If $$\Delta x$$ is the uncertainty in position and $$\Delta v$$ is the uncertainty in velocity of a particle are equal, the correct expression for uncertainty in momentum for the same particle is

The number of radial nodes and angular nodes of a $$4 f$$-orbital are respectively

Lithium shows diagonal relationship with element '$$X$$' and aluminium with $$Y . X$$ and $$Y$$ respectively are

The correct order of the metallic character of the elements $$\mathrm{Be}, \mathrm{Al}, \mathrm{Na}, \mathrm{K}$$ is

Choose the correct option from the following.

The bond lengths of $$\mathrm{C}_2, \mathrm{~N}_2$$ and $$\mathrm{B}_2$$ molecules are $$X_1, X_2$$ and $$X_3 \mathrm{~pm}$$ respectively. The correct order of their bond lengths is

Among the gases a, b, c , d, e and f, the gases that show only positive deviation from ideal behaviour at all pressures in the graph are

The statement related to law of definite proportions is

What are the oxidation states of three Br atoms in $$\mathrm{Br}_3 \mathrm{O}_8$$ molecule?

Identify the reaction/process in which the entropy increases.

State $$1 \rightleftharpoons$$ State $$2 \rightleftharpoons$$ State 3 $$\left(\begin{array}{l}T=300 \mathrm{~K} \\ p=15 \mathrm{bar} \\ 1 \mathrm{~mol}\end{array}\right)\left(\begin{array}{l}T=300 \mathrm{~K} \\ p=10 \mathrm{bar} \\ 1 \mathrm{~mol}\end{array}\right)\left(\begin{array}{l}T=300 \mathrm{~K} \\ p=5 \mathrm{bar} \\ 1 \mathrm{~mol}\end{array}\right)$$

Above shows a cyclic process. Calculate the total work done during one complete cycle. [Assume a single step to reach the next state].

The formation of ammonia from its constituent elements is an exothermic reaction. The effect of increased temperature on the reaction equilibrium is

Equal volumes of 0.5 N acetic acid and 0.5 N sodium acetate are mixed. What is the pH of resultant solution? ($$\mathrm{p} K_a$$ of acetic acid $$=4.75$$)

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

$$\begin{aligned} & \underline{X}+D_2 \mathrm{O} \longrightarrow C_2 D_2+P \\ & \underline{Y}+D_2 \mathrm{O} \longrightarrow C D_4+Q \end{aligned}$$

Assertion (A) MgSO$$_4$$ is readily soluble in water.

Reason (R) The greater hydration enthalpy of Mg$$^{2+}$$ ions overcome is lattice enthalpy.

Identify A and B from the following reaction,

$$\mathrm{NaNO}_3 \xrightarrow{\Delta} x A+y B$$

Identify the correct statements about boron.

I. It has high melting point.

II. It has high density.

III. It has high electrical conductivity.

IV. B-10 isotope of it has high ability to absorb neutrons.

Which of the following tetrahalides does not exist?

The correct order of acidity of the following compounds is

The compound or ion which is not aromatic in the following is

The number of network solids and ionic solids in the list given below is respectively. $$\mathrm{H}_2 \mathrm{O}$$ (ice), $$\mathrm{AlN}, \mathrm{Cu}, \mathrm{CaF}_2$$, diamond, MgO , $$\mathrm{CCl}_4, \mathrm{ZnS}, \mathrm{Ag}, \mathrm{NaCl}, \mathrm{SiO}_2$$

If molten NaCl contains $$\mathrm{SrCl}_2$$ as impurity, crystallisation can generate

At $$T(\mathrm{~K}) \times \mathrm{g}$$ of a non-volatile solid (molar mass $$78 \mathrm{~g} \mathrm{~mol}^{-1}$$) when added to 0.5 kg water, lowered its freezing point by $$1.0^{\circ} \mathrm{C}$$. What is $$x$$ (in g)? ($$K_f$$ of water at $$T(\mathrm{~K})=1.86 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}$$)

Assertion (A) Blood cells collapse when suspended in saline water.

Reason (R) Cell membrane dissolves in saline water.

The reduction potential of hydrogen electrode at $$25^{\circ} \mathrm{C}$$ in a neutral solution is ($$p_{\mathrm{H}_2}=1$$ bar)

. The rate constant for a zero order reaction $$A \longrightarrow$$ products is $$0.0030 \mathrm{~mol} \mathrm{~L}^{-1} \mathrm{~S}^{-1}$$. How long it will take for the initial concentration of $$A$$ to fall from 0.10 M to 0.075 M ?

The diameters range of colloidal particles is approximately.

Photographic plates are prepared by coating emulsion of which of the following in gelatin?

What are $$x$$ and $$y$$ in the following reaction?

$$x \mathrm{~Pb}_3 \mathrm{O}_4 \longrightarrow y \mathrm{PbO}+\mathrm{O}_2$$

Assertion (A) HCl gas is dried by passing through concentrated H$$_2$$SO$$_4$$.

Reason (R) HCl gas reacts with NH$$_3$$ that gives white fumes.

The catalyst used in the manufacture of polyethylene is a mixture of

Which of the following is correct related to the colours of $$\mathrm{TiCl}_3(X)$$ and $$\left[\mathrm{Ti}\left(\mathrm{H}_2 \mathrm{O}\right)_6\right] \mathrm{Cl}_3(Y)$$ ?

Which hormone tends to increase the blood glucose level in human?

Which of the following molecules is eliminated during peptide bond formation?

Identify the major product formed from the following.

When 1-chlorobutane is treated with aqueous KOH it gives P. However, when it is treated with alcoholic KOH it gives Q. Identify the products P and Q respectively

Identify the major product formed in the following reaction sequence

Arrange the following in increasing order of their reactivity for nucleophilic addition reaction.

(A)

(B)

(C)

(D)

In the presence of peroxide, styrene reacts with HBr to give $$X$$. When $$X$$ is reacted with magnesium in dry ether followed by $$\mathrm{CO}_2$$ and hydrolysis gave $$Y$$. Treatment of Y with $$\mathrm{PCl}_5$$ and then next with $$\mathrm{H}_2$$. $$\mathrm{Pd}-\mathrm{BaSO}_4$$ gave Z . What is Z ?

Arrange the following in decreasing order of their pK_b values

A. ,$$\mathrm{CH}_3 \mathrm{NH}_2$$

B. $$(\mathrm{CH}_3)_3 \mathrm{~N}$$

C.

D.

The range of the real valued function $$f(x)=\sqrt{\frac{x^2+2 x+8}{x^2+2 x+4}}$$ is

If $$f(x)=\sqrt{2-x^2}$$ and $$g(x)=\log (1-x)$$ are two real valued functions, then the domain of the function $$(f+g)(x)$$ is

For $$i=1,2,3$$ and $$j=1,23$$ If $$a_i^2+b_i^2+c_i^2=1, a_i a_j+b_i b_j+c_i c_j=0, \forall i \neq j$$ and $$A=\left[\begin{array}{lll}a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3\end{array}\right]$$, then $$\operatorname{det}\left(A A^T\right)=$$

If $$A=\frac{1}{7}\left[\begin{array}{ccc}3 & -2 & 6 \\ -6 & -3 & 2 \\ -2 & 6 & 3\end{array}\right]$$, then

If $$A=\left[\begin{array}{cc}\alpha^2 & 5 \\ 5 & -\alpha\end{array}\right]$$ and $$\operatorname{det}\left(A^{10}\right)=1024$$, then $$\alpha=$$

Let $$A=\left[\begin{array}{ccc}5 & \sin ^2 \theta & \cos ^2 \theta \\ -\sin ^2 \theta & -5 & 1 \\ \cos ^2 \theta & 1 & 5\end{array}\right]$$. Then, maximum value of $$\operatorname{det}(A)$$ is

$$i z^3+z^2-z+i=0 \Rightarrow|z|=$$

If $$\frac{x-1}{3+i}+\frac{y-1}{3-i}=i$$, then the true statement among the following is

If the identity $$\cos ^4 \theta=a \cos 4 \theta+b \cos 2 \theta+c$$ holds for some $$a, b, c \in Q$$ then $$(a, b, c)=$$

The number of integer solutions of the equation $$|1-i|^x=2^x$$ is

If $$f(x)=a x^2+b x+c$$ for some $$a, b, c \in R$$ with $$a+b+c=3$$ and $$f(x+y)=f(x)+f(y)+x y, \forall x, y \in R$$. Then, $$\sum_\limits{n=1}^{10} f(n)=$$

The number of positive real roots of the equation $$3^{x+1}+3^{-x+1}=10$$ is

The number of real roots of the equation $$\sqrt{\frac{x}{1-x}}+\sqrt{\frac{1-x}{x}}=\frac{13}{6}$$ is

If $$4^x-3^{x-1 / 2}=3^{x+1 / 2}-2^{2 x-1}$$, then the value of $$x$$ is

The total number of permutations of $$n$$ different things taken not more than $$r$$ at a time, when each thing may be repeated any number of times is

How many chords can be drawn through 21 points on a circle?

If a polygon of $$n$$ sides has 560 diagonals, then $$n=$$

A person writes letters to 6 friends and addresses the corresponding envelopes. In how many ways can the letters be placed in the envelopes so that at least two of them are in the wrong envelopes? Notation $$D_n=n!\left(\sum_\limits{i=0}^n \frac{(-1)^i}{i!}\right)$$

If $$\frac{x^4+24 x^2+28}{\left(x^2+1\right)^3}=\frac{A x+B}{x^2+1}$$ $$+\frac{C x+D}{\left(x^2+1\right)^2}+\frac{E x+F}{\left(x^2+1\right)^3},$$ then the value of $$A+B+C+D+E+F=$$

The value of $$\frac{\sin \theta+\sin 3 \theta}{\cos \theta+\cos 3 \theta}$$ is

If $$(1+\tan 1^{\circ})(1+\tan 2^{\circ}) \ldots(1+\tan 45^{\circ})=2^n,$$ then $$n=$$

$$\frac{\cos \theta}{1-\tan \theta}+\frac{\sin \theta}{1-\cot \theta}=$$

In a $$\triangle A B C$$, if $$a \neq b, \frac{a \cos A-b \cos B}{a \cos B-b \cos A}+\cos C=$$

If $$\operatorname{cosech} x=\frac{4}{5}$$, then $$\sinh x=$$

The value of $$\frac{1+\tan \mathrm{h} x}{1-\tan \mathrm{h} x}$$ is

If in a $$\triangle A B C, a=2, b=3$$ and $$c=4$$, then $$\tan (A / 2)=$$

If the angles of a $$\triangle A B C$$ are in the ratio $$1: 2: 3$$, then the corresponding sides are in the ratio

In a $$\triangle A B C, r_1 \cot \frac{A}{2}+r_2 \cot \frac{B}{2}+r_3 \cot \frac{C}{2}=$$

The point of intersection of the lines $$\mathbf{r}=2 \mathbf{b}+t(6 \mathbf{c}-\mathbf{a})$$ and $$\mathbf{r}=\mathbf{a}+s(\mathbf{b}-3 \mathbf{c})$$ is

In quadrilateral $$A B C D, \mathbf{A B}=\mathbf{a}, \mathbf{B C}=\mathbf{b}$$. $$\mathbf{D A}=\mathbf{a}-\mathbf{b}, M$$ is the mid-point of $$B C$$ and $$X$$ is a point on DM such that, $$\mathbf{D X}=\frac{4}{5}$$ DM. Then, the points $$A, X$$ and $$C$$.

The vectors $$3 \mathbf{a}-5 \mathbf{b}$$ and $$2 \mathbf{a}+\mathbf{b}$$ are mutually perpendicular and the vectors $$a+4 b$$ and $$-\mathbf{a}+\mathbf{b}$$ are also mutually perpendicular, then the acute angle between $$\mathbf{a}$$ and $$\mathbf{b}$$ is

Let $$\mathbf{a}=x \hat{i}+y \hat{j}+z \hat{k}$$ and $$x=2 y$$. If $$|\mathbf{a}|=5 \sqrt{2}$$ and a makes an angle of $$135^{\circ}$$ with the Z-axis, then $$\mathbf{a}=$$

Let $$\mathbf{a}, \mathbf{b}, \mathbf{c}$$ be the position vectors of the vertices of a $$\triangle A B C$$. Through the vertices, lines are drawn parallel to the sides to form the $$\Delta A^{\prime} B^{\prime} C^{\prime}$$. Then, the centroid of $$\Delta A^{\prime} B^{\prime} C^{\prime}$$ is

The mean deviation about the mean for the following data.

$$5,6,7,8,6,9,13,12,15 \text { is }$$

A box contains 100 balls, numbered from 1 to 100 . If 3 balls are selected one after the other at random with replacement from the box, then the probability that the sum of the three numbers on the balls selected from the box is an odd number, is

In a lottery, containing 35 tickets, exactly 10 tickets bear a prize. If a ticket is drawn at random, then the probability of not getting a prize is

A bag contains 7 green and 5 black balls. 3 balls are drawn at random one after the other. If the balls are not replaced, then the probability of all three balls being green is

If $$x$$ is chosen at random from the set $$\{1,2,3, 4\}$$ and $$y$$ is chosen at random from the set $$\{5,6,7\}$$, then the probability that $$x y$$ will be even is

The discrete random variables $$X$$ and $$Y$$ are independent from one another and are defined as $$X \sim B(16,0.25)$$ and $$Y \sim P(2)$$. Then, the sum of the variance of $$X$$ and $$Y$$ is

If 6 is the mean of a Poisson distribution, then $$P(X \geq 3)=$$

A stick of length $$r$$ units slides with its ends on coordinate axes. Then, the locus of the mid-point of the stick is a curve whose length is

The least distance from origin to a point on the line $$y=x+3$$ which lies at a distance of 2 units from $$(0,3)$$ is

Starting from the point $$A(-3,4)$$, a moving object touches $$2 x+y-7=0$$ at $$B$$ and reaches the point $$C(0,1)$$. If the object travels along the shortest path, the distance between $$A$$ and $$B$$ is

Suppose a triangle is formed by $$x+y=10$$ and the coordinate axes. Then, the number of points $$(x, y)$$ where $$x$$ and $$y$$ are natural numbers, lying inside the triangle is

If the lines represented by $$a x^2+2 h x y+b y^2+2 g x+2 f y+c=0$$ intersect on the $$X$$-axis, which of the following is in general incorrect?

For $$\alpha \in\left[0, \frac{\pi}{2}\right]$$, the angle between the lines represented by $$[x \cos \theta-y] [(\cos \theta+\tan \alpha) x-(1-\cos \theta \tan \alpha) y]=0$$ is

The locus of centers of the circles, possessing the same area and having $$3 x-4 y+4=0$$ and $$6 x-8 y-7=0$$ as their common tangent, is

For any two non-zero real numbers $$a$$ and $$b$$ if this line $$\frac{x}{a}+\frac{y}{b}=1$$ is a tangent to the circle $$x^2+y^2=1$$, then which of the following is true?

The length of the intercept on the line $$4 x-3 y-10=0$$ by the circle $$x^2+y^2-2 x+4 y-20=0$$ is

The pole of the line $$\frac{x}{a}+\frac{y}{b}=1$$ with respect to the circle $$x^2+y^2=c^2$$ is

If the tangent at the point $$P$$ on the circle $$x^2+y^2+6 x+6 y=2$$ meets the straight line $$5 x-2 y+6=0$$ at a point $$Q$$ on the $$Y$$-axis, then the length of $$P Q$$ is

Suppose a parabola with focus at $$(0,0)$$ has $$x-y+1=0$$ as its tangent at the vertex. Then, the equation of its directrix is

The eccentric angle of a point on the ellipse $$x^2+3 y^2=6$$ lying at a distance of 2 units from its centre is

Let origin be the centre, $$( \pm 3,0)$$ be the foci and $$\frac{3}{2}$$ be the eccentricity of a hyperbola. Then, the line $$2 x-y-1=0$$

The locus of a variable point whose chord of contact w.r.t. the hyperbola $$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$$ subtends a right angle at the origin is

If the point $$(a, 8,-2)$$ divides the line segment joining the points $$(1,4,6)$$ and $$(5,2,10)$$ in the ratio $$m: n$$, then $$\frac{2 m}{n}-\frac{a}{3}=$$

If $$(a, b, c)$$ are the direction ratios of a line joining the points $$(4,3,-5)$$ and $$(-2,1,-8)$$, then the point $$P(a, 3 b, 2 c)$$ lies on the plane

The $$x$$-intercept of a plane $$\pi$$ passing through the point $$(1,1,1)$$ is $$\frac{5}{2}$$ and the perpendicular distance from the origin to the plane $$\pi$$ is $$\frac{5}{7}$$. If the $$y$$-intercept of the plane $$\pi$$ is negative and the $$z$$-intercept is positive, then its $$y$$-intercept is

Let $$f: R^{+} \longrightarrow R^{+}$$ be a function satisfying $$f(x)-x=\lambda$$ (constant), $$\forall x \in R^{+}$$ and $$f(x f(y))=f(x y)+x, \forall x, y, \in R^{+}$$. Then, $$\lim _\limits{x \rightarrow 0} \frac{(f(x))^{1 / 3}-1}{(f(x))^{1 / 2}-1}=$$

$$\begin{aligned} & \text { If } \lim _{x \rightarrow 0} \frac{|x|}{\sqrt{x^4+4 x^2+5}}=k \\ & \lim _{x \rightarrow 0} x^4 \sin \left(\frac{1}{3 \sqrt{x}}\right)=l \text {. Then, } k+l= \end{aligned}$$

If $$\lim _\limits{n \rightarrow \infty} x^n \log _e x=0$$, then $$\log _x 12=$$

If $$f(x)=\cot ^{-1}\left(\frac{x^x+x^{-x}}{2}\right)$$, then $$f^{\prime}(1)=$$

If $$f(x)=\operatorname{Max}\{3-x, 3+x, 6\}$$ is not differentiable at $$x=a$$, and $$x=b$$, then $$|a|+|b|=$$

If $$x^3-2 x^2 y^2+5 x+y-5=0$$, then at $$(\mathrm{l}, \mathrm{l}), y^{\prime \prime}(\mathrm{l})=$$

If the curves $$y=x^3-3 x^2-8 x-4$$ and $$y=3 x^2+7 x+4$$ touch each other at a point $$P$$, then the equation of common tangent at $$P$$ is

If $$a x+b y=1$$ is a normal to the parabola $$y^2=4 p x$$, then the condition is

The maximum value of $$f(x)=\frac{x}{1+4 x+x^2}$$ is

The minimum value of $$f(x)=x+\frac{4}{x+2}$$ is

The condition that $$f(x)=a x^3+b x^2+c x+d$$ has no extreme value is

Assertion (A) If $$I_n=\int \cot ^n x d x$$, then $$I_6+I_4=\frac{-\cot ^5 x}{5}$$

Reason (R) $$\int \cot ^n x d x=\frac{-\cot ^{n-1} x}{n} -\int \cot ^{n-2} x d x$$

If $$I_n=\int \tan ^n x d x$$, and $$I_0+I_1+2 I_2+2 I_3+2 I_4 +I_5+I_6=\sum_\limits{k=1}^n \frac{\tan ^k x}{k}$$, then $$n=$$

$$\int \frac{e^{\cot x}}{\sin ^2 x}(2 \log \operatorname{cosec} x+\sin 2 x) d x=$$

The parametric form of a curve is $$x=\frac{t^3}{t^2-1} y=\frac{t}{t^2-1}$$, then $$\int \frac{d x}{x-3 y}=$$

$$\int_0^1 a^k x^k d x=$$

Let $$\alpha$$ and $$\beta(\alpha<\beta)$$ are roots of $$18 x^2-9 \pi x+\pi^2=0, f(x)=x^2, g(x)=\cos x$$. Then, $$\int_\alpha^\beta x(g \circ f(x)) d x=$$

$$\int_0^\pi x\left(\sin ^2(\sin x)+\cos ^2(\cos x)\right) d x=$$

$$\lim _\limits{n \rightarrow \infty}\left(\frac{1}{1^5+n^5}+\frac{2^4}{2^5+n^5}+\frac{3^4}{3^5+n^5}+\ldots+\frac{n^4}{n^5+n^5}\right)=$$

If the solution of $$\frac{d y}{d x}-y \log _e 0.5=0, y(0)=1$$, and $$y(x) \rightarrow k$$, as $$x \rightarrow \infty$$, then $$k=$$

At any point $$(x, y)$$ on a curve if the length of the subnormal is $$(x-1)$$ and the curve passes through $$(1,2)$$, then the curve is a conic. A vertex of the curve is

$$y=A e^x+B e^{-2 x}$$ satisfies which of the following differential equations?

If $$N_A, N_B$$ and $$N_C$$ are the number of significant figures in $$A=0.001204 \mathrm{~m}, B=43120000 \mathrm{~m}$$ and $$C=1.200 \mathrm{~m}$$ respectively, then

A car covers a distance at speed of $$60 \mathrm{~km} \mathrm{~h}^{-1}$$. It returns and comes back to the original point moving at a speed of $$v$$. If the average speed for the round trip is $$48 \mathrm{~kmh}^{-1}$$, then the magnitude of $$v$$ is

A car travels with a speed of $$40 \mathrm{~km} \mathrm{~h}^{-1}$$. Rain drops are falling at a constant speed vertically. The traces of the rain on the side windows of the car make an angle of $$30^{\circ}$$ with the vertical. The magnitude of the velocity of the rain with respect to the car is

A projectile with speed $$50 \mathrm{~ms}^{-1}$$ is thrown at an angle of $$60^{\circ}$$ with the horizontal. The maximum height that can be reached is (acceleration due to gravity $$=10 \mathrm{~ms}^{-2}$$)

Two rectangular blocks of masses 40 kg and 60 kg are connected by a string and kept on a frictionless horizontal table. If a force of 1000 N is applied on 60 kg block away from 40 kg block, then the tension in string is

A ball of mass 0.5 kg moving horizontally at $$10 \mathrm{~ms}^{-1}$$ strikes a vertical wall and rebounds with speed $$v$$. The magnitude of the change in linear momentum is found to be $$8.0 \mathrm{~kg}-\mathrm{~ms}^{-1}$$. The magnitude of $$v$$ is

A mass of 1 kg falls from a height of 1 m and lands on a massless platform supported by a spring having spring constant $$15 \mathrm{~Nm}^{-1}$$ as shown in the figure. The maximum compression of the spring is (acceleration due to gravity $$=10 \mathrm{~ms}^{-2}$$)

A bead of mass 400 g is moving along a straight line under a force that delivers a constant power 1.2 W to the bead. If the bead is initially at rest, the speed it attains after 6 s in $$\mathrm{ms}^{-1}$$

Masses $$m\left(\frac{1}{3}\right)^N \frac{1}{N}$$ are placed at $$x=N$$, when $$N=2,3,4 \ldots \infty$$. If the total mass of the system is $$M$$, then the centre of mass is

Consider a disc of radius $$R$$ and mass $$M$$. A hole of radius $$\frac{R}{3}$$ is created in the disc, such that the centre of the hole is $$\frac{R}{3}$$ away from centre of the disc. The moment of inertia of the system along the axis perpendicular to the disc passing through the centre of the disc is

A hydrometer executes simple harmonic motion when it is pushed down vertically in a liquid of density $$\rho$$. If the mass of hydrometer is $$m$$ and the radius of the hydrometer tube is $$r$$, then the time period of oscillation is

An object undergoing simple harmonic motion takes 0.5 s to travel from one point of zero velocity to the next such point. The angular frequency of the motion is

A projectile is thrown straight upward from the earth's surface with an initial speed $$v=\alpha v_e$$ where $$\alpha$$ is a constant and $$v_e$$ is the escape speed. The projectile travels upto a height 800 km from earth's surface, before it comes to rest. The value of the constant $$\alpha$$ is (radius of the earth $$=6400 \mathrm{~km}$$)

Same tension is applied to the following four wires made of same material. The elongation is longest in

A cone with half the density of water is floating in water as shown in figure. It is depressed down by a small distance $$\delta(\ll< H)$$ and released. The frequency of simple harmonic oscillations of the cone is

Statement (A) When the temperature increases the viscosity of gases increases and the viscosity of liquids decreases.

Statement (B) Water does not wet an oily glass because cohesive force of oil is less than that of water.

Statement (C) A liquid will wet a surface of a solid, if the angle of contact is greater than $$90^{\circ}$$.

A sphere of surface area $$4 \mathrm{~m}^2$$ at temperature 400 K and having emissivity 0.5 is located in an environment of temperature 200 K. The net rate of energy exchange of the sphere is (Stefan Boltzmann constant, $$\sigma=5.67 \times 10^{-8} \mathrm{Wm}^{-2} \mathrm{~K}^4)$$

A Carnot engine operates between a source and a sink. The efficiency of the engine is $$40 \%$$ and the temperature of the sink is $$27^{\circ} \mathrm{C}$$. If the efficiency is to be increased to $$50 \%$$, then the temperature of the source must be increased by

A car engine has a power of 20 kW. The car makes a roundtrip of 1 h. If the thermal efficiency of the engine is $$40 \%$$ and the ambient temperature is 300 K . The energy generated by fuel combustion is

The number of vibrational degree of freedom of a diatomic molecule is

A body is suspended from a string of length 1 m and mass 2 g. The mass of the body to produce a fundamental mode of 100 Hz frequency in the string is (Acceleration due to gravity $$=10 \mathrm{~ms}^{-2}$$)

Electrostatic force between two identical charges placed in vacuum at distance of $$r$$ is F. A slab of width $$\frac{r}{5}$$ and dielectric constant 9 is inserted between these two charges, then the force between the charges is

A ray is incident from a medium of refractive index 2 into a medium of refractive index 1. The critical angle is

An electric dipole with dipole moment $$5 \times 10^{-7} \mathrm{C}-\mathrm{m}$$ is in the electric field of $$2 \times 10^4 \mathrm{NC}^{-1}$$ at an angle of $$60^{\circ}$$ with the direction of the electric field. The torque acting on the dipole is

A capacitor of capacitance $$C_1=1 \mu \mathrm{F}$$ is charged using a 9 V battery. $$C_1$$ is, then removed from the battery and connected to capacitors $$C_2$$ and $$C_3$$ of $$2 \mu \mathrm{F}$$ and $$3 \mu \mathrm{F}$$, respectively as shown in the figure. Find the charge on $$C_3$$ after equilibrium has reached is

Two positive point charges of $$10 \mu \mathrm{C}$$ and $$12 \mu \mathrm{C}$$ are placed 10 cm apart in air. The work done to bring them 6 cm closer is

Current density in a cylindrical wire of radius $$R$$ varies with radial distance as $$\beta\left(r+r_0\right)^2$$. The current through the section of the wire shown in the figure is

A cell can supply currents of 1 A and 0.5 A via resistances of $$2.5 \Omega$$ and $$10 \Omega$$, respectively. The internal resistance of the cell is

Two infinitely long wires each carrying the same current and pointing in $$+y$$ direction are placed in the $$x y$$-plane, at $$x=-2 \mathrm{~cm}$$ and $$x=1 \mathrm{~cm}$$. An electron is fired with speed $$u$$ from the origin making an angle of $$+45^{\circ}$$ from the $$X$$-axis. The force on the electron at the instant it is fired is

[$$B_0$$ is the magnitude of the field at origin due to the wire at $$x=1 \mathrm{~cm}$$ alone].

Two electrons, $$e_1$$ and $$e_2$$ of mass $$m$$ and charge $$q$$ are injected into the perpendicular direction of the magnetic field $$B$$ such that the kinetic energy of $$e_1$$ is double than that of $$e_2$$. The relation of their frequencies of rotation, $$f_1$$ and $$f_2$$ is

A compass needle oscillates 20 times per minute at a place where the dip is $$45^{\circ}$$ and the magnetic field is $$B_1$$. The same needle oscillates 30 times per minute at a place where the dip is $$30^{\circ}$$ and magnetic field is $$B_2$$. Then, $$B_1: B_2$$ is

A plane electromagnetic wave travels in free space along $$Z$$-axis. At a particular point in space, the electric field along $$X$$-axis is $$8.7 \mathrm{~Vm}^{-1}$$. The magnetic field along $$Y$$-axis is

A coil of inductance 0.1 H and resistance $$110 \Omega$$ is connected to a source of 110 V and 350 Hz . The phase difference between the voltage maximum and the current maximum is

If the average power per unit area delivered by an electromagnetic wave is $$9240 \mathrm{~Wm}^{-2}$$. then the amplitude of the oscillating magnetic field in EM wave is

A beam of light with intensity $$10^{-3} \mathrm{~Nm}^{-2}$$ and cross-sectional area $$20 \mathrm{~cm}^2$$ is incident on a fully reflective surface at angle $$45^{\circ}$$. Then, the force exerted by the beam on the surface is

The metal which has the highest work function in the following is

Energy of a stationary electron in the hydrogen atom is $$E=\frac{13.6}{n^2} \mathrm{~eV}$$, then the energies required to excite the electron in hydrogen atom to (a) its second excited state and (b) ionised state, respectively.

The graph of $$\ln \left(\frac{R}{R_0}\right)$$ versus $\ln A$ is where $$R$$ is radius of a nucleus, $$A$$ is its mass number, and $$R_0$$ is constant

Output of following logic circuit is

The maximum number of TV signals, that can be transmitted along a co-axial cable is