III. Identify the set(s) which is (are) correctly matched.

I. Staggered ethane > eclipsed ethane ..........torsional strain.

II. 2, 2-dimethylbutane > 2-methylpentane ......... boiling point

III. cis-but-2-ene > trans-but-2-ene ......... dipole moment

What are $B$ and $C$ respectively in the following set of reactions?

The crystal system with edge lengths $a \neq b \neq c$ and axial angles $\alpha=\beta=\gamma=90^{\circ}$ is ' $x$ ' and number of Bravais lattices for it is ' $y$ '. $x$ and $y$ are

A solution is prepared by adding 124 g of ethylene glycol (molar mass $=62 \mathrm{~g} \mathrm{~mol}^{-1}$ ) to $x \mathrm{~g}$ of water to get 10 m solution. What is the value of $x$ (in g )?

The following graph is obtained for an ideal solution containing a non-volatile solute $x$-and $y$-axis represent, respectively

Observe the following statements about dry cell

I. It is a primary battery.

II. Zinc vessel acts as cathode.

III. A paste of moist $\mathrm{NH}_4 \mathrm{Cl}, \mathrm{MnO}_2$ and $\mathrm{ZnCl}_2$, is present between two electrodes

IV. The potential of this cell is 1.5 V .

The correct statements are

For a reaction, the graph of $\ln k$ (on $y$-axis) and $\frac{1}{T}$ (on $x$-axis) is a straight line with a slope $-2 \times 10^4 \mathrm{~K}$. The activation energy of the reaction (in $\mathrm{kJ} \mathrm{mol}^{-1}$ ) is ( $R=8.3 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}$ )

Match the following

The critical micelle concentration (CMC) of a soap solution is $5 \times 10^{-4} \mathrm{~mol} \mathrm{~L}^{-1}$. Identify the correct statements about this solution.

I. The micelle is stable if the soap solution concentration is $10^{-7} \mathrm{~mol} \mathrm{~L}^{-1}$.

II. The micelle is stable if the soap solution concentration is higher than $5 \times 10^{-4} \mathrm{~mol} \mathrm{~L}^{-1}$.

III. Micelles are also known as associated colloids.

The metal purified by Mond process is $X$. The number of unpaired electrons in $X$ is

Complete hydrolysis of xenon hexafluoride gives HF along with compound $X$. The hybridisation in $X$ is

$\mathrm{KMnO}_4$ oxidises hydrogen sulphide in acidic medium, The number of moles of $\mathrm{KMnO}_4$ which react with one mole of hydrogen sulphide is

Identify the set which does not have ambidentate ligand(s)

The number of linear and crosslinked polymers in the following respectively are

Novolac, Nylon 6,6, Bakelite, PVC, melamine

Which of the following represents the correct structure of $\beta-D-(-)$ - fructofuranose?

Which of the following statement is not correct for glucose?

The synthetic detergent used in tooth paste is of type $X$. Animal starch is $Y . X$ and $Y$ respectively are

What are $X$ and $Y$ respectively in the following sets of reactions?

Identify the two reactions $A(\mathrm{I} \rightarrow \mathrm{II})$ and $B(\mathrm{I} \rightarrow \mathrm{III})$ respectively in the following set of reactions.

An alcohol, $X\left(\mathrm{C}_5 \mathrm{H}_{12} \mathrm{O}\right)$ in the presence of $\mathrm{Cu} / 573 \mathrm{~K}$ gives $Y\left(\mathrm{C}_5 \mathrm{H}_{10}\right)$. The reactants required for the preparation of $X$ are

A carbonyl compound $X\left(\mathrm{C}_8 \mathrm{H}_8 \mathrm{O}\right)$ undergoes disproportionation with conc. KOH on heating. Product of $X$ with $\mathrm{Zn}-\mathrm{Hg} / \mathrm{HCl}$ is $Y$ and product of $X$ with $\mathrm{NaBH}_4$ is Z . What are $Y$ and Z respectively?

What is the major product $Y$ in the following reaction sequence?

What are $X$ and $Y$ respectively in the following set of reactions?

If $f: R \rightarrow A$, defined by $f(x)=\cos x+\sqrt{3} \sin x-1$ is an onto function then $A=$

Let $g(x)=1+x-[x]$ and ${ }^{\prime}$

$$ f(x)= \begin{cases}-1, & x<0 \\ 0, & x=0,[x] \text { denotes the greatest integer less } \\ 1, & x>0\end{cases} $$

than or equal to $x$. Then for all $x, f(g(x))=$

The remainder obtained when $(2 m+1)^{2 n}(m, n \in N)$ is divided by 8 is

A value of $\theta$ lying between 0 and $\pi / 2$ and satisfying $\left|\begin{array}{ccc}1+\sin ^2 \theta & \cos ^2 \theta & 4 \sin 4 \theta \\ \sin ^2 \theta & 1+\cos ^2 \theta & 4 \sin 4 \theta \\ \sin ^2 \theta & \cos ^2 \theta & 1+4 \sin 4 \theta\end{array}\right|=0$ is

If the system of equations $2 x+p y+6 z=8$, $x+2 y+q z=5$ and $x+y+3 z=4$ has infinitely many solutions, then $p=$

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

If $z$ and $w$ are two non-zero complex numbers such that $|z w|=1$ and $\arg z-\arg w=\frac{\pi}{2}$, then $\bar{z} w=$

Let $z$ satisfy $|z|=1, z=1-\bar{z}$ and $\operatorname{Im}(z)>0$

Statement $\mathbf{I} z$ is a real number

Statement II Principal argument of $z$ is $\frac{\pi}{3}$.

Then,

If $w_1$ and $w_2$ are two non-zero complex numbers and ${ }a, b$ are non-zero real numbers such that $\left|a w_1+b w_2\right|=\left|a w_1-b w_2\right|$, then $\frac{w_1}{w_2}$ is

If $\alpha$ is the common root of the quadratic equations $x^2-5 x+4 a=0, x^2-2 a x-8=0$, where $a \in R$, then the value $\alpha^4-\alpha^3+68$ is

If $\alpha, \beta$ are the roots of $x^2-5 \gamma x-6 \delta=0$ and $\gamma, \delta$ are the roots of $x^2-5 \alpha x-6 \beta=0$, then $\alpha+\beta+\gamma+\delta=$

The equation $x^{\frac{3}{4}\left(\log _2 x\right)^2+\log _2 x-\frac{5}{4}}=\sqrt{2}$ has

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3+p x^2+q x+r=0$, then $(\alpha+\beta)(\beta+\gamma)(\gamma+\alpha)=$

An eight digit number divisible by 9 is to be formed using digits from 0 to 9 without repeating the digits. The number of ways in which this can be done is

$$ \sum_{r=1}^{15} r^2\left(\frac{{ }^{15} C_r}{{ }^{15} C_{r-1}}\right)= $$

A string of letters is to be formed by using 4 letters from all the letters of the word "MATHEMATICS". The number of ways this can be done such that two letters are of same kind and the other two are of different kind is

$$ \frac{1}{81^n}-{ }^{2 n} C_1 \frac{10}{81^n}+{ }^{2 n} C_2 \frac{10^2}{81^n}-\ldots+\frac{10^{2 n}}{81^n}= $$

If $x$ is positive real number and the first negative term in the expansion of $(1+x)^{\frac{27}{5}}$ is $t_k$, then $k=$

If $\frac{x^2}{\left(x^2+2\right)\left(x^4-1\right)}=\frac{A}{x^2-1}+\frac{B}{x^2+1}+\frac{C}{x^2+2}$, then $A+B-C=$

If $\cos x+\sin x=\frac{1}{2}$ and $0

If $\sin \theta+2 \cos \theta=1$ and $\theta$ belongs to 4 th quadrant (not lying on the coordinate axes), then $7 \cos \theta+6 \sin \theta=$

If $A$ and $B$ are acute angles satisfying $3 \cos ^2 A+2 \cos ^2 B=4$ and $\frac{3 \sin A}{\sin B}=\frac{2 \cos B}{\cos A}$, then $A+2 B=$

Statement I In the interval $[0,2 \pi]$, the number of common solutions of the equations $2 \sin ^2 \theta-\cos 2 \theta=0$ and $2 \cos ^2 \theta-3 \sin \theta=0$ is two.

Statement II The number of solutions of $2 \cos ^2 \theta-3 \sin \theta=0$ in $[0, \pi]$ is two.

The equation $\cos ^{-1}(1-x)-2 \cos ^{-1} x=\frac{\pi}{2}$ has

If $\sinh ^{-1}(2)+\sinh ^{-1}(3)=\alpha$, then $\sinh \alpha=$

In $\triangle A B C$, if $A, B, C$ are in arithmetic progression, then

$$ \sqrt{a^2-a c+c^2} \cdot \cos \left(\frac{A-C}{2}\right)= $$

If in $\triangle A B C, B=45^{\circ}, a=2(\sqrt{3}+1)$ and area of $\triangle A B C$ is $6+2 \sqrt{3}$ sq. units, then the side $b=$

In a $\triangle A B C$, if $\sin ^2 B=\sin A$ and $2 \cos ^2 A=3 \cos ^2 B$, then the triangle is

$P$ is the circumcentre of $\triangle A B C$. If the position vectors of $A, B, C$ and $P$ are $\mathbf{a}, \mathbf{b}, \mathbf{c}, \frac{\mathbf{a}+\mathbf{b}+\mathbf{c}}{4}$ respectively, then the position vector of the orthocentre of this triangle is

If the position vectors of $A, B, C, D$ are $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}, 2 \hat{\mathbf{i}}-\hat{\mathbf{j}}, \hat{\mathbf{i}}+\hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ and $4 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}$ respectively, then the quadrilateral $A B C D$ is a

The set of all real values of $c$ so that the angle between the vectors $\mathbf{a}=c x \hat{\mathbf{i}}-6 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ and $\mathbf{b}=x \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+2 c x \hat{\mathbf{k}}$ is an obtuse angle for all real $x$ is

Let $\mathbf{a}=2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+3 \hat{\mathbf{k}}, \mathbf{b}=3 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\mathbf{c}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ be three vectors. If $\mathbf{r}$ is a vector such that $\mathbf{r} \times \mathbf{a}=\mathbf{r} \times \mathbf{b}$ and $\mathbf{r} \cdot \mathbf{c}=18$, then the magnitude of the orthogonal projection of $4 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}-\hat{\mathbf{k}}$ on $\mathbf{r}$ is

If $\mathbf{u}, \mathbf{v}, \mathbf{w}$ are non-coplanar vectors and $p, q$ are real numbers, then the equality $[3 \mathbf{u} p \mathbf{v} p \mathbf{w}]-[p \mathbf{v} \mathbf{w} q \mathbf{u}]-[2 \mathbf{w} q \mathbf{v} q \mathbf{u}]=0$ holds for

If $\sum_{i=1}^9\left(x_i-5\right)=9$ and $\sum_{i=1}^9\left(x_i-5\right)^2=45$, then the standard deviation of the nine observations $x_1, x_2, \ldots, x_9$ is

Two students appeared simultaneously for an entrance exam. If the probability that the first student gets qualified in the exam is $\frac{1}{4}$ and the probability that the second student gets qualified in the same exam is $\frac{2}{5}$, then the probability that atleast one of them gets qualified in that exam is

For three events $A, B$ and $C$ of a sample space, $P$ (exactly one of $A$ or $B$ occurs ) $=P$ (exactly one of $B$ or $C$ occurs) $=P($ exactly one of $C$ or $A$ occurs $)=\frac{1}{4}$. If probability of all the three events occurring simultaneously is $\frac{1}{16}$, then the probability that atleast one of the events occur is

$A$ bag $P$ contains 4 red and 5 black balls another bag Q contains 3 red and 6 black balls. If one ball is drawn at random from bag $P$ and two balls are drawn from bag $Q$, then the probability that out of the three balls drawn two are black and one is red, is

On every evening, a student either watches TV or reads a book. The probability of watching TV is $\frac{4}{5}$ If he watches TV, the probability that he will fall asleep is $\frac{3}{4}$ and it is $\frac{1}{4}$ when he reads a book. If the student is found to be asleep on an evening the probability that he watched the TV is

Let $X$ be the random variable taking values $1,2, \ldots n$ for a fixed positive integer $n$. If $P(X=k)=\frac{1}{n}$ for $1 \leq k \leq n$, then the variance of $X$ is

A radar system can detect an enemy plane in one out of ten consecutive scans.

The probability that it can detect an enemy plane atleast twice in four consecutive scans is

The locus of the third vertex of a right-angled triangle, the ends of whose hypotenuse are $(1,2)$ and $(4,5)$ is

The coordinate axes are rotated about the origin in the counter clockwise direction through an angle $60^{\circ}$. If a and $b$ are the intercepts made on the new axes by a straight line whose equation referred to the original axes is $x+y=1$, then $\frac{1}{a^2}+\frac{1}{b^2}=$

The image of a point $(2,-1)$ with respect to the line $x-y+1=0$ is

If a straight line is at a distance of 10 units from the origin and the perpendicular drawn from the origin to it makes an angle $\frac{\pi}{4}$ with the negative $X$-axis in the negative direction, then the equation of that line is

If one of the lines given by the pair of lines $3 x^2-2 y^2+a x y=0$ is making an angle $60^{\circ}$ with $X$-axis, then $a=$

$A$ straight line passing through the origin $O$ meets the parallel lines $4 x+2 y=9$ and $2 x+y+6=0$ at the points $P$ and $Q$ respectively. Then, the point $O$ divides the line segment $P Q$ in the ratio

A circle is drawn with its centre at the focus of the parabola $y^2=2 p x$ such that it touches the directrix of the parabola. Then, a point of intersection of the circle and the parabola is

A circle touches both the coordinate axes and the straight line $L \equiv 4 x+3 y-6=0$ in the first quadrant. If this circle lies below the line $L=0$, then the equation of that circle is

If the smallest circle through the points of intersection of $x^2+y^2=a^2$ and $x \cos \alpha+y \sin \alpha=p, 0

If the lines $3 x-4 y+4=0$ and $6 x-8 y-7=0$ are the tangents to the same circle, then the area of that circle (in sq. units) is

Circles are drawn through the point $(2,0)$ to cut intercepts of length 5 units on the $X$-axis. If their centre lie in the first quadrant, then their equation is

If the locus of a point that divides a chord of slope 2 of the parabola $y^2=4 x$ internally in the ratio $1: 2$ is a parabola, then its vertex is

Assertion (A) The length of the latus rectum of an ellipse is 4 . The focus and its corresponding directrix are respectively $(1,-2)$ and $3 x+4 y-15=0$. Then, its eccentricity is $\frac{1}{2}$.

Reason $(\mathrm{R})$ Length of the perpendicular drawn from focus of an ellipse to its corresponding directrix is $\frac{a\left(1-e^2\right)}{e}$.

Then, which one of the following is correct?

If the eccentricity of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ passing through the point $(4,6)$ is 2 , then the equation of the tangent to this hyperbola at $(4,6)$ is

A hyperbola passes through the point $P(\sqrt{2}, \sqrt{3})$ and has foci at $( \pm 2,0)$. Then, the point that lies on the tangent drawn to this hyperbola at $P$ is

The circumradius of the triangle formed by the points $(2,-1,1),(1,-3,-5)$ and $(3,-4,-4)$ is

Let $A(2,3,5), B(-1,3,2)$ and $C(\lambda, 5, \mu)$ be the vertices of $\triangle A B C$. If the median through the vertex $A$ is equally inclined to the coordinate axes, then

Equation of the plane passing through the origin and perpendicular to the planes $x+2 y-z=1$ and $3 x-4 y+z=5$ is

$$\mathop {\lim }\limits_{x \to {\pi \over 4}} \frac{2 \sqrt{2}-(\cos x+\sin x)^3}{1-\sin 2 x}= $$

Let $[x]$ denote the greatest integer less than or equal to $x$. Then,

$$ \lim _{x \rightarrow 2^{+}}\left(\frac{[x]^3}{3}-\left[\frac{x}{3}\right]^3\right)= $$

If the function $f$ defined by

$$ f(x)=\left\{\begin{array}{cc} \frac{1-\cos 4 x}{x^2}, & x<0 \\ a, & x=0 \\ \frac{\sqrt{x}}{\sqrt{16+\sqrt{x}}-4}, & x>0 \end{array}\right. $$

is continuous at $x=0$, then $a=$

The domain of the derivative of the function $f(x)=\frac{x}{1+|x|}$ is

If $x=\sqrt{2^{\operatorname{cosec}^{-1} t}}$ and $y=\sqrt{2^{\sec ^{-1} t}},|t| \geq 1$, then $\frac{d y}{d x}=$

If $(a+\sqrt{2} b \cos x)(a-\sqrt{2} b \cos y) =a^2-b^2$, where $a>b>0$, then at $\left(\frac{\pi}{4}, \frac{\pi}{4}\right), \frac{d y}{d x}=$

Consider the quadratic equation $a x^2+b x+c=0$, where $2 a+3 b+6 c=0$ and let $g(x)=\frac{a x^3}{3}+\frac{b x^2}{2}+c x$

Statement I The given quadratic equation $a x^2+b x+c=0$ has atleast one root in $(0,1)$.

Statement II Rolle's theorem is applicable to $g(x){\text {on }}$ [0, 1].

Then

The difference between the absolute maximum and absolute minimum values of the function $f(x)=2 x^3-15 x^2+36 x-30$ on $[-1,4]$ is

If $f(x)=x e^{x(1-x)}, x \in R$, then $f(x)$ is

The angle between the curves $y^2=x$ and $x^2=y$ at the point $(1,1)$ is

If $\int \frac{5 \tan x}{\tan x-2} d x=a x+b \log |\sin x-2 \cos x|+c$, then $a+b=$

$$ \int x \cos ^{-1}\left(\frac{1-x^2}{1+x^2}\right) d x(x>0)= $$

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

$$ \int \sin ^{-1}\left(\sqrt{\frac{x}{a+x}}\right) d x= $$

If $\int \frac{x}{x \tan x+1} d x=\log f(x)+k$, then $f\left(\frac{\pi}{4}\right)=$

$$ \int_0^1 \frac{2 x+5}{x^2+3 x+2} d x= $$

The area (in sq units) of the region given by $R=\left\{(x, y) ; \frac{y^2}{2} \leq x \leq y+4\right\}$ is

$$ \int_0^1 x^{\frac{5}{2}}(1-x)^{\frac{3}{2}} d x= $$

$$ \lim _{n \rightarrow \infty}\left[\begin{array}{c} \frac{1}{n^2} \sec ^2 \frac{1}{n^2}+\frac{2}{n^2} \sec ^2 \frac{4}{n^2}+\frac{3}{n^2} \sec ^2 \\ \frac{9}{n^2}+\ldots+\frac{1}{n^2} \sec ^2 1 \end{array}\right]= $$

The general solution of the differential equation $\left(x \sin \frac{y}{x}\right) d y=\left(y \sin \frac{y}{x}-x\right) d x$ is

The general solution of the differential equation $\cos (x+y) d y=d x$ is

If $A x^3+B x y=4$ ( $A$ and $B$ are arbitrary constants) is the general solution of the differential equation $F(x) \frac{d^2 y}{d x^2}+G(x) \frac{d y}{d x}-2 y=0$, then $F(l)+G(l)=$

The physical quantity having the dimensions of the square root of the ratio of the kinetic energy and surface tension is

If the displacement ( $s$ in metre) of a moving particle in terms of time $(t$ in second $) s=t^3-6 t^2+18 t+9$, then the minimum velocity attained by the particle is

If a force $(\beta \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}) \mathrm{N}$ acting on a body displaces it through $(2 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+1 \hat{\mathbf{k}}) \mathrm{m}$, then the work done by the force on the body is

If two bodies $A$ and $B$ are projected with same velocity but with different angles $\theta_1$ and $\theta_2$ respectively with the horizontal such that both will have same range, then the ratio of times of flight of the bodies $A$ and $B$ is

The apparent weight of a girl of mass 30 kg when she is in a lift moving vertically upwards with an acceleration of $2 \mathrm{~ms}^{-2}$ is (Acceleration due to gravity $=10 \mathrm{~ms}^{-2}$ )

If a stone of mass 0.5 kg tied to one end of a wire is whirled in a circular path of radius 2 m with a speed $40 \mathrm{rev} / \mathrm{min}$ in a horizontal plane, then the tension in the wire is nearly

A body is projected vertically upwards with a velocity of $20 \mathrm{~ms}^{-1}$. If the potential energy of the body at a height of 5 m from the ground is 100 J , then the kinetic energy of the body at a height of 10 m from the ground is

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

A body falls freely on to a hard horizontal surface. If the coefficient of restitution between the surface and the body is 0.8 , then the ratio of the maximum height to which the body rises after second impact and the initial height of the body is

Two bodies of masses $M$ and $4 M$ initially at rest, start moving towards each other due to their mutual attraction. The velocity of their centre of mass when the first body attains a velocity $v_0$ is

The angular velocity of a body changes from $6 \mathrm{rad} \mathrm{s}^{-1}$ to $21 \mathrm{rad} \mathrm{s}^{-1}$ in a time of 1.5 s . If the moment of inertia of the body is $\mathrm{g} \mathrm{m}^2$, then the rate of change of angular momentum of the body is

If the displacement of a particle executing simple harmonic motion is given by $x=0.5 \cos (125.6 t)$, then the time period of oscillation of the particle is nearly (Here, $x$ is displacement in metre and $t$ is time in second)

The amplitude of a damped harmonic oscilator becomes $50 \%$ of its initial value in a time of 12 s . If the amplitude of the oscillator at a time of 36 s is $x \%$ of its initial amplitude, then the value of $x$ is

The escape velocity of a body from a planet of mass $M$ and radius $R$ is $14 \mathrm{~km} \mathrm{~s}^{-1}$. The escape velocity of the body from another planet having same mass and diameter 8 R (in $\mathrm{km} \mathrm{s}^{-1}$ ) is

The stress-strain graph of two wires $A$ and $B$ is shown in the figure. If $Y_A$ and $Y_B$ are Young's moduli of materials of wires $A$ and $B$ respectively, then

If two soap bubbles each of radius 2 cm combine in vacuum under isothermal conditions, then the radius of the new bubble formed is

A rectangular slab consists of two cubes of copper and brass of equal sides having thermal conductivities in the ratio $4: 1$. If the free face of brass is at $0^{\circ} \mathrm{C}$ and that of copper is at $100^{\circ} \mathrm{C}$, then the temperature of their interface is

The efficiency of a Carnot's heat engine is $\frac{1}{3}$. If the temperature of the source is decreased by $50^{\circ} \mathrm{C}$ and the temperature of the sink is increased by $25^{\circ} \mathrm{C}$, the efficiency of the engine becomes $\frac{3}{16}$. The initial temperature of the sink is

The change in internal energy of given mass of a gas, when its volume changes from $V$ to $3 V$ at constant pressure $p$ is

( $\gamma=$ Ratio of the specific heat capacities of the gas)

A monoatomic gas at a pressure of 100 kPa expands adiabatically such that its final volume becomes 8 times its initial volume. If the work done during the process is 180 J , then the initial volume of the gas is

If a gaseous mixture consists of 3 moles of oxygen and 4 moles of argon at an absolute temperature $T$, then the total internal energy of the mixture is (neglect vibrational modes and $R=$ Universal gas constant)

A sound wave of frequency 500 Hz travels between two points $X$ and $Y$ separated by a distance of 600 m in a time of 2 s . The number of waves between the points $X$ and $Y$ are

A ray of light incidents at an angle of $60^{\circ}$ on the first face of a prism. The angle of the prism is $30^{\circ}$ and its second face is silvered. If the light ray inside the prism retraces its path after reflection from the second face, then the refractive index of the material of the prism is

In an experiment, two polariods are arranged such that the intensity of the polarised light emerged from the second polaroid is $37.5 \%$ of the intensity of the unpolarised light incident on the first polaroid. Then the angle between the axes of the two polaroids is

If two particles $A$ and $B$ of charges $1.6 \times 10^{-19} \mathrm{C}$ and $3.2 \times 10^{-19} \mathrm{C}$ respectively are separated by a distance of 3 cm in air, then the magnitude of electrostatic force on particle $A$ due to particle $B$ is

If four charges $+12 \mathrm{nC},-20 \mathrm{nC},+32 \mathrm{nC}$ and -15 nC are arranged at the four vertices of a square of side $\sqrt{2} \mathrm{~m}$, then the net electric potential at the centre of the square due to these four charges is

Four capacitors are connected as shown in the figure. If $C_1, C_2, C_3$ and $C_4$ are in the ratio of $1: 2: 3: 4$, then the ratio of the charges on the capacitors $C_2$ and $C_4$ is

In the given circuit, the internal resistance of the cell is zero. If $i_1$ and $i_2$ are the readings of the ammeter when the key $(K)$ is opened and closed respectively, then $i_1: i_2=$

In a meter bridge, the null point is located at 20 cm from left end of the wire when resistances $R$ and $S$ are connected in the left and right gaps respectively. If the resistance $S$ is shunted with $60 \Omega$ resistance, the null point shifted by 5 cm , then the values of $R$ and $S$ are respectively

If a wire of length ' $L$ ' carrying a current ' $i$ ' is bent in the shape of a semi-circular arc as shown in the figure, then the magnetic field at centre of the arc is

A galvanometer having 30 divisions has a current sensitivity of $0.0625 \frac{d i \nu}{\mu A}$. If it is converted into a voltmeter to read a maximum of 6 V , then the resistance of that voltmeter is

If the given figure shows the relation between magnetic field ( $B$-along $Y$-axis) and magnetic intensity ( $H$-along $X$-axis) of a ferromagnetic material, then the point that represents coercivity of the material is

A coil having 100 square loops each of side 10 cm is placed such that its plane is normal to a magnetic field, which is changing at a rate of $0.7 \mathrm{Ts}^{-1}$. The emf induced in the coil is

An AC source of internal resistance $10^3 \Omega$ is connected to a transformer. The ratio of the number of turns in the primary to the number of turns in the secondary to match the source to a load resistance of $10 \Omega$ is

If $11 \%$ of the power of a 200 W bulb is converted to visible radiation, then the intensity of the light at a distance of 100 cm from the bulb is

The de-Broglie wavelength associated with an electron accelerated through a potential difference of $\frac{200}{3} \mathrm{~V}$ is nearly

The ratio of the shortest wavelengths of Bracket and Balmer series of hydrogen atom is

If the binding energy per nucleon of deuteron $\left({ }_1 \mathrm{H}^2\right)$ is 1.15 MeV and an $\alpha$-particle has a binding energy of 7.1 MeV per nucleon, then the energy released per nucleon in the given reaction is

$$ { }_1 \mathrm{H}^2+{ }_1 \mathrm{H}^2 \rightarrow{ }_2 \mathrm{He}^4+\mathrm{Q} $$

In a transistor, if the collector current is $98 \%$ of emitter current, then the ratio of the base and collector currents is

In the given circuit, if $A=0, B=1$ and $C=1$ are inputs, then the values of $y_1$ and $y_2$ are respectively

In amplitude modulation, if a message signal of 5 kHz 2 is modulated by a carrier wave of frequency 900 kHz , then the frequencies of the side bands are