The solution from the following with highest depression in freezing point/lowest freezing point is

Choose the correct statements about the hydrides of group 15 elements.

A. The stability of the hydrides decreases in the order $\mathrm{NH}_3>\mathrm{PH}_3>\mathrm{AsH}_3> \mathrm{SbH}_3>\mathrm{BiH}_3$.

B. The reducing ability of the hydride increases in the order $\mathrm{NH}_3<\mathrm{PH}_3<\mathrm{AsH}_3 <\mathrm{SbH}_3<\mathrm{BiH}_3$.

C. Among the hydrides, $\mathrm{NH}_3$ is strong reducing agent while $\mathrm{BiH}_3$ is mild reducing agent.

D. The basicity of the hydrides increases in the order $\mathrm{NH}_3<\mathrm{PH}_3<\mathrm{AsH}_3< \mathrm{SbH}_3<\mathrm{BiH}_3$.

Choose the most appropriate from the options given below :

m-chlorobenzaldehyde on treatment with 50% KOH solution yields

The coordination geometry around the manganese in decacarbonyldimanganese $(0)$ is

Given below are two statements:

Statement - I: Along the period, the chemical reactivity of the elements gradually increases from group 1 to group 18 .

Statement - II: The nature of oxides formed by group 1 elements is basic while that of group 17 elements is acidic.

In the light of the above statements, choose the most appropriate from the options given below:

Number of spectral lines obtained in $\mathrm{He}^{+}$ spectra, when an electron makes transition from fifth excited state to first excited state will be

If a substance '$A$' dissolves in solution of a mixture of '$B$' and '$C$' with their respective number of moles as $\mathrm{n}_{\mathrm{A}}, \mathrm{n}_{\mathrm{B}}$ and $\mathrm{n}_{\mathrm{C}_3}$. Mole fraction of $\mathrm{C}$ in the solution is

Alkaline oxidative fusion of $\mathrm{MnO}_2$ gives "A" which on electrolytic oxidation in alkaline solution produces B. A and B respectively are

Salicylaldehyde is synthesized from phenol, when reacted with

Products A and B formed in the following set of reactions are

The total number of correct statements, regarding the nucleic acids is _________.

A. RNA is regarded as the reserve of genetic information

B. DNA molecule self-duplicates during cell division

C. DNA synthesizes proteins in the cell

D. The message for the synthesis of particular proteins is present in DNA

E. Identical DNA strands are transferred to daughter cells.

Total number of species from the following which can undergo disproportionation reaction is ________.

$\mathrm{H}_2 \mathrm{O}_2, \mathrm{ClO}_3^{-}, \mathrm{P}_4, \mathrm{Cl}_2, \mathrm{Ag}, \mathrm{Cu}^{+1}, \mathrm{~F}_2, \mathrm{NO}_2, \mathrm{K}^{+}$

Given below are two statements: One is labelled as Assertion A and the other is labelled as Reason R:

Assertion A: $\mathrm{H}_2 \mathrm{Te}$ is more acidic than $\mathrm{H}_2 \mathrm{~S}$.

Reason R: Bond dissociation enthalpy of $\mathrm{H}_2 \mathrm{Te}$ is lower than $\mathrm{H}_2 \mathrm{~S}$.

In the light of the above statements, choose the most appropriate from the options given below:

The orange colour of $\mathrm{K}_2 \mathrm{Cr}_2 \mathrm{O}_7$ and purple colour of $\mathrm{KMnO}_4$ is due to

Reduction potential of ions are given below:

$$\begin{array}{ccc} \mathrm{ClO}_4^{-} & \mathrm{IO}_4^{-} & \mathrm{BrO}_4^{-} \\ \mathrm{E}^{\circ}=1.19 \mathrm{~V} & \mathrm{E}^{\circ}=1.65 \mathrm{~V} & \mathrm{E}^{\circ}=1.74 \mathrm{~V} \end{array}$$

The correct order of their oxidising power is :

Number of metal ions characterized by flame test among the following is ________.

$\mathrm{Sr}^{2+}, \mathrm{Ba}^{2+}, \mathrm{Ca}^{2+}, \mathrm{Cu}^{2+}, \mathrm{Zn}^{2+}, \mathrm{Co}^{2+}, \mathrm{Fe}^{2+}$

The products A and B formed in the following reaction scheme are respectively

The correct stability order of carbocations is

Options:

C)

Which among the following purification methods is based on the principle of "Solubility" in two different solvents?

IUPAC name of following compound is :

Given below are two statements:

Statement - I: Since Fluorine is more electronegative than nitrogen, the net dipole moment of $\mathrm{NF}_3$ is greater than $\mathrm{NH}_3$.

Statement - II: In $\mathrm{NH}_3$, the orbital dipole due to lone pair and the dipole moment of $\mathrm{NH}$ bonds are in opposite direction, but in $\mathrm{NF}_3$ the orbital dipole due to lone pair and dipole moments of N-F bonds are in same direction.

In the light of the above statements, choose the most appropriate from the options given below:

A and B formed in the following reactions are:

$$\begin{aligned} & \mathrm{CrO}_2 \mathrm{Cl}_2+4 \mathrm{NaOH} \rightarrow \mathrm{A}+2 \mathrm{NaCl}+2 \mathrm{H}_2 \mathrm{O}, \\ & \mathrm{A}+2 \mathrm{HCl}+2 \mathrm{H}_2 \mathrm{O}_2 \rightarrow \mathrm{B}+3 \mathrm{H}_2 \mathrm{O} \end{aligned}$$

The molecule / ion with square pyramidal shape is

Given below are two statements:

Statement - I: High concentration of strong nucleophilic reagent with secondary alkyl halides which do not have bulky substituents will follow $\mathrm{S}_{\mathrm{N}}{ }^2$ mechanism.

Statement - II: A secondary alkyl halide when treated with a large excess of ethanol follows $\mathrm{S}_{\mathrm{N}}{ }^1$ mechanism.

In the light of the above statements, choose the most appropriate from the options given below:

Let $y=f(x)$ be a thrice differentiable function in $(-5,5)$. Let the tangents to the curve $y=f(x)$ at $(1, f(1))$ and $(3, f(3))$ make angles $\pi / 6$ and $\pi / 4$, respectively with positive $x$-axis. If 27 \int_\limits1^3\left(\left(f^{\prime}(t)\right)^2+1\right) f^{\prime \prime}(t) d t=\alpha+\beta \sqrt{3} where $\alpha, \beta$ are integers, then the value of $\alpha+\beta$ equals

Number of geometrical isomers possible for the given structure is/are _________.

Let $f(x)=(x+3)^2(x-2)^3, x \in[-4,4]$. If $M$ and $m$ are the maximum and minimum values of $f$, respectively in $[-4,4]$, then the value of $M-m$ is

Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{b}|=1$ and $|\vec{b} \times \vec{a}|=2$. Then $|(\vec{b} \times \vec{a})-\vec{b}|^2$ is equal to

Bag A contains 3 white, 7 red balls and Bag B contains 3 white, 2 red balls. One bag is selected at random and a ball is drawn from it. The probability of drawing the ball from the bag A, if the ball drawn is white, is

The area of the region enclosed by the parabola $(y-2)^2=x-1$, the line $x-2 y+4=0$ and the positive coordinate axes is _________.

The number of symmetric relations defined on the set $\{1,2,3,4\}$ which are not reflexive is _________.

Let $S_n$ be the sum to $n$-terms of an arithmetic progression $3,7,11$, If 40<\left(\frac{6}{n(n+1)} \sum_\limits{k=1}^n S_k\right)<42, then $n$ equals ________.

An electron revolving in $n^{\text {th }}$ Bohr orbit has magnetic moment $\mu_n$. If $\mu_n \alpha \cdot n^x$, the value of $x$ is

When a potential difference $V$ is applied across a wire of resistance $R$, it dissipates energy at a rate $W$. If the wire is cut into two halves and these halves are connected mutually parallel across the same supply, the energy dissipation rate will become:

For the photoelectric effect, the maximum kinetic energy $\left(E_k\right)$ of the photoelectrons is plotted against the frequency $(v)$ of the incident photons as shown in figure. The slope of the graph gives

If three moles of monoatomic gas $\left(\gamma=\frac{5}{3}\right)$ is mixed with two moles of a diatomic gas $\left(\gamma=\frac{7}{5}\right)$, the value of adiabatic exponent $\gamma$ for the mixture is

If mass is written as $m=k \mathrm{c}^{\mathrm{P}} G^{-1 / 2} h^{1 / 2}$ then the value of $P$ will be : (Constants have their usual meaning with $k a$ dimensionless constant)

A big drop is formed by coalescing 1000 small identical drops of water. If $E_1$ be the total surface energy of 1000 small drops of water and $E_2$ be the surface energy of single big drop of water, then $E_1: E_2$ is $x: 1$ where $x=$ ________.

A simple pendulum is placed at a place where its distance from the earth's surface is equal to the radius of the earth. If the length of the string is $4 m$, then the time period of small oscillations will be __________ s. [take $g=\pi^2 m s^{-2}$]

$\mathrm{NO}_2$ required for a reaction is produced by decomposition of $\mathrm{N}_2 \mathrm{O}_5$ in $\mathrm{CCl}_4$ as by equation

$2 \mathrm{~N}_2 \mathrm{O}_{5(\mathrm{~g})} \rightarrow 4 \mathrm{NO}_{2(\mathrm{~g})}+\mathrm{O}_{2(\mathrm{~g})}$

The initial concentration of $\mathrm{N}_2 \mathrm{O}_5$ is $3 \mathrm{~mol} \mathrm{~L}^{-1}$ and it is $2.75 \mathrm{~mol} \mathrm{~L}^{-1}$ after 30 minutes.

The rate of formation of $\mathrm{NO}_2$ is $\mathrm{x} \times 10^{-3} \mathrm{~mol} \mathrm{~L}^{-1} \mathrm{~min}^{-1}$, value of $\mathrm{x}$ is _________. (nearest integer)

Two reactions are given below:

$$\begin{aligned} & 2 \mathrm{Fe}_{(\mathrm{s})}+\frac{3}{2} \mathrm{O}_{2(\mathrm{~g})} \rightarrow \mathrm{Fe}_2 \mathrm{O}_{3(\mathrm{~s})}, \Delta \mathrm{H}^{\circ}=-822 \mathrm{~kJ} / \mathrm{mol} \\ & \mathrm{C}_{(\mathrm{s})}+\frac{1}{2} \mathrm{O}_{2(\mathrm{~g})} \rightarrow \mathrm{CO}_{(\mathrm{g})}, \Delta \mathrm{H}^{\circ}=-110 \mathrm{~kJ} / \mathrm{mol} \end{aligned}$$

Then enthalpy change for following reaction $3 \mathrm{C}_{(\mathrm{s})}+\mathrm{Fe}_2 \mathrm{O}_{3(\mathrm{~s})} \rightarrow 2 \mathrm{Fe}_{(\mathrm{s})}+3 \mathrm{CO}_{(\mathrm{g})}$ is _______ $\mathrm{kJ} / \mathrm{mol}$.

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function defined by $f(x)=\frac{x}{\left(1+x^4\right)^{1 / 4}}$, and $g(x)=f(f(f(f(x))))$. Then, $18 \int_0^{\sqrt{2 \sqrt{5}}} x^2 g(x) d x$ is equal to

Let $R=\left(\begin{array}{ccc}x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z\end{array}\right)$ be a non-zero $3 \times 3$ matrix, where $x \sin \theta=y \sin \left(\theta+\frac{2 \pi}{3}\right)=z \sin \left(\theta+\frac{4 \pi}{3}\right) \neq 0, \theta \in(0,2 \pi)$. For a square matrix $M$, let trace $(M)$ denote the sum of all the diagonal entries of $M$. Then, among the statements:

(I) Trace $(R)=0$

(II) If trace $(\operatorname{adj}(\operatorname{adj}(R))=0$, then $R$ has exactly one non-zero entry.

Let $a$ and $b$ be real constants such that the function $f$ defined by $f(x)=\left\{\begin{array}{ll}x^2+3 x+a & , x \leq 1 \\ b x+2 & , x>1\end{array}\right.$ be differentiable on $\mathbb{R}$. Then, the value of \int_\limits{-2}^2 f(x) d x equals

Let $\mathrm{f}: \mathbb{R} \rightarrow \mathbb{R}$ be defined as $f(x)=a e^{2 x}+b e^x+c x$. If $f(0)=-1, f^{\prime}\left(\log _e 2\right)=21$ and $\int_0^{\log _e 4}(f(x)-c x) d x=\frac{39}{2}$, then the value of $|a+b+c|$ equals

Let $L_1: \vec{r}=(\hat{i}-\hat{j}+2 \hat{k})+\lambda(\hat{i}-\hat{j}+2 \hat{k}), \lambda \in \mathbb{R}$,

$L_2: \vec{r}=(\hat{j}-\hat{k})+\mu(3 \hat{i}+\hat{j}+p \hat{k}), \mu \in \mathbb{R} \text {, and } L_3: \vec{r}=\delta(\ell \hat{i}+m \hat{j}+n \hat{k}), \delta \in \mathbb{R}$

be three lines such that $L_1$ is perpendicular to $L_2$ and $L_3$ is perpendicular to both $L_1$ and $L_2$. Then, the point which lies on $L_3$ is

Let $\vec{a}=\hat{i}+\alpha \hat{j}+\beta \hat{k}, \alpha, \beta \in \mathbb{R}$. Let a vector $\vec{b}$ be such that the angle between $\vec{a}$ and $\vec{b}$ is $\frac{\pi}{4}$ and $|\vec{b}|^2=6$. If $\vec{a} \cdot \vec{b}=3 \sqrt{2}$, then the value of $\left(\alpha^2+\beta^2\right)|\vec{a} \times \vec{b}|^2$ is equal to

Let $P$ be a point on the hyperbola $H: \frac{x^2}{9}-\frac{y^2}{4}=1$, in the first quadrant such that the area of triangle formed by $P$ and the two foci of $H$ is $2 \sqrt{13}$. Then, the square of the distance of $P$ from the origin is

Consider the system of linear equations $x+y+z=5, x+2 y+\lambda^2 z=9, x+3 y+\lambda z=\mu$, where $\lambda, \mu \in \mathbb{R}$. Then, which of the following statement is NOT correct?

If the domain of the function $f(x)=\log _e\left(\frac{2 x+3}{4 x^2+x-3}\right)+\cos ^{-1}\left(\frac{2 x-1}{x+2}\right)$ is $(\alpha, \beta]$, then the value of $5 \beta-4 \alpha$ is equal to

The number of real solutions of the equation $x\left(x^2+3|x|+5|x-1|+6|x-2|\right)=0$ is _________.

Let a line passing through the point $(-1,2,3)$ intersect the lines $L_1: \frac{x-1}{3}=\frac{y-2}{2}=\frac{z+1}{-2}$ at $M(\alpha, \beta, \gamma)$ and $L_2: \frac{x+2}{-3}=\frac{y-2}{-2}=\frac{z-1}{4}$ at $N(a, b, c)$. Then, the value of $\frac{(\alpha+\beta+\gamma)^2}{(a+b+c)^2}$ equals __________.

A block of ice at $-10^{\circ} \mathrm{C}$ is slowly heated and converted to steam at $100^{\circ} \mathrm{C}$. Which of the following curves represent the phenomenon qualitatively:

Three blocks $A, B$ and $C$ are pulled on a horizontal smooth surface by a force of $80 \mathrm{~N}$ as shown in figure

The tensions T$_1$ and T$_2$ in the string are respectively :

A particle of charge '$-q$' and mass '$m$' moves in a circle of radius '$r$' around an infinitely long line charge of linear charge density '$+\lambda$'. Then time period will be given as :

(Consider $k$ as Coulomb's constant)

Projectiles A and B are thrown at angles of $45^{\circ}$ and $60^{\circ}$ with vertical respectively from top of a $400 \mathrm{~m}$ high tower. If their ranges and times of flight are same, the ratio of their speeds of projection $v_A: v_B$ is :

[Take $g=10 \mathrm{~ms}^{-2}$]

If the total energy transferred to a surface in time $\mathrm{t}$ is $6.48 \times 10^5 \mathrm{~J}$, then the magnitude of the total momentum delivered to this surface for complete absorption will be:

A block of mass $1 \mathrm{~kg}$ is pushed up a surface inclined to horizontal at an angle of $60^{\circ}$ by a force of $10 \mathrm{~N}$ parallel to the inclined surface as shown in figure. When the block is pushed up by $10 \mathrm{~m}$ along inclined surface, the work done against frictional force is :

$\left[g=10 \mathrm{~m} / \mathrm{s}^2\right]$

The current of $5 \mathrm{~A}$ flows in a square loop of sides $1 \mathrm{~m}$ is placed in air. The magnetic field at the centre of the loop is $X \sqrt{2} \times 10^{-7} T$. The value of $X$ is _________.

In an experiment to measure the focal length $(f)$ of a convex lens, the magnitude of object distance $(x)$ and the image distance $(y)$ are measured with reference to the focal point of the lens. The $y$-$x$ plot is shown in figure.

The focal length of the lens is ________ $\mathrm{cm}$.

The $\mathrm{pH}$ of an aqueous solution containing $1 \mathrm{M}$ benzoic acid $\left(\mathrm{pK}_{\mathrm{a}}=4.20\right)$ and $1 \mathrm{M}$ sodium benzoate is 4.5. The volume of benzoic acid solution in $300 \mathrm{~mL}$ of this buffer solution is _________ $\mathrm{mL}$. (given : $\log 2=0.3$)

Number of complexes which show optical isomerism among the following is ________.

$\text { cis- }\left[\mathrm{Cr}(\mathrm{ox})_2 \mathrm{Cl}_2\right]^{3-},\left[\mathrm{Co}(\text {en})_3\right]^{3+}, \text { cis- }\left[\mathrm{Pt}(\text {en})_2 \mathrm{Cl}_2\right]^{2+}, \text { cis- }\left[\mathrm{Co}(\text {en})_2 \mathrm{Cl}_2\right]^{+}, \text {trans- }\left[\mathrm{Pt}(\text {en})_2 \mathrm{Cl}_2\right]^{2+}, \text { trans- }\left[\mathrm{Cr}(\mathrm{ox})_2 \mathrm{Cl}_2\right]^{3-}$

Let $f: \mathbb{R}-\{0\} \rightarrow \mathbb{R}$ be a function satisfying $f\left(\frac{x}{y}\right)=\frac{f(x)}{f(y)}$ for all $x, y, f(y) \neq 0$. If $f^{\prime}(1)=2024$, then

If $x^2-y^2+2 h x y+2 g x+2 f y+c=0$ is the locus of a point, which moves such that it is always equidistant from the lines $x+2 y+7=0$ and $2 x-y+8=0$, then the value of $g+c+h-f$ equals

In an examination of Mathematics paper, there are 20 questions of equal marks and the question paper is divided into three sections : $A, B$ and $C$. A student is required to attempt total 15 questions taking at least 4 questions from each section. If section $A$ has 8 questions, section $B$ has 6 questions and section $C$ has 6 questions, then the total number of ways a student can select 15 questions is __________.

Consider two circles $C_1: x^2+y^2=25$ and $C_2:(x-\alpha)^2+y^2=16$, where $\alpha \in(5,9)$. Let the angle between the two radii (one to each circle) drawn from one of the intersection points of $C_1$ and $C_2$ be $\sin ^{-1}\left(\frac{\sqrt{63}}{8}\right)$. If the length of common chord of $C_1$ and $C_2$ is $\beta$, then the value of $(\alpha \beta)^2$ equals _______.

Let \alpha=\sum_\limits{k=0}^n\left(\frac{\left({ }^n C_k\right)^2}{k+1}\right) and \beta=\sum_\limits{k=0}^{n-1}\left(\frac{{ }^n C_k{ }^n C_{k+1}}{k+2}\right) If $5 \alpha=6 \beta$, then $n$ equals _______.

If 50 Vernier divisions are equal to 49 main scale divisions of a traveling microscope and one smallest reading of main scale is $0.5 \mathrm{~mm}$, the Vernier constant of traveling microscope is

Choose the correct statement for processes A & B shown in figure.

In a nuclear fission reaction of an isotope of mass $M$, three similar daughter nuclei of same mass are formed. The speed of a daughter nuclei in terms of mass defect $\Delta M$ will be :

A beam of unpolarised light of intensity $I_0$ is passed through a polaroid $A$ and then through another polaroid $B$ which is oriented so that its principal plane makes an angle of $45^{\circ}$ relative to that of $A$. The intensity of emergent light is:

A vector has magnitude same as that of $\vec{A}=3 \hat{i}+4 \hat{j}$ and is parallel to $\vec{B}=4 \hat{i}+3 \hat{j}$. The $x$ and $y$ components of this vector in first quadrant are $x$ and 3 respectively where $x=$ _________.

2-chlorobutane $+\mathrm{Cl}_2 \rightarrow \mathrm{C}_4 \mathrm{H}_8 \mathrm{Cl}_2$ (isomers)

Total number of optically active isomers shown by $\mathrm{C}_4 \mathrm{H}_8 \mathrm{Cl}_2$, obtained in the above reaction is _________.

Let $a$ and $b$ be be two distinct positive real numbers. Let $11^{\text {th }}$ term of a GP, whose first term is $a$ and third term is $b$, is equal to $p^{\text {th }}$ term of another GP, whose first term is $a$ and fifth term is $b$. Then $p$ is equal to

For $\alpha, \beta \in(0, \pi / 2)$, let $3 \sin (\alpha+\beta)=2 \sin (\alpha-\beta)$ and a real number $k$ be such that $\tan \alpha=k \tan \beta$. Then, the value of $k$ is equal to

If $z$ is a complex number, then the number of common roots of the equations $z^{1985}+z^{100}+1=0$ and $z^3+2 z^2+2 z+1=0$, is equal to

Suppose $2-p, p, 2-\alpha, \alpha$ are the coefficients of four consecutive terms in the expansion of $(1+x)^n$. Then the value of $p^2-\alpha^2+6 \alpha+2 p$ equals

Let $A(\alpha, 0)$ and $B(0, \beta)$ be the points on the line $5 x+7 y=50$. Let the point $P$ divide the line segment $A B$ internally in the ratio $7:3$. Let $3 x-25=0$ be a directrix of the ellipse $E: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and the corresponding focus be $S$. If from $S$, the perpendicular on the $x$-axis passes through $P$, then the length of the latus rectum of $E$ is equal to,

Let $Y=Y(X)$ be a curve lying in the first quadrant such that the area enclosed by the line $Y-y=Y^{\prime}(x)(X-x)$ and the co-ordinate axes, where $(x, y)$ is any point on the curve, is always $\frac{-y^2}{2 Y^{\prime}(x)}+1, Y^{\prime}(x) \neq 0$. If $Y(1)=1$, then $12 Y(2)$ equals __________.