$a, b, c, d$ are electromagnetic radiations. Frequencies of $a, b$ are $3 \times 10^{15} \mathrm{~Hz}, 2 \times 10^{14} \mathrm{~Hz}$, respectively, whereas wavelength of $c, d$ are $400 \mathrm{~nm}, 750 \mathrm{~nm}$, respectively. The increasing order of their energies is

The number of electrons with magnetic quantum number, $m_l=0$ in the elements with atomic numbers $Z=24$ and $Z=29$ are respectively.

Which of the following orders is not correct for the given property?

$$ \text { Match the following } $$

The correct answer is

$$ \text { Which of the following sets are correctly matched? } $$

$$ \begin{array}{llcc} \hline & \text { Molecule } & \begin{array}{c} \text { Number of lone } \\ \text { pair of electrons } \\ \text { on central atom } \end{array} & \text { Hybridisation } \\ \hline \text { (I) } & \mathrm{PCl}_3 & 1 & s p^3 \\ \hline \text { (II) } & \mathrm{SO}_2 & 2 & s p^3 \\ \hline \text { (III) } & \mathrm{SF}_4 & 2 & s p^3 d^2 \\ \hline \text { (IV) } & \mathrm{ClF}_3 & 2 & s p^3 d \\ \hline \end{array} $$

The correct equation for one mole of a real gas is $a, b$ are constants)

$A$ and $B$ are ideal gases. At $T(\mathrm{~K}), 2 \mathrm{~L}$ of ' $A^{\prime}$ 'with a pressure of 1 bar is mixed with 4 L of ' $B$ ' with a pressure $p_B$ bar in a 100 L flash. The pressure exerted by gaseous mixture is 0.1 bar. What is the value of $p_B$ in bar?

The mass of a mixture containing NaCl and NaBr is 4.0 g . If Na is $30 \%$ of the total mixture, the composition of NaCl in the mixture is $(\mathrm{Na}=23 \mathrm{u}, \mathrm{Cl}=35.5 \mathrm{u}, \mathrm{Br}=80 \mathrm{u})$

The number of extensive and intensive properties in the list given below is respectively, density, enthalpy, mass, temperature, volume, pressure

One mole of ethanol ( $l$ ) was completely burnt in oxygen to form $\mathrm{CO}_2(\mathrm{~g})$ and $\mathrm{H}_2 \mathrm{O}(l)$. What is the $\Delta_r H^{\circ}$ (in $\mathrm{kJ} \mathrm{mol}^{-1}$ ) for this reaction?

(The standard enthalpy of formation $\left(\Delta_f H^{\circ}\right)$ of $\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}(l), \mathrm{CO}_2(g)$ and $\mathrm{H}_2 \mathrm{O}(l)$ is respectively $-277,-393$ and $-286 \mathrm{~kJ} \mathrm{~mol}^{-1}$ )

For the following given equilibrium reaction $\frac{K_c}{K_p}$ is equal to 1076 at $T(\mathrm{~K})$. What is the value of $T$ (in K )?

$$ \begin{aligned} & \left(R=0.082 \mathrm{~L}-\mathrm{atm} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\right) \\ & \mathrm{N}_2(\mathrm{~g})+3 \mathrm{H}_2(\mathrm{~g}) \rightleftharpoons 2 \mathrm{NH}_3(\mathrm{~g}) \end{aligned} $$

The molar solubility of $\mathrm{PbI}_2$ in $0.2 \mathrm{MPb}\left(\mathrm{NO}_3\right)_2$ solution in terms of $K_{s p}$ (solubility product) is

Which of the following property is less for $\mathrm{D}_2 \mathrm{O}$ than $\mathrm{H}_2 \mathrm{O}$ ?

Identify the correct statement from the following.

A. Among alkali metal ions, $\mathrm{Li}^{+}$has highest hydration enthalpy.

B. Boiling point of alkali metals increases from Li to Cs .

C. Density of K is less than that of Na and Rb .

D. Li has strong tendency to form superoxide.

The correct answer is

The correct order of electronegativity of group 13 elements is

Identify the correct statements.

I. CO reduces the oxygen carrying ability of blood

II. Producer gas contains CO and $\mathrm{N}_2$

III. $\mathrm{C}-\mathrm{O}$ bond length in $\mathrm{CO}_2$ is 115 pm

The incorrect statement from the following is

IUPAC names of the given compounds (I) and (II) are respectively

$$ \text { Identify the most stable carbocation from the following } $$

A metal crystallises in simple cubic lattice. The volume of one unit cell is $6.4 \times 10^7 \mathrm{pm}^3$. What is the radius of the metal atom in pm ?

What is the approximate molality of $10 \%(w / w)$ aqueous glucose solution?

(Molar mass of glucose $=180 \mathrm{~g} \mathrm{~mol}^{-1}$ )

The van't Hoff factor for 0.5 m aqueous $\mathrm{CH}_2 \mathrm{FCOOH}$ solution is 1.075 . What is the experimentally observed $\Delta T_f$ (in K ) for this solution?

( $K_f=1.86 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}$ )

$$ \text { Match the following } $$

$$ \begin{array}{llll} \hline & \begin{array}{l} \text { List-I (Symbol of } \\ \text { electrical property) } \end{array} & & \text { List-I (Units) } \\ \hline \text { (A) } & \Lambda_{\mathrm{m}} & \text { (I) } & \mathrm{Scm}^{-1} \\ \hline \text { (B) } & \mathrm{G} & \text { (II) } & \mathrm{m}^{-1} \\ \hline \text { (C) } & \mathrm{K} & \text { (III) } & \mathrm{Scm}^2 \mathrm{~mol}^{-1} \\ \hline \text { (D) } & \mathrm{G}^* & \text { (IV) } & \mathrm{S} \\ \hline \end{array} $$

The correct answer is

The following graph is obtained for a first order reaction $(A \rightarrow P)$. The activation energy ( $E_a$ in $\left.\mathrm{kJ} \mathrm{mol}^{-1}\right)$ and heat of reaction $\left(|\Delta H|\right.$ in $\left.\mathrm{kJ} \mathrm{mol}^{-1}\right)$ for this reaction are respectively

$\left(x=\right.$ reaction coordinate; $y=E$ in $\left.\mathrm{kJ} \mathrm{mol}^{-1}\right)$

$$ \text { Match the following } $$

$$ \begin{array}{llll} \hline & \begin{array}{l} \text { List-I } \\ \text { (Sol) } \end{array} & & \begin{array}{c} \text { List-II } \\ \text { (Method of preparation) } \end{array} \\ \hline \text { (A) } & \mathrm{As}_2 \mathrm{~S}_3 & \text { I. } & \text { Bredig's arc method } \\ \hline \text { (B) } & \mathrm{Au} & \text { II. } & \text { Oxidation } \\ \hline \text { (C) } & \mathrm{S} & \text { III. } & \text { Hydrolysis } \\ \hline \text { (D) } & \mathrm{Fe}(\mathrm{OH})_3 & \text { IV. } & \text { Double decomposition } \\ \hline \end{array} $$

The correct answer is

Which of the following enzymatic reaction is not correctly matched with enzyme shown against it in brackets?

Which of the following methods is useful for producing semiconductor grade metals of high purity?

Observe the following

$\mathrm{P}_4+\mathrm{SOCl}_2 \rightarrow$ Products

$\mathrm{P}_4+\mathrm{SO}_2 \mathrm{Cl}_2 \rightarrow$ Products

In both the reactions, a common product ' $x$ ' is obtained. The number of lone pair of electrons on the central atom of $x$ is

The IUPAC name of the complex shown below is $\mathrm{K}_3\left[\mathrm{Co}(\mathrm{ox})_3\right]$

Identify the ion (hydrated in solution) which is not correctly matched with its spin only magnetic moment (in BM) given in brackets

Which one of the statements, regarding $X$ is not correct?

3-Hydroxybutanoic acid +3 -Hydroxypentanoic acid $\rightarrow X$

Identify the essential amino acids from the following

I. Leucine

II. Tyrosine

III. Cysteine

IV. Histidine

$$ \text { Which of the following represents nucleoside of RNA? } $$

Which of the following is not an antibiotic?

What are the major products $X$ and $Y$ respectively in the following set of reactions?

Which of the following will undergo methylation with $\mathrm{CH}_3 \mathrm{Cl}$ /anhy. $\mathrm{AlCl}_3$ ?

I. Aniline

II. Chlorobenzene

III. Benzoic acid

IV. Anisole

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

$$ \text { Match the following } $$

$$ \begin{array}{llll} \hline & \text { List-I (Compound) } & & \text { List-II }\left(\mathrm{pK}_{\mathrm{a}}\right) \\ \hline \text { (A) } & \text { p-nitrophenol } & \text { (I) } & 15.9 \\ \hline \text { (B) } & \text { Phenol } & \text { (II) } & 7.1 \\ \hline \text { (C) } & \text { Ethanol } & \text { (IIII) } & 10.0 \\ \hline \text { (D) } & \text { p-cresol } & \text { (IV) } & 10.2 \\ \hline & & \text { (N) } & 8.3 \\ \hline \end{array} $$

The correct answer is

The structures of succinic acid $(x)$ and malonic acid $(y)$ respectively are

Benzyl amine can be prepared from which of the following reactions?

1. Let [ $x$ ] represent the greatest integer less than or equal to $x,\{x\}=x-[x] \sqrt{2}=1.414$ and $\sqrt{3}=1.732$. If $f(x)=\left\{x+\left[\frac{x}{1+x^2}\right]\right\}$ is a real valued function, then $f(\sqrt{2})+f(-\sqrt{3})=$

If the range of the function $f(x)=-3 x-3$ is $\{3,-6,-9,-18\}$, then which one of the following is not in the domain of $f$ ?

$\frac{1}{3 \cdot 5}+\frac{1}{5 \cdot 7}+\frac{1}{7 \cdot 9}+\ldots$ to 24 terms $=$

If $B$ is the inverse of a third order matrix $A$ and det $B=k$, then $(\operatorname{adj}(\operatorname{adj} \mathrm{A}))^{-1}=$

If $A=\left[\begin{array}{lll}2 & 2 & 1 \\ 1 & 3 & 1 \\ 1 & 2 & 2\end{array}\right]$ and $\alpha, \beta, \gamma$ are the roots of the equation represented by $|A-x I|=0$, then $\alpha^2+\beta^2+\gamma^2=$

If the values of $x, y$ and $z$ which satisfy the equations $2 x-3 y+2 z+15=0,3 x+y-z+2=0$ and $x-3 y-3 z+8=0$ simultaneously are $\alpha, \beta$ and $\gamma$ respectively, then

If $x=3-2 \sqrt{3} \mathrm{i}$, then $x^4-12 x^3+54 x^2-108 x-54=$

$z_1, z_2, z_3$ represent the vertices $A, B, C$ of a $\triangle A B C$ respectively in the argand plane. If $\left|z_1-z_2\right|=\sqrt{25-12 \sqrt{3}},\left|\frac{z_1-z_3}{z_2-z_3}\right|=\frac{3}{4}$ and $\angle A C B=30^{\circ}$, then the area (in sq units) of that triangle is

The product of the four values of the complex number $(1+i)^{3 / 4}$ is

If the difference of the roots of the equation $x^2-7 x+10=0$ is same as the difference of the roots of the equation $x^2-17 x+k=0$, then a divisor of $k$ is $x^2-7 x+10=0$

The product of all the real roots of the equation $|x|^2-5|x|+6=0$

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

After the roots of the equation $6 x^3+7 x^2-4 x-2=0$ are diminished by $h$, if the transformed equation does not contain $x$ term, then the product of all the possible value of $h$ is

The number of integers greater than 6000 that can be formed by using the digits $0,5,6,7,8$ and 9 without repetition is

The number of distinct quadratic equations $a x^2+b x+c=0$ with unequal real roots that can be formed by choosing the coefficients $a, b, c(a \neq b \neq c)$ from the set $\{0,1,2,4\}$ is

The number of ways of dividing 15 persons into 3 groups containing 3,5 and 7 persons so that two particular persons are not included into the 5 persons groups is

The coefficient of $x^{10}$ in the expansion of $\left(x+\frac{2}{x}-5\right)^{12}$ is

Let $S_1=\sum\limits_{j=1}^{10} j(j-1) \cdot{ }^{10} C_j, S_2=\sum\limits_{j=1}^{10} j \cdot{ }^{10} C_j$ and

$$ S_3=\sum\limits_{j=1}^{10} j^2 \cdot{ }^{10} C_j $$

Assertion (A) $S_3=55 \times 2^9$

Reason (R) $S_1=90 \times 2^8$ and $S_2=10 \times 2^8$

If $\frac{2 x^4-3 x^2+4}{\left(x^2+1\right)\left(x^2+2\right)}=a+\frac{p x+q}{x^2+1}+\frac{m x+n}{x^2+2}$, then $\frac{n}{q}=$

$$ \begin{aligned} & \left(4 \cos ^2 \frac{\pi}{20}-1\right)\left(4 \cos ^2 \frac{3 \pi}{20}-1\right) \\ & \left(4 \cos ^2 \frac{5 \pi}{20}+1\right)\left(4 \cos ^2 \frac{7 \pi}{20}-1\right)\left(4 \cos ^2 \frac{9 \pi}{20}-1\right)= \end{aligned} $$

If $A$ and $B$ are the values such that $(A+B)$ and $(A-B)$ are not odd multiples of $\frac{\pi}{2}$ and $2 \tan (A+B)=3 \tan (A-B)$, then $\sin A \cos A=$

If $\cos ^3 80^{\circ}+\cos ^3 40^{\circ}-\cos ^3 20^{\circ}=k$, then $\frac{4 k}{3}=$

The number of solutions of the equation $4 \cos 2 \theta \cos 3 \theta=\sec \theta$ in the interval $[0,2 \pi]$ is

$$ \tan \left(2 \tan ^{-1}\left(\frac{1}{3}\right)+\tan ^{-1}\left(\frac{1}{7}\right)\right)= $$

$$ \tanh ^{-1}\left(\frac{1}{3}\right)+\operatorname{coth}^{-1}(3)= $$

In a $\triangle A B C$, if $A=30^{\circ}$ and $\frac{b}{(\sqrt{3}+1)^2+2(\sqrt{2}-1)} =\frac{c}{(\sqrt{3}+1)^2-2(\sqrt{2}-1)}$, then $B$

In $\triangle A B C$ is the line joining the circumcentre and the incentre is parallel to $B C$, then $\cos B+\cos C=$

In a $\triangle A B C$, if $r_1: r_2=3: 4$ and $r_2: r_3=2: 3$, then $a:$$b:$$c$=

Let $(x, y) \in R \times R$ and $\mathbf{a}=x \hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}, \mathbf{b}=6 \hat{\mathbf{i}}-y\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ be two vectors. If

$$ |\mathbf{a} \times \mathbf{b}|^2+|\mathbf{a} \cdot \mathbf{b}|^2=f(x) g(y), \text { then } f(x)+g(y)-46=0 $$

represents

30. Line $L_1$ passes through the point $\hat{\mathbf{i}}+\hat{\mathbf{j}}$ and $\hat{\mathbf{k}}-\hat{\mathbf{i}}$. Line $L_2$ passes through the point $\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ and is parallel to the vector $\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$. If $x \hat{\mathbf{i}}+y \hat{\mathbf{j}}+z \hat{\mathbf{k}}$ is the point of intersection of the lines $L_1$ and $L_2$, then $(y-x)=$

$\mathbf{a} \cdot \mathbf{b}$ and $\mathbf{c}$ are the position vectors of three non-collinear points on a plane. If

$$ \alpha=[\mathbf{a b c}] \text { and } \mathbf{r}=\mathbf{a} \times \mathbf{b}-\mathbf{c} \times \mathbf{b}-\mathbf{a} \times \mathbf{c} \text {, then }\left|\frac{\alpha}{\mathbf{r}}\right| $$

represents

If $P=(\mathbf{a} \times \hat{\mathbf{i}})^2+(\mathbf{a} \times \hat{\mathbf{j}})^2+(\mathbf{a} \times \hat{\mathbf{k}})^2$ and $Q=(\mathbf{a} \cdot \hat{\mathbf{i}})^2+(\mathbf{a} \cdot \hat{\mathbf{j}})^2+(\mathbf{a} \cdot \hat{\mathbf{k}})^2$, then

$\mathbf{a}=\hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}}, \mathbf{b}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}, \mathbf{c}=2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}$ are three vectors. If $\mathbf{r}$ is a vector such that $\mathbf{r} \cdot \mathbf{a}=0, \mathbf{r} \cdot \mathbf{c}=3$ and $\left[\begin{array}{ll}\mathbf{r} & \mathbf{a} \\ \mathbf{b}\end{array}\right]=0$, then $|\mathbf{r}|=$

The mean deviation from the median for the following data is

$$ \begin{array}{llllll} \hline x_1 & 9 & 3 & 7 & 2 & 5 \\ \hline f_1 & 1 & 6 & 2 & 8 & 4 \\ \hline \end{array} $$

A company representative is distributing 5 identical samples of a product among 12 houses in a row such that each house gets at most one sample. The probability that no two consecutive house get one sample is

38. $A$ and $B$ are two independent events of a random experiment and $P(A)>P(B)$.

If the probability that both $A$ and $B$ occurs is $\frac{1}{6}$ and neither of them occurs is $\frac{1}{3}$, then the probability of the occurance of $B$ is

Two dice are thrown and the sum of the numbers appeared on the dice is noted. If $A$ is the event of getting a prime number as their sum and $B$ is the event of getting a number greater than 8 as their sum, then $P(A \cap \bar{B})=$

A family consists of 8 persons. If 4 persons are chosen a random and they are found to be 2 men and 2 women, then the probability that there are equal number of men and women in that family is

The number of trials conducted in a binomial distribution is 6 . If the difference between the mean and variance of this variate is $\frac{27}{8}$, then the probability of getting atmost 2 successes is

Let $X \sim B(n, p)$ with mean $\mu$ and variance $\sigma^2$. If $\mu=2 \sigma^2$ and $\mu+\sigma^2=3$, then $P(X \leq 3)=$

If $A(\cos \alpha, \sin \alpha), B(\sin \alpha,-\cos \alpha), C(1,2)$ are the vertices of a $\triangle A B C$, then the locus of its centroid is

If the axes are translated to the orthocentre of the triangle formed by the points $\mathrm{A}(7,5), \mathrm{B}(-5,-7)$ and $C(7,-7)$, then the coordinates of the incentre of the triangle in the new system are

The angle made by a line $L$ with positive $X$-axis measured in the positive direction is $\frac{\pi}{6}$ and the intercept made by $L$ on $Y$-axis is negative. IF $L$ is at a distance of 5 units from the origin, then the perpendicular distance from the point $(1,-\sqrt{3})$ to the line $L$ is

$L_1$ and $L_2$ are two lines having slopes 2 and $-\frac{1}{2}$ respectively. If both $L_1$ and $L_2$ are concurrent with the lines $x-y+2=0$ and $2 x+y+3=0$, then sum of the absolute values of the intercepts made by the lines $L_1$ and $L_2$ on the coordinate axes is

The lines $L_1: y-x=0$ and $L_2: 2 x+y=0$ intersect the line $L_3: y+2=0$ at $P$ and $Q$ respectively. The bisector of the angle between $L_1$ and $L_2$ divides the line segment $P Q$ internally at $R$.

Statement $I P R: R Q=2 \sqrt{2}: \sqrt{5}$

Statement II In any triangle, bisector of an angle divides that triangle into two similar triangles

If $2 x^2+3 x y-2 y^2-5 x+2 f y-3=0$ represents a pair of straight lines, then one of the possible values of $f$ is

A circle passing through origin cuts the coordinate axes is $A$ and $B$. If the straight line $A B$ passes through a fixed point $\left(x_1, y_1\right)$, then the locus of the centre of the circle is

If $(\alpha, \beta)$ is the external centre of similitude of the circles $x^2+y^2=3$ and $x^2+y^2-2 x+4 y+4=0$, then $\frac{\beta}{\alpha}=$

The equation of the circle touching the lines $|x-2|+|y-3|=4$ is

If the chord joining the points $(1,2)$ and $(2,-1)$ on a circle subtends an angle of $\frac{\pi}{4}$ at any point on its circumference, then the equation of such a circle is

The equation of the circle which cuts all the three circles $4(x-1)^2+4(y-1)^2=1,4(x+1)^2+4(y-1)^2$ and $4(x+1)^2+4(y+1)^2=1$ orthogonally is

If the normal chord drawn at the point $\left(\frac{15}{2}, \frac{15}{\sqrt{2}}\right)$ to the parabola $y^2=15 x$ subtends an angle $\theta$ at the vertex of the parabola, then $\sin \frac{\theta}{3}+\cos \frac{2 \theta}{3}-\sec \frac{4 \theta}{3}=$

If a tangent having slope $\frac{1}{3}$ to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b)$ is a normal to the circle $(x+1)^2+(y+1)^2=1$, then $a^2$ lies in the interval

Let $P(a \sec \theta, b \tan \theta)$ and $Q(a \sec \phi, b \tan \phi)$, where $\theta+\phi=\frac{\pi}{2}$ be two points on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ If $(h, k)$ is the point of intersection of the normals drawn at $P$ and $Q$ then $K=$

If the angle between the asymptotes of a hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is $2 \tan ^{-1}\left(\frac{2}{3}\right)$ and $a^2-b^2=45$, then $a b=$

The point in the $X Y$ - plane which is equidistant from the points $A(2,0,3), B(0,3,2)$ and $C(0,0,1)$ has the coordinates

If the direction ratio of two lines $L_1$ and $L_2$ are given by $(1,-2,2)$ and $(-2,3,-6)$ respectively, then the direction ratios of the line which is perpendicular to the linesh and $L_2$ are

If the image of the point $A(1,1,1)$ with respect to the plane $4 x+2 y+4 z+1=0$ is $B(\alpha, \beta, \gamma)$, then $\alpha+\beta+\gamma=$

$$ \mathop {\lim }\limits_{x \to 0} \frac{x+2 \sin x+3 \tan x-\tan ^3 x}{\sqrt{x^2+2 \sin x+\tan x+3}-\sqrt{\sin ^2 x-2 \tan x-x+3}} $$

$$ \mathop {\lim }\limits_{x \to \infty } \frac{(3-x)^{25}(6+x)^{35}}{(12+x)^{38}(9-x)^{22}}= $$

If a real valued function

$$ f(x)=\left\{\begin{array}{cc} \log (1+[x]), & x \geq 0 \\ \sin ^{-1}[x], & -1 \leq x<0 \\ k([x]+|x|), & x<-1 \end{array}\right. $$

is continuous at $x=-1$, then $k=$

If $y=\sin ^{-1}\left(\frac{2 x}{1+x^2}\right)$ and $\left(\frac{d^2 y}{d x^2}\right)_{x=2}=k$, then $25 k=$

If $f(x)=x^{\sec ^{-1} x}$, then $f^{\prime}(2)=$

If $f(x)=\sec ^{-1}\left(\frac{1}{2 x^2-1}\right)$ and $g(x)=\tan ^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right)$, then the derivative of $f(x)$ with respect to $g(x)$ is

If the tangent of the curve $x y+a x+b y=0$ at $(1,1)$ makes an angle $\tan ^{-1} 2$ with $X$-axis, then $\frac{a b}{a+b}=$

If the displacement $S$ of a particle travelling along a straight line in $t$ seconds is given by $S=2 t^3+2 t^2-2 t-3$, then the time taken (in second) by the particle to change its direction is

If the function $f(x)=x^3+b x^2+c x-6$ satisfies all the conditions of Rolle's theorem in $[1,3]$ and $f^{\prime}\left(\frac{2 \sqrt{3}+1}{\sqrt{3}}\right)=0$, then $b c=$

If $P(\alpha, \beta)$ is a point on the curve $9 x^2+4 y^2=144$ in the first quadrant and the minimum area of the triangle formed by the tangent of the curve at $P$ with the coordinate axis is $S$, then

$$ \int(\log 2 x)^3 d x= $$

$$ \int \frac{x+1}{(x-2) \sqrt{1-x}} d x= $$

$$ \int \frac{1}{1+x+x^2} d x= $$

If $\int \frac{d x}{(x \tan x+1)^2}=f(x)+C$, then $\lim\limits_{x \rightarrow \frac{\pi}{2}} f(x)=$

$$ \int \sin ^3 x \cos ^2 x d x= $$

$$\mathop {\lim }\limits_{n \to \infty } \frac{\pi}{2 n}\left[\sin \frac{\pi}{2 n}+\sin \frac{2 \pi}{2 n}+\sin \frac{3 \pi}{2 n}+\ldots+\sin \frac{\pi}{2}\right]= $$

$$ \int_0^\pi\left(\sin ^5 x \cos ^3 x+\sin ^4 x \cos ^4 x+\sin ^3 x \cos ^4 x\right) d x= $$

$$ \int_0^1 \frac{x^4+1}{x^6+1} d x= $$

The area of the region (in sq units) bounded by the curves $x^2+y^2=16$ and $y^2=6 x$ is

If $a$ and $b$ are arbitrary constants, then the differential equation corresponding to the family of curves $y=\tan (a x+b)$ is

The general solution of the differential equation $x y(y+2) d y+\left(y^3-1\right) d x=0$ is

The general solution of the differential equation $\left(1+\sin ^2 x\right) \frac{d y}{d x}+y \sin 2 x=\cos x+\sin ^2 x \cos x$ is

If force $=\frac{\alpha}{\operatorname{density}+\beta^3}$, then the dimensional formulae of $\alpha$ and $\beta$ are respectively

The displacement $(x)$ and time $(t)$ graph of a particle moving along a straight line is shown in the figure. The average velocity of the particle in the time of 10 s is

If the horizontal range of a body projected with a velocity ' $u$ ' is 3 times the maximum height reached by it, then the range of the body is

( $g=$ Acceleration due to gravity)

If the velocity at the maximum height of a projectile projected at an angle of $45^{\circ}$ is $20 \mathrm{~ms}^{-1}$, then the maximum height reached by the projectile is (Acceleration due to gravity $=10 \mathrm{~ms}^{-2}$ )

A body of mass ' $m$ ' moving along a straight line collides with a stationary body of mass ' $2 m^{\prime}$. After collision if the two bodies move together with the same velocity, then the fraction of kinetic energy lost in the process is

If a body of mass 2 kg moving with initial velocity of $4 \mathrm{~ms}^{-1}$ is subjected to a force of 3 N for a time of 2 s normal to the direction of its initial velocity, then the resultant velocity of the body is

If a constant force of $(2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}) \mathrm{N}$ acting on a body of mass 5 kg displaces it from $(3 \hat{\mathbf{i}}-4 \hat{\mathbf{k}}) \mathrm{m}$ to $(2 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}})$ m , then the work done by the force on the body is

A motor can pump 7560 kg of water per hour from a well of depth 100 m . If the efficiency of the pump is $70 \%$, then power of the pump is

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

A circular dise of diameter 0.8 m and mass 4 kg is rolling on a smooth horizontal plane. If 2.56 N m torque is acting on the disc, then its angular acceleration is

A solid sphere and a solid cylinder have same mass and same radius. The ratio of the moment of inertia of the solid sphere about its diameter and the moment of inertia of the solid cylinder about its axis is

A particle is executing simple harmonic motion with amplitude $A$. The ratio of the kinetic energies of the particle when it is at displacements of $\frac{A}{4}$ and $\frac{A}{2}$ from the mean position is

If the potential energy of a particle of mass 0.1 kg moving along $X$-axis is $5 x(x-4) \mathrm{J}$, then the speed of the particle is maximum at a position of

The potential energy of a satellite of mass ' $m$ ' revolving around the Earth at a height of $R_e$ from the surface of the Earth is

( $R_e=$ Radius of Earth, $\mathrm{g}=$ acceleration due to gravity)

The elastic potential energy stored in a copper rod of length one metre and area of cross-section $1 \mathrm{~mm}^2$ when stretched by 1 mm is

(Young's modulus of copper $=1.2 \times 10^{11} \mathrm{Nm}^{-2}$ )

When the temperature increases, the viscosity of

If a body cools from a temperature of $62^{\circ} \mathrm{C}$ to $50^{\circ} \mathrm{C}$ in 10 minutes and to $42^{\circ} \mathrm{C}$ in the next 10 minutes, then the temperature of the surroundings is

If the ratio of universal gas constant and specific heat capacity at constant volume of a gas is given by 0.67 , then the gas is

The internal energy of 4 moles of a monoatomic gas at a temperature of $77^{\circ} \mathrm{C}$ is

( $R=$ Universal gas constant)

If 5.6 litres of a monoatomic gas at STP is adiabatically compressed to 0.7 litres, then the work done on the gas is nearly ( $R=$ Universal gas constant)

If the rms speed of the molecules of a diatomic gas at a temperature of 322 K is $2000 \mathrm{~ms}^{-1}$, then the gas is (Universal gas constant $=8.31 \mathrm{Jmol}^{-1} \mathrm{~K}^{-1}$ )

The equation of a transverse wave propagating along a stretched string of length 80 cm is $y=1.5 \sin \left\{\left(5 \times 10^{-3} x\right)+20 t\right\}$, here ' $x$ ' and ' $y$ ' are in cm and the time ' $t$ ' is in second. If the mass of the string is 3 g , then the tension in the string is 80 cm

When an object is placed infront of a convex mirror at a distance ' $u$ ' from the pole of the mirror such that the size of the image is ' $n$ ' times that of the object. Then, the object distance ' $u=$

A narrow slit of width 2 mm is illuminated with a monochromatic light of wavelength 500 nm . If the distance between the slit and the screen is 1 m , then first minima are separated by a distance of

The force between two conducting spheres of same radius having charges $+8 \mu \mathrm{C}$ and $-4 \mu \mathrm{C}$ separated by some distance in air is $F$. If the spheres are connected by a conducting wire and after some time the wire is removed, then the magnitude of the force between the two conducting spheres is

In space the electric potential varies as $V=20|\mathbf{r}|$ volt. where $\mathbf{r}=x \hat{\mathbf{i}}+y \hat{\mathbf{j}}+z \hat{\mathbf{k}}$ is the position vector. Then, electric field in $\left(\mathrm{NC}^{-1}\right)$ at the point $(4 \mathrm{~m}, 3 \mathrm{~m},-5 \mathrm{~m})$ is

A capacitor of capacitance $2 \mu \mathrm{~F}$ is charged with the help of a 60 V battery. After disconnecting the battery, if this capacitor is connected in parallel with another uncharged capacitor of capacitance $l \mu \mathrm{~F}$, then the potential difference across the plates of $2 \mu \mathrm{~F}$ capacitor is

The readings of the voltmeter and ammeter in the circuit shown in the diagram are respectively

When two identical batteries of internal resistance $1 \Omega$ each are connected in series across a resistor $R$, the rate of heat produced in $R$ is $P_1$. When the same batteries are connected in parallel across $R$, the rate of heat produced is $P_2$. If $P_1=2.25 P_2$, then the value of $R$ is

The magnetic field at the centre of a long solenoid having 400 turns per unit length and carrying a current ' $i$ ' is $6.24 \times 10^{-2} \mathrm{~T}$. The magnetic field at the centre of another long solenoid having 200 turns per unit length and carrying a current $\frac{i}{2}$ is

If a proton of kinetic energy 8.35 MeV enters a uniform magnetic field of 10 T at right angles to the direction of the field, then the force acting on the proton is

(Mass of proton $=1.67 \times 10^{-27} \mathrm{~kg}$ and Charge of proton $=1.6 \times 10^{-19} \mathrm{C}$ )

A sample of a ferromagnetic iron in the shape of a cube of side $1.0 \mu \mathrm{~m}$ contains $8.7 \times 10^{28}$ atoms per cubic metre and the magnetic dipole moment of each iron atom is $93 \times 10^{-24} \mathrm{Am}^2$. Then, the maximum possible magnetic dipole moment (in $\mathrm{Am}^2$ ) of the sample is nearly

When current in a coil changes from 2 A to 5 A in time of 0.3 s , if the emf induced in the coil is 40 mv , then the self inductance of the coil is

In a series LCR circuit, the voltages across the capacitor, resistor and inductor are in the ratio $2: 3: 6,$ if the voltage of the AC source in the circuit is 240 V , then the voltage across the inductor is

If a 10 W bulb emits electromagnetic waves uniformly in all directions, then the intensity of light at a distance 0.5 m from the source is nearly

The ratio of de-Broglie wavelengths associated with thermal neutrons at temperatures $127^{\circ} \mathrm{C}$ and $352^{\circ} \mathrm{C}$ is

The ratio of the time periods of the revolution of the electrons in the second and third excited states of hydrogen atom is

If the surface areas of two nucleii are in the ratio $9: 47$, then the ratio of their mass number is

$$ \text { In the given options, the diode that is forward biased is } $$

In a common emitter transistor amplifier the resistance of collector is $3 \mathrm{k} \Omega$. If the current amplification factor is 100 and the base resistance is $2 \mathrm{k} \Omega$, then the power gain of the transistor is

The layer of the atmosphere that reflects low frequency (LF) electromagnetic waves during day time only is