If $$\Delta x$$ is the uncertainty in position and $$\Delta v$$ is the uncertainty in velocity of a particle are equal,

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{~p

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

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

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

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

What are $$X$$ and $$Y$$ respectively in the following reactions? $$\begin{aligned} & \underline{X}+D_2 \mathrm{O} \long

Assertion (A) MgSO$$_4$$ is readily soluble in water. Reason (R) The greater hydration enthalpy of Mg$$^{2+}$$ ions over

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 ele

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)

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

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}=

. The rate constant for a zero order reaction $$A \longrightarrow$$ products is $$0.0030 \mathrm{~mol} \mathrm{~L}^{-1}

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}+\math

Assertion (A) HCl gas is dried by passing through concentrated H$$_2$$SO$$_4$$. Reason (R) HCl gas reacts with NH$$_3$$

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{

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. I

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 fo

Arrange the following in decreasing order of their pKb values A. ,$$\mathrm{CH}_3 \mathrm{NH}_2$$ B. $$(\mathrm{CH}_3)_3

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)(

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[\be

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

Let $$A=\left[\begin{array}{ccc}5 & \sin ^2 \theta & \cos ^2 \theta \\ -\sin ^2 \theta & -5 & 1 \\ \cos ^2 \theta & 1 &

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

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 $$f(x+y)=f(x)+f(y)+x y, \forall x, y \in R$$. Th

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 r

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 plac

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}

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

In quadrilateral $$A B C D, \mathbf{A B}=\mathbf{a}, \mathbf{B C}=\mathbf{b}$$. $$\mathbf{D A}=\mathbf{a}-\mathbf{b}, M$

The vectors $$3 \mathbf{a}-5 \mathbf{b}$$ and $$2 \mathbf{a}+\mathbf{b}$$ are mutually perpendicular and the vectors $$a

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 $

Let $$\mathbf{a}, \mathbf{b}, \mathbf{c}$$ be the position vectors of the vertices of a $$\triangle A B C$$. Through the

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 replacemen

In a lottery, containing 35 tickets, exactly 10 tickets bear a prize. If a ticket is drawn at random, then the probabili

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

If $$x$$ is chosen at random from the set $$\{1,2,3, 4\}$$ and $$y$$ is chosen at random from the set $$\{5,6,7\}$$, the

The discrete random variables $$X$$ and $$Y$$ are independent from one another and are defined as $$X \sim B(16,0.25)$$

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

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

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

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

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

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 co

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

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

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

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 l

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

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$

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

The $$x$$-intercept of a plane $$\pi$$ passing through the point $$(1,1,1)$$ is $$\frac{5}{2}$$ and the perpendicular di

Let $$f: R^{+} \longrightarrow R^{+}$$ be a function satisfying $$f(x)-x=\lambda$$ (constant), $$\forall x \in R^{+}$$ a

$$\begin{aligned} & \text { If } \lim _{x \rightarrow 0} \frac{|x|}{\sqrt{x^4+4 x^2+5}}=k \\ & \lim _{x \rightarrow 0} x

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

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

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

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

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

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

$$\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

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

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

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

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

A car covers a distance at speed of $$60 \mathrm{~km} \mathrm{~h}^{-1}$$. It returns and comes back to the original poin

A car travels with a speed of $$40 \mathrm{~km} \mathrm{~h}^{-1}$$. Rain drops are falling at a constant speed verticall

A projectile with speed $$50 \mathrm{~ms}^{-1}$$ is thrown at an angle of $$60^{\circ}$$ with the horizontal. The maximu

Two rectangular blocks of masses 40 kg and 60 kg are connected by a string and kept on a frictionless horizontal table.

A ball of mass 0.5 kg moving horizontally at $$10 \mathrm{~ms}^{-1}$$ strikes a vertical wall and rebounds with speed $$

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

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

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 m

Consider a disc of radius $$R$$ and mass $$M$$. A hole of radius $$\frac{R}{3}$$ is created in the disc, such that the c

A hydrometer executes simple harmonic motion when it is pushed down vertically in a liquid of density $$\rho$$. If the m

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

A projectile is thrown straight upward from the earth's surface with an initial speed $$v=\alpha v_e$$ where $$\alpha$$

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

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

A sphere of surface area $$4 \mathrm{~m}^2$$ at temperature 400 K and having emissivity 0.5 is located in an environment

A Carnot engine operates between a source and a sink. The efficiency of the engine is $$40 \%$$ and the temperature of t

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 \%$

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

Electrostatic force between two identical charges placed in vacuum at distance of $$r$$ is F. A slab of width $$\frac{r}

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

A capacitor of capacitance $$C_1=1 \mu \mathrm{F}$$ is charged using a 9 V battery. $$C_1$$ is, then removed from the ba

Two positive point charges of $$10 \mu \mathrm{C}$$ and $$12 \mu \mathrm{C}$$ are placed 10 cm apart in air. The work do

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

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

Two infinitely long wires each carrying the same current and pointing in $$+y$$ direction are placed in the $$x y$$-plan

Two electrons, $$e_1$$ and $$e_2$$ of mass $$m$$ and charge $$q$$ are injected into the perpendicular direction of the m

A compass needle oscillates 20 times per minute at a place where the dip is $$45^{\circ}$$ and the magnetic field is $$B

A plane electromagnetic wave travels in free space along $$Z$$-axis. At a particular point in space, the electric field

A coil of inductance 0.1 H and resistance $$110 \Omega$$ is connected to a source of 110 V and 350 Hz . The phase differ

If the average power per unit area delivered by an electromagnetic wave is $$9240 \mathrm{~Wm}^{-2}$$. then the amplitud

A beam of light with intensity $$10^{-3} \mathrm{~Nm}^{-2}$$ and cross-sectional area $$20 \mathrm{~cm}^2$$ is incident

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

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

Output of following logic circuit is

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