The wavelength associated with the electron moving in the first orbit of hydrogen atom with velocity $$2.2 \times 10^6 \mathrm{~ms}^{-1}$$ (in nm) is

$$\left(m_e=9.0 \times 10^{-31} \mathrm{~kg}, h=6.6 \times 10^{-34} \mathrm{Js}\right)$$

The energy required (in eV) to excite an electron of H -atom from the ground state to the third state is

An element '$$X$$' with the atomic number 13 forms a complex of the type $$[\mathrm{XCl}(\mathrm{H}_2 \mathrm{O})_5]^{2+}$$. The covalency and oxidation state of $$X$$ in it are respectively

In which of the following oxides of three elements $$X, Y$$ and $$Z$$ are correctly arranged in the increasing order of acidic nature. The electronic configurations of $$X, Y$$ and $$Z$$ are $$[\mathrm{Ne}] 3 s^2 3 p^1, [\mathrm{Ne}] 3 s^2 3 p^5$$. [Ne] $$3 s^2$$ respectively

In the Lewis dot structure of carbonate ion shown under the formal charges on the oxygen atoms 1, 2 and 3 are respectively

The set of species having only fractional bond order values is

Identify the correct variation of pressure and volume of a real gas $$(A)$$ and an ideal gas ($$B$$) at constant temperature. $$(y=p ; x=V)$$

What are the oxidation numbers of S atoms in $$\mathrm{S}_4 \mathrm{O}_6^{2-}$$ ?

50 g of a substance is dissolved in 1 kg of water at $$+90^{\circ} \mathrm{C}$$. The temperature is reduced to $$+10^{\circ} \mathrm{C}$$. The density is increased from 1.1 to $$1.15 \mathrm{~g} \mathrm{~cc}^{-1}$$. What is the % change of molarity of the solution?

Identify the correct statements from the following.

I. At 0 K , the entropy of pure crystalline materials approach zero.

II. Entropy for the process, $$\mathrm{H}_2 \mathrm{O}(\mathrm{l}) \longrightarrow \mathrm{H}_2 \mathrm{O}(\mathrm{g})$$ decreases.

III. Gibb's energy is a state function.

Use the data from table to estimate the enthalpy of formation of $$\mathrm{CH}_3 \mathrm{CHO}$$.

At 500 K , for the reaction $$\mathrm{N}_2(\mathrm{~g})+3 \mathrm{H}_2(\mathrm{~g}) \rightleftharpoons 2 \mathrm{NH}_3(\mathrm{~g})$$, the $$K_p$$ is $$0.036 \mathrm{~atm}^{-2}$$. What is its $$K_C$$ in $$\mathrm{L}^2 \mathrm{~mol}^{-1}$$ ? $$\left(R=0.082 \mathrm{~L}^2\right.$$ atom $$\left.\mathrm{mol}^{-1} \mathrm{~K}^{-1}\right)$$.

The relative basic strength of the compounds is correctly shown in the option.

Calcium carbide $$+\mathrm{D}_2 \mathrm{O} \longrightarrow \underline{X}+\mathrm{Ca}(\mathrm{OD})_2$$. The hybridisation of carbon atom(s) in $$X$$

Identify the correct statements from the following.

(A) $$\mathrm{BeSO}_4$$ is soluble in water.

(B) BeO is an amphoteric oxide.

(C) CO can be obtained by heating $$\mathrm{CaCO}_3$$ at $$1070-1270 \mathrm{~K}$$.

The two major constituents of Portland cement are

Identify the $$P$$ and $$Q$$ of the following reaction

$$P+Q \longrightarrow\left[B(O H)_4\right]^{-}+\mathrm{H}_3 \mathrm{O}^{+}$$

Identify the species, which does not exist?

The IUPAC name of the following compound is

The gaseous mixture used for welding of metals is

An element crystallising in fcc lattice has a density of $$8.92 \mathrm{~g} \mathrm{~cm}^{-3}$$ and edge length of $$3.61 \times 10^{-8} \mathrm{~cm}$$. What is the atomic weight of element? $$\left(N=6.022 \times 10^{23} \mathrm{~mol}^{-1}\right)$$

Which of the following statement is correct for fcc lattice?

0.05 mole of a non-volatile solute is dissolved in 500 g of water. What is the depression in freezing point of resultant solution?

$$\left(K_f\left(\mathrm{H}_2 \mathrm{O}\right)=1.86 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}\right)$$

Which of the following form an ideal solution?

I. Chloroethane and bromoethane

II. Benzene and toluene

III. $$n$$-hexane and $$n$$-heptane

IV. Phenol and aniline

96.5 amperes current is passed through the molten $$\mathrm{AlCl}_3$$ for 100 seconds. The mass of aluminium deposited at the cathode is (atomic weight of $$\mathrm{Al}=27 \mathrm{u}$$)

The rate constant of a reaction at 500 K and 700 K are $$0.02 \mathrm{~s}^{-1}$$ and $$0.2 \mathrm{~s}^{-1}$$ respectively. The activation energy of the reaction (in $$\left.\mathrm{kJ} \mathrm{mol}^{-1}\right)$$ is $$\left(R=8.3 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}\right)$$

Match the following

The correct order of coagulating power of the following ions to coagulate the positive sol is

$$\mathrm{\mathop {{{[Fe{{(CN)}_6}]}^{4 - }}}\limits_I ,\mathop {C{l^ - }}\limits_{II} ,\mathop {SO_4^{2 - }}\limits_{III}}$$

Assertion (A) 16 th group elements have higher ionisation enthalpy values than 15 th group elements in the corresponding periods.

Reason (R) 15 th group elements have half-filled stable electronic configurations.

Assertion (A) Fluorine has smaller negative electron gain enthalpy than chlorine.

Reason (R) The electron-electron repulsion is higher in chlorine than in fluorine.

Identify the isoelectronic pair of ions from the following.

The homoleptic complex in the following is

Carbohydrates are stored in plants and animals in which of the following forms respectively?

Glycylalanine is a dipeptide of which amino acids?

Identify the major product formed from the following reaction.

Identify the major product of the following reaction.

Identify the product(s) formed in the following reaction.

Which compound is formed on catalytic hydrogenation of carbon monoxide at high $$p$$ and high $$T$$ in presence of $$\mathrm{ZnO}-\mathrm{Cr}_2 \mathrm{O}_3$$ catalyst?

Identify the major product from the following reaction sequence.

Arrange the following in decreasing order of their boiling points.

$$f(x)=\log \left(\left(\frac{2 x^2-3}{x}\right)+\sqrt{\frac{4 x^4-11 x^2+9}{|x|}}\right) \text { is }$$

Let $$f: R-\left\{\frac{-1}{2}\right\} \rightarrow R$$ be defined by $$f(x)=\frac{x-2}{2 x+1}$$. If $$\alpha$$ and $$\beta$$ satisfy the equation $$f(f(x))=-x$$, then $$4\left(\alpha^2+\beta^2\right)=$$

If $$A=\left[\begin{array}{lll}3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1\end{array}\right]$$, then $$A A^T$$ is a

If $$A X=D$$ represents the system of simultaneous linear equations $$x+y+z=6, 5 x-y+2 z=3$$ and $$2 x+y-z=-5$$, then (Adj $$A$$) $$D=$$

If $$A=\left[\begin{array}{ll}1 & 0 \\ 2 & 1\end{array}\right], B=\left[\begin{array}{ll}1 & 3 \\ 0 & 1\end{array}\right]$$, then $$\operatorname{det}\left(A^6+B^6\right)=$$

Let $$G(x)=\left[\begin{array}{ccc}\cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1\end{array}\right]$$. If $$x+y=0$$ then $$G(x) G(y)=$$

By simplifying $$i^{18}-3 i^7+i^2\left(1+i^4\right)(i)^{22}$$, we get

The values of $$x$$ for which $$\sin x+i \cos 2 x$$ and $$\cos x-i \sin 2 x$$ are conjugate to each other are

The locus of a point $$z$$ satisfying $$|z|^2=\operatorname{Re}(z)$$ is a circle with centre

If $$\sin ^4 \theta \cos ^2 \theta=\sum_\limits{n=0}^{\infty} a_{2 n} \cos 2 n \theta$$, then the least $$n$$ for which $$a_{2 n}=0$$ is

If $$S=\left\{m \in R: x^2-2(1+3 m) x+7(3+2 m)=0\right.$$ has distinct roots}, then the number of elements in $$S$$ is

$$4^x-3^{x-\frac{1}{2}}=3^{x+\frac{1}{2}}-2^{2 x-1} \Rightarrow x=$$

The sum of the real roots of the equation $$x^4-2 x^3+x-380=0$$ is

If one root of the cubic equation $$x^3+36=7 x^2$$ is double of another, then the number of negative roots are

$$\text { If } 10{ }^n C_2=3^{n+1} C_3 \text {, then the value of } n \text { is }$$

There are 10 points in a plane, out of these 6 are collinear. If $$N$$ is the total number of triangles formed by joining these points, then $$N=$$

205 students take an examination of whom 105 pass in English, 70 students pass in Mathematics and 30 students pass in both. How many students fail in both subjects?

In an examination, the maximum marks for each of three subjects is $$n$$ and that for the fourth subject is $$2 n$$. The number of ways in which candidates can get $$3 n$$ marks is

$$\frac{2 x^2+1}{x^3-1}=\frac{A}{x-1}+\frac{B x+C}{x^2+x+1} \Rightarrow 7 A+2 B+C=$$

If $$\sin \theta=-\frac{3}{4}$$, then $$\sin 2 \theta=$$

$$\begin{aligned} & \frac{1}{\sin 1^{\circ} \sin 2^{\circ}}+\frac{1}{\sin 2^{\circ} \sin 3^{\circ}}+\ldots +\frac{1}{\sin 89^{\circ}+\sin 90^{\circ}}= \end{aligned}$$

Which of the following trigonometric values are negative?

I. $$\sin \left(-292^{\circ}\right)$$

II. $$\tan \left(-190^{\circ}\right)$$

III. $$\cos \left(-207^{\circ}\right)$$

IV. $$\cot \left(-222^{\circ}\right)$$

$$\text { If } \sin \theta+\operatorname{cosec} \theta=4, \text { then } \sin ^2 \theta+\operatorname{cosec}^2 \theta=$$

$$\sin ^2 \frac{2 \pi}{3}+\cos ^2 \frac{5 \pi}{6}-\tan ^2 \frac{3 \pi}{4}=$$

If $$2 \cosh 2 x+10 \sinh 2 x=5$$, then $$x=$$

In any $$\triangle A B C, \frac{\cos A}{a}+\frac{\cos B}{b}+\frac{\cos C}{c}=$$

In a $$\triangle A B C$$, if $$r_1=36, r_2=18$$ and $$r_3=12$$, then $$s=$$

In a $$\triangle A B C, a=6, b=5$$ and $$c=4$$, then $$\cos 2 A=$$

a, b, c are non-coplanar vectors. If $$\mathbf{a}+3 \mathbf{b}+4 \mathbf{c}=x(\mathbf{a}-2 \mathbf{b}+3 \mathbf{c})+y(\mathbf{a}+5 \mathbf{b}-2 \mathbf{c}) +z(6 \mathbf{a}+14 \mathbf{b}+4 \mathbf{c}) \text {, then } x+y+z=$$

Three vectors of magnitudes $$a, 2 a, 3 a$$ are along the directions of the diagonals of 3 adjacent faces of a cube that meet in a point. Then, the magnitude of the sum of those diagonals is

If $$\mathbf{a}$$ is collinear with $$\mathbf{b}=3 \hat{i}+6 \hat{j}+6 \hat{k}$$ and $$\mathbf{a} \cdot \mathbf{b}=27$$, then $$|\mathbf{a}|=$$

Let $$a, b$$ and $$c$$ be unit vectors such that $$a$$ is perpendicular to the plane containing $$\mathbf{b}$$ and $$\mathbf{c}$$ and angle between $$\mathbf{b}$$ and $$\mathbf{c}$$ is $$\frac{\pi}{3}$$. Then, $$|\mathbf{a}+\mathbf{b}+\mathbf{c}|=$$

Let $$\mathbf{F}=2 \hat{i}+2 \hat{j}+5 \hat{k}, A=(1,2,5), B=(-1,-2,-3)$$ and $$\mathbf{B A} \times \mathbf{F}=4 \hat{i}+6 \hat{j}+2 \lambda \hat{k}$$, then $$\lambda=$$

If the mean deviation of the data $$1,1+d 1+2 d, \ldots, 1+100 d,(d>0)$$ from their mean is 255, then '$$d$$' is equal to

The probability of getting a sum 9 when two dice are thrown is

If $$A$$ and $$B$$ are two events such that $$P(B) \neq 0$$ and $$P(B) \neq 1$$, then $$P(\bar{A} \mid \bar{B})$$ is

Two brothers $$X$$ and $$Y$$ appeared for an exam. Let $$A$$ be the event that $$X$$ has passed the exam and $$B$$ is the event that $$Y$$ has passed. The probability of $$A$$ is $$\frac{1}{7}$$ and of $$B$$ is $$\frac{2}{9}$$. Then, the probability that both of them pass the exam is

A bag contains 4 red and 3 black balls. A second bag contains 2 red and 3 black balls. One bag is selected at random. If from the selected bag, one ball is drawn at random, then the probability that the ball drawn is red, is

In a Binomial distribution, if '$$n$$' is the number of trials and the mean and variance are 4 and 3 respectively, then $$2^{32} p\left(X=\frac{n}{2}\right)=$$

For a Poisson distribution, if mean $$=l$$, variance $$=m$$ and $$l+m=8$$, then $$e^4[1-P(X>2)]=$$

The locus of mid-points of points of intersection of $$x \cos \theta+y \sin \theta=1$$ with the coordinate axes is

Suppose $$P$$ and $$Q$$ lie on $$3 x+4 y-4=0$$ and $$5 x-y-4=0$$ respectively. If the mid-point of $$P Q$$ is $$(1,5)$$, then the slope of the line passing through $$P$$ and $$Q$$ is

The length of intercept of $$x+1=0$$ between the lines $$3 x+2 y=5$$ and $$3 x+2 y=3$$ is

Suppose that the three points $$A, B$$ and $$C$$ in the plane are such that their $$x$$-coordinates as well as $$y$$-coordinates are in GP with the same common ratio. Then, the points $$A, B$$ and $$C$$

Suppose the slopes $$m_1$$ and $$m_2$$ of the lines represented by $$a x^2+2 h x y+b y^2=0$$ satisfy $$3\left(m_1-m_2\right)-7=0$$ and $$m_1 m_2-2=0$$. Then, which of the following is true?

Suppose that the sides passing through the vertex $$(\alpha, \beta)$$ of a triangle are bisected at right angles by the lines $$y^2-8 x y-9 x^2=0$$. Then, the centroid of the triangle is

The radius of the circle having. $$3 x-4 y+4=0$$ and $$6 x-8 y-7=0$$ as its tangents is

A circle is such that $$(x-2) \cos \theta+(y-2) \sin \theta=1$$ touches it for all values of $$\theta$$. Then, the circle is

The least distance of the point $$(10,7)$$ from the circle $$x^2+y^2-4 x-2 y-20=0$$ is

Suppose that the $$x$$-coordinates of the points $$A$$ and $$B$$ satisfy $$x^2+2 x-a^2=0$$ and their $$y$$-coordinates satisfy $$y^2+4 y-b^2=0$$. Then, the equation of the circle with $$A B$$ as its diameter is

The radical centre of the three circles $$x^2+y^2-1=0, x^2+y^2-8 x+15=0$$ and $$x^2+y^2+10 y+24=0$$ is

Which of the following represents a parabola?

If the angle between the straight lines joining the foci and the ends of the minor axis of the ellipse $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ is $$90^{\circ}$$, then it eccentricity

The locus of point of intersection of tangents at the ends of normal chord of the hyperbola $$x^2-y^2=a^2$$ is

If $$e_1$$ and $$e_2$$ are the eccentricities of the hyperbola $$16 x^2-9 y^2=1$$ and its conjugate respectively. Then, $$3 e_1=$$

If P divides the line segment joining the points $$A(1,2,-1)$$ and $$B(-1,0,1)$$ externally in the ratio 1 : 2 and $$Q=(1,3,-1)$$, then $$PQ=$$

If the direction cosines of a line are $$\left(\frac{a}{\sqrt{83}}, \frac{5}{\sqrt{83}}, \frac{c}{\sqrt{83}}\right)$$ and $$c-a=4$$, then $$ca=$$

Let the plane $$\pi$$ pass through the point (1, 0, 1) and perpendicular to the planes $$2x + 3y - z = 2$$ and $$x - y + 2z = 1$$. Let the equation of the plane passing through the point (11, 7, 5) and parallel to the plane $$\pi$$ be $$ax + by - z - d = 0$$. Then, $${a \over b} + {b \over d} = $$

$$\lim _\limits{x \rightarrow-\infty} \log _e(\cosh x)+x=$$

If $$a, b$$ and $$c$$ are three distinct real numbers and $$\lim _\limits{x \rightarrow \infty} \frac{(b-c) x^2+(c-a) x+(a-b)}{(a-b) x^2+(b-c) x+(c-a)}=\frac{1}{2}$$, then $$a+2 c=$$

$$\lim _\limits{x \rightarrow-\infty} \frac{3|x|-x}{|x|-2 x}-\lim _\limits{x \rightarrow 0} \frac{\log \left(1+x^3\right)}{\sin ^3 x}=$$

If $$3 f(\cos x)+2 f(\sin x)=5 x$$, then $$f^{\prime}(\cos x)+f^{\prime}(\sin x)=$$

Assertion (A) $$\frac{d}{d x}\left(\frac{x^2 \sin x}{\log x}\right)=\frac{x^2 \sin x}{\log x}\left(\cot x+\frac{2}{x}-\frac{1}{x \log x}\right)$$

Reason (R) $$\frac{d}{d x}\left(\frac{u v}{w}\right)=\frac{u v}{w}\left[\frac{u^{\prime}}{u}+\frac{v^{\prime}}{v}+\frac{w^{\prime}}{w}\right]$$

If $$x=f(\theta)$$ and $$y=g(\theta)$$, then $$\frac{d^2 y}{d x^2}=$$

If the normal drawn at a point $$P$$ on the curve $$3 y=6 x-5 x^3$$ passes through $$(0,0)$$, then the positive integral value of the abscissa of the point $$P$$ is

The line joining the points $$(0,3)$$ and $$(5,-2)$$ is a tangent to the curve $$y=\frac{c}{x+1}$$, then $$c=$$

$$y=x^3-a x^2+48 x+7$$ is an increasing function for all real values of $$x$$, then $$a$$ lies in the interval

If $$a, b>0$$, then minimum value of $$y=\frac{b^2}{a-x}+\frac{a^2}{x}, 0< x< a$$ is

The point on the curve $$y=x^2+4 x+3$$ which is closest to the line $$y=3 x+2$$ is

$$\int \frac{3 x+4}{x^3-2 x+4} d x=\log f(x)+C \Rightarrow f(3)=$$

$$\int \frac{e^{\tan ^{-1} x}}{1+x^2}\left[\left(\sec ^{-1} \sqrt{1+x^2}\right)^2+\cos ^{-1}\left(\frac{1-x^2}{1+x^2}\right)\right] d x=$$

$$\int \frac{d x}{(x-3)^{\frac{4}{5}}(x+1)^{\frac{6}{5}}}=$$

If $$I_n=\int\left(\cos ^n x+\sin ^n x\right) d x$$ and $$I_n-\frac{n-1}{n} I_{n-2} =\frac{\sin x \cos x}{n} f(x)$$, then $$f(x)=$$

Let $$T>0$$ be a fixed number. $$f: R \rightarrow R$$ is a continuous function such that $$f(x+T)=f(x), x \in R$$ If $$I=\int_\limits0^T f(x) d x$$, then $$\int_\limits0^{5 T} f(2 x) d x=$$

$$\int_\limits1^3 x^n \sqrt{x^2-1} d x=6 \text {, then } n=$$

[ . ] represents greatest integer function, then $$\int_{-1}^1(x[1+\sin \pi x]+1) d x=$$

$$\begin{aligned} & \lim _{n \rightarrow \infty}\left[\frac{n}{(n+1) \sqrt{2 n+1}}+\frac{n}{(n+2) \sqrt{2(2 n+2)}}\right. \\ & \left.+\frac{n}{(n+3) \sqrt{3(2 n+3)}}+\ldots n \text { terms }\right]=\int_\limits0^1 f(x) d x \end{aligned}$$

then $$f(x)=$$

The general solution of the differential equation $$\frac{d y}{d x}=\cos ^2(3 x+y)$$ is $$\tan ^{-1}\left(\frac{\sqrt{3}}{2} \tan (3 x+y)\right)=f(x)$$. Then, $$f(x)=$$

If the general solution of the differential equation $$\cos ^2 x \frac{d y}{d x}+y=\tan x$$ is $$y=\tan x-1+C e^{-\tan x}$$ satisfies $$y\left(\frac{\pi}{4}\right)=1$$, then $$C=$$

Assertion (A) Order of the differential equations of a family of circles with constant radius is two.

Reason (R) An algebraic equation having two arbitrary constants is general solution of a second order differential equation.

The energy of $$E$$ of a system is function of time $$t$$ and is given by $$E(t)=\alpha t-\beta t^3$$, where $$\alpha$$ and $$\beta$$ are constants. The dimensions of $$\alpha$$ and $$\beta$$ are

A student is at a distance 16 m from a bus when the bus begins to move with a constant acceleration of $$9 \mathrm{~m} \mathrm{~s}^{-2}$$. The minimum velocity with which the student should run. towards the bus so as the catch it is $$\alpha \sqrt{2} \mathrm{~ms}^{-1}$$. The value of $$\alpha$$ is

The component of a vector $$\mathbf{P}=3 \hat{i}+4 \hat{j}$$ along the direction $$(\hat{i}+2 \hat{j})$$ is

A projectile is launched from the ground, such that it hits a target on the ground which is 90 m away. The minimum velocity of projectile to hit the target is (acceleration due to gravity $$=10 \mathrm{~ms}^{-2}$$)

If two vectors $$\mathbf{A}$$ and $$\mathbf{B}$$ are mutually perpendicular, then the component of $$\mathbf{A}-\mathbf{B}$$ along the direction of $$\mathbf{A}+\mathbf{B}$$ is

A body is travelling with $$10 \mathrm{~ms}^{-1}$$ on a rough horizontal surface. It's velocity after 2 s is $$4 \mathrm{~ms}^{-1}$$. The coefficient of kinetic friction between the block and the plane is (acceleration due to gravity $$=10 \mathrm{~ms}^{-2}$$)

A small disc of mass $$m$$ slides down with initial velocity zero from the top $$(A)$$ of a smooth hill of height $$H$$ having a horizontal portion $$(BC)$$ as shown in the figure. If the height of the horizontal portion of the hill is $$h$$, then the maximum horizontal distance covered by the disc from the point $$D$$ is

A block of mass 50 kg is pulled with a constant speed of $$4 \mathrm{~ms}^{-1}$$ across a horizontal floor by an applied force of 500 N directed $$30^{\circ}$$ above the horizontal. The rate at which the force does work on the block in watt is

Ball $$A$$ of mass 1 kg moving along a straight line with a velocity of $$4 \mathrm{~ms}^{-1}$$ hits another ball $$B$$ of mass 3 kg which is at rest. After collision, they stick together and move with the same velocity along the same straight line. If the time of impact of the collision is 0.1 s then the force exerted on $$B$$ is

A solid cylinder of radius $$R$$ is at rest at a height $$h$$ on an inclined plane. If it rolls down then its velocity on reaching the ground is

A particle is executing simple harmonic motion with an instantaneous displacement $$x=A \sin ^2\left(\omega t-\frac{\pi}{4}\right)$$. The time period of oscillation of the particle is

If the amplitude of a lightly damped oscillator decreases by $$1.5 \%$$ then the mechanical energy of the oscillator lost in each cycle is

Statement (A) Two artificial satellites revolving in the same circular orbit have same period of revolution.

Statement (B) The orbital velocity is inversely proportional to the square root of radius of the orbit.

Statement (C) The escape velocity of the body is independent of the altitude of the point of projection.

Two wires $$A$$ and $$B$$ of same cross-section are connected end to end. When same tension is created in both wires, the elongation in $$B$$ wire is twice the elongation in $$A$$ wire. If $$L_A$$ and $$L_B$$ are the initial lengths of the wires $$A$$ and $$B$$ respectively, then (Young's modulus of material of wire $$A=2 \times 10^{11} \mathrm{~Nm}^{-2}$$ and Young's modulus of material of wire $$B=1.1 \times 10^{11} \mathrm{~Nm}^{-2}$$).

5 g of ice at $$-30^{\circ} \mathrm{C}$$ and 20 g of water at $$35^{\circ} \mathrm{C}$$ are mixed together in a calorimeter. The final temperature of the mixture is (Neglect heat capacity of the calorimeter, specific heat capacity of ice $$=0.5 \mathrm{cal} \mathrm{g}^{-1}{ }^{\circ} \mathrm{C}^{-1}$$ and latent heat of fusion of ice $$=80 \mathrm{cal} \mathrm{g}^{-1}$$ and specific heat. capacity of water $$=1 \mathrm{cal} \mathrm{g}^{-1}{ }^{\circ} \mathrm{C}^{-1}$$)

A hydraulic lift is shown in the figure. The movable pistons $$A, B$$ and $$C$$ are of radius $$10 \mathrm{~cm}, 100 \mathrm{~m}$$ and 5 cm respectively. If a body of mass 2 kg is placed on piston $$A$$, the maximum masses that can be lifted by piston $$B$$ and $$C$$ are respectively.

An iron sphere having diameter $$D$$ and mass $$M$$ is immersed in hot water so that the temperature of the sphere increases by $$\delta T$$. If $$\alpha$$ is the coefficient of linear expansion of the iron then the change in the surface area of the sphere is

The work done by a Carnot engine operating between 300 K and 400 K is 400 J. The energy exhausted by the engine is

The slopes of the isothermal and adiabatic $$p-V$$ graphs of a gas are by $$S_I$$ and $$S_A$$ respectively. If the heat capacity ratio of the gas is $$\frac{3}{2}$$, then $$\frac{S_I}{S_A}=$$

The number of rotational degrees of freedom of a diatomic molecule

Two cars are moving towards each other at the speed of $$50 \mathrm{~ms}^{-1}$$. If one of the cars blows a horn at a frequency of 250 Hz , the wave length of the sound perceived by the driver of the other car is

(Speed of sound in air $$=350 \mathrm{~ms}^{-1}$$)

A needle is lying at the bottom of a water tank of height 12 cm. The apparent depth of the needle measured by a microscope is 9 cm . If the water is replaced by a liquid of refractive index of 1.5 of same height, the distance through which the microscope has to be moved to focus the needle again is

Young's double slit experiment is conducted with monochromatic light of wavelength 5000$$\mathop A\limits^o $$, with slit separation of 3 mm and observer at 20 cm away from the slits. If a 1 mm transparent plate is placed infront of one of the slits, the fringes shift by 6 mm . The refractive index of the transparent plate is

A large number of positive charges each of magnitude $$q$$ are placed along the $$X$$-axis at the origin and at every 1 cm distance in both the directions. The electric flux through a spherical surface of radius 2.5 cm centred at the origin is

The capacitance between the points A and B in the following figure.

The electric field in a region of space is given as $$\mathbf{E}=\left(5 \mathrm{NC}^{-1}\right) x \hat{i}$$. Consider point $$A$$ on the $$Y$$-axis at $$y=5 \mathrm{~m}$$ and point $$B$$ on the $$X$$-axis at $$x=2 \mathrm{~m}$$. If the potentials at points $$A$$ and $$B$$ are $$V_A$$ and $$V_B$$ respectively, then $$\left(V_B-V_A\right)$$ is

In the given circuit values of $$I_1, I_2, I_3$$ are respectively

The resistance of wire at $$0^{\circ} \mathrm{C}$$ is $$20 \Omega$$. If the temperature coefficient of the resistance is $$5 \times 10^{-3}{ }^{\circ} \mathrm{C}^{-1}$$. The temperature at which the resistance will be double of that at $$0^{\circ} \mathrm{C}$$ is

An electron having kinetic energy of 100 eV circulates in a path of radius 10 cm in a magnetic field. The magnitude of magnetic field $$|\mathbf{B}|$$ is approximately [Mass of electron $$=0.5 \mathrm{~MeV} \mathrm{c}^{-2}$$, where c is the velocity of light].

A particle of mass $$2.2 \times 10^{-30} \mathrm{~kg}$$ and charge $$1.6 \times 10^{-19} \mathrm{C}$$ is moving at a speed of $$10 \mathrm{~km} \mathrm{~s}^{-1}$$ in a circular path of radius 2.8 cm inside a solenoid. The solenoid has $$25 \frac{\text { turns }}{\mathrm{cm}}$$ and its magnetic field is perpendicular to the plane of the particle's path. The current in the solenoid is

(Take, $$\mu_0=4 \pi \times 10^{-7} \mathrm{~Hm}^{-1}$$)

Two short magnets of equal dipole moments $$M$$ are fastened perpendicularly at their centres. The magnitude of the magnetic field at a distance $$d$$ from the centre on the bisector of the right angle is ($$\mu_0=$$ Permeability of free space)

A circular loop of wire of radius 14 cm is placed in magnetic field directed perpendicular to the plane of the loop. If the field decreases at a steady rate of $$0.05 \mathrm{~Ts}^{-1}$$ in some interval, then the magnitude of the emf induced in the loop is

An $$R-L-C$$ circuit consists of a $$150 \Omega$$ resistor, $$20 \mu \mathrm{F}$$ capacitor and a 500 mH inductor connected in series with a 100 V AC supply. The angular frequency of the supply voltage is $$400 \mathrm{rad} \mathrm{s}^{-1}$$. The phase angle between current and the applied voltage is

The magnetic field in a plane electromagnetic wave is given as $$\mathbf{B}=\left(3 \times 10^{-7} \mathrm{~T}\right) \sin \left(3 \times 10^4 x+9 \times 10^{12} t\right) \hat{j}$$

The electric field of this wave is given as

In Young's double slit experiment the slits are 3 mm apart and are illuminated by light of two wavelengths $$3750 \mathop A\limits^o$$ and $$7500 \mathop A\limits^o$$. The screen is placed at 4 m from the slits. The minimum distance from the common central bright fringe on the screen at which the bright fringe of one interference pattern due to one wavelength coincide with the bright fringe of the other is

The following statement is correct in the case of photoelectric effect

An electron in the hydrogen atom excites from 2nd orbit to 4th orbit then the change in angular momentum of the electron is (Planck's constant $$h=6.64 \times 10^{-34} \mathrm{~J}-\mathrm{s}$$)

Choose the correct statement of the following

A ancient discovery found a sample, where $$75 \%$$ of the original carbon ($$\mathrm{C}^{14}$$) remains. Then the age of the sample is $$\binom{T_{\frac{1}{2}}\left(C^{14}\right)=5730 \text { years, } \ln 0.5=-0.7}{\ln (0.75)=-0.3} $$

Frequencies in the UHF range normally propagate by means of