Observe the following statements.

Statement -I Rutherford model of an atom cannot explain the stability of an atom.

Statement-II The wavelength of X-rays is higher than the wavelength of microwaves.

The correct answer is

In hydrogen atom, an electron is transferred from an orbit of radius 1.3225 nm to another orbit of radius 0.2116 nm . What is the energy (in J ) of emitted radiation?

Identify the correct orders regarding atomic radii

(i) $\mathrm{Cl}>\mathrm{F}>\mathrm{Li}$

(ii) $\mathrm{P}>\mathrm{C}>\mathrm{N}$

(iii) $\mathrm{Tm}>\mathrm{Sm}>\mathrm{Eu}$

(iv) $\mathrm{Sr}>\mathrm{Ca}>\mathrm{Mg}$

$$ \text { Match the following } $$

$$ \begin{array}{cccc} \hline & \text { List-I (Elements) } & & \text { List-II (Group) } \\ \hline \text { A } & \mathrm{Mn}, \mathrm{Tc}, \mathrm{Re} & \text { I } & 12 \\ \hline \text { B } & \mathrm{Zn}, \mathrm{Cd}, \mathrm{Hg} & \text { II } & 4 \\ \hline \text { C } & \mathrm{Ti}, \mathrm{Zr}, \mathrm{Hf} & \text { III } & 17 \\ \hline \text { D } & \mathrm{Ga}, \mathrm{In}, \mathrm{Tl} & \text { IV } & 7 \\ \hline & & \text { V } & 13 \\ \hline \end{array} $$

The correct answer is

The atomic numbers of the elements $X, Y, Z$ are $a, a+1, a+2$ respectively. $Z$ is an alkali metal. The nature of bonding in the compound formed by $X$ and $Z$ is

The sets of molecules in which central atom has no lone pair of electrons are

I. $\mathrm{SnCl}_2, \mathrm{NH}_3, \mathrm{SF}_4$

II. $\mathrm{HgCl}_2, \mathrm{SO}_3, \mathrm{SF}_6$

III. $\mathrm{BeCl}_2 \mathrm{BF}_3, \mathrm{PCl}_5$

IV. $\mathrm{ClF}_3, \mathrm{BrF}_5, \mathrm{XeF}_6$

The isobars of one mole of an ideal gas were obtained at three different pressure ( $p_1, p_2$ and $p_3$ ). The slopes of these isobars are $m_1, m_2$ and $m_3$ respectively. If $p_1 < p_2 < p_3$, then the correct relation of the slopes is

100 mL of $0.05 \mathrm{M} \mathrm{Cu}^{2+}$ aqueous solution is added to IL of 0.1 M KI solution. The number of moles of $\mathrm{I}_2$ and $\mathrm{Cu}_2 \mathrm{I}_2$ formed are respectively.

The $C_p$ of an ideal gas is $10314 \mathrm{Jmol}^{-1} \mathrm{~K}^{-1}$. One mole of this gas is expanded against a constant pressure of $p \mathrm{~atm}$. The change in temperature during expansion is 1.0 K . The value of $q$ (in J ) and $\Delta H$ (in $\mathrm{Jmol}^{-1}$ ) are respectively.

At $T(\mathrm{~K}), K_p$ value for the reaction,

$$ 2 \mathrm{AO}_2(\mathrm{~g})+\mathrm{O}_2(\mathrm{~g}) \rightleftharpoons 2 \mathrm{AO}_3(\mathrm{~g}) \text { is } 4 \times 10^{10}, $$

What is the $K_p^{\prime}$ value for

$$ 2 \mathrm{AO}_2(\mathrm{~g})+\frac{3}{2} \mathrm{O}_2 \rightleftharpoons 3 \mathrm{AO}_3(\mathrm{~g}) \text { at } T(\mathrm{~K}) $$

A sample of water contains $\mathrm{Mg}\left(\mathrm{HCO}_3\right)_2$ and $\mathrm{Ca}\left(\mathrm{HCO}_3\right)_2$ On boiling this water, these hydrogen carbonates are removed as precipitates. The precipitates are

Which of the following statements is not correct?

The order of negative standard potential values of Li, $\mathrm{Na}, \mathrm{K}$ is

In which of the following reactions, hydrogen is evolved?

I. Reaction of sodium borohydride with iodine

II. Oxidation of diborane

III. Reaction of boron trifluoride with sodium hydride

IV. Hydrolysis of diborane

Which of the following statements is not correct regarding the gas evolved by the reaction of dilute HCl on $\mathrm{CaCO}_3$ ?

Observe the following statements

Statement I The carbon containing components of photochemical smog are acrolein, methanal and PAN. Statement II The number of greenhouse gases in the list given below is 5 .

$$ \mathrm{CH}_4, \mathrm{CO}_2, \mathrm{NO}, \mathrm{H}_2 \mathrm{O}(l), \mathrm{H}_2 \mathrm{O}(g), \mathrm{O}_2, \mathrm{O}_3 $$

The correct answer is

The condensed, bond line and complete formulae of $n$-butane are respectively.

' $X$ ' is the isomer of $\mathrm{C}_6 \mathrm{H}_{14}$. It has four primary carbons and two tertiary carbons, ' $X$ ' can be prepared from which of the following reactions?

$$ \text { What are } X \text { and } Y \text { in the following reaction sequence? } $$

$$ \text { Isopentane } \xrightarrow{\mathrm{KMnO}_4} X \xrightarrow[358 \mathrm{~K}]{20 \% \mathrm{H}_3 \mathrm{PO}_4} Y $$

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

A metal ( $M$ ), crystallises in fcc lattice with edge length of $4.242 \mathop {\rm{A}}\limits^{\rm{o}}$. What is the radius of $M$ atom (in $\mathop {\rm{A}}\limits^{\rm{o}}$ )?

A solid mixture weighing 5 g contains equal number of moles of $\mathrm{Na}_2 \mathrm{CO}_3$ and NaHCO . This solid mixture was dissolved in 1 L of water. What is the volume (in mL ) of 0.1 M HCl required to completely react with this 1 L mixture solution?

At 298 K the equilibrium constant for the reaction $M(s)+2 \mathrm{Ag}^{+}(a q) \longrightarrow M^{2+}(a q)+2 \mathrm{Ag}(s)$ is $10^{15}$. What is the $E_{\text {cell }}^{\ominus}$ (in V) for this reaction?

$$ \left(\frac{2.303 R T}{F}\right)=0.06 \mathrm{~V} $$

$A \rightarrow P$ is a first order reaction. At 300 K this reaction was started with $[A]=0.5 \mathrm{~mol} \mathrm{~L}^{-1}$.

The rate constant of reaction was $0.125 \mathrm{~min}^{-1}$. The same reaction was started separately with $[A]=1 \mathrm{molL}^{-1}$ at 300 K . The rate constant (in $\mathrm{min}^{-1}$ ) now is

Observe the following reaction

I. $\mathrm{CO}(\mathrm{g})+\mathrm{H}_2(\mathrm{~g}) \xrightarrow{X} \mathrm{HCHO}(\mathrm{g})$

II. $\mathrm{CO}(\mathrm{g})+3 \mathrm{H}_2(\mathrm{~g}) \xrightarrow{\gamma} \mathrm{CH}_4(\mathrm{~g})+\mathrm{H}_2 \mathrm{O}(\mathrm{g})$

The catalysts $X$ and $Y$ in the above reactions are respectively

Composition of siderite ore is

Which of the following gives more number of oxides on reacting with HCl ?

The number of lone pairs of electrons on the central atom of $\mathrm{XeO}_3, \mathrm{XeOF}_4$ and $\mathrm{XeF}_6$ respectively is

Which of the following statements is not correct?

The pair of ions with paramagnetic nature and same number of electrons is

$$ \text { Observe the following complex ions } $$

$$ \begin{array}{cccc} \hline\left[\mathrm{Mn}(\mathrm{CN})_6\right]^{3-} & {\left[\mathrm{Fe}(\mathrm{CN})_6\right]^{3-}} & {\left[\mathrm{CoF}_6\right]^{3-}} & {\left[\mathrm{Co}\left(\mathrm{C}_2 \mathrm{O}_4\right)_3\right]^{3-}} \\ \hline A & B & C & D \\ \hline \end{array} $$

Identify the option in which the unpaired electrons in the complex ions are in correct increasing order

Amino acid ' $X$ ' contains phenolic hydroxy group and amino acid ' $Y$ ' contains amide group. ' $X$ ' and ' $Y$ ' respectively are

The chemical $X$ is used in the prevention of heart attack. The structure of $X$ is

$$ \text { Observe the following reactions } $$

The correct order of reactivity of $X, Y, Z$ towards $\mathrm{S}_{\mathrm{N}} 1$ reaction is

An alochol $X\left(\mathrm{C}_5 \mathrm{H}_{12} \mathrm{O}\right)$ produces turbidity instantly with conc. $\mathrm{HCl} / \mathrm{ZnCl}_2$. Isomer $(Y)$ of $X$ undergoes dehydration with conc. $\mathrm{H}_2 \mathrm{SO}_4$ at $443 \mathrm{~K}, X$ and $Y$ respectively are

Toluene on reaction with reagent $A$ gives $X$. This $(X)$ forms 2, 4 dinitrophenylhydrazone and reduces ammoniacal silver nitrate solution. Reaction of toluene with another reagent $B$ forms $Y$, which dissolves in $\mathrm{NaHCO}_3$ with evolution of $\mathrm{CO}_2$. What are $A$ and $B$ respectively?

$$ \begin{aligned} &\text { What are } X \text { and } Y \text { in the following reaction sequence? }\\ &\mathrm{C}_5 \mathrm{H}_{12} \mathrm{O} \xrightarrow[573 \mathrm{~K}]{\mathrm{Cu}} \mathrm{C}_5 \mathrm{H}_{10} \xrightarrow[\text { (ii) } \mathrm{Zn}+\mathrm{H}_2 \mathrm{O}]{\text { (i) } \mathrm{O}_3} X+Y \end{aligned} $$

$$ \text { Observe the following set of reactions } $$

$$ \text { What are } X, Y \text { and } Z \text { respectively } $$

Consider the following set of reactions

What are $A$ and $B$ respectively?

The domain and range of $f(x)=\frac{1}{\sqrt{|x|-x^2}}$ are $A$ and $B$ respectively. Then $A \cup B=$

A function $f: R \rightarrow R$ defined by

$$ f(x)=\left\{\begin{array}{c} 2 x+3, x \leq \frac{4}{3} \\ -3 x^2+8 x, x>\frac{4}{3} \end{array}\right. \text { is } $$

If $2^{4 n+3}+3^{3 n+1}$ is divisible by $P$ for all natural numbers $n$, then $P$ is

A is a $3 \times 3$ matrix satisfying $A^3-5 A^2+7 A+I=0$ If $A^5-6 A^4+12 A^3-6 A^2+2 A+2 I=l A+m I$, then $l+m=$

If $A=\left[\begin{array}{lll}0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & x & 1\end{array}\right], A^{-1}=\frac{1}{2}\left[\begin{array}{ccc}1 & -1 & 1 \\ -8 & 6 & 2 y \\ 5 & -3 & 1\end{array}\right]$, then the point $(x, y)$ lies on the curve represented by the equation.

Consider a homogeneous system of three linear equations in three unknowns represented by $A X=0$.

If $X=\left[\begin{array}{c}l \\ m \\ 0\end{array}\right], l \neq 0, m \neq 0, l, m \in R$ represents an infinite number of solutions of this system, then rank of $A$ is

The number of real values of ' $a$ ' for which the system of equations $2 x+3 y+a z=0, x+a y-2 z=0$ and $3 x+y+3 z=0$ has non-trivial solution is

If the eight vertices of a regular octagon are given by the complex number $\frac{1}{x_j-2 i}(j=1,2,3,4,5,6,7,8)$, then the radius of the circumcircle of the octagon is

If $\left|Z_1-3-4 i\right|=5$ and $\left|Z_2\right|=15$, then the sum of the maximum and minimum values of $\left|Z_1-Z_2\right|$ is

If $Z=r(\cos \theta+i \sin \theta),\left(\theta \neq-\frac{\pi}{2}\right)$ is solution of $x^3=i$, then $r^9(\cos \theta+i \sin \theta)^9=x^{3-}=i$

If $\omega \neq 1$ is a cube root of unity, then one root among the 7th roots of $(1+\omega)$ is

$f(x)=x^2-2(4 k-1) x+g(k)>0, \forall x \in R$ and for $k \in(a, b)$. If $g(k)=15 k^2-2 k-7$, then

If local maximum of $f(x)=\frac{a x+b}{(x-1)(x-4)}$ exists at $(2,-1)$, then $a+b=$

If $1+2 i$ is a root of the equation $x^4-3 x^3+8 x^2-7 x+5=0$, then sum of the squares of the other roots is

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3+\frac{a}{2} x+b=0$ and $(\alpha-\beta)(\alpha-\gamma),(\beta-\alpha)(\beta-\gamma),(\gamma-\alpha),(\gamma-\beta)$ are the roots of the equation

$(y+a)^3+K(y+a)^2+L=0$, then $\frac{L}{K}=$

All the letters of the word MOTHER are arranged in all possible ways and the resulting words (may or may not have meaning) are arranged as in the dictionary. The number of words that appear after the word MOTHER is

The number of positive integral solution of $\frac{1}{x}+\frac{1}{y}=\frac{1}{2025}$ is

The number of positive integral solutions of $x y z=60$ is

Numerically greatest term in the expansion of $(3 x-4 y)^{23}$ when $x=\frac{1}{6}$ and $y=\frac{1}{8}$ is

Let $K$ be the number of rational terms in the expansion of $(\sqrt{2}+\sqrt[3]{3})^{6144}$. If the coefficient of $x^P(P \in N)$ in the expansion of $\frac{1}{(1+x)\left(1+x^2\right)\left(1+x^4\right)\left(1+x^8\right)\left(1+x^{16}\right)}$ is $\alpha_p$, then $\alpha_k-\alpha_{k+1}-\alpha_{k-1}=$

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

If $\tan \left(\frac{\pi}{4}+\frac{\alpha}{2}\right)=\tan ^3\left(\frac{\pi}{4}+\frac{\beta}{2}\right)$, then $\frac{3+\sin ^2 \beta}{1+3 \sin ^2 \beta}=$

If $P=\sin \frac{2 \pi}{7}+\sin \frac{4 \pi}{7}+\sin \frac{8 \pi}{7}$ and $Q=\cos \frac{2 \pi}{7}+\frac{4 \pi}{7}+\cos \frac{8 \pi}{7}$, then the point $(P, Q)$ lies on the circle of radius

If $\cos \alpha=\frac{l \cos \beta+m}{l+m \cos \beta}$, then $\left(\frac{\tan \frac{\alpha}{2}}{\tan \frac{\beta}{2}}\right)^2=$

If $a, b$ are real numbers and $\alpha$ is a real roots of $x^2+12+3 \sin (a+b x)+6 x=0$, then the value of $\cos (a+b \alpha)$ for the least positive value of $a+b \alpha$ is

The number of real solution of $\tan ^{-1} x+\tan ^{-1} 2 x=\frac{\pi}{4}$ is

Consider the following statements

Statement $\mathrm{I} \cosh ^{-1} x=\tanh ^{-1} x$ has no solution

Statement II $\cosh ^{-1} x=\operatorname{coth}^{-1} x$ has only one solution

The correct answer is

If the angular bisector of the angle $A$ of the $\triangle A B C$ meets its circumcircle at $E$ and the opposite side $B C$ at $D$, then $D E \cos \frac{A}{2}=$

In a $\triangle A B C, a=5, b=4$ and $\tan \frac{C}{2}=\sqrt{\frac{7}{9}}$, then its inradius $r=$

Two adjacent sides of a triangle are represented by the vectors $2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ and $2 \sqrt{3} \hat{\mathbf{i}}-2 \sqrt{3} \hat{\mathbf{j}}+\sqrt{3} \hat{\mathbf{k}}$. Then, the least angle of the triangle and perimeter of the triangle are respectively.

A plane $\pi_1$ contains the vectors $\hat{\mathbf{i}}+\hat{\mathbf{j}}$ and $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}$. Another plane $\pi_2$ contains the vectors $2 \hat{\mathbf{i}}-\hat{\mathbf{j}}$ and $3 \hat{\mathbf{i}}+2 \hat{\mathbf{k}}$. $\mathbf{a}$ is a vectors parallel to the line of intersection of $\pi_1$ and $\pi_2$. If the angle $\theta$ between $\mathbf{a}$ and $\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ is acute, then $\theta=$

In a quadrilateral $A B C D, \mathbf{A}=\frac{2 \pi}{3}$ and $A C$ is the bisector of angle $\mathbf{A}$. If $15|\mathbf{A C}|=5|\mathbf{A D}|=3|\mathbf{A B}|$, then angle between $\mathbf{A B}$ and $\mathbf{B C}$ is

$\mathbf{a}, \mathbf{b}, \mathbf{c}$ are three non- coplanar and mutually perpendicular vectors of same magnitude $K . r$ is any vectors satisfying $\mathbf{a} \times((\mathbf{r}-\mathbf{b}) \times \mathbf{a})+\mathbf{b} \times((\mathbf{r}-\mathbf{c}) \times \mathbf{b})+\mathbf{c} \times((\mathbf{r}-\mathbf{a}) \times \mathbf{c})=\mathbf{0}$, then $\mathbf{r}=$

Consider the following

Assertion (A) The two lines $\mathbf{r}=\mathbf{a}+t(\mathbf{b})$ and $\mathbf{r}=\mathbf{b}+s(\mathbf{a})$ intersect each other.

Reason (R) The shortest distance between the lines $\mathbf{r}=\mathbf{p}+t(\mathbf{q})$ and $\mathbf{r}=\mathbf{c}+s(\mathbf{d})$ is equal to the length of projection of the vector ( $\mathbf{p}-\mathbf{c}$ ) on ( $\mathbf{q} \times \mathbf{d}$ )

The correct answer is

The mean deviation about median of the numbers $3 x, 6 x, 9 x, \ldots .81 x$ is 91 , then $|x|=$

Functions are formed from the set $A=\left\{a_1, a_2, a_3\right\}$ to another set $B=\left\{b_1, b_2, b_3, b_4, b_5\right\}$. If a function is selected at random, then probability, that it is a non-one function is

$A$ and $B$ are two events of a random experiment such that $P(B)=0.4, P(A \cap \bar{B})=0.5, P(A \cup B)+P\left(\frac{B}{A \cup \bar{B}}\right)=1.15$ then $P(A)=$

There are two boxes each containing 10 balls. In each box, few of them are black balls and rest are white. A ball is drawn at random from one of the boxes and found that it is black. If the probability that the black ball drawn is from the second box is $\frac{1}{5}$, then number of black balls in the first box is

In a shelf there are three mathematics and two physics books. A student takes a book randomly. If he randomly takes, successively for three time by replacing the book already taken every time, then the mean of the number of mathematics books which is treated as random variable is

In possion distribution, if $\frac{P(x=5)}{P(X=2)}=\frac{1}{7500}$ and $\frac{P(X=5)}{P(X=3)}=\frac{1}{500}$, then the mean of the distribution is

$A(2,0), B(0,2), C(-2,0)$ are three points. Let $a, b, c$ be the perpendicular distances from a variable point $P$ on to the lines $A B, B C$ and $C A$ respectively. If $a, b, c$ are in arithmetic progression, then the locus of $P$ is

When the coordinate axes are rotated about the origin through an angle $\frac{\pi}{4}$ in the positive direction, the equation $a x^2+2 h x y+b y^2=c$ is transformed to $25 x^2+9 y^2=225$, then $(a+2 h+b-\sqrt{c})^2=$

$y-x=0$ is the equation of a side of a $\triangle A B C$. The orthocentre and circumcentre of the $\triangle A B C$ are respectively $(5,8)$ and $(2,3)$. The reflection of orthocentre with respect to any side of the triangle lies on its circumcircle. Then, the radius of the circumcircle of the triangle is

Two families of lines are given by $a x+b y+c=0$ and $4 a^2+9 b^2-c^2-12 a b=0$. Then, the line common to both the families is

Two non-parallel sides of a rhombus are parallel to the lines $x+y-1=0$ and $7 x-y-5=0$. If $(1,3)$ is the centre of the rhombus and one of its vertices $A(\alpha, \beta)$ lies on $15 x-5 y=6$, then one of the possible values of $(\alpha+\beta)$ is

If the equations $3 x^2+2 h x y-3 y^2=0$ and $3 x^2+2 h x y-3 y^2+2 x-4 y+c=0$ represent the four sides of a square, then $\frac{h}{c}=$

The radius of the circle having three chords along Y-axis, the line $y=x$ and the line $2 x+3 y=10$

Among the chords of the circle $x^2+y^2=75$, the number of chords having their mid-points on the line $x=8$ and having their slopes as integers is

The equation of the circle which touches the circle $S \equiv x^2+y^2-10 x-4 y+19=0$ at the point $(2,3)$ internally and having radius equal to half of the radius of the circle $S=0$ is

If $P\left(\frac{7}{5}, \frac{6}{5}\right)$ is the inverse point of $A(1,2)$ with respect to a circle with centre $C(2,0)$, then the radius of that circle is

If the circle $S=0$ intersect the three circle

$$ \begin{aligned} & S_1 \equiv x^2+y^2+4 x-7=0 \\ & S_2 \equiv x^2+y^2+y=0 \text { and } S_3 \equiv x^2+y^2+\frac{3}{2} x+\frac{5}{2} y-\frac{9}{2}=0 \end{aligned} $$

orthogonally, then radical axis of $S=0$ and $S_1=0$ is

If a tangent of the circle $x^2+y^2+2 x+2 y+1=0$ is radical axis of the circles $x^2+y^2+2 g x+2 f y+c=0$ and $2 x^2+2 y^2+3 x+8 y+2 c=0$, then

If the angle between the tangents drawn to the parabola $y^2=4 x$ from the points on the line $4 x-y=0$ is $\frac{\pi}{3}$, then the sum of the abscissae of all such points is

The normal at a point on the parabola $y^2=4 x$ passes through a point $P$. Two more normals to this parabola also pass through $P$. If the centroid of the triangle formed by the feet of these three normals is $G(2,0)$, then the abscissa of $P$ is

The circumcenter of the equilateral triangle having the three points $\theta_1, \theta_2, \theta_3$ lying on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ as its vertices is $(r, s)$. Then, the average of $\cos \left(\theta_1-\theta_2\right)$, $\cos \left(\theta_2-\theta_3\right)$ and $\cos \left(\theta_3-\theta_1\right)$ is

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,(b>a)$ is an ellipse with eccentricity $\frac{1}{\sqrt{2}}$. If the angle of intersection between the ellipse and parabola $y^2=4 a x$ is $\theta$, then the coordinates of the point $\frac{2 \theta}{3}$ on the ellipse is

The number of common tangents that can drawn to the curves $\frac{x^2}{16}-\frac{y^2}{9}=1$ and $x^2+y^2=16$ is

Let $A(\alpha, 4,7)$ and $B(3, \beta, 8)$ be two points in space. If $Y Z$ plane and $Z X$-plane respectively divide the line segment joining the points $A$ and $B$ in the ratio $2: 3$ and $4: 5$, then the point $C$ which divides $A B$ in the ratio $\alpha: \beta$ externally is

The direction ratios of the line bisecting the angle between the $X$-axis and the line having direction ratios $(3,-1,5)$ are

If the plane $-4 x-2 y+2 z+\alpha=0$ is at a distance of two units from the plane $2 x+y-z+1=0$, then the product of all the possible values of $\alpha$ is

$$ \mathop {\lim }\limits_{x \to 0} \frac{\sqrt{\cos x}-\sqrt[3]{\cos x}}{\sin ^2 x}= $$

Let $f:[-1,2] \rightarrow R$ be defined by $f(x)=\left[x^2-3\right]$ where $[$. denotes greatest integer function, then the number of points of discontinuity for the function $f$ in $(-1,2)$ is

If $f(x)=\left\{\begin{array}{cc}x^2\left|\cos \frac{\pi}{2}\right|, & x \neq 0 \\ 0, & x=0\end{array}\right.$, then at $x=2, f(x)$ is

The set of all values of $x$ for which $f(x)=\| x|-1|$ is differentiable is

$$ \begin{aligned} &\text { If } y=f(x)^{g(x)} \text { and } \frac{d y}{d x}=y\left[H(x) f^{\prime}(x)+G(x) g^{\prime}(x)\right] \text {, then }\\ &\int \frac{G(x) H(x) f^{\prime}(x)}{g(x)} d x= \end{aligned} $$

If $x=t-\sin t, y=1-\cos t$ and $\frac{d^2 y}{d x^2}=-1$ at $t=k, k>0$ then $\lim _{i \rightarrow K} \frac{y}{x}=$

For the curve $\frac{x^n}{a^n}+\frac{y^n}{b^n}=2,(n \in N$ and $n>1)$ the line $\frac{x}{a}+\frac{y}{b}=2$ is

The height of a cone with semi-vertical angle $\frac{\pi}{3}$ is increasing at the rate of 2 units $/ \mathrm{min}$. The rate at which the radius of the cone is to be decreased so as to have a fixed volume always is

The function $f(x)=2 x^3-9 a x^2+12 a^2 x+1$ where $a>0$ attains its local maximum and local minimum at $p$ and $q$ respectively. If $p^2=q$, then $a=$

Consider all functions given in List I in the interval [1,3]. The list II has the value of ' $c$ ' obtained by applying Lagrange's mean value theorem on the function of List I . Match the function and values of ' c '

$$ \begin{array}{llll} \hline & \text { List I } & & \text { List II } \\ \hline \text { A } & |x-1| & \text { I } & 2 \log \left(e^3+e^2\right) \\ \hline \text { B } & \log x & \text { II } & 2 \\ \hline \text { C } & x^2+x+1 & \text { III } & \log _3 e^2 \\ \hline \text { D } & e^x & \text { IV } & \sqrt{2} \\ \hline & & \text { V } & \log \left(\frac{e^3-e}{2}\right) \\ \hline \end{array} $$

If the percentage error in the radius of a circle is 3 , then the percentage error in its area is

$$ \begin{aligned} I_1 & =\int \frac{e^x}{e^{4 x}+e^{2 x}+1} d x, I_2 \\ & =\int \frac{e^{-x}}{e^{-4 x}+e^{-2 x}+1} d x, \text { then } I_2-I_1= \end{aligned} $$

If $\int \frac{1}{x} \sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}} d x=2 f(x)-2 \sin ^{-1} \sqrt{x}+c$, then $f(x)=$

$$ \begin{aligned} & \int \frac{3 x+2}{4 x^2+4 x+5} d x=A \log \\ & \left(4 x^2+4 x+5\right)+B \tan ^{-1}\left(\frac{2 x+1}{2}\right)+C, \text { then } A+B= \end{aligned} $$

Consider the following

Assertion

$$ \begin{aligned} & \text { (A) } \begin{array}{r} \sqrt{x-3}\left(\sin ^{-1}(\log x)+\cos ^{-1}\right. \\ (\log x) d x=\frac{\pi}{3}(x-3)^{3 / 2}+c \end{array} \end{aligned} $$

Reason $(\mathrm{R}) \sin ^{-1}(f(x))+\cos ^{-1}(f(x))=\frac{\pi}{2},|f(x)|<1$

The correct answer is

$$ \lim _{n \rightarrow \infty} \frac{\left(2 n(2 n-1) \ldots .(n+2)(n+1)^{1 / n}\right.}{n}= $$

The area of the region bounded by $y=x^3, X$-axis, $x=-2$ and $x=4$ is

If $\int_0^{\frac{\pi}{2}} \tan ^{14}\left(\frac{x}{2}\right) d x=2\left[\sum_{n=1}^7 f(n)-\frac{\pi}{4}\right]$, then $f(n)=$

The differential equation of the family of all circles of radius ' $a$ ' is

If the general solution of $\left(1+y^2\right) d x=\left(\tan ^{-1} y-x\right) d y$ is $x=f(y)+c e^{-\tan ^{-1} y}$, then $f(y)=$

The force of mutual attraction between any two objects by virtue of their masses is

The error in the measurement of force acting normally on a square plate is $3 \%$. If the error in the measurement of the side of the plate is $1 \%$, then the error in the determination of the pressure acting on the plate is

For a particle moving along a straight line path, the displacements in third and fifth seconds of its motion are 10 m and 18 m respectively. The speed of the particle at time $t=4 \mathrm{~s}$ is

The vertical displacement ( $y$ in metre) of a projectile in term of its horizontal displacement ( $x$ in metre) is given by $y=\left(\sqrt{3} x-0.2 x^2\right)$. The time of flight of the projectile is

(Acceleration due to gravity $=10 \mathrm{~ms}^{-2}$ )

A block of mass $\sqrt{2} \mathrm{~kg}$ is placed on a rough horizontal surface. A force ' $F$ ' acting upwards at an angle of $45^{\circ}$ with the horizontal causes the block to start motion. If the coefficient of static friction between the surface and the block is 0.25 , the magnitude of the force ' $F$ ' is

(Acceleration due to gravity $=10 \mathrm{~ms}^{-2}$ )

If the kinetic energy of a body moving with a velocity of $(2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-4 \hat{\mathrm{k}}) \mathrm{ms}^{-1}$ is 87 J , then the mass of the body is

A body of mass 0.5 kg is supplied with a power ' $P$ ' (in watt) which varies with time ' $f$ ' (in second) as $P=3 t^2+3$. If the velocity of the body at time $t=0$ is zero, then the velocity of the body at time $t=3 \mathrm{~s}$ is

A solid sphere of mass 2 kg and radius 0.5 m is rolling without slipping on a horizontal surface. The ratio of the rotational and translational kinetic energies of the sphere is

If the length of a thin uniform rod is ' $L$ ' and the radius of gyration of the rod about an axis perpendicular to its length and passing through one end is $K$, then $K: L=$

The force ( $F$ in newton) acting on a particle of mass 90 g executing simple harmonic motion is given by $F+0.04 \pi^2 y=0$, where $y$ is displacement of the particle in metre. If the amplitude of the particle is $\frac{6}{\pi} \mathrm{~m}$, then the maximum velocity of the particle is

Which of the following is incorrect about the gravitational force between two bodies?

A steel rod with a circular cross-section of diameter 1 cm and another steel rod with a square cross-section of side 1 cm have equal mass. If the two rods are subjected to same tension, the ratio of the elongations of the two rods is

A cube of side 40 cm is floating with $\frac{1}{4}$ th of its volume immersed in water. When a circular disc is placed on the cube, it floats with $\frac{2}{5}$ th of its volume immersed in water. The mass of the disc is

The maximum length of water column that can stay without falling in a vertically held capillary tube of diameter 1 mm and open at both the ends is (Acceleration due to gravity $=10 \mathrm{~ms}^{-2}$ and surface tension of water $=0.07 \mathrm{Nm}^{-1}$ )

A steel pendulum clock manufactured at $32^{\circ} \mathrm{C}$ and working at $47^{\circ} \mathrm{C}$ is nearly

(Coefficient of linear expansion of steel $=12 \times 10^{-6} /{ }^{\circ} \mathrm{C}$ )

A metal metre scale that is accurate up to 0.5 mm is made at a temperature of $25^{\circ} \mathrm{C}$. The range of temperatures within which it can be used is (Coefficient of linear expansion of the metal $=10^{-5} /{ }^{\circ} \mathrm{C}$ )

A Carnot engine uses diatomic gas as a working substance. During the adiabatic expansion part of the cycle, if the volume of the gas becomes 32 times its initial volume, then the efficiency of the engine is

The ratio of the average translational kinetic energies of hydrogen and oxygen at the same temperature is

The air columns in two tubes closed at one end vibrating in their fundamental modes produce 2 beats per second. The number of beats produced per second when the same tubes are vibrated in their fundamental mode with their both ends open are

A car moving towards a cliff emits sound of frequency ' $n$ '. If the difference in frequencies of the horn and its echo heard by the driver of the car is $10 \%$ of ' $n$ ', then the speed of the car is nearly

(Speed of sound in air is $336 \mathrm{~ms}^{-1}$ )

A straight metal rod of length 6 cm is placed along the principal axis of a concave mirror of focal length 9 cm such that the end of the rod closer to the mirror is at a distance of 15 cm from the pole of the mirror. The length of the image of the rod is

A ray of light incidents at an angle of $9.3^{\circ}$ on one face of a small angle prism of refracting angle $6^{\circ}$. If the ray of light emerges normally from the second face, the refractive index of the material of the prism is

The distance for which ray optics becomes a good approximation for an aperture of 0.3 cm and a light of wavelength $6000 \mathop {\rm{A}}\limits^{\rm{o}}$ is

The electrostatic force between two charges kept in air is $F$. If $30 \%$ of the space between the charges is filled with a medium, then the electrostatic force between the charges becomes $\frac{F}{2.56}$. The dielectric constant of the medium is

729 small identical spheres each charged to an electric potential 3V combine to form a bigger sphere. The electric potential of the bigger sphere is

For the circuit shown in the figure, the current through $6 \Omega$ resistor connected between the junctions $A$ and $B$ is

The area of cross-section of a potentiometer wire is $6 \times 10^{-7} \mathrm{~m}^2$. The potential difference per unit length of the potentiometer wire when it is connected to a cell of negligible internal resistance and a resistor in series is $0.15 \mathrm{Vm}^{-1}$. If the current through potentiometer wire is 0.3 A , then the resistivity of the material of the potentiometer wire is

As shown in the figure, a uniform straight wire of length $30 \sqrt{3} \mathrm{~cm}$ is bent in the form of an equilateral triangle $A B C$. A uniform magnetic field $2 T$ is applied parallel to the side $B C$. If the current through the wire is 2 A , the magnitude of the force on the side $A C$ is ( $\bar{B}$ represents the direction of the magnetic field)

A proton moving with a velocity of $8 \times 10^5 \mathrm{~ms}^{-1}$ enters a uniform magnetic fleld normal to the direction of the magnetic field. If the radius of the circular path of the proton in the magnetic field is 8.3 cm , then the magnitude of the magnetic field is

(Charge of proton $=1.6 \times 10^{-19} \mathrm{C}$ and mass of the proton $=1.66 \times 10^{-27} \mathrm{~kg}$ )

At a certain place in the magnetic meridian the Earth's magnetic field is twice its vertical component. The ratio of horizontal component of Earth's magnetic field and the total magnetic field of the Earth at the place is

A coil of resistance $16 \Omega$ is placed with its plane perpendicular to a uniform magnetic field whose flux ( $\phi$ in $10^{-3}$ weber) changes with time ( $t$ in second) as $\phi=5 t^2+4 t+2$. The induced current at time $t=6$ second is

The small energy losses in transformers due to eddy currents can be reduced by

If the electric field of a plane electromagnetic wave is $E_z=60 \sin \left(0.5 \times 10^3 x+1.5 \times 10^{11} t\right) \mathrm{Vm}^{-1}$, then the magnetic field of the wave is

In a photoelectric experiment, the slope of the graph drawn between stopping potential along $Y$-axis and frequency of incident radiation along $X$-axis is (Planck's constant $=6.6 \times 10^{-34} \mathrm{Js}$ )

The maximum wavelength of incident radiation required to ionize a hydrogen atom in its ground state is nearly

When an element ${ }_{90}^{232} \mathrm{Th}$ decays into ${ }_{82}^{208} \mathrm{~Pb}$, the number of $\alpha$ and $\beta^{-}$particles emitted respectively are

During the disintegration of a radioactive nucleus of mass number 208 at rest, two alpha particles each with kinetic energy $E$ are emitted. The total kinetic energy of the emitted alpha particles and the daughter nucleus after the disintegration is

The current amplification factor of a transistor in common emitter configuration is 80 . If the emitter current is 2.43 mA , then the base current is

The negative feedback in an amplifier

If the frequencies of the carrier wave and message signal are 1 MHz and 28 kHz respectively, then the frequencies of the side bands are