If the radius and energy of the second Bohr orbit of hydrogen atom is $r_2$ and $E_2$. respectively. The radius and energy of the third Bohr orbit will be $\_\_\_\_$ respectively.

When a radiation of 300 nm is shined on five metals, namely $\mathrm{Li}, \mathrm{Mg}, \mathrm{Ag}, \mathrm{Cu}$ and K , the number of metals that show photoelectric effect are

The correct order of ionic radii for the given species is

Which of the following set of properties generally decreases along a period?

The correct order of the bond angles of the compounds $\mathrm{SiCl}_4, \mathrm{BF}_3, \mathrm{BeCl}_2$ and $\mathrm{SF}_6$ is

Identify all the species that do not exist $\mathrm{H}_2^{+}, \mathrm{He}_2^{2+}, \mathrm{Li}_2^{2-}, \mathrm{Ne}_2, \mathrm{Be}_2^{-}, \mathrm{He}_2$

The compressibility factor of a real gas at high pressure is

The compressibility factor $(\mathrm{Z})$ is lower for $\mathrm{NH}_3$ and $\mathrm{CO}_2$ gases than that of $\mathrm{N}_2$ gas because

Combustion of 10 mL of a gaseous hydrocarbon gives 40 mL of $\mathrm{CO}_2$ and 50 mL of water vapour under the same conditions. The molecular formula of the hydrocarbon is

The number of moles of ferrous oxalate oxidised by one mole of $\mathrm{KMnO}_4$ in acidic medium is

An air bag on adiabatic expansion undergoes $5 \%$ increase in its volume. The percentage change in pressure is $\left[\gamma_{\text {air }}=1.4\right]$

The $K_p$ value at equilibrium of $\mathrm{SO}_3$ formation reaction from $\mathrm{SO}_2(g)$ and $\mathrm{O}_2(g)$ is $5 \mathrm{~atm}^{-1}$. What is the equilibrium partial pressure of $\mathrm{O}_2$ if the equilibrium pressure of $\mathrm{SO}_2$ and $\mathrm{SO}_3$ are equal?

$$ \text { Match the following } $$

$$ \text { The correct match is } $$

The correct statements among the following are

(A) $\mathrm{BeH}_2$ and $\mathrm{MgH}_2$ are polymeric in nature.

(B) LiH is unreactive to oxygen, at moderate temperatures.

(C) $\mathrm{BeH}_2$ and $\mathrm{MgH}_2$ possesses significant covalent character.

(D) The stability of alkali hydrides follows the order $\mathrm{LiH}<\mathrm{NaH}< \mathrm{KH}<\mathrm{RbH}<\mathrm{CsH}$.

The correct order of melting points of the following salts is

$$ \begin{array}{lll} \text { LiCl } & \text { LiF } & \text { Lil } \\ \text { I } & \text { II } & \text { III } \end{array} $$

$\mathrm{Al}+$ aq. NaOH (excess) $\longrightarrow P+Q \cdot P$ and $Q$ are

Among the following elements, $X$ exhibits maximum catenation and $Y$ is the least abundant on earth. $X$ and $Y$ elements are

For the following radicals, the correct order of their stability is

The order of decreasing reactivity towards an electrophilic reagent, for the following compounds, is

(i) benzene

(iii) chlorobenzene

(ii) toluene

(iv) phenol

\text { The major product of the following reactions is }

If the length of the body diagonal of a FCC unit cell is $x \mathop {\rm{A}}\limits^{\rm{o}}$, the distance between two octahedral voids in the cell in $\mathop {\rm{A}}\limits^{\rm{o}}$ is

Calculate the quantity of $\mathrm{CO}_2$ required to prepare 1 L of soda water when the soda water was packed under 2 atm of $\mathrm{CO}_2$.

[Henry's law constant for $\mathrm{CO}_2$ is $1.67 \times 10^8 \mathrm{~Pa}$ ]

Which of the following substances show the highest colligative properties?

In two separate experiments, the same quantity of electricity was passed through silver and gold solutions. [Assume ' $l$ ' constant]. The amounts of Ag and Au deposited are 2.15 and 1.31 g , respectively. The valency of gold is

[atomic mass of $\mathrm{Ag}=107.9$; $\mathrm{Au}=197$ ]

Which of the following is a zero order reaction?

The coagulation of 200 mL of a positive colloid took place when 0.73 g of HCl was added to it without changing the volume much. The flocculation value of HCl for the colloid is

The oxoacid of phosphorus which contains $4 \mathrm{P}-\mathrm{O}-\mathrm{H}, 2 \mathrm{P}=\mathrm{O}$ and one $\mathrm{P}-\mathrm{O}-\mathrm{P}$ bond is

Group 16 elements are also called

For the reaction, $\mathrm{Br}_2+\mathrm{F}_2$ (excess) ⟶ P , the molecular formula and structure of P , respectively, are

In the following reaction, $a, b, p, q, r$ and $s$ are $a \mathrm{XeF}_4+b \mathrm{H}_2 \mathrm{O} \longrightarrow p \mathrm{Xe}+q \mathrm{XeO}_3+r \mathrm{HF}+s \mathrm{O}_2$

The increase in the atomic radii of the third (5d) series of transition elements is very small, which may be accounted for the filling of ' $X$ ' orbitals before ${ }^{\prime} Y^{\prime}$ orbitals $X$ and $Y$ are

A metal complex absorbed orange light. The colour in which it appears is

The major product of the following reaction is

$$ \text { Glucose } \frac{\text { (i) } \mathrm{HI}, \Delta}{\text { (ii) } \mathrm{Mo}_2 \mathrm{O}_3, 773 \mathrm{~K}, 10 \cdot 20 \mathrm{am}} $$

$$ \text { The major product in the following reaction is } $$

Arrange the following phenols in decreasing order of their $\mathrm{p} K_a$.

(A) Phenol

(B) ortho - nitrophenol

(C) meta - nitrophenol

(D) para - nitrophenol

Decreasing order of reactivity of the following compounds in the Williamson's ether synthesis is

(II) $\mathrm{Cl}-\mathrm{CH}_2-\mathrm{CH}=\mathrm{CH}_2$

(III) $\mathrm{Cl}-\mathrm{CH}_2-\mathrm{CH}_2-\mathrm{CH}_3$

$$ \text { Identify } X \text { and } Y \text { in the reactions, given below } $$

A primary alcohol was reacted with pyridinium chlorochromate (PCC), which resulted in a product $P$. The product $P$ on treatment with ammonical silver nitrate solution produces

The major product of the following reactions

In the following reaction, the suitable starting reagent $P$ is

Let $f: A \rightarrow B$ be defined as $f(x)=\frac{1}{2}-\tan \left(\frac{\pi x}{2}\right)$ and $g: B \rightarrow C$ be defined as $g(x)=\sqrt{3+4 x-4 x^2}$. If $A, B$ and $C$ are subsets of $R$ and $f$ is an onto function, then the range of the function $f(x)$ is

If $D$ is the domain and $G$ is the range of the real valued function $f(x)=\sqrt{\frac{1-x^2}{1+x^2}}$, then $D \cap G=$

Let $A=\left[\begin{array}{ll}0 & 1 \\ 1 & k\end{array}\right], k \in R$ and $A^3=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]$. If $d=228$, then $b+c=$

Let $A$ and $B$ be two $3 \times 3$ matrices and $C$ be a $3 \times 3$ unit matrix such that $A B-C$ is a non-singular matrix. Let $D=(A B-C)^{-1}$. Then, consider the following statements.

Statement I $\operatorname{det}(B A)=\operatorname{det}(B A-C) \operatorname{det}(B D A)$

Statement II $A B D=D A B$

Which of the above statements is (are) true?

Let $A=\left[\begin{array}{ccc}0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0\end{array}\right], B=\left[\begin{array}{lll}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{array}\right]$, then $\left(A^{-1} B\right)^{-1}+\left(A B^{-1}\right)^{-1}=$

Let $\alpha, \beta$ and $\gamma$ be real numbers.

If $\left[\begin{array}{ccc}7 & 5 & \alpha \\ \beta & 2 & 11 \\ 3 & \gamma & 1\end{array}\right]\left[\begin{array}{l}1 \\ 3 \\ 2\end{array}\right]=\left[\begin{array}{c}\alpha+\beta \\ -2 \alpha+\beta-2 \gamma \\ \alpha+2 \beta+3 \gamma\end{array}\right]$, then $100+\frac{2 \alpha+11 \beta}{\gamma}=$

If $z=\alpha+i \beta$ satisfies the equation $|z|-z=1+2 i$ and $|z|=\sqrt{\alpha^2+\beta^2}$, then $z \bar{z}=$

If $-i$ and $\alpha$ are the roots of the equation $i z^2-2(i+1) z+(2-i)=0, \tan \theta=\frac{-1}{2}$ and $\theta \in 4$ th quadrant, then $5^3 \cos 6 \theta=$

If $1, \alpha_1, \alpha_2, \alpha_3, \ldots \alpha_{n-1}$ are $n$th roots of unity then $\sum\limits_{1 \le i < f \le n - 1}^{} {} {a_i}{a_j} = $

Let $f(x)=A x^2+B x, g(x)=L x^2+M x+N$. Given that $f(2)-g(2)=1, f(3)-g(3)=4, f(4)-g(4)=9$. Then, a root of $f(x)-g(x)=0$ is

If $f(x)=\frac{2 x-3}{(x-2)(x-3)}$ is a real valued function, then the value that $f(x)$ does not take is

If $\alpha, \beta$ and $\gamma$ are the roots of the equation $5 x^3-3 x^2+2 x-4=0$, then $\Sigma \alpha^2 \beta^2=$

If $\alpha, \beta$ and $\gamma$ are the roots of the equation $x^3+4 x^2-9 x-36=0$ such that $\alpha+\beta=0$, then $\alpha^2+2 \beta^2+3 \gamma^2=$

If $(2-i)$ is one of the roots of the equation $x^4-9 x^3+31 x^2-49 x+30=0$ and $\alpha, \beta(\alpha<\beta)$ are its real roots, then $2 \alpha-\beta=$

If $m$ and $M$ are respectively, the smallest and greatest rational roots of the equation $6 x^6-25 x^5+31 x^4-31 x^2+25 x-6=0$, then $M-m=$

The number of ways of arranging the letters of the word LINEAR so that the letters N and R do not come together and E and A come together is

15 lines are concurrent at a point $P$. A line $L$ is not passing through $P$ intersects all the 15 lines and forms triangles with them. Then, the number of triangles having $L$ as one of its side is

The expansion of $(a+x)^n$ contains 15 terms. When $x=1$ the ratio of the neighbouring terms to the middle term in this expansion is 16 . Then, the positive integral value of ' $a$ ' is

If $\frac{d}{d x}\left(\frac{2 x+1}{(x+1)^2(x-2)}\right)=\frac{A}{(x-2)^2}+\frac{B}{(x+1)^3}+\frac{C}{(x+1)^2}$, then $A+B+C=$

If $\frac{x^2-2}{\left(x^2+1\right)\left(x^2+3\right)}=\frac{A x+B}{x^2+1}+\frac{C x+D}{x^2+3}$, then $D=$

Let $\alpha$ be the period of $3 \sin \frac{\pi x}{3}-\cos \frac{\pi x}{2}+\tan \frac{\pi x}{4}, \beta$ be the period of $\sin ^2\left(\frac{\pi}{7}+\frac{x}{4}\right)-\sin ^2\left(\frac{\pi}{7}-\frac{x}{4}\right)$, and $\gamma$ be the period of $\cos ^4 x+\sin ^4 x$. Then, $\frac{\alpha \gamma}{\beta}=$

If $\theta$ does not lie in the second quadrant and $\tan \theta=\frac{-3}{4}$, then $\tan \frac{\theta}{2}+\sin 2 \theta=$

$$ \cos ^2 76^{\circ}+\sin ^2 46^{\circ}+\sin 76^{\circ} \cos 46^{\circ}= $$

If $x=\log _e\left(\cot \left(\frac{\pi}{4}+\theta\right)\right)$, then $\lim _{\theta \rightarrow 0} \frac{\theta}{(\sinh x)(\cosh x)}=$

If $e^{i t}=\cos t+i \sin t$ and $e^{-i t}=\cos t-i \sin t$, then $\cosh (x+i y)-\cosh (x-i y)=$

In a $\triangle A B C, A D$ and $B E$ are medians. If $A D=4, \angle D A B=\frac{\pi}{6}$ and $\angle A B E=\frac{\pi}{3}$, then the area of $\triangle A B C$ is

If $S$ is the circumentre of a $\triangle A B C, a=5, b=6, c=9$ and $S B=\frac{27}{4 \sqrt{2}}$, then $\sin 2 C=$

In a $\triangle A B C$, if $\frac{r}{r_1}=\frac{1}{2}$, then $4 \tan \frac{A}{2}\left(\tan \frac{B}{2}+\tan \frac{C}{2}\right)=$

If $2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}, \hat{\mathbf{i}}-3 \hat{\mathbf{j}}-5 \hat{\mathbf{k}}$ are the position vectors of the points $\mathbf{A}$ and $\mathbf{B}$ respectively, $\mathbf{C}$ divides $\mathbf{A B}$ in the ratio $2: 3$ and $\mathbf{M}$ is the mid-point of $A B$, then 5 (position vector of $\mathbf{C})-2($ position vector of $\mathbf{M})=$

30. If $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ are the non-coplanar vectors and $\mathbf{a}-2 \mathbf{b}+3 \mathbf{c},-4 \mathbf{a}+5 \mathbf{b}-6 \mathbf{c}, x \mathbf{a}-9 \mathbf{b}+z \mathbf{c}$ are collinear points, then $2 x-z=$

The point which lies on the plane passing through the point $\hat{\mathbf{i}}-2 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}, 3 \hat{\mathbf{i}}-\hat{\mathbf{j}}+4 \hat{\mathbf{k}},-3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}-5 \hat{\mathbf{k}}$ is

If $\mathbf{a}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ and $\mathbf{b}=2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+\hat{\mathbf{k}}$, then the component of $\mathbf{b}$ perpendicular to $\mathbf{a}$ is

If the angle between the planes $\mathbf{r} \cdot(11 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\alpha \hat{\mathbf{k}})=7$ and $\mathbf{r} \cdot(2 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}-2 \hat{\mathbf{k}})=5$ is $\frac{\pi}{2}$, then $\alpha=$

If $2 \mathbf{i}+3 \mathbf{j}-4 \mathbf{k}$ and $-\mathbf{i}+2 \mathbf{j}+\mathbf{k}$ are the two diagonals of a parallelogram, then the area of the parallelogram in square units is

There are $n$ observations and all of them are negative numbers. The ascending order of these observations is $x_1, x_2, \ldots . x_n$. If the signs of the first term and last term in that order are changed, then the range of the data is

A bag contains 3 white and 6 red balls. Four balls are drawn at a time randomly. Then, the probability of getting at least two red balls is

$A$ and $B$ are two independent events $P(A)=\frac{2}{5}, P(B)=\frac{1}{3}$.

Match the following :

The correct match is

Two players $A$ and $B$ are alternately throwing a coin and a die together. $A$ player who first throws head and 6 wins the game. If $A$ starts the game, then the probability that $B$ wins the game is

If two dice are thrown and if $X$ denotes the sum of the numbers that show up on the faces of the dice, then the mean of the random variable $X$ is

In a university campus, the probability that a person chosen at random is an engineering student is $\frac{1}{5}$. The probability of having atmost two engineering students in a sample of 8 people is

If $A(1,1), B(-1,1)$ and $C(-1,-1)$ are three points and a point $P$ moves such that $(P A)^2=(P B)^2+(P C)^2$, then the equation of the locus of $P$ is

If $x^2=8 a y$ is the transformed equation of $x^2-4 y+6 x+15=0$ when the origin is shifted to the point $(\alpha, \beta)$ by translation of axes, then $2 \alpha+8 \beta^2=$

If a straight line $L$ passing through the point $(5,-3)$ is inclined at an angle of $60^{\circ}$ to the line $\sqrt{3} x+y-9=0$ and $L$ intersects $X$-axis, then the equation of $L$ is

Let $\alpha, \beta$ and $\gamma$ be three non-zero real constants and $a, b$ and $c$ be three arbitrary real numbers which satisfy $\alpha a+\beta b+\gamma c=0$. Then, the point of concurrence of the family of lines $a x+b y+c=0$ is

If the algebraic sum of the perpendicular distances from the points $(2,0),(0,2)$ and $(1,1)$ to a variable line is zero, then the variable line always passes through a fixed point. The coordinates of that point are

For $a, b, c \in R$, if $6 a^2-3 b^2-c^2+7 a b-a c+4 b c=0$ and $|a|+|b| \neq 0$, then all the lines given by $a x+b y+c=0$ are

If $\theta$ is the acute angle between the pair of lines $H \equiv a x^2-x y+b y^2=0, \tan \theta=5$ and $(1,-1)$ is a point on $H=0$, then $a^2+a b+b^2=$

The equation of the pair of straight lines passing through the point $(2,3)$ and perpendicular to the pair of lines $3 x^2-4 x y+5 y^2=0$ is $a x^2+2 h x y+b y^2+2 g x+2 f y+c=0$, then $a+b+c+f+g+h=$

If $f(x, y)=0$ is the combined equation of the lines joining the origin to the points where the line $4 x-6 y-2=0$ meets the curve $3 x^2-4 x y+5 y^2-2 x+y-6=0$, then $\frac{f(1,-1)}{f(-1,-1)}=$

The radius of the circle passing through the points $(-1,1),(2,-1)$ and $(1,0)$ is

If $A=(0,-2)$ and $B$ is any point on the circle $x^2+y^2-2 x-2 y+1=0$, then the maximum value of $(\mathbf{A B})^2$ is

If $(\alpha, \beta)$ is the pole of the line $3 x-5 y+6=0$ with respect to the circle $x^2+y^2-10 x+14 y+46=0$, then $\alpha+\beta=$

$O(0,0)$ and $A(1,0)$ are centres of two units circles $C_1$ and $C_2$, respectively. $C_3$ is also a unit circle having its centre above $X$ - axis and passing through $O$ and $A$. The equation of the common tangent to $C_1$ and $C_3$ which does not intersect the circle $C_2$ is

If the circles $x^2+y^2-16 x-20 y+164=r^2(r>0)$ and $x^2+y^2-8 x-14 y+29=0$ intersect in two distinct points, then the maximum possible integral value of $r$ is

If the circle $x^2+y^2-6 x-12 y+1=0$ cuts another circle $C$ orthogonally and the centre of the circle $C$ is $(-4,2)$, then its radius of

Let $L L^{\prime}$ be the latusrectum and $P Q$ be the focal chord of the parabola $y^2=16 x$. If $P=(1,4)$ and $P, L$ lie in the same quadrant, then $L Q=$

If $P\left(\frac{1}{2}, 4\right)$ and $Q$ are the ends of a focal chord of the parabola $y^2=32 x$ and $S$ is the focus of the parabola, then $S Q=$

Statement I The equation of the directrix of the ellipse $4 x^2+y^2-8 x-4 y+4=0$ is $3 y=6-4 \sqrt{3}$

Statement II The equation of the latusrectum of the ellipse $x^2+4 y^2-4 x-8 y+4=0$ is $y=2+\sqrt{3}$

Which of the above statement(s) is (are) true?

If $S$ is the focus of the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ lying on the positive $X$ - axis and $P(\theta)$ is a point on the ellipse such that $S P=1$, then $\cos \theta=$

A hyperbola having its centre at the origin is passing through the point $(5,2)$ and has transverse axis of length 8 along the $X$-axis. Then, the eccentricity of its conjugate hyperbola is

If $e_1$ is the eccentricity of the hyperbola $x=\sec \theta$, $y=\sqrt{2} \tan \theta$ and $e_2$ is the eccentricity of the hyperbola $x=\sqrt{2} \sec \theta$ and $y=\tan \theta$, then $\frac{e_2^2}{e_1^2}=$

$A(27,-243,81)$ is a point in space, $B, C$ and $D$ are images of $A$ with respect to $X Y, Y Z$ and $Z X$ planes respectively. If the centroid of the $\triangle B C D$ is $(\alpha, \beta, \gamma)$, then $\alpha+\beta+\gamma=$

Let $A(2,5,7)$ be the image of the point $B(1,-2,3)$ with respect to a plane $\pi$. Let $C$ be the point where $A B$ meets the plane $\pi$. Let $D=(2,1,6)$. Then, the direction cosines of $C D$ are

If a plane $x+y+z-5=0$ intersects the line joining $A(1,1,1)$ and $B(2,2,2)$ at $P$, then $A P: P B=$

$$ \mathop {\lim }\limits_{x \to 2}\left[\left(x^2-4 x+4\right) \cos \left(\frac{2}{x-2}\right)+\frac{x^2-4}{x^3-2 x-4}\right]= $$

$$ \lim _{x \rightarrow 0} \frac{\tan 2 x-2 \tan x}{(1-\cos x)\left(2^x-1\right)}= $$

$$ \frac{d}{d x}\left[\left(x^{\frac{5}{2}}-x^{\frac{3}{2}}+1\right)\left(x^2-3 x+5\right)\right]= $$

The value of $\frac{d}{d x}\left[\log \left(\sin \sqrt{\frac{x^2+1}{x^2+2}}\right)\right]$ when $x=\sqrt{2}$, is

If $f(x)=\frac{1+\sec x}{2(\sec x-1)}$ for $0

The equation of the normal to the curve $\sin y=\sqrt{3} x \sin \left(\frac{\pi}{6}+y\right)$ at $x=0$, is

Assertion (A) The curves $y^2=4 x$ and $x^2=-2 y$ intersect at $(1,2)$ orthogonally.

Reason (R) If the product of the slopes of the tangents drawn to two curves at their point of intersection is -1 , then the curves are said to cut each other orthogonally.

Let $f(x)=\left\{\begin{array}{cc}1+6 x-3 x^2 & x \leq 1 \\ x+\log _2\left(b^2+7\right) & x>1\end{array}\right.$. Then, the set of all possible values of $b$ such that $f(1)$ is the maximum value of $f(x)$ is

Let $g(x)$ be the anti-derivative of $f(x)$. Then, the function for which $\log _e\left(1+(g(x))^2\right)+c$ is an anti-derivative is

If $f(x)=\int\left[\tan ^2 x+\cot ^2 x+\frac{4\left(\sin ^3 x+\cos ^3 x\right)}{\sin ^2 2 x}\right] d x$ and $f\left(\frac{\pi}{4}\right)=0$, then $3\left[f\left(\frac{\pi}{6}\right)+2\right]=$

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

$\int_0^3\left(\sin \left(\frac{\pi}{3} x\right)-\cos \left(\frac{\pi}{3} x\right)\right) d x=$

$$ \int_0^{\pi / 2} \sin ^4 \theta \cos ^3 \theta d \theta= $$

Statement I The differential equation corresponding to the family of circles having their centres on $Y$-axis and fixed radius $k$ is $\left(x^2-k^2\right)\left(\frac{d y}{d x}\right)^2+x^2=0$

Statement II The differential equation corresponding to the family of circles passing through the origin and having their centres on $X$-axis is $x^2-y^2+2 x y \frac{d y}{d x}=0$

Which of the above statements is (are) true?

If $m$ and $n$ are respectively the order and the degree of the differential equation representing the family of curves $y^2-5 a x-5 a^{3 / 2}=0(a>0$ is a parameter), then the value of $m-n$ is

The general solution of $\left(\left(1+x^2\right) y \sin x-2 x y\right) d x-\log y^{1+x^2} d y=0$ is

The range of the nuclear force is

Select the physical quantities in Column I and Column II having same dimensions

A rocket moves straight upward with zero initial velocity and with an acceleration $20 \mathrm{~m} / \mathrm{s}^2$. It runs out of fuel and stops accelerating at the end of 5th second. It reaches a maximum height and falls back to the earth. The speed when it hits the ground is (take $g=10 \mathrm{~m} / \mathrm{s}^2$ )

Two cars, at a certain instant, are 50 km apart on a line running from south to north. The one farther north is moving west at $25 \mathrm{~km} / \mathrm{h}$. The other is moving towards north at $25 \mathrm{~km} / \mathrm{h}$. How long do they take to reach their distance of closest approach?

85. A particle initially at origin starts moving in $X Y$ - plane has velocity component $\mathbf{v}=(6+2 t) \hat{\mathbf{i}}+(4+2 \sqrt{3 t}) \hat{\mathbf{j}} \mathrm{m} / \mathrm{s}$. Acceleration of the particle in $\mathrm{m} / \mathrm{s}^2$ is $[x, y$ are measured in meters, $t$ in seconds, respectively

A bullet is fired at time $t=0$ with velocity $20 \mathrm{~m} / \mathrm{s}$ and at an initial angle of $30^{\circ}$ with the horizontal. The angle between the displacement vector and the horizontal after time 0.1 s is (assume $g=10 \mathrm{~m} / \mathrm{s}^2$ ).

A constant horizontal force $\mathbf{F}$ of magnitude 10 N is applied to a block $A$ and this produces an acceleration of magnitude $20 \mathrm{~m} / \mathrm{s}^2$. If this block $A$ is then kept against another block $B$ of mass 1.5 kg as shown in figure and a force $F^{\prime}$ of 20 N is applied, find the force on the block $B$. Neglect friction

A body of mass $m$ slides down along a frictionless inclined plane from height $h$ and just completes motion in a vertical circle of radius 2 m after reaching the bottom. What is the value of $h$ ? [Use, $g=10 \mathrm{~m} / \mathrm{s}^2$ ]

Particle $A$ moving with a velocity $v=10 \mathrm{~m} / \mathrm{s}$ experienced a head on collision with a stationary particle $B$ of the same mass. As a result of collision, the kinetic energy of the system decreased by $1 \%$. The speed of particle $A$ after collision is

A metre stick is balanced on the knife edge at its centre. When four coins, each of mass 2 g are put one on top of the other at 10.0 cm mark, the stick it found to be balanced at 46.0 cm mark. The mass of the metre stick is

The amplitude of a damped oscillator varies with time as $A(t)=A_0 \exp (-b t / 2 \mathrm{~m})$, where $b=70 \mathrm{~g} / \mathrm{s}$ and $m=200$ g. How long does it take for the mechanical energy to drop to one-fourth of its initial value?

[Take, $\ln 2=0.7$ ]

Four particles each of mass $m$ are placed at four vertices of a rectangle having side length as $3 l_0$ and $4 l_0$. The potential energy of the system in $\frac{G m^2}{l_0}$ is

Two wires of same length having radius of 2 mm and 1.5 mm respectively, are loaded with same weights. Extension of the second wire is double than that of the first wire. What is the ratio of the Young's modulus of the first wire to that of the second wire?

Consider an increase of $1 \%$ in each of radius of artery, viscosity of blood and density of blood, respectively. The percentage change in flow rate of blood in artery is

A metal cube of side 10 cm rests on a film of a liquid of thickness 0.2 mm . If upon applying a horizontal force $\mathbf{F}$ of magnitude 0.1 N . The cube slides with a constant speed of $0.08 \mathrm{~m} / \mathrm{s}$, then the coefficient of viscosity is

176 g of $\mathrm{CO}_2$ can change its temperature from $0^{\circ} \mathrm{C}$ to $30^{\circ} \mathrm{C}$ by absorbing 3600 J of thermal energy. Molar specific heat of $\mathrm{CO}_2\left(\right.$ in $\left.\mathrm{J} \mathrm{mol}^{-1} \mathrm{~K}^{-1}\right)$ is

A solution consists of ether and 5.0 g of water at $0^{\circ} \mathrm{C}$. If the ether evaporates completely to freeze the water, then the mass of the ether in the solution is

Assertion (A) Heat and work are modes of energy transfer to a system resulting in change in its internal energy.

Reason (R) Heat and work in thermodynamics are state variables.

The correct option among the following is

An ideal gas at pressure $p_0$ undergoes an isothermal expansion until its volume is 8.0 times its initial volume. The gas is slowly and adiabatically compressed back to its original volume. If the adiabatic constant of the gas is $\gamma=\frac{4}{3}$, then the ratio of the average kinetic energy per molecule in this final state to that, in the initial state is

At what temperature is the root mean square rms speed of neon gas atoms is equal to the rms speed of helium gas atoms at $-33^{\circ} \mathrm{C}$ ?

(Atomic mass of $\mathrm{Ne}=20.2 \mathrm{u}$, and that of $\mathrm{He}=4.0 \mathrm{u}$ )

A wire of length 0.4 m stretched at both ends vibrates 250 times per second. If the length of the wire is increased by 0.1 m and the stretching force is reduced to $1 / 4$ th of its original value, then the new frequency is

A spherical glass is attached to a rigid wall as shown in the figure. An observer located at point $O$ is looking at a point $A$ on the wall. The refractive index of the glass is 1.5 and that of air is 1.0 . The distances are $O A=8 \mathrm{~cm}$, $X A=3 \mathrm{~cm}$. If the radius of curvature of spherical glass surface is $R=5 \mathrm{~cm}$, then the apparent distance of $A$ from observer $O$ is

In a double slit experiment performed in air, the angular width of a fringe is found to be $0.15^{\circ}$ on a screen placed 80 cm away. The wavelength of light is used 490 nm . The angular width of the fringe, if the entire apparatus is immersed in a medium of refractive index $\frac{5}{3}$ is

$6 \mu \mathrm{C}$ charge is placed at the centre of a cube. What will be the electric flux at each face of the cube?

$$ \left[\text { Take, } \frac{1}{4 \pi \varepsilon_0}=9 \times 10^9 \mathrm{~N}-\mathrm{m}^2 \mathrm{C}^2\right] $$

There are two thin wire rings, each of radius $R$, whose axes coincide. The charges of the rings are $q$ and $-q$. The magnitude of potential difference between the centres of the rings separated by a distance $\sqrt{3} R$ is

Statement I The temperature coefficient of resistance for most of metals in pure form is positive.

Statement II A metal wire 2 mm in diameter carries a charge of $360 \pi \mathrm{C}$ in two hours. If the metal contains $5 \times 10^{22}$ free electrons $/ \mathrm{cm}^3$, then drift velocity of the electrons in the wire is $6.25 \times 10^{10} \mathrm{~m} / \mathrm{s}$.

Statement III Semiconductors like pure germanium does not obey Ohm's law for all range of electric field values.

Which of the following is correct?

A cylindrical resistor of radius 7.0 mm and length 4.0 cm is made of material that has a resistivity of $10^{-6} \Omega-\mathrm{m}$. If the energy is dissipated at rate 1.54 W in the resistor, then the current density is

Statement I A uniform electric field and a uniform magnetic field are pointed in the same direction. If an electron is projected in the same direction, the electron velocity will decrease in magnitude.

Statement II Two infinite long parallel wires are carrying current in the same direction. The magnetic field at a point mid-way between the wires is zero.

Statement III No net force acts on a rectangular coil carrying a steady current, when suspended in a uniform magnetic field.

Which of the following is correct?

Two parallel conductor each 50 m long, separated by 0.2 m experience a force of 1 N . If the current in first conductor is twice that of the second conductor, then what is the current in the second conductor?

$$ \left(\mu=4 \pi \times 10^{-7}\right) $$

The magnitude of axial field due to a bar magnet at a distance of 1 m , is found to be $5 \times 10^{-8} \mathrm{~T}$. The magnetic moment of the bar magnet is $\left(\mu_0=4 \pi \times 10^{-7}\right)$

The magnetic flux through the triangular loop shown in the figure below

Where a uniform magnetic field of strength 2 T points perpendicularly into the plane of the triangle is

The $Q$ value of a series $L-C-R$ circuit with $L=2 \mathrm{H}$, $C=32 \mu \mathrm{~F}, R=20 \Omega$ is

A laser beam has intensity $2.1 \times 10^{15} \mathrm{~W} / \mathrm{m}^2$. The amplitude of magnetic field in the beam in approximately is

In a photoelectric experiment, the wavelength of the light incident on the metal is changed from 200 nm to 400 nm . The decrease in the stopping potential is close to

[use $h c=1240 \mathrm{eV}$-nm, where $h=$ Planck's constant and $c$ is velocity of light]

The de-Broglie wavelength of an electron with kinetic energy of 320 eV is (take, $h=6.0 \times 10^{-34}$ SI unit, mass of electron $=m_c=9.0 \times 10^{-31} \mathrm{~kg}$, charge of an electron $=1.6 \times 10^{-19} \mathrm{C}$ )

Considering the Bohr's model of hydrogen atom, the ratio of velocities of electrons orbiting in the 4th orbit to that in the 9 th orbit is

What is the mass number of the nucleus having radius equal to $\frac{1}{3}$ of that of ${ }^{189} \mathrm{Os}$ ?

The number of silicon atoms per $\mathrm{m}^3$ is $5 \times 10^{28}$. This is doped with $4.5 \times 10^{21}$ atoms $/ \mathrm{m}^3$ of arsenic. The ratio of number of electrons to number of holes after doping is (take $n_i=$ number of thermally generated electrons $=1.5 \times 10^{16} / \mathrm{m}^3$ )

The output of the following circuit is equivalent to $.......$ gate

Which of the following statements is not true?