Assuming that the incident radiation is capable of ejecting photoelectrons from all the given metals, the lowest kinetic energy of the ejected photoelectron is observed with which of the given metals?

If the energies of two light radiations $$E_1$$ and $$E_2$$ are $$25 \mathrm{~eV}$$ and $$100 \mathrm{~eV}$$ respectively, then their respective wavelengths $$\lambda_1$$ and $$\lambda_2$$ would be in the ratio $$\lambda_1: \lambda_2$$

The electronic configuration of $$\mathrm{Fe}^{3+}$$ is (atomic number of $$\mathrm{Fe}=26$$ )

The element with outer electronic configuration $$(n-1) d^2 n s^2$$, where $$n=4$$, would belong to

Choose the correct option regarding the following statements

Statement 1 Nitrogen has lesser electron gain enthalpy than oxygen.

Statement 2 Oxygen has lesser ionisation enthalpy than nitrogen.

Among the given configurations, identify the element which does not belong to the same family as the others?

Which compound among the following has the highest dipole moment?

How many among the given species have a bond order of 0.5 ?

$$\mathrm{H}_2^{+}, \mathrm{He}_2^{+}, \mathrm{He}_2^{-}, \mathrm{B}_2^{+}, \mathrm{F}_2^{-}, \mathrm{Be}_2^{2-}$$

Match the following.

Among the following the maximum deviation from ideal gas behaviour is expected from

Two flasks $$A$$ and $$B$$ have equal volumes. $$A$$ is maintained at $$300 \mathrm{~K}$$ and $$B$$ at $$600 \mathrm{~K}$$. Equal masses of $$\mathrm{H}_2$$ and $$\mathrm{CO}_2$$ are taken in flasks $$A$$ and $$B$$ respectively. Find the ratio of total KE of gases in flask $$A$$ to that of $$B$$.

If 0.2 moles of sulphuric acid is poured into 250 mL of water, calculate the concentration of the solution.

For the redox reaction

$$\mathrm{MnO}_4^{-}+\mathrm{C}_2 \mathrm{O}_4^{2-}+\mathrm{H}^{+} \longrightarrow \mathrm{Mn}^{2+}+\mathrm{CO}_2+\mathrm{H}_2 \mathrm{O}$$,

the correct coefficients of the reactants for the balanced reaction are respectively

When the temperature of 2 moles of an ideal gas is increased by 20$$^\circ$$C at constant pressure. Find the work involved in the process.

Using the data provided, find the value of equilibrium constant for the following reaction at $$298 \mathrm{~K}$$ and $$1 \mathrm{~atm}$$ pressure.

$$\begin{aligned} \mathrm{NO}(g)+\frac{1}{2} \mathrm{O}_2(g) \rightleftharpoons & \mathrm{NO}_2(g) \\ \Delta_f H \mathrm{Y}[\mathrm{NO}(g)] & =90.4 \mathrm{~kJ} \mathrm{~mol}^{-1} \\ \Delta_f H \mathrm{Y}\left[\mathrm{NO}_2(g)\right] & =32.48 \mathrm{~kJ} \mathrm{~mol}^{-1} \\ \Delta S Y a t ~298 \mathrm{~K} & =-70.8 \mathrm{~JK}^{-1} \mathrm{~mol}^{-1} \end{aligned}$$

$$[\operatorname{antilog}(0.50)=3162 \text { ] }$$

Calculate the pOH of 0.10 M HCl solution.

Which among the following pairs is not an acidic buffer?

Assertion (A) The colour of old lead paintings can be restored by washing them with a dilute solution of $$\mathrm{H}_2 \mathrm{O}_2$$.

Reason (R) Hydrogen peroxide reduces $$\mathrm{PbS}$$ to $$\mathrm{Pb}$$.

In the preparation of baking soda, H$$_2$$O and CO$$_2$$ in ratio ......... is used to react with Na$$_2$$CO$$_3$$.

The structure of diborane B$$_2$$H$$_6$$ is given below. Identify the bond angles of x and y. In diborane, ........... are commonly known as banana-bonds.

The incorrect statement(s) among the following is/are

Which among the following has the highest concentration of PAN?

What is the IUPAC name of $$\mathrm{CH}_3 \mathrm{CH}\left(\mathrm{CH}_2 \mathrm{CH}_3\right) \mathrm{CHO}$$ ?

Among the following, in which type of chromatography, both stationary and mobile phases are in liquid state?

The product formed when a hydrocarbon $$X$$ of molecular formula $$\mathrm{C}_6 \mathrm{H}_{10}$$ is reacted with sodamide is subjected to ozonolysis, followed by hydrolysis with $$\mathrm{Zn} / \mathrm{H}_2 \mathrm{O}_2$$ and upon further oxidation gave two carboxylic acids, of which one is optically active. The hydrocarbon '$$X$$' is

The fcc crystal contains how many atoms in each unit cell?

Which condition is not satisfied by an ideal solution?

A solution of urea (molar mass $$60 \mathrm{~g} \mathrm{~mol}^{-1}$$ ) boils at $$100.20^{\circ} \mathrm{C}$$ at the atmospheric pressure, if $$K_f$$ and $$K_b$$ for water are 1.86 and $$0.512 \mathrm{~K}$$ $$\mathrm{kg} \mathrm{mol}^{-1}$$ respectively. The freezing point of the solution will be

At $$291 \mathrm{~K}$$, saturated solution of $$\mathrm{BaSO}_4$$ was found to have a specific conductivity of $$3.648 \times 10^{-6} \mathrm{ohm}^{-1} \mathrm{~cm}^{-1}$$ and that of water being used is $$1.25 \times 10^{-6} \mathrm{ohm}^{-1} \mathrm{~cm}^{-1}$$. If the ionic conductances of $$\mathrm{Ba}^{2+}$$ and $$\mathrm{SO}_4^{2-}$$ are 110 and $$136.6 \mathrm{ohm}^{-1} \mathrm{~cm}^2 \mathrm{~mol}^{-1}$$ respectively. The solubility of $$\mathrm{BaSO}_4$$ at $$291 \mathrm{~K}$$ will be [Atomic masses of $$\mathrm{Ba}=137, \mathrm{~S}=32, \mathrm{O}=16]$$

Find the emf of the following cell reaction. Given, $$E_{\mathrm{Cr}^{3+} / \mathrm{Cr}^{2+}}^{\Upsilon}=-0.72 \mathrm{~V}$$ and $$E_{\mathrm{Fe}^{2+} / \mathrm{Fe}}^{\Upsilon}= -0.42 \mathrm{~V}$$ at $$25^{\circ} \mathrm{C}$$ is $$\mathrm{Cr}\left|\mathrm{Cr}^{3+}(0.1 \mathrm{M})\right| \mid \mathrm{Fe}^{2+} (0.1 \mathrm{M}) \mid \mathrm{Fe}$$

For $$\mathrm{C{r_2}O_7^{2 - } + 14{H^ + } + 6{e^ - }\buildrel {Yields} \over \longrightarrow 2C{r^{3 + }} + 7{H_2}O,{E^\Upsilon } = 1.33}$$ V at $$[C{r_2}O_7^{2 - }] = 4.5$$ millimole, $$[C{r^{3 + }}] = 1.5$$ millimole and $$E = 1.067$$ V, then calculate the pH of the solution.

The protective power of a lyophilic colloidal sol is expressed in terms of

Due to $$p \pi-p \pi$$ bonding interactions, nitrogen for $$\mathrm{N}_2$$. But phosphorus forms .................. and does not form a diatomic molecule.

Identify the incorrect statements among the following?

(i) $$\mathrm{SF}_6$$ does not react with water.

(ii) $$\mathrm{SF}_6$$ is $$s p^3 d$$ hybridised.

(iii) $$\mathrm{S}_2 \mathrm{O}_3^{2-}$$ is a linear ion.

(iv) There is no $$\pi$$-bonding in $$\mathrm{SO}_4^{2-}$$ ion.

Which statements among the following are correct about helium?

(i) Liquid helium is used to sustain powerful superconducting magnets.

(ii) Liquid helium is useful to carry low temperature experiments.

(iii) It is a heavy gas.

(iv) It is a flammable gas.

The magnetic moment of Fe$$^{2+}$$ is ........ BM.

Which of the following statement is not correct?

Assertion (A) An optically active amino acid can exist in three forms depending on the pH of the solution.

Reason (R) Amino acids contain both acidic and basic groups, they exist as Zwitter ion in aq. medium, anionic form in acidic medium and cationic form in basic medium.

The IUPAC name of diacetone alcohol is.

Identify the product of the following reaction.

The real valued function $$f(x)=\frac{x}{e^x-1}+\frac{x}{2}+1$$ defined on $$R /\{0\}$$ is

The domain of the function $$f(x)=\frac{1}{[x]-1}$$, where $$[x]$$ is greatest integer function of $$x$$ is

Let $$f: R \rightarrow R$$ be a function defined by $$f(x)=\frac{4^x}{4^x+2}$$, what is the value of $$f\left(\frac{1}{4}\right)+2 f\left(\frac{1}{2}\right)+f\left(\frac{3}{4}\right)$$ is equal to

For what natural number $$n \in N$$, the inequality $$2^n > n+1$$ is valid?

The value of $$\left|\begin{array}{ccc}b+c & a & a \\ b & c+a & b \\ c & c & a+b\end{array}\right|$$ is

Let $$A, B, C, D$$ be square real matrices such that $$C^T=D A B, D^{\mathrm{T}}=A B C$$ and $$S=A B C D$$, then $$S^2$$ is equal to

$$A=\left[\begin{array}{ccc}a^2 & 15 & 31 \\ 12 & b^2 & 41 \\ 35 & 61 & c^2\end{array}\right]$$ and $$B=\left[\begin{array}{ccc}2 a & 3 & 5 \\ 2 & 2 b & 8 \\ 1 & 4 & 2 c-3\end{array}\right]$$ are two matrices such that the sum of the principal diagonal elements of both $$A$$ and $$B$$ are equal, then the product of the principal diagonal elements of $$B$$ is

Let $$a, b$$ and $$c$$ be such that $$b+c \neq 0$$ and $$\begin{aligned} & \left|\begin{array}{ccc} a & a+1 & a-1 \\ -b & b+1 & b-1 \\ c & c-1 & c+1 \end{array}\right| \\ & +\left|\begin{array}{ccc} a+1 & b+1 & c-1 \\ a-1 & b-1 & c+1 \\ (-1)^{n+2} a & (-1)^{n-1} b & (-1)^n c \end{array}\right|=0 \text {, } \\ & \end{aligned}$$

then the value of $$n$$ is

If $$|z-2|=|z-1|$$, where $$z$$ is a complex number, then locus $$z$$ is a straight line

If $${\left( {{{1 + i} \over {1 - i}}} \right)^m} = 1$$, then m cannot be equal to

If $$1+x^2=\sqrt{3} x$$, then $$\sum_{n=1}^{24}\left(x^n-\frac{1}{x^n}\right)^2$$ is equal to

If $$\alpha$$ and $$\beta$$ are the roots of $$11 x^2+12 x-13=0$$, then $$\frac{1}{\alpha^2}+\frac{1}{\beta^2}$$ is equal to (approximately close to)

The value of $$a$$ for which the equations $$x^3+a x+1=0$$ and $$x^4+a x^2+1=0$$ have a common root is

If $$a$$ is a positive integer such that roots of the equation $$7 x^2-13 x+a=0$$ are rational numbers, then the smallest possible value of $$a$$ is

Let $$p$$ and $$q$$ be the roots of the equation $$x^2-2 x+A=0$$ and let $$r$$ and $$s$$ be the roots of the equation $$x^2-18 x+B=0$$. If $$p < q < r < s$$ are in AP then the values of $$A$$ and $$B$$ are

For $$1 \leq r \leq n, \frac{1}{r+1}\left\{{ }^n P_{r+1}-{ }^{(n-1)} P_{r+1}\right\}$$ is equal to

In how many ways 4 balls can be picked from 6 black and 4 green coloured balls such that at least one black ball is selected?

In how many ways can 9 examination papers be arranged so, that the best and the worst papers are never together?

Which of the following is partial fraction of $$\frac{-x^2+6 x+13}{(3 x+5)\left(x^2+4 x+4\right)}$$ is equal to

In a $$\triangle A B C$$, if $$3 \sin A+4 \cos B=6$$ and $$4 \sin B+3 \cos A=1$$, then $$\sin (A+B)$$ is equal to

$$\tan \alpha+2 \tan 2 \alpha+4 \tan 4 \alpha+8 \cot 8 \alpha$$ is equal to

If $$f(x)=\frac{\cot x}{1+\cot x}$$ and $$\alpha+\beta=\frac{5 \pi}{4}$$, then the value of $$f(\alpha) f(\beta)$$ is equal to

If $$\theta \in[0,2 \pi]$$ and $$\cos 2 \theta=\cos \theta+\sin \theta$$, then the sum of all values of $$\theta$$ satisfying the equation is

For how many distinct values of $$x$$, the following $$\sin \left[2 \cos ^{-1} \cot \left(2 \tan ^{-1} x\right)\right]=0$$ holds?

In $$\triangle A B C$$, suppose the radius of the circle opposite to an angle $$A$$ is denoted by $$r_1$$, similarly $$r_2 \leftrightarrow$$ angle $$B, r_3 \leftrightarrow$$ angle $$C$$. If $$r_1=2, r_2=3, r_3=6$$, what is the value of $$r_1+r_2+r_3-r=$$ (R - radius of the circum circle).

In $$\triangle A B C \cdot \frac{a+b+c}{B C+A B}+\frac{a+b+c}{A C+A B}=3$$, then $$\tan \frac{C}{8}$$ is equal to

Which of the following vector is equally inclined with the coordinate axes?

If $$\hat{\mathbf{i}}+4 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}, \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$$, and $$3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}}$$ are position vectors of $$A, B$$ and $$C$$ respectively and if $$D$$ and $$E$$ are mid points of sides $$B C$$ and $$A C$$, then $$\mathbf{D E}$$ is equal to

$$X$$ intercept of the plane containing the line of intersection of the planes $$x-2 y+z+2=0$$ and $$3 x-y-z+1=0$$ and also passing through $$(1,1,1)$$ is

If $$\mathbf{a}$$ and $$\mathbf{b}$$ are two vectors such that $$\frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}||\mathbf{b}|} < 0$$ and $$|\mathbf{a} \cdot \mathbf{b}|=|\mathbf{a} \times \mathbf{b}|$$ then the angle between the vectors $$\mathbf{a}$$ and $$\mathbf{b}$$ is

Let $$L_1$$ (resp, $$L_2$$ ) be the line passing through $$2 \hat{\mathbf{i}}-\hat{\mathbf{k}}$$ (resp. $$2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-3 \hat{\mathbf{k}})$$ and parallel to $$3 \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$$ ( resp. $$\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{\mathbf{k}}$$ ). Then the shortest distance between the lines $$L_1$$ and $$L_2$$ is equal to

Let $$\mathbf{a}, \mathbf{b}$$ and $$\mathbf{c}$$ be three-unit vectors and $$\mathbf{a} \cdot \mathbf{b}=\mathbf{a} \cdot \mathbf{c}=0$$. If the angle between $$\mathbf{b}$$ and $$\mathbf{c}$$ is $$\frac{\pi}{3}$$. Then $$[\mathbf{a b c}]^2$$ is equal to

Let $$x$$ and $$y$$ are real numbers. If $$\mathbf{a}=(\sin x) \hat{\mathbf{i}}+(\sin y) \hat{\mathbf{j}}$$ and $$\mathbf{b}=(\cos x) \hat{\mathbf{i}}+(\cos y) \hat{\mathbf{j}}$$, then $$|\mathbf{a} \times \mathbf{b}|$$ is

The mean and variance of $$n$$ observations $$x_1, x_2, x_3, \ldots . . x_n$$ are 5 and 0 respectively. If $$\sum_{i=1}^n x_i^2=400$$, then the value of $$n$$ is equal to

Mean of the values $$\sin ^2 10 Y, \sin ^2 20 Y, \sin ^2 30 Y, \ldots \ldots \ldots ., \sin ^2 90 Y$$ is

P speaks truth in 70% of the cases and Q in 80% of the cases. In what percent of cases are they likely to agree in stating the same fact

If $$A$$ and $$B$$ are two events with $$P(A \cap B)=\frac{1}{3}, P(A \cup B)=\frac{5}{6}$$ and $$P\left(A^C\right)=\frac{1}{2}$$, then the value of $$P\left(B^C\right)$$ is

A coin is tossed 2020 times. The probability of getting head on 1947th toss is

A discrete random variable X takes values 10, 20, 30 and 40. with probability 0.3, 0.3, 0.2 and 0.2 respectively. Then the expected value of X is

Let $$X$$ be a random variable which takes values $$1,2,3,4$$ such that $$P(X=r)=K r^3$$ where $$r=1,2,3,4$$ then

Given, two fixed points $$A(-2,1)$$ and $$B(3,0)$$. Find the locus of a point $$P$$ which moves such that the angle $$\angle A P B$$ is always a right angle.

When the coordinate axes are rotated through an angle 135$$\Upsilon$$, the coordinates of a point $$P$$ in the new system are known to be $$(4,-3)$$. Then find the coordinates of $$P$$ in the original system.

If $$A(4,7), B(-7,8)$$ and $$C(1,2)$$ are the vertices of $$\triangle A B C$$, then the equation of perpendicular bisector of the side $$A B$$ is

The ratio in which the straight line $$3 x+4 y=6$$ divides the join of the points $$(2,-1)$$ and $$(1,1)$$ is

Find the equation of a line passing through the point $$(4,3)$$, which cuts a triangle of minimum area from the first quadrant.

If the orthocenter of the triangle formed by the lines $$2 x+3 y-1=0, x+2 y+1=0$$ and $$a x+b y-1=0$$ lies at origin, then $$\frac{1}{a}+\frac{1}{b}$$ is equal to

The equation $$8 x^2-24 x y+18 y^2-6 x+9 y-5=0$$ represents a

Find the angle between the pair of lines represented by the equation $$x^2+4 x y+y^2=0$$.

If the acute angle between lines $$a x^2+2 h x y+b y^2=0$$ is $$\frac{\pi}{4}$$, then $$4 h^2$$ is equal to

The angle between the lines represented by $$\cos \theta(\cos \theta+1) x^2-\left(2 \cos \theta+\sin ^2 \theta\right) x y+(1-\cos \theta) y^2=0$$ is

The equations of the tangents to the circle $$x^2+y^2=4$$ drawn from the point $$(4,0)$$ are

If $$P(-9,-1)$$ is a point on the circle $$x^2+y^2+4 x+8 y-38=0$$, then find equation of the tangent drawn at the other end of the diameter drawn through $$P$$

Find the equation of a circle whose radius is 5 units and passes through two points on the $$X$$-axis, which are at a distance of 4 units from the origin

If a foot of the normal from the point $$(4,3)$$ to a circle is $$(2,1)$$ and $$2 x-y-2=0$$, is a diameter of the circle, then the equation of circle is

The length of the tangent from any point on the circle $$(x-3)^2+(y+2)^2=5 r^2$$ to the circle $$(x-3)^2+(y+2)^2=r^2$$ is 16 units, then the area between the two circles in square units is

The equation of the circle, which cuts orthogonally each of the three circles

$$\begin{aligned} & x^2+y^2-2 x+3 y-7=0, \\ & x^2+y^2+5 x-5 y+9=0 \text { and } \\ & x^2+y^2+7 x-9 y+29=0 \end{aligned}$$

Find the equation of the parabola which passes through (6, $$-$$2), has its vertex at the origin and its axis along the Y-axis.

In an ellipse, if the distance between the foci is 6 units and the length of its minor axis is 8 units, then its eccentricity is

If the points (2, 4, $$-$$1), (3, 6, $$-$$1) and (4, 5, $$-$$1) are three consecutive vertices of a parallelogram, then its fourth vertex is

$$A(-1,2-3), B(5,0,-6)$$ and $$C(0,4,-1)$$ are the vertices of a $$\triangle A B C$$. The direction cosines of internal bisector of $$\angle B A C$$ are

If the projections of the line segment AB on xy, yz and zx planes are $$\sqrt{15},\sqrt{46},7$$ respectively, then the projection of AB on Y-axis is

Find the equation of the plane passing through the point $$(2,1,3)$$ and perpendicular to the planes $$x-2 y+2 z+3=0$$ and $$3 x-2 y+4 z-4=0$$.

If $$f(x)=\left\{\begin{array}{cc}\frac{e^{\alpha x}-e^x-x}{x^2}, & x \neq 0 \\ \frac{3}{2}, & x=0\end{array}\right.$$

Find the value of $$\alpha$$ for which the function $$f$$ is continuous

The value of $$k(k > 0)$$, for which the function $$f(x)=\frac{\left(e^x-1\right)^4}{\sin \left(\frac{x^2}{k^2}\right) \log \left(1+\frac{x^2}{2}\right)}$$, where $$x \neq 0$$ and $$f(0)=8$$

If $$\log \left(\sqrt{1+x^2}-x\right)=y\left(\sqrt{1+x^2}\right)$$, then $$\left(1+x^2\right) \frac{d y}{d x}+x y$$ is equal to

If $$f^{\prime \prime}(x)$$ is continuous at $$x=0$$ and $$f^{\prime \prime}(0)=4$$, then find the following value. $$\lim _\limits{x \rightarrow 0} \frac{2 f(x)-3 f(2 x)+f(4 x)}{x^2}$$ is equal to

If $$y=e^{x^2+e^{x^2+e^{x^2+\cdots \infty}}}$$, then $$\frac{d y}{d x}$$ is equal to

$$\frac{d}{d x}\left[\tan ^{-1}\left(\frac{\cos x}{1+\sin x}\right)\right]$$ is equal to

The maximum value of $$f(x)=\sin (x)$$ in the interval $$\left[\frac{-\pi}{2}, \frac{\pi}{2}\right]$$ is

Given, $$f(x)=x^3-4x$$, if x changes from 2 to 1.99, then the approximate change in the value of $$f(x)$$ is

If the curves $$\frac{x^2}{a^2}+\frac{y^2}{4}=1$$ and $$y^3=16 x$$ intersect at right angles, then $$a^2$$ is equal to

Let $$x$$ and $$y$$ be the sides of two squares such that, $$y=x-x^2$$. The rate of change of area of the second square with respect to area of the first square is

If $$f^{\prime \prime}(x)$$ is a positive function for all $$x \in R, f^{\prime}(3)=0$$ and $$g(x)=f\left(\tan ^2(x)-2 \tan (x)+4\right)$$ for $$0 < x <\frac{\pi}{2}$$, then the interval in which $$g(x)$$ is increasing is

$$\int \frac{1+x+\sqrt{x+x^2}}{\sqrt{x}+\sqrt{1+x}} d x$$ is equal to

$$\int(\cos x) \log \cot \left(\frac{x}{2}\right) d x$$ is equal to

$$\int \sqrt{e^{4 x}+e^{2 x}} d x$$ is equal to

If $$\int \frac{1}{1+\sin x} d x=\tan (f(x))+c$$, then $$f^{\prime}(0)$$ is equal to

If $$\int_0^{\pi / 2} \tan ^n(x) d x=k \int_0^{\pi / 2} \cot ^n(x) d x$$, then

$$\int_0^2 x e^x d x$$ is equal to

If $$x^2+y^2=1$$, then

The speed of ripples $$(v)$$ on water surface depends on surface tension $$(\sigma)$$, density $$(\rho)$$ and wavelength $$(\lambda)$$. Then, the square of speed $$(v)$$ is proportional to

An object travelling at a speed of 36 km/h comes to rest in a distance of 200 m after the brakes were applied. The retardation produced by the brakes is

A ball is projected upwards. Its acceleration at the highest point is

A 500 kg car takes a round turn of radius 50 m with a velocity of 36 km/h. The centripetal force acting on the car is

A motor cyclist wants to drive in horizontal circles on the vertical inner surface of a large cylindrical wooden well of radius $$8.0 \mathrm{~m}$$, with minimum speed of $$5 \sqrt{5} \mathrm{~ms}^{-1}$$. The minimum value of coefficient of friction between the tyres and the wall of the well must be $$\left(g=10 \mathrm{~ms}^{-2}\right)$$

Two blocks $$A$$ and $$B$$ of masses $$4 \mathrm{~kg}$$ and $$6 \mathrm{~kg}$$ are as shown in the figure. A horizontal force of $$12 \mathrm{~N}$$ is required to make $$A$$ slip over $$B$$. Find the maximum horizontal force $$F_B$$ that can be applied on $B$, so that both $$A$$ and $$B$$ move together (take, $$g=10 \mathrm{~ms}^{-2}$$ )

What is the shape of the graph between speed and kinetic energy of a body?

A quarter horse power motor runs at a speed of 600 rpm. Assuming 60% efficiency, the work done by the motor in one rotation is

A particle of mass m is projected with a velocity u making an angle $$\theta$$ with the horizontal. The magnitude of angular momentum of the projectile about the point of projection when the particle is at its maximum height is

A sphere of mass m is attached to a spring of spring constant k and is held in unstretched position over an inclined plane as shown in the figure. After letting the sphere go, find the maximum length by which the spring extends, given the sphere only rolls.

A girl of mass M stands on the rim of a frictionless merry-go-round of radius R and rotational inertia I, that is not moving. She throws a rock of mass m horizontally in a direction that is tangent to the outer edge of the merry-go-round. The speed of the rock, relative to the ground is v. Afterwards, the linear speed of the girl is

A block of mass $$\mathrm{l} \mathrm{kg}$$ is fastened to a spring of spring constant of $$100 ~\mathrm{Nm}^{-1}$$. The block is pulled to a distance $$x=10 \mathrm{~cm}$$ from its equilibrium position $$(x=0 \mathrm{~cm})$$ on a frictionless surface, from rest at $$t=0$$. The kinetic energy and the potential energy of the block when it is $$5 \mathrm{~cm}$$ away from the mean position is

The scale of a spring balance which can measure from 0 to $$15 \mathrm{~kg}$$ is $$0.25 \mathrm{~m}$$ long. If a body suspended from this balance oscillates with a time period $$\frac{2 \pi}{5} \mathrm{~s}$$, neglecting the mass of the spring, find the mass of the body suspended.

The distance through which one has to dig the Earth from its surface, so as to reach the point where the acceleration due to gravity is reduced by 40% of that at the surface of the Earth, is (radius of Earth is 6400 km)

Infinite number of masses each of 3kg are placed along a straight line at the distances of 1 m, 2m, 4m, 8m, ...... from a point O on the same line. If G is the universal gravitational constant, then the magnitude of gravitational field intensity at O is

Young's modulus of a wire is $$2 \times 10^{11} \mathrm{Nm}^{-2}$$. If an external stretching force of $$2 \times 10^{11} \mathrm{~N}$$ is applied to a wire of length $$L$$. The final length of the wire is (cross-section = unity)

Identify the incorrect statement regarding Reynold's number $$\left(R_e\right)$$.

Expansion during heating

Match the following.

Which of the following is not a reversible process?

Which one of the graphs below best illustrates the relationship between internal energy U of an ideal gas and temperature T of the gas in K?

A refrigerator with coefficient of performance 0.25 releases 250 J of heat to a hot reservoir. The work done on the working substance is

A vessel has 6 g of oxygen at pressure p and temperature 400 K. A small hole is made in it, so that oxygen leaks out. How much oxygen leaks out if the final pressure is p/2 and temperature is 300 K?

Match the following.

Light of wavelength $$300 \mathrm{~nm}$$ in medium $$A$$ enters into medium $$B$$ through a plane surface. If the frequency of light is $$5 \times 10^{14} \mathrm{~Hz}$$ and the ratio of speed in medium $$A$$ to that in medium $$B$$ is $$4 / 5$$, the absolute refractive index of medium $$B$$ is

In Young’s double slit experiment, the separation between the slits is halved and the distance between the screen is doubled. The fringe width is

Gauss's law helps in

Charge on the outer sphere is $$q$$ and the inner sphere is grounded. The charge on the inner sphere is $$q^{\prime}$$, for $$\left(r_2 > r_1\right)$$. Then,

Four capacitors with capacitances $$C_1=l \propto \mathrm{F}, C_2=1.5 \propto \mathrm{F}, C_3=2.5 \propto \mathrm{F}$$ and $$C_4=0.5 \propto \mathrm{F}$$ are connected as shown and are connected to a $$30 \mathrm{~V}$$ source. The potential difference between points $$a$$ and $$b$$ is

In a potentiometer of 10 wires, the balance point is obtained on the 6th wire. To shift the balance point to 8th wire, we should

In a co-axial, straight cable, the central conductor and the outer conductor carry equal currents in opposite directions. The magnetic field is zero

The magnetic field, of a given length of wire for single turn coil, at its centre is B, then its value for two turns coil for the same wire is

A solenoid of length $$60 \mathrm{~cm}$$ with 15 turns per $$\mathrm{cm}$$ and area of cross-section $$4 \times 10^{-3} \mathrm{~m}^2$$ completely surrounds another co-axial solenoid of same length and area of cross-section $$2 \times 10^{-3} \mathrm{~m}^2$$ with 40 turns per $$\mathrm{cm}$$. Mutual inductance of the system is

A bulb of resistance $$280 \Omega$$ is supplied with a 200 V AC supply. What is the peak current?

The magnetic field of a plane electromagnetic wave is given by $$B=(400 \propto \mathrm{T})\sin \left[\left(4.0 \times 10^{15} \mathrm{~s}^{-1}\right)\left(t-\frac{x}{c}\right)\right]$$. Average energy density corresponding to the electric field is

The de-Broglie wavelength associated with a proton under the influence of an electric potential of 100 V is

The ionisation potential of hydrogen atom is 13.6 V. How much energy need to be supplied to ionise the hydrogen atom in the first excited state?

Which of the following statement is correct?

The length of germanium rod is $$0.925 \mathrm{~cm}$$ and its area of cross-section is $$1 \mathrm{~mm}^2$$. If for germanium $$n_i=2.5 \times 10^{19} \mathrm{~m}^{-3}, \propto_n=0.19 \mathrm{~m}^2 / \mathrm{V}-\mathrm{s}, \propto_e=0.39 \mathrm{~m}^2 / \mathrm{V}$$-s, then the resistance of the rod is

In an amplitude modulated signal, the maximum amplitude is $$15 \mathrm{~V}$$ and minimum amplitude is $$5 \mathrm{~V}$$. The amplitude of modulating wave will be