Which of the following statements is not true about Thomson's model of atom?

Which of the following statements is not the result of cathode ray discharge tube experiment with perforated anode?

The highest oxidation state observed in first row transition metals is

Based on the quantum numbers, what will be the maximum number of element for sixth period of the periodic table?

Let's assume the $\mathrm{C}_1 \equiv \mathrm{C}_2$ bond is acetylene is along $Z$-axis. Find out the correct combination of atomic orbitals with non-zero overlapping.

Which of the following molecules is not paramagnetic in nature?

Identify the correct observation with respect to the given graphs.

A gas is present at a pressure of 2 atm . What should be the increase in pressure, so that the volume of the gas can be decreased to $\frac{1}{4}$ th of the initial volume at constant temperature?

Identify the law for which the following statement is true.

"Equal volume of all gases at same temperature and pressure should contain equal number of molecules".

What is the equivalent weight of $\mathrm{KMnO}_4$ in acidic medium? (Molecular weight of $\mathrm{KMnO}_4=158 \mathrm{~g}$ )

What will be the $\Delta U$ value, when one mole of oxygen $\left(\mathrm{O}_2\right)$ is going from $-20^{\circ} \mathrm{C}$ to $40^{\circ} \mathrm{C}$ at constant volume? (Molar heat capacity for oxygen $\simeq 20.8 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}$ )

Calculate the molar solubility of calcium hydroxide $\mathrm{Ca}(\mathrm{OH})_2$ in 0.10 M NaOH solution. The ionic product of calcium hydroxide is $5.5 \times 10^{-6}$.

Match the given ionisation constant values with the corresponding acids.

$$ \begin{array}{lll} \hline \text { A. } & \mathrm{HI} & \text { (i) } 3.2 \times 10^9 \\ \hline \text { B. } & \mathrm{HF} & \text { (ii) } 3.5 \times 10^{-4} \\ \hline \text { C. } & \mathrm{HCl} & \text { (iii) } 1.3 \times 10^6 \\ \hline \text { D. } & \mathrm{HBr} & \text { (iv) } 1.0 \times 10^9 \\ \hline \end{array} $$

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

With reference to the redox properties of hydrogen peroxide $\left(\mathrm{H}_2 \mathrm{O}_2\right)$, which of these reactions are feasible?

(i) $2 \mathrm{Fe}^{2+} 2 \mathrm{H}^{+}+\mathrm{H}_2 \mathrm{O}_2 \longrightarrow 2 \mathrm{Fe}^{3+}+2 \mathrm{H}_2 \mathrm{O}$

(ii) $2 \mathrm{MnO}_4^{-}+6 \mathrm{H}^{+}+5 \mathrm{H}_2 \mathrm{O}_2 \longrightarrow 2 \mathrm{Mn}^{2+}+8 \mathrm{H}_2 \mathrm{O}+5 \mathrm{O}_2$

(iii) $2 \mathrm{Fe}^{2+}+\mathrm{H}_2 \mathrm{O}_2 \longrightarrow 2 \mathrm{Fe}^{3+}+2 \mathrm{OH}^{-}$

(iv) $2 \mathrm{MnO}_4^{-}+3 \mathrm{H}_2 \mathrm{O}_2 \longrightarrow 2 \mathrm{MnO}_2+2 \mathrm{H}_2 \mathrm{O} +3 \mathrm{O}_2+2 \mathrm{OH}^{-} $

Which halide of alkaline earth metals is covalent in nature and can be soluble in organic solvent, such as ethanol?

An aqueous solution of borax is

Which among the following is/are primary component(s) of the synthesis gas?

Which of the following is air pollutant?

$$ \text { Hybridisations of carbon-2 in } P \text { and } Q \text { are respectively. } $$

A benzene derivative did not produce white precipitate with the ammonical silver nitrate solution but decolorised the cold dilute alkaline $\mathrm{KMnO}_4$ solution. The compound is

Which of the following can be used as the test for unsaturation with regard to colour change of reaction?

In a bcc lattice having the edge length of 200 pm , the cation has the radius of 70 pm . The radius ratio of $\frac{r^{+}}{r^{-}}$is (Given, $\sqrt{2}=1.4, \sqrt{3}=1.7$ and $\sqrt{6}=2.4$ )

An aqueous solution of $98 \%(w / w) \mathrm{H}_2 \mathrm{SO}_4$ has density of $1.02 \mathrm{~g} / \mathrm{cc}$. The molality of the solution is

25 mL of 0.1 N NaOH solution neutralises 12.5 mL of HCl solution. The amount of water needed to convert 500 mL of such HCl solution to 0.1 N is

The maximum work that can be obtained from the following cells is

$$ X\left|X^{2+}(a q) \| Y^{+}(a q)\right| Y $$

Given, $E_{X^{2+} / X}^{\circ}=-1.7 \mathrm{~V}, E_{Y^{2+} / Y}^{\circ}=0.8 \mathrm{~V}$

The specific rate constant of decomposition of a compound is given by $\ln k=5.0-\frac{12000}{T}$. The activation energy of decomposition for this compound at 300 K is

Which of the following statements is correct for chemisorption?

$$ \text { Match the following. } $$

$$ \begin{array}{llll} \hline & \text { Ores } & & \text { Composition } \\ \hline \text { A. } & \text { Calamine } & \text { (i) } & \mathrm{CuFeS}_2 \\ \hline \text { B. } & \text { Chalcopyrite } & \text { (ii) } & \mathrm{ZnCO}_3 \\ \hline \text { C. } & \text { Bauxite } & \text { (iii) } & \mathrm{Fe}_2 \mathrm{O}_3 \\ \hline \text { D. } & \text { Haematite } & \text { (iv) } & \mathrm{Al}_2 \mathrm{O}_3 \cdot 2 \mathrm{H}_2 \mathrm{O} \\ \hline \end{array} $$

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

The molecular formula of metaphosphoric acid is

The final acid product obtained during the synthesis of $\mathrm{H}_2 \mathrm{SO}_4$ by contact process is

Among the following series of transition metal ions, the one in which all the metal ions have $3 d^2, 3 p^6$ electronic configuration is

(Atomic number, $\mathrm{Ti}=22, \mathrm{~V}=23, \mathrm{Cr}=24$, $\mathrm{Mn}=25$ )

The correct match for complex with its magnetic behaviour in the following is

Which of the following are synthetic rubbers?

(i) Terylene

(ii) Buna-S

(iii) Buna-N

(iv) Neoprene

(v) Polyacrylonitrile

$$ \text { Choose the correct Zwitter ionic form for aspartic acid. } $$

Paracetamol is

The major products $P$ and $Q$ formed in the following reactions are respectively.

$$ \text { The major product formed in the following reactions is :} $$

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

What is the product $R$ in the following reaction sequence?

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

If $f(x)=x-\frac{1}{x}, x \neq 0$, then $3 f(x)=$

Let $[\cdot]$ denote greatest integer function. If $f(x)=[x]$ and $g(x)=3\left[\frac{x}{3}\right]$, then the set of all real $x$ such that $f(x)=g(x)$ is

If $S_n$ is the sum of the first $n$ terms of the series $1^2+2 \times 2^2+3^2+2 \times 4^2+5^2+2 \times 6^2+\ldots \infty$, then, when $n$ is even $S_n=$

Let $A=\left[\begin{array}{ccc}1 & 4 & 2 \\ 2 & -1 & 4 \\ -3 & 7 & -6\end{array}\right]$ and $B=\left[b_{i j}\right]_{3 \times 3}$ with $b_{11}=2$, $b_{13}=-2, b_{12}=0$ is such that $A B=\left[\begin{array}{ccc}2 & 14 & -4 \\ 4 & 1 & -8 \\ -6 & 15 & 12\end{array}\right]$, then $|B|+\operatorname{trace}(B)=$

A is a $m \times n$ matrix of rank 4 . If A contains an $m$ th order non singular sub matrix and $A^T A$ is a $7 \times 7$ matrix, then the number of rows of $A$ is

If $C$ and $D$ are two $n \times n$ non-singular matrices over the set of real number $\mathbf{R}$ such that $C D=-D C$, then $n$ is

If $z_1=x_1+i y_1, z_2=x_2+i y_2, z_3=x_1+\frac{i x_2}{2}, z_4=2 y_1+i y_2$ are complex numbers such that $\left|z_1\right|=1,\left|z_2\right|=2$ and $\operatorname{Re} \left(\begin{array}{ll}z_1 & z_2\end{array}\right)=0$, then

Assertion (A) If $z$ is a complex number such that $|z| \geq 3$, then the least value of $\left|z+\frac{3}{z}\right|$ is 1 .

Reason (R) $\left|z_1-z_2\right| \leq\left|z_1\right|+\left|z_2\right|$, for any two complex numbers $z_1, z_2$

The correct option among the following is

$$ \text { If }\left(\frac{\cos \theta+i \sin \theta}{\sin \theta+i \cos \theta}\right)^{2020}+\left(\frac{1+\cos \theta+i \sin \theta}{1-\cos \theta+i \sin \theta}\right)^{2021}=x+i y, $$

then the value of $x+y$ at $\theta=\frac{\pi}{2}$ is

If $\omega$ is a complex cube root of unity, then $\sum_{x=1}^{10}\left((\omega x+2)\left(\omega^2 x+2\right)-3\right)$

If $\alpha$ and $\beta$ are the real roots of the equation $\sqrt{\frac{5 x}{x-2}}+\sqrt{\frac{x-2}{5 x}}=\frac{29}{10}$ and $\alpha>\beta$, then $\sqrt{\alpha^2-11^4 \beta^2}=$

The minimum value of $\frac{9 \cdot 3^{2 x}+6 \cdot 3^x+4}{9 \cdot 3^{2 x}-6 \cdot 3^x+4}$ is

$p$ is non-zero real number. If the equation whose roots are the squares of the roots of the equation $x^3-p x^2+p x-1=0$ is identical with the given equation, then $p=$

If the roots of the equation, $8 x^3+6 p x^2+3 q x-27=0$ are in a geometric progression, then $q^2+9 p^2+6 p q+q / p=$

If $x$ and $y$ represent the number of arrangements of the letters of word ATRAPATRAM such that (i) all A's are together and (ii) no two A's are together respectively, then $x+y$

Numbers between 1 and 10,000 are formed using the digits 2 and 3 only once and the digit 4 twice. If the numbers thus formed are arranged in increasing order and $x, y$ represent the ranks of 4324 and 324 respectively then $x-y=$

If the 9th and 10th terms are the numerically greatest terms in the expansion of $(5 x-6 y)^n$ when $x=2 / 5$ and $y=1 / 2$, then the absolute value of the middle terms of that expansion is

$$ 1-\frac{3}{16}+\frac{1 \cdot 4}{1 \cdot 2}\left(\frac{3}{16}\right)^2-\frac{1 \cdot 4 \cdot 7}{1 \cdot 2 \cdot 3}\left(\frac{3}{16}\right)^3+\ldots $$

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

Let $a$ be maximum value of $(3 \cos \theta-4 \sin \theta)$ and $\theta \neq \frac{n \pi}{2}$. If $\alpha=a \sin ^2 \theta \cdot \cos ^3 \theta$ and $\beta=a \sin ^3 \theta \cdot \cos ^2 \theta$, then $\sqrt{\frac{\left(\alpha^2+\beta^2\right)^5}{(\alpha \beta)^4}}=$

If $A$ does not belong to the first quadrant, $B$ does not belong to the second quadrant, $\sin A=\frac{11}{61}$ and $\cos B=\frac{-7}{25}$, then $A-B$ and $A+B$ lie respectively in the quadrants

If $\cos \left(\frac{\pi}{4}-x\right) \cos 2 x+\sin x \sin 2 x \sec x =\cos x \sin 2 x \sec x+\cos \left(\frac{\pi}{4}+x\right) \cos 2 x$, then a possible value of $\sec x$ is

The general solution of the equation $(\sqrt{3}-1) \sin \theta+(\sqrt{3}+1) \cos \theta=2$ is

If $\sin ^{-1}\left(\frac{12}{x}\right)+\sin ^{-1}\left(\frac{5}{x}\right)=\frac{\pi}{2}$, then $x=$

$$ \log (9+3 \sqrt{2}(2+\sqrt{5})+4 \sqrt{5})= $$

In a $\triangle A B C,\left(b^2-c^2\right) \cot A+\left(c^2-a^2\right) \cot B=$

In a $\triangle A B C, \frac{\Delta^2}{a^2+b^2+c^2}\left(\frac{1}{r_1^2}+\frac{1}{r_2^2}+\frac{1}{r_3^2}+\frac{1}{r^2}\right)=$

If $R: r_1: r=5: 12: 2$, then $r+r_3+r_2-r_1=$

If $12 \hat{\mathbf{i}}-12 \hat{\mathbf{j}}-18 \hat{\mathbf{k}},-3 \hat{\mathbf{i}}-6 \hat{\mathbf{j}}-9 \hat{\mathbf{k}}$ and $3 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}-24 \hat{\mathbf{k}}$ be the position vectors of the vertices $A, B$ and $C$ respectively of $\triangle A B C$, then the position vector of the incentre of $\triangle A B C$ is

Let $\Pi$ be a plane containing the points $(0,-5,-1),(1,-2,5),(-3,5,0)$ and $L$ be a line passing through the point $(0,-5,-1)$ and parallel to the vector $\hat{\mathbf{i}}+5 \hat{\mathbf{j}}-6 \hat{\mathbf{k}}$. Then the length of the projection of the unit normal vector to the plane $\Pi$ on the line $L$ is

For non-coplanar vectors $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$, if the point of intersection of the line $\mathbf{r}=\mathbf{a}+t(\mathbf{b}-\mathbf{c})$ and the plane $\mathbf{r}=\mathbf{b}+\mathbf{c}+x(\mathbf{a}-\mathbf{b})+y(\mathbf{c}+\mathbf{a})$ is $l \mathbf{a}+m \mathbf{b}+n \mathbf{c}$, then $3 l+4 m+2 n=$

If the orthocentre of the triangle whose vertices are $2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}, 5 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ and $3 \hat{\mathbf{i}}+5 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ is $x \hat{\mathbf{i}}+y \hat{\mathbf{j}}+z \hat{\mathbf{k}}$, then

If the vectors $\mathbf{A B}=p \hat{\mathbf{i}}+q \hat{\mathbf{j}}+r \hat{\mathbf{k}}, \mathbf{A C}=s \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}$, $\mathbf{C B}=3 \hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ from $\triangle A B C$, then the values of $p, q, r$ and $s$ such that the area of that $\triangle A B C$ is $5 \sqrt{6}$ are

Let $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ be three unit vectors such that $\mathbf{a} \times(\mathbf{b} \times \mathbf{c})=\frac{1}{\sqrt{2}}(\mathbf{b}+\mathbf{c})$ and $\mathbf{b}$ is not parallel to $\mathbf{c}$. If $\alpha$ and $\beta$ are the angles between $\mathbf{a}, \mathbf{b}$ and $\mathbf{a}, \mathbf{c}$ respectively then $\alpha-\beta=$

Assertion (A) Variance of $4 x_1, 4 x_2, \ldots, 4 x_n$ is 16 times the variance of $x_1, x_2, x_3, \ldots, x_n$

Reason (R) If $y=a x+b$, then variance of $y$ is a $($ variance of $x)+b$

The correct option among the following is

If $\alpha, \beta$ are respectively the mean deviation about the mean and variance of the first five prime numbers, then the ordered pair ( $\alpha, \beta$ )

If $A_1, A_2, \ldots, A_{15}$ are the events of a random experiment, then which one of the following is true?

In an examination there are four Yes/No type of questions. The probability that the answer by the student to a question without guess to be correct is $2 / 3$. The probability that a student guesses a correct answer is $1 / 2$. A student writes the examination either by without guessing answers to all the 4 questions or by guessing answers to all 4 questions. The probability that he attempt the exam by guessing answers to all questions is $3 / 7$. Given that a student answered at least 3 questions correctly, the probability that he answered all the questions without guessing is

Four boxes $A, B, C$ and $D$ contain 5000, 3000, 2000 and 1000 fuses respectively. The percentages of defective fuses in these boxes are $3 \%, 2 \%, 1 \%$ and $0.5 \%$ respectively. If a fuse selected at random from one of the boxes is found to be defective, then the probability that it has come from box $D$ is

A die is thrown thrice. If getting 1 or 6 in a single throw is considered as success, then the variance of the number of successes is

In a hospital, on an average if there are 35 births in a weak, then the probability that there will be less than 3 births in a day, is

Let $A(2,1)$ be a point and equation of the straight line $L$ be $x-y=0$. Let $a$ and $b$ respectively represent the distances from a variable point $P(\alpha, \beta)$ to $A$ and to the line $L$. If $C$ is distance of the point $A$ from origin such that $a=b c$, then locus of $P$ is

The point $(4,1)$ undergoes the following transformations successively :

(i) Reflection is the line $x-y=0$

(ii) Shifting through a distance of 2 units along the positive $X$-axis

(iii) Projection on $X$-axis

The coordinates of the point in its final position are

A function $f: \mathbf{R} \rightarrow \mathbf{R}$ is such that $f(\mathrm{l})=2$ and $f(x+y)=f(x) \cdot f(y) \forall x, y$. The area (in square units) enclosed by the lines $2|x|+5|y| \leq 4$ expressed interms of $f(1), f(2)$ and $f(4)$ is

Two straight lines $3 x+4 y=5$ and $4 x-3 y=15$ intersect at the point $A$. The equations of the lines passing through $(1,2)$ and intersecting the given lines at $B$ and $C$ such that $A B=A C$ are

The equation of a line making an angle $60^{\circ}$ with the line $x+y-3=0$ and passing through the point $(1,1)$ is

Let $P$ be the pair of lines represented by $2 x^2-5 x y+2 y^2+6 x-3 y=0$ and consider the following independent statements

(i) $\alpha$ is the $x$ coordinate of the point of intersection of the pair of lines $P$.

(ii) $\beta$ is the slope of one of the lines of $P$ passing through origin.

(iii) $\gamma$ is the constant term in the equation of the pair of angular bisectors of $P$.

Then,

The combined equation of the diagonals of the parallelogram formed by the lines

$$ \left(7 x^2-4 x y+8 y^2\right)^2+(4 x-8 y-32)\left(7 x^2-4 x y+8 y^2\right)=0 $$

is

If the origin lies on a diameter of the circle $x^2+y^2-4 x-2 y-4=0$, then the equation of the circle passing through the end points of that diameter and the point $(1,2)$ is

If $\alpha \neq-4$ and $(2, \alpha)$ is the mid-point of a chord of the circle $x^2+y^2-4 x+8 y+6=0$, then the values of the $y$-intercept of the chord lie in the interval

$C_1$ and $C_2$ are the external and internal centres of similitude of the circles $x^2+y^2-2 x+4 y+1=0$ and $x^2+y^2+4 x-6 y+12=0$. If the radius of the circle having $C_1 C_2$ as its diameters is $r$, then $\frac{9}{2} r=$

Suppose the circle $S: x^2+y^2+2 g x+2 f y+c=0$ cuts orthogonally the two circles $S^{\prime}: x^2+y^2-4 x-6 y+11=0$ and $S^{\prime \prime}: x^2+y^2-10 x-4 y+21=0$. If the centre of $S=0$ lies on the bisector of the angle between the positive coordinate axes, then $2 g+2 f+c=$

If the circle $S_1: x^2+y^2=16$ intersects another circle $S_2$ of radius 5 units such that the common chord is of maximum length and slope $\frac{3}{4}$, then the centre of the circle $S_2$ is

For the parabola $y=\frac{h^3}{3} x^2+\frac{h^2}{2} x-h+\frac{3}{4 h^3}$, if the equation of directrix is $y=k$, then $k: h$

The equation of the common tangent of the parabolas $x^2=108 y$ and $y^2=32 x$ is

The ellipse having its foci $(0, \pm 1)$ and major axis of length $\sqrt{5}$ is

An ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ with eccentricity $\frac{2 \sqrt{2}}{3}$ is inscribed in a circle $x^2+y^2=18$ such that the length of its major axis is equal to the diameter of this circle. The locus of the poles of all the tangents of the circle with respect to the ellipse is

If the circle $x^2+y^2=a^2$ intersects the hyperbola $x y=b^2$ at four points $\left(x_1, y_1\right),\left(x_2, y_2\right),\left(x_3, y_3\right),\left(x_4, y_4\right)$, then $y_1 \quad y_2 \quad y_3 y_4=$

If the line passing through the points $(a, 2,-4)$ and $(5,3, b)$ crosses the $Z X$-plane at the point $(-a+2 b, 0, a+b)$, then $14 a+7 b$

The direction cosines of the normal to the plane containing the lines having direction ratios $1,2,1$ and 4,5, -3 are

The foot of the perpendicular drawn from the point $(1,1,1)$ to the plane $\pi_1$ is $(1,3,5)$. If $(2,2,-1),(3,4,2)$, $(3,3,0)$ are three points on the plane $\pi_2$, then the angle between the planes $\pi_1$ and $\pi_2$ is

$$ \mathop {\lim }\limits_{x \to 0} \frac{1-\cos \left(x^2+\pi(x+2)\right)}{x^2}= $$

The value of ' $a$ ' for which the function

$f(x)=\left\{\begin{array}{cl}\frac{1-\cos 4 x}{x^2}, & x<0 \\ \frac{a}{\sqrt{x}}, & x=0 \text { is continuous at } x=0, \text { is } \\ \frac{\sqrt{16+\sqrt{x}}-4}{\sqrt{16+}} & \end{array}\right.$

$$ \begin{aligned} & \text { If } f(x)=\tan ^{-1}\left(\frac{1}{\sin ^2 x+\sin x+1}\right) \\ & \quad+\tan ^{-1}\left(\frac{1}{\sin ^2 x+3 \sin x+3}\right)+\tan ^{-1} \end{aligned} $$

$\left(\frac{1}{\sin ^2 x+5 \sin x+7}\right)+\ldots+$ upto 10 terms, then $f^{\prime}(0)=$

If $\alpha$ is such a minimum value for which the inverse of $f(x)=x^2+3 x-3$ exists in $[\alpha, \infty)$ and $g$ is the inverse of the $f$, then at $x=\alpha+\frac{5}{2}, \frac{d g}{d x}$

If $y=e^{a x}(\cos b x+\sin b x)$ satisfies the equation $\frac{d^2 y}{d x^2}-K \frac{d y}{d x}+L y=0$, then $L+b K=$

If $\frac{k}{\alpha^3}$ is the length of the sub normal at any point $P(\alpha, y)$ on the curve $x^2-a^2=\frac{x^2 y^2}{a^2}$, then $k=$

A tank in the shape of a rectangular parallelopiped has volume 27 cubic meters. This tank is filled with water such that the rate of change of level of the water is thrice the rate of change water quantity falling in the tank, then the height of the tank (in meters) is

Let $f:[2,5] \rightarrow \mathbf{R}$ be a differentiatiable function and $\frac{f(5)}{f(2)}=1$. If there is a $c \in(2,5)$ such that $c f^{\prime}(c)=2 f(c)-2 c^3$, then $f(x)=$

Let $f:[2,5] \rightarrow \mathbf{R}$ be a differentiatiable function and $\frac{f(5)}{f(2)}=1$. If there is a $c \in(2,5)$ such that $c f^{\prime}(c)=2 f(c)-2 c^3$, then $f(x)=$

$$ \text { Match the functions of List I with the items of List II. } $$

For $x \in\left(\frac{3 \pi}{4}, \pi\right), \int(\sqrt{1+\sin 2 x}+\sqrt{1-\sin 2 x}) d x=$

$$ \begin{aligned} & \text { If } \int \frac{x^2\left(x \sec ^2 x+\tan x\right)}{(x \tan x+1)^2} d x=A \log (|x \sin x+\cos x|) \\ & +B \frac{f(x)}{(x \tan x+1)}+C \text {, then } f(A+B)= \end{aligned} $$

$$ \text { If } \begin{aligned} & \int x^3(\log x)^2 d x=x^4\left[A(\log x)^2+B(\log x)\right. \\ &+C \log e]+K, \text { then } A+B+C \end{aligned} $$

$$ \begin{aligned} & \text { If } \int \frac{9 x+15}{x^3-6 x-9} d x=A \log |g(x)| \\ & \quad+B \log |f(x)|+C, \text { then } \frac{(A-B) g(4)}{f(-1)}= \end{aligned} $$

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

$$ \int_0^{\pi / 2} \frac{d x}{4+5 \sin x} $$

The area (in square units) of the region enclosed between the parabola $y^2=2 x$ and the line $y=4 x-1$

The differential equation for which $y=a x^2+b x+c$ is the general solution is

The general solution of the differential equation

$(3 y-7 x+7) d x+(7 y-3 x+3) d y=0$ is

The general solution of the differential equation $(3 y-7 x+7) d x+(7 y-3 x+3) d y=0$ is

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

In atomic scale the weakest force in nature is

In five successive measurements, the mass of a ball is measured to be $2.61 \mathrm{~g}, 2.58 \mathrm{~g}, 2.40 \mathrm{~g}, 2.73 \mathrm{~g}$ and 2.80 g . The absolute error in the measurement is

A particle is moving along the $Y$-axis. The position of the particle from the origin as a function of time $(t)$ is given as $y(t)=10 t e^{-2 t}$. How far is the particle from the origin when it stops momentarily? ( $y$ is given in units of metre and $t$ is in units of second)

Two cars $A$ and $B$ are moving with speeds $v_A=120 \mathrm{km} / \mathrm{h}$ and $v_B=50 \mathrm{~km} / \mathrm{h}$ respectively in the directions as indicated by the arrow in the figure below. What is the relative speed of the car $B$ with respect to car $A$ ?

Initial velocity with which a body is projected is $10 \mathrm{~m} / \mathrm{s}$ from the base of an inclined plane as shown in the given figure. If the angle of projection is $60^{\circ}$ with the horizontal, then the range $R$ is [take, $g=10 \mathrm{~m} / \mathrm{s}^2$ ]

A projectile is fired at an angle of $45^{\circ}$ with the horizontal. Elevation angle of the projectile at its highest point as seen from the point of projection is

A point $P$ is moving in uniform circular motion with radius 3 m . Let at some instant the acceleration of the point is $\quad \mathbf{a}=(6 \hat{\mathbf{i}}-4 \hat{\mathbf{j}}) \mathrm{m} / \mathrm{s}^2$, the position vector is $\mathbf{r}$ and velocity vector is $\mathbf{v}$. Choose the correct statement.

A block of mass 3 kg is pressed against a vertical wall by applying a force $F$ at an angle $30^{\circ}$ to the horizontal as shown in the figure. As a result, the block is prevented from falling down. If the coefficient of static friction between the block and wall is $\sqrt{3}$, then the value of $F$ is (use, $g=10 \mathrm{~m} / \mathrm{s}^2$ )

A force $\mathbf{F}=(2 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}) \mathrm{N}$ is applied on an object of mass $M$. What is the work done by this force in moving the object horizontally along the $X$-axis by 3 m ?

A ball of mass $m=1 \mathrm{~kg}$ is thrown from the top of a building with initial velocity $\mathbf{v}=(20 \mathrm{~m} / \mathrm{s}) \hat{\mathbf{i}}+(24 \mathrm{~m} / \mathrm{s}) \hat{\mathbf{j}}$ at time $t=0$. The change in the potential energy of the ball between $t=0$ and $t=6 \mathrm{~s}$, if the ball does not hit the ground, then (assume, $g=10 \mathrm{~m} \mathrm{~s}^2$ )

A solid sphere and a solid cylinder, each of mass $M$ and radius $R$ are rolling with a linear speed on a flat surface without slipping. Let $L_1$ be magnitude of the angular momentum of the sphere with respect to a fixed point along the path of the sphere. Likewise $L_2$ be the magnitude of angular momentum of the cylinder with respect to the same fixed point along its path. The ratio $L_1 / L_2$ is

An object of mass 2 kg is hanging from a rope that is wrapped around a pulley of radius 25 cm . The mass of pulley is 2 kg . Find the acceleration of the object. (Assume, pulley to be a solid disk $g=10 \mathrm{~m} / \mathrm{s}^2$ )

A point mass oscillates along the $X$-axis according to the law $x=x_0 \cos \left(\omega t-\frac{\pi}{4}\right)$. If the acceleration of the particle is written as $a=A \cos (\omega t-\delta)$, then

The graph correctly represents the variation of acceleration due to gravity $(g)$ with radial distance from the centre of the earth (radius of the earth $=R_e$ ) is

The length of a metal wire is found to be $L_1$ and $L_2$ when the tension of $T_1$ and $T_2$ are applied to it respectively. The natural length of the wire is

A cubical block of wood having mass of 160 g has a metal piece fastened underneath as shown in the figure. Find the maximum mass of the metal piece which will allow the block to float in water. Specific gravity of wood is 0.8 and that metal is 10 and density of water $=1 \mathrm{~g} / \mathrm{cc}$.

A solid of 2 kg mass absorbs 50 kJ when its temperature is raised from $20^{\circ} \mathrm{C}$ to $70^{\circ} \mathrm{C}$. The specific heat capacity of this solid in unit of $\mathrm{J} / \mathrm{kg}{ }^{\circ} \mathrm{C}$ is

A solid cylinder of radius $r_1=2.5 \mathrm{~cm}$, length $l_1=5.0 \mathrm{~cm}$ and temperature $40^{\circ} \mathrm{C}$ is suspended in an environment of temperature $60^{\circ} \mathrm{C}$. The thermal radiation transfer rate for cylinder is 1.0 W . If the cylinder is stretched until its radius becomes $r_2=0.50 \mathrm{~cm}$, the thermal radiation transfer rate is changed to

Five moles of an ideal gas has pressure $p_0$, volume $V_0$ and temperature $T_0$. The gas is expanded to volume $3 V_0$ along a path, so that the pressure $p$ is changed as function of volume $V$ as $p=p_0\left(V / V_0\right)$. The pressure is then reduced to $p_0$ maintaining the volume constant. The gas undergoes an isobaric compression till the volume and temperature become $V_0$ and $T_0$, respectively. The total work done by the gas during the entire process is

How many rotational degrees of freedom does a rigid diatomic molecule have?

The transverse displacement $y(x, t)$ of a wave on a string is given by $y(x, t)=e^{-\left(a x^2+b t^2+2 \sqrt{a b x} t\right)}$. This represents a

A telescope has an objective of focal length 100 cm and an eye-piece of focal length 5 cm . The magnifying power of the telescope is

A convex lens and a concave lens, each with focal length of 4 cm are separated by a distance of 6 cm along their axis. An object is placed 8 cm before the convex lens. The distance between the object and its image is

The limit of resolution of a telescope is $3.0 \times 10^{-7} \mathrm{rad}$. Assuming that it is used to see the light of wavelength 525 nm from a star, what should be the diameter of the objective?

Two negative charges of equal magnitude are located in $x y$-plane as shown below in the figure. The direction of the electric field at point $P$ is

An infinite non-conducting sheet has a surface charge density $2 \times 10^{-7} \mathrm{C} / \mathrm{m}^2$ on one side. The distance between two equipotential surfaces whose potential differ by 90 V is (assume, $\frac{1}{4 \pi \varepsilon_0}=9 \times 10^9 \frac{\mathrm{Nm}^2}{\mathrm{C}^2}$ )

A cylindrical wire $P$ has resistance $10 \Omega$. A second wire $Q$ has length and diameter half that of $P$. If the material of both the wires is same, then resistance of wire $Q$ is

Find the current in the circuit.

Two tangent galvanometers $A$ and $B$ have coils of radii 8 cm and 16 cm respectively and having resistance of 8 $\Omega$ each. They are connected in parallel with a cell of emf 4 V and negligible internal resistance. The deflections produced in the tangent galvanometers $A$ and $B$ are $30^{\circ}$ and $60^{\circ}$, respectively. If $A$ has 2 turns, then $B$ must have

A particle of charge $q$ and mass $m$ moves in a circular orbit of radius $r$ with angular speed $\omega$. The ratio of the magnitude of its magnetic moment to that of its angular momentum is

Two short magnets of equal dipole moments $M$ are fastened perpendicularly at their centres which lies at origin. Let two magnets lie along $X$-axis and $Y$-axis, respectively.

The magnitude of the magnetic field at a distance $R$ from the centre on the $Y$-axis is $\frac{\mu_0}{4 \pi} \frac{M_0}{R^3}$. Assuming, $R \gg l$ (magnet length), the magnitude of $M$ is

A varying current in a coil changes from 10 A to zero in 1.5 s . If the average emf induced in the coil is 200 V , the self-inductance of the coil is

An alternating voltage $\varepsilon=30 \sin 200 t$ (in volts) is applied to the circuit below. The amplitude of the current through the circuit is

A parallel-plate capacitor with circular plates is being discharged. The radius of the circular plate is 10 cm . A circular loop of radius 20 cm is concentric with the capacitor and located halfway between the plates. If the electric field between the plates is charging at the rate $3.6 \times 10^{12} \mathrm{~V} /(\mathrm{ms})$, then the displacement current through the loop is

$$ \text { (Assume } \frac{1}{4 \pi \varepsilon_0}=9 \times 10^9 \mathrm{Nm}^2 / \mathrm{C}^2 \text { ) } $$

Let $v_1$ and $v_2$ be the maximum velocities of the emitted electrons when the surface of a metal is illuminated with light waves of energy $E_1=4 \mathrm{eV}$ and $E_2=2.5 \mathrm{eV}$,respectively. If the work function of the metal is 2 eV , then the ratio $\frac{v_1}{v_2}$ is

The wavelength of a spectral line emitted by hydrogen atom in the Balmer series is $\frac{16}{3 R}$

( $R$ is Rydberg constant). What is the value of the principal quantum number of the state from which the transition takes place?

The half-life of a radioactive sample is 5 s . If the initial mass of the sample is 60 g , then the time required to reduce the sample to 7.5 g is

A Zener diode is connected to battery and a load resistance as shown below

The currents $I, I_Z$ and $I_L$ respectively are

A semiconductor is doped with phosphorous atoms as impurity. The impurity levels created in the semiconductor are close to the

Assertion (A) Television signals are received through sky-wave propagation.

Reason (R) The ionosphere reflects electromagnetic waves of frequencies in the range ( $3-30) \mathrm{MHz}$.

Choose the correct option.