The wavenumber of first spectral line of Lyman series of $\mathrm{He}^{+}$ion is $x \mathrm{~m}^{-1}$. What is the wave

The uncertainty in determination of position of a small ball of mass 10 g is $10^{-33} \mathrm{~m}$. With what $\%$ of a

In which of the following ionic pairs, second ion is smaller in size than the first ion?

The set of elements which obey the general electronic configuration $(n-1) d^{1-10} n s^2$ is?

Identify the set of molecules which are not in the correct order of their dipole moments

Match the following. .tg {border-collapse:collapse;border-spacing:0;} .tg td{border-color:black;border-style:solid;bor

At 400 K , an ideal gas is enclosed in a $0.5 \mathrm{~m}^3$ vessel at pressure of 203 kPa . What is the change in tempe

Match the following. .tg {border-collapse:collapse;border-spacing:0;} .tg td{border-color:black;border-style:solid;

The standard enthalpy of combustion of C (graphite). $\mathrm{H}_2(g)$ and $\mathrm{CH}_3 \mathrm{OH}(l)$ respectively a

Observe the following species. (i) $\mathrm{NH}_3$ (ii) $\mathrm{AlCl}_3$ (iii) $\mathrm{SnCl}_4$ (iv) $\mathrm{CO}_2$ (

The normality of 20 volume solution of hydrogen peroxide is

Consider the following reactions. $$ \begin{array}{r} \mathrm{Cs}+\mathrm{O}_2 \text { (excess) } \rightarrow X \\ \math

Choose the correct statement from the following. I. In vapour phase $\mathrm{BeCl}_2$ exists as chlorobridge dimer. II.

Observe the following reactions (not balanced) $$ \begin{aligned} & \mathrm{BF}_3+\mathrm{LiAlH}_4 \xrightarrow{\left(\m

$\mathbf{Assertion (A)}$ Silicones are used for water proofing of fabrics. $\mathbf{Reason (R)}$ The repeating unit in s

Acrolein $(X)$ is one of the chemicals formed when $\mathrm{O}_3$ and $\mathrm{NO}_2$ react with unburnt hydrocarbons pr

An organic compound containing phosphorous on oxidation with $\mathrm{Na}_2 \mathrm{O}_2$ gives a compound ' $X$ '. This

The correct IUPAC name of the structure given below is

The major product ' $Y^{\prime}$ in the given sequence of reactions is $$ \mathrm{C}_3 \mathrm{H}_7 \mathrm{OH} \xrighta

Compound ' $A$ ' on heating with sodalime gives propane. Identify the compound ' $A$ '.

An element with molar mass $2.7 \times 10^{-2} \mathrm{~kg} \mathrm{~mol}^{-1}$ forms a cubic unit cell with edge length

At $300 \mathrm{~K}, 0.06 \mathrm{~kg}$ of an organic solute is dissolved in 1 kg water. The vapour pressure of solution

The molar conductivity of 0.02 M solution of an electrolyte is $124 \times 10^{-4} \mathrm{~S} \mathrm{~m}^2 \mathrm{~mo

The decomposition of benzene diazonium chloride is a first order reaction. The time taken for its decomposition to $\fra

10 mL of 0.5 M NaCl is required to coagulate 1 L of $\mathrm{Sb}_2 \mathrm{~S}_3 \mathrm{sol}$ in 2 hours time. The floc

Kaolinite is a silicate mineral of metal ' $X$ ' and calamine is a carbonate mineral of metal ' $Y^{\prime}, X$ and $Y$

$\mathrm{NH}_2 \mathrm{CONH}_2+2 \mathrm{H}_2 \mathrm{O} \rightarrow[\mathrm{X}] \rightleftharpoons 2 \mathrm{NH}_3+\mat

Among the hydrides $\mathrm{NH}_3, \mathrm{PH}_3$ and $\mathrm{BiH}_3$, the hydride with highest boiling point is $X$ an

Xenon (VI) fluoride on complete hydrolysis gives an oxide of xenon ' $O$ '. The total number of $\sigma$ and $\pi$-bonds

In which of the following ions the spin only magnetic moment is lowest?

Identify the complex ion with electronic configuration $t_{2 g}^3 e_g^2$.

Identify the structure of the polymer 'P' formed in the given reaction Caprolactam $\xrightarrow[\mathrm{H}_2 \mathrm{O}

Which of the following vitamin is also called pyridoxine?

The number of -OH groups present in the structures of bithionol, terpineol and chloroxylenol is respectively

Conversion of $X$ to $Y$ in the above reaction is

$$ \mathrm{CH}_3 \mathrm{CH}_2 \mathrm{OH} \xrightarrow[443 \mathrm{~K}]{\text { Conc. } \mathrm{H}_2 \mathrm{SO}_4} X \

Arrange the following in the increasing order of pKa values

What is ' $C$ ' in the following reaction sequence?

Identify the products $R$ and $S$ in the reaction sequence given $\left(\mathrm{CH}_3\right)_3 \mathrm{COH} \xrightarro

In the given reaction sequence sequence, $Z$ is

If the real valued function $f(x)=\sin ^{-1}\left(x^2-1\right)-3 \log _3\left(3^x-2\right)$ is not defined for all $x \i

If $f$ is a real valued function from $A$ onto $B$ defined by $f(x)=\frac{1}{\sqrt{|x-|x||}}$, then $A \cap B=$

Among the following four statements, the statement which is not true, for all $n \in N$ is

If $A=\left[\begin{array}{lll}x & y & y \\ y & x & y \\ y & y & x\end{array}\right]$ is a matrix such that $5 A^{-1}=\le

If $A=\left[\begin{array}{lll}9 & 3 & 0 \\ 1 & 5 & 8 \\ 7 & 6 & 2\end{array}\right]$ and $A A^T-A^2=\left[\begin{array}{

If $a \neq b \neq c, \Delta_1=\left[\begin{array}{lll}1 & a^2 & b c \\ 1 & b^2 & c a \\ 1 & c^2 & a b\end{array}\right]$

The system of equations $x+3 y+7=0$, $3 x+10 y-3 z+18=0$ and $3 y-9 z+2=0$ has

If $x$ and $y$ are two positive real numbers such that $x+i y=\frac{13 \sqrt{-5+12 i}}{(2-3 i)(3+2 i)}$, then $13 y-26 x

If $z=x+i y$ and if the point $P$ represents $z$ in the argand plane, then the locus of $z$ satisfying the equation $|z-

One of the values of $(-64 i)^{5 / 6}$ is

If $\alpha, \beta$ are the roots of the equation $x+\frac{4}{x}=2 \sqrt{3}$, then $\frac{2}{\sqrt{3}}\left|\alpha^{2024}

$\alpha, \beta$ are the real roots of the equation $12 x^{\frac{1}{3}}-25 x^{\frac{1}{6}}+12=0$. If $\alpha>\beta$, then

If the expression $7+6 x-3 x^2$ attains its extreme value $\beta$ at $x=\alpha$, then the sum of the squares of the root

$\alpha, \beta$ and $\gamma$ are the roots of the equation $x^3+3 x^2-10 x-24=0$. If $\alpha>\beta>\gamma$ and $\alpha^3

$\alpha, \beta$ and $\gamma$ are the roots of the equation $8 x^3-42 x^2+63 x-27=0$. If $\beta

All the letters of word 'COLLEGE' are arranged in all possible ways and all the seven letter words (with or without mean

If all the possible 3-digit numbers are formed using the digits $1,3,5,7$ and 9 without repeating any digit, then the nu

A question paper has 3 parts $A, B$ and $C$. Part $A$ contains 7 questions, part $B$ contains 5 questions and Part Ccont

If $p$ and $q$ are the real numbers such that the 7 th term in the expansion of $\left(\frac{5}{p^3}-\frac{3 q}{7}\right

If $T_4$ represents the 4 th term in the expansion of $\left(5 x+\frac{7}{x}\right)^{\frac{-3}{2}}$ and $x \notin\left[-

If $\frac{2 x^3+1}{2 x^2-x-6}=a x+b+\frac{A}{P x-2}+\frac{B}{2 x+q}$, then 51 apB $=$

$\tan A=\frac{-60}{11}$ and $A$ does not lie in the 4th quadrant. $\sec B=\frac{41}{9}$ and $B$ does not lie in the 1st

If $\tan A+\tan B+\cot A+\cot B=\tan A \tan B-\cot A \cot B$ and $0^{\circ}

If $\cos ^2 84^{\circ}+\sin ^2 126^{\circ}-\sin 84^{\circ} \cos 126^{\circ}=K$ and $\cot A+\tan A=2 K$, then the possibl

The equation that is satisfied by the general solution of the equation $4-3 \cos ^2 \theta=5 \sin \theta \cos \theta$ is

If $\sin ^{-1}(4 x)-\cos ^{-1}(3 x)=\frac{\pi}{6}$, then $x=$

If $\sin h^{-1}(-\sqrt{3})+\cos ^{-1}(2)=K$, then $\cosh K=$

In triangle $A B C$, if $a=4, b=3$ and $c=2$, then $2(a-b \cos C)(a-c \sec B)=$

In $\triangle A B C$, if $A=45^{\circ}, C=75^{\circ}$ and $R=\sqrt{2}$, than $r=$

$P$ and $Q$ are the points of trisection of the segment $A B$. If $2 \hat{\mathbf{i}}-5 \hat{\mathbf{j}}+3 \hat{\mathbf{

The position vector of the point of intersection of the line joining the points $\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{

If $\mathbf{a}=4 \hat{\mathbf{i}}+5 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}$ and $\mathbf{b}=6 \hat{\mathbf{i}}-2 \hat{\math

A plane $\pi_1$ passing through the point $3 \hat{\mathbf{i}}-7 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}$ is perpendicular to

$\mathbf{a}=2 \hat{\mathbf{i}}-\hat{\mathbf{j}}, \mathbf{b}=2 \hat{\mathbf{j}}-\hat{\mathbf{k}}$ and $\mathbf{c}=2 \hat{

If $M_1$ is the mean deviation from the mean of the discrete data $44,5,27,20,8,54,9,14,35$ and $M_2$ is the mean deviat

If two dice are thrown, then the probability of getting co-prime numbers on the dice is

If two cards are drawn at random simultaneously from a well shuffled pack of 52 playing cards, then the probability of g

Bag $P$ contains 3 white, 2 red, 5 blue balls and bag $Q$ contains 2 white, 3 red, 5 blue balls. A ball is chosen at ran

If the probability distribution of a random variable $X$ is as follow, then the variance of $X$ is .tg {border-collaps

The mean of a binomial variate $X \sim B(n, p)$ is 1 . If $n>2$ and $P(X=2)=\frac{27}{128}$, then the variance of the di

If the distance from a variable point $P$ to the point $(4,3)$ is equal to the perpendicular distance from $P$ to the li

$(0, k)$ is the point to which the origin is to be shifted by the translation of the axes so as to remove the first degr

$\beta$ is the angle made by the perpendicular drawn from origin to the line $L \equiv x+y-2=0$ with the positive $X$-ax

The line $2 x+y-3=0$ divides the line segment joining the points $A(1,2)$ and $B(-2,1)$ in the ratio $a: b$ at the point

If $Q$ and $R$ are the images of the point $P(2,3)$ with respect to the lines $x-y+2=0$ and $2 x+y-2=0$ respectively, th

If $(2,-1)$ is the point of intersection of the pair of lines $2 x^2+a x y+3 y^2+b x+c y-3=0$, then $3 a+2 b+c=$

$(1, k)$ is a point on the circle passing through the points $(-1,1),(0,-1)$ and $(1,0)$. If $k \neq 0$, then $k=$

If the tangents $x+y+k=0$ and $x+a y+b=0$ drawn to the circle $S=x^2+y^2+2 x-2 y+1=0$ are perpendicular to each other an

If $(h, k)$ is the internal centre of similitude of the circles $x^2+y^2+2 x-6 y+1=0$ and $x^2+y^2-4 x+2 y+4=0$, then $4

The slope of a common tangent to the circles $x^2+y^2-4 x-8 y+16=0$ and $x^2+y^2-6 x-16 y+64=0$ is

$x^2+y^2+2 x-6 y-6=0$ and $x^2+y^2-6 x-2 y+k=0$ are two intersecting circles and $k$ is not an integer. If $\theta$ is t

If $(p, q)$ is the centre of the circle which cuts the three circles $x^2+y^2-2 x-4 y+4=0, x^2+y^2+2 x-4 y+1=0$ and $x^2

If the focal chord of the parabola $x^2=12 y$, drawn through the point $(3,0)$ intersects the parabola at the points $P$

If the normal drawn at the point $P(9,9)$ on the parabola $y^2=9 x$ meets the parabola again at $Q(a, b)$, then $2 a+b=$

The length of the latus rectum of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b)$ is $\frac{8}{3}$. If the distance

$S$ is the focus of the ellips $\frac{x^2}{25}+\frac{y^2}{b^2}=1,(b

The slope of the tangent drawn from the point $(1,1)$ to the hyperbola $2 x^2-y^2=4$ is

$A(2,3, k), B(-1, k,-1)$ and $C(4,-3,2)$ are the vertices of $\triangle A B C$. If $A B=A C$ and $k>0$, then $\triangle

If $A(1,2,-3), B(2,3,-1)$ and $C(3,1,1)$ are the vertices of $\triangle A B C$, then $\left|\frac{-\cos A}{\cos B}\right

If $a, b$ and $c$ are the intercepts made on $X, Y, Z$-axes respectively by the plane passing through the points $(1,0,-

If $\lim \limits_{x \rightarrow 4} \frac{2 x^2+(3+2 a) x+3 a}{x^3-2 x^2-23 x+60}=\frac{11}{9}$, then $\lim \limits_{x \r

If the function $$ f(x)= \begin{cases}\frac{\tan a(x-1)}{x-1}, & \text { if } 04\end{cases} $$ domain, then $6 a+9 b^4=$

If $y=\log \left[\tan \sqrt{\frac{2^x-1}{2^x+1}}\right], x>0$, then $\left(\frac{d y}{d x}\right)_{x=1}=$

If $y=\cos ^{-1}\left(\frac{6 x-2 x^2-4}{2 x^2-6 x+5}\right)$, then $\frac{d y}{d x}=$

If $\log y=y^{\log x}$, then $\frac{d y}{d x}=$

If $y=a \cos 3 x+b e^{-x}$, then $y^{\prime \prime}(3 \sin 3 x-\cos 3 x)=$

The approximate value of $\sec 59^{\circ}$ obtained by taking $1^{\circ}$ $=0.0174$ and $\sqrt{3}=1.732$ is

The equation of the normal drawn to the curve $y^3=4 x^5$ at the point $(4,16)$ is

A point $P$ is moving on the curve $x^3 y^4=2^7$. The $x$-coordinate of $P$ is decreasing at the rate of 8 units per sec

If the function $f(x)=x^3+a x^2+b x+40$ satisfies the conditions of Rolle's theorem on the interval $[-5,4]$ and $-5,4$

If $x$ and $y$ are two positive integers such that $x+y=24$ and $x^3 y^5$ is maximum, then $x^2+y^2=$

$\int \sqrt{4 \cos ^2 x-5 \sin ^2 x} \cos x d x=$

$\int\left(\frac{4 \tan ^4 x+3 \tan ^2 x-1}{\tan ^2 x+4}\right) d x=$

$\int\left(\frac{\left(\sin ^4 x+2 \cos ^2 x-1\right) \cos x}{(1+\sin x)^6}\right) d x=$

$\int(\log x)^3 d x=$

$\int_0^\pi\left(\sin ^3 x+\cos ^2 x\right)^2 d x=$

$\int_{\frac{-\pi}{8}}^{\frac{\pi}{8}} \frac{\sin ^4(4 x)}{1+e^{4 x}} d x=$

The area of the region enclosed by the curves $y^2=4(x+1)$ and $y^2=5(x-4)$ is

If $A$ and $B$ are arbitrary constants, then the differential equation having $y=A e^{-x}+B \cos x$ as its general solut

The general solution of the differential equation $\frac{d y}{d x}+\frac{\sin (2 x+y)}{\cos x}+2=0$ is

Which of the following statement regarding nature of physical laws is not correct?

The internal and external diameters of hollow cylinder measured with vernier callipers are $(5.73 \pm 0.01) \mathrm{cm}$

A body moving with uniform acceleration, travels a distance of 25 m in the fourth second and 37 m in the sixth second. T

A body is projected from the ground at an angle of $\tan ^{-1}(\sqrt{7})$ with the horizontal. At half of the maximum he

An aircraft executes a horizontal loop of radius 9 km at a constant speed of $540 \mathrm{kmh}^{-1}$. The wings of the a

A body thrown vertically upwards from the ground reaches a maximum height $h$. The ratio of the kinetic and potential en

A ball of mass 1.2 kg moving with a velocity of $12 \mathrm{~ms}^{-1}$ makes one-dimensional collision with anothe stati

An alphabet $T$ made of two similar thin uniform metal plates of each length $L$ and width $a$ is placed on a horizontal

A solid sphere and a disc of same mass $M$ and radius $R$ - are kept such that their curved surfaces are in contact and

If a body dropped freely from a height of 20 m reaches the surface of a planet with a velocity of $31.4 \mathrm{~ms}^{-1

Two stars of masses $M$ and $2 M$ that are at a distance $d$ apart, are revolving one around another. The angular veloci

A block of mass 2 kg is tied to one end of a 2 m long metal wire of $1.0 \mathrm{~mm}^2$ area of cross-section and rotat

A big liquid drop splits into $n$ similar small drops under isothermal conditions, then in the process

A wooden cube of side 10 cm floats at the interface between water and oil with its lower surface 3 cm below the interfac

37 g of ice at $0^{\circ} \mathrm{C}$ temperature is mixed with 74 g of water at $70^{\circ} \mathrm{C}$ temperature. Th

The thickness of a uniform rectangular metal plate is 5 mm and the area of each surface is $5 \mathrm{~cm}^5$. In steady

The ratio of the specific heat capacities of a gas is 1.5 . When the gas undergoes adiabatic process, its volume is doub

A vessel contains hydrogen and nitrogen gases in the ratio $2: 3$ by mass. If the temperature of the mixture of the gase

The difference between the fundamental frequencies of an open pipe and a closed pipe of same length is 100 Hz . The diff

The displacement equations of sound waves produced by two sources are given by $y_1=5 \sin 400 \pi t$ and $y_2=8 \sin 40

When an object of height 12 cm is placed at a distance from a convex lens, an image of height 18 cm is formed on a scree

A thin plano-convex lens of focal length 73.5 cm has a circular aperture of diameter 8.4 cm . If the refractive index of

In Young's double slit experiment, intensity of light at a point on the screen, where the path difference becomes $\lamb

Two point charges $-10 \mu \mathrm{C}$ and $-5 \mu \mathrm{C}$ are situated on $X$-axis at $x=0$ and $x=\sqrt{2} \mathrm

A $10 \mu \mathrm{~F}$ capacitor is charged by a 100 V battery. It is disconnected from the battery and is connected to

A conductor of length 1.5 m and area of cross-section $3 \times 10^{-5} \mathrm{~m}^2$ has electrical resistance of $15

The relation between the current $i$ (in ampere) in a conductor and the time $t$ (in second) is $i=12 t+9 t^2$. The char

A long straight rod of diameter 4 mm carries a steady current $i$. The current is uniformly distributed across its cross

A straight wire of length 20 cm carrying a current of $\frac{3 .}{\pi^2} \mathrm{~A}$ is bent in the form of a circle. T

A circular coil carrying a current of 2.5 A is free to rotate about an axis in its plane perpendicular to an external fi

A circular coil of area $200 \mathrm{~cm}^2$ and 50 turns is rotating about its vertical diameter with an angular speed

An inductor and a resistor are connected in series to an AC source of 10 V . If the potential difference across the indu

If the peak value of the magnetic field of an electromagnetic wave is $30 \times 10^{-9} \mathrm{~T}$, then the peak val

The de-Broglie wavelength of a proton is twice the de-Broglie wavelength of an alpha particle. The ratio of the kinetic

The ratio of the centripetal accelerations of the electron in two successive orbits of hydrogen is $81: 16$. Due to ${ }

The operation of a nuclear reactor is said to be critical when the value of neutron multiplication factor $K i s$

An $\alpha$-particle of energy $E$ is liberated during the decay of a nucleus of mass number 236. The total energy relea

The voltage gain of a transistor in common emitter configuration is 160 . The resistances in base and collector sides of

Normally a capacitor is connected across the output terminals of a rectifier to

The process of the loss of strength of a signal while propagating through a medium is