Maximum number of electrons in a subshell with $n=4$ and $l=3$ is

From the following sets of quantum numbers, which set is possible?

What is the correct order of atomic radius of $\mathrm{Al}, \mathrm{Na}, \mathrm{B}$ and Be ?

Find the element, which displays greater ability to form $p \pi-p \pi$ multiple bonds to itself and to other elements in the same block.

Which of the sulphur compound follows the octet rule?

$\Delta_r H$ of which reaction correctly represents the lattice enthalpy of $\mathrm{NaCl}(s)$ ?

The volume of a given amount of gas at $27^{\circ} \mathrm{C}$ at constant pressure is $420 \mathrm{~cm}^3$. If the temperature is reduced by $20^{\circ} \mathrm{C}$ at constant pressure, what will be the volume of the gas?

The compressibility factor $Z=\frac{p V}{n R T}$ for hydrogen gas a 273 K and 1 atm pressure is

In the following reaction mixtures,

I. $\mathrm{Cu} / \mathrm{CuSO}_4+\mathrm{Ag} / \mathrm{Ag}_2 \mathrm{SO}_4$

II. $\mathrm{Zn} / \mathrm{ZnSO}_4+\mathrm{Cu} / \mathrm{CuSO}_4$

copper respectively undergoes

The molecular mass of sucrose $\left(\mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11}\right)$ is

In which of the processes, the entropy will decrease

For a given equilibrium reaction, addition of inert argon gas at constant volume can shift the equilibrium in

$$ \mathrm{N}_2(g)+3 \mathrm{H}_2(g) \rightleftharpoons 2 \mathrm{NH}_3(g) $$

For the given equilibrium reaction

$$ \mathrm{H}_2(\mathrm{~g})+\mathrm{I}_2(\mathrm{~g}) \rightleftharpoons 2 \mathrm{HI}(\mathrm{~g}) $$

Choose the correct equation to calculate $K_p$.

When calcium carbide is reacted with heavy water, which of the following product(s) will be formed

I. $\mathrm{CD}_4$

II. $\mathrm{C}_2 \mathrm{D}_2$

III. $\mathrm{Ca}(\mathrm{OD})_2$

IV. $\mathrm{Ca}_2 \cdot \mathrm{D}_2 \mathrm{O}$

Which alkali metal emits blue colour light in flame test?

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

Glass reacts with HF to produce

Which of the following molecules react with haemoglobin of the blood to produce toxic effect?

$$ \text { The IUPAC name of the following compound is } $$

When $\mathrm{CH}_3 \mathrm{Br}$ and $\mathrm{C}_2 \mathrm{H}_5 \mathrm{Br}$ are subjected to Wurtz reaction, the maximum number of possible alkane product(s) formed is/are

Maximum number of bromine atoms present in the final product $P$ upon complete bromination are

A cubic structure is formed where atoms of element $X$ are occupied at corner of cube and also at face centers. Atoms of element $Y$ are present at body center and at the edge centers. If all the atoms are removed along a plane passing through the middle of the cube (bisecting the four edges), the formula will become

pH of 0.05 M aqueous solution of diethyl amine is 10 . The value of equilibrium constant is

The observed molar mass determined for $\mathrm{Na}_2 \mathrm{SO}_4$ by freezing point depression method is $50.0 \mathrm{~g} / \mathrm{mol}$. The molecular weight of $\mathrm{Na}_2 \mathrm{SO}_4$ is 142 . What will be the degree of dissociation $\alpha$ for $\mathrm{Na}_2 \mathrm{SO}_4$ in water?

Copper is to be electrodeposited on a nickel block of $(20 \times 5) \mathrm{cm}^2$ area by using $\mathrm{CuSO}_4$ as electrolyte. How much quantity of electricity is needed to deposit a 3.6 $\mu \mathrm{m}$ layer of copper?

[Atomic weight of $\mathrm{Cu}=63.5 \mathrm{~mol}^{-1}$ Density of $\mathrm{CuSO}_4=8.9 \mathrm{~g} / \mathrm{cc}$ ]

The half-life of a first order reaction varies with temperature according to

The charge of " $\mathrm{SnO}_2-\mathrm{sol}^{\prime}$ in alkaline and acidic medium are respectively

(Hint $\mathrm{SnO}_2$ is amphoteric oxide)

Which one of the following statements is true about the froth flotation method?

When cold and dilute NaOH reacts with $\mathrm{Cl}_2$, which of the following is formed?

The hybridisation of Xe in $\mathrm{XeO}_3$ is

Which of the following is a diamagnetic ion?

Among the following Cr(III) complexes, which one will have the highest octahedral crystal field splitting?

Which one among the following is a semi-synthetic polymer?

Considering the facts that

I. sucrose forms glycosidic linkage between $\mathrm{C}_1$ of glucose and $\mathrm{C}_2$ fructose, while

II. lactose forms glycosidic linkage between $\mathrm{C}_1$ of galactose and $\mathrm{C}_4$ of glucose.

Choose the correct statement.

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

$$ \begin{array}{llll} \hline & \text { List-I } & & \text { List-II } \\ \hline \text { A. } & \text { Codeine } & \text { I. } & \text { Antiseptic } \\ \hline \text { B. } & \text { Dettol } & \text { II. } & \text { Antibiotic } \\ \hline \text { C. } & \text { Tetracycline } & \text { III. } & \text { Narcotic analgesics } \\ \hline \end{array} $$

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

The major product in the following reaction sequence is

$$ \text { Match the following } $$

$$ \begin{array}{llll} \hline & \begin{array}{c} \text { List-I } \\ \text { (Reaction) } \end{array} & & \begin{array}{c} \text { List-II } \\ \text { (Product) } \end{array} \\ \hline \text { A. } & \text { Ketone }+\mathrm{NaBH}_4 & \text { I. } & 1^{\circ} \text { - alcohol } \\ \hline \text { B. } & \text { Ester }+\mathrm{LiAlH}_4 & \text { II. } & 3^{\circ} \text { - alcohol } \\ \hline \text { C. } & R \mathrm{Mg} X+\text { Ketone } & \text { III. } & 2^{\circ} \text { - alcohol } \\ \hline & & \text { IV. } & \text { Alkane } \\ \hline \end{array} $$

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

The reaction of one mole of 3-ethyl-3-methylpenta-1, 4-diene with $\mathrm{O}_3$ and then with $\mathrm{Zn} / \mathrm{H}_2 \mathrm{O}$, gives

Which of the following reactions gives carboxylate ion in their reaction mixture?

The product formed when aniline is treated with chloroform in the presence of KOH (alcoholic) is

Given that for any $n \in \mathbf{N}$ there exist an odd integer $q$ and a non-negative integer $r$ such that, $n$ can be written uniquely as $n=q \times 2^r$.

Let $f: \mathbf{N} \rightarrow \mathbf{N} \times \mathbf{N}$ be function defined by $f(n)=\left(r+1, \frac{q+1}{2}\right)$. Then,

2. If $f: \mathbf{R} \rightarrow \mathbf{R}$ be defined by $f(x)=x+2|x+1|+2|x-1|$, then the element in the co-domain, which has unique pre image in the domain is

If $f(1)=3$, and $f(n+1)-f(n)=3\left(4^n-1\right)$, then $\forall n \in \mathbf{N}$, $f(n)=$

For a square matrix $B$ of order 3 , if $B^T=B^{-1}$ and $|B|=1$, then $|B-I|=$

For $\alpha, \beta \in[0,2 \pi]$ and $\gamma \in[0, \pi)$ consider the system of equations

$$ \begin{aligned} & 2 \sin \alpha-\cos \beta+3 \tan \gamma=3 \\ & 4 \sin \alpha+2 \cos \beta-2 \tan \gamma=2 \\ & 6 \sin \alpha-3 \cos \beta+\tan \gamma=9 \end{aligned} $$

Then, which one of the following is true?

$$ \text { The rank of } A=\left[\begin{array}{ccc} 1 & x & x+1 \\ 2 x & x^2-x & x^2+x \\ 3 x(x-1) & x\left(x^2-3 x+2\right) & x\left(x^2-1\right) \end{array}\right] \text { is } $$

$z_1, z_2$ are two fixed points on the Argand plane. If $z$ is a complex number such that $\left|z-z_1\right|+\left|z-z_2\right|=\lambda$, then the locus of $z$ is

If the four points $A, B, C, D$ in the Argand plane represented respectively by the complex numbers $2+i, 4+3 i, 2+5 i, 3 i$ lie on a circle, then the centre of the circle is

The roots of the equation $(x-1)^5=32(x+1)^5$ are

If $\omega$ is a non-real cube root of unity and $x=\omega^2-\omega-3$, then the value of $x^4+6 x^3+10 x^2-12 x-19$ is

The number of integral values of $x$ satisfying $9 x-2<(x+2)^2<12 x-3$ is

Sum of the modulii of the complex roots of the equation $\left(x^2+\frac{1}{x^2}\right)-5\left(x+\frac{1}{x}\right)+6=0$ is

If $2+\sqrt{3}$ is a root of the equation $f(x)=x^4+2 x^3-16 x^2-22 x+7=0$, then which one of the following is not a root of $f(x)=0$ ?

Assertion (A) If $a_1, a_2, \ldots, a_n$ are the $n$ distinct roots of the equation $x^n-2=0$, then $1+\left(1-a_1\right)\left(1-a_2\right) \ldots \left(1-a_{n-1}\right)\left(1-a_n\right)=0$

Reason (R) If $\alpha_1, \alpha_2, \ldots, \alpha_n$ are the roots of $f(x) \equiv p_0 x^n+p_1 x^{n-1}+p_2 x^{n-2}+\ldots+p_n=0$, then the roots of

$$ f(g(x))=0 \text { are } \mathrm{g}^{-1}\left(\alpha_i\right), i=1,2,3, \ldots, n $$

The correct option among the following is

Let $S_r=\{x, y, z) / x+y+z=11, x \geq r, y \geq r$, $z \geq r, x, y, z, r$ are integers $\}$ and $n\left(S_r\right)$ represents the number of elements in $S_r$. Then $n\left(S_{2)}+n\left(S_3\right)+n\left(S_4\right)=\right.$

A certain question paper contains three parts $A, B, C$ with four questions in part $A$, five questions in part $B$ and six questions in part $C$. A student is required to answer seven questions choosing at least two questions from each part. Then the total number of different ways a student can choose his seven questions for answering, is

$p, q$ are two prime numbers. For $n=p q$, if the expansion $\left(\sqrt[4]{x^{-5}}+2 \sqrt[5]{x^5}\right)^n$ contains non-zero coefficient of $x^{-n}$ and $x^0$, then the least value of such $n$ is

The binomial expansion $(7+3 x)^{-2 / 5}$ is valid for all $x$ in the interval $\left(\frac{-7}{3}, \frac{7}{3}\right)$ and if the 4 th term of its expansion is $k x^3$, then $\left(7^{12 / 5} k\right)=$

Let $H(x)=3 x^4+6 x^3-2 x^2+1$ and $g(x)$ be a polynomial of degree one. If

$\frac{H(x)}{(x-1)(x+1)(x-2)}=f(x)+\frac{g(x)}{(x-1)(x+1)(x-2)}$ then

$H(-1)+2 H(2)-3 H(1)=$

The smallest positive value of $x$ (in degrees) for which $\tan \left(x+100^{\circ}\right)=\tan \left(x+50^{\circ}\right) \tan (x) \tan \left(x-50^{\circ}\right)$ is

For $n \in \mathbf{N}$, if $f(n)=(\cos n x)(\sec x)^n$ and $g(n)=(\sin n x)(\sec x)^n$, then $f(2020)-f(2019)+(\tan x) g(2019)=$

$\theta$ and $\alpha$ lie in $Q_3$. If $\cos (\theta-\alpha), \cos \theta, \cos (\theta+\alpha)$ are in harmonic progression, then $\cos \theta \sec \frac{\alpha}{2}=$

If the possible solution of the equation $2 \cos ^2 x+3 \sin x-3=0$ constitute two unequal angles of a triangle, then the third angle of that triangle is

In $\triangle A B C$ if $\angle C=\frac{\pi}{2}$ then

$\tan ^{-1}\left(\frac{a}{b+c}\right)+\tan ^{-1}\left(\frac{b}{c+a}\right)+\tan ^{-1}\left(\frac{c}{a+b}\right)=$

If $\sinh \left(2 \tanh ^{-1} x\right)=\frac{11}{60}$, then $x=$

If the sides of a triangle are in the ratio $\sqrt{3}: \sqrt{5}: \sqrt{8+\sqrt{15}}$, then the largest angle in that triangle is

In a $\triangle A B C$, if $\tan A: \tan B: \tan C=1: 2: 3$ and $\sin A: \sin B: \sin C=\sqrt{5}: 2 \sqrt{2}: k$, then $k=$

In $\triangle A B C$, if $R=\frac{65}{8}, r r_1=42$ and $r_1-r=6.5$, then $s(s-a)=$

Let $A B C D$ be a parallelogram and $E$ be the mid-point of $A B$. If $P$ is the point of intersection of $D E$ and $A C$, then $\frac{D P}{P E}+\frac{A P}{P C}=$

A vector $\mathbf{a}$ has components $2 p$ and 1 with respect to a two dimensional rectangular cartesian system. This system is rotated through a certain angle about the origin in the counter-clockwise direction. If $\mathbf{a}$ has components $p+1$ and 1 with respect to the new system, then

If $\mathbf{a}=2 \mathbf{u}+3 \mathbf{v}+7 \mathbf{w}, b=\mathbf{u}+\mathbf{v}-2 \mathbf{w}$ and $\mathbf{c}=-\mathbf{u}-2 \mathbf{v}-3 \mathbf{w}$ then $\left|\frac{[\mathbf{u} \mathbf{v} \mathbf{w}]}{[\mathbf{a} \mathbf{b} \mathbf{c}]}\right|(\mathbf{a}+\mathbf{b}+\mathbf{c})=$

Let $\mathbf{V}=2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}$ and $\mathbf{W}=\hat{\mathbf{i}}+3 \hat{\mathbf{k}}$. If $\mathbf{U}$ is a unit vector, then the maximum value of $[\mathbf{U} \mathbf{V} \mathbf{W}]$ is

$L_1$ is a line passing through the points with position vectors $\hat{\mathbf{i}}-2 \hat{\mathbf{j}}-\hat{\mathbf{k}}$ and $4 \hat{\mathbf{i}}-3 \hat{\mathbf{k}} . L_2$ is a line passing through the points with position vectors $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}$ and $2 \hat{\mathbf{i}}-4 \hat{\mathbf{j}}-5 \hat{\mathbf{k}}$. Then the distance between $L_1$ and $L_2$ is

The mean and standard deviation of 100 observations $x_1, x_2, \ldots, x_{100}$ were calculated as 40 and 5.1 respectively by a student who took by mistake 50 instead of 40 for one observation. Then the correct value of $\sum_{i=1}^{100} x_i^2=$

The coefficient of variation of the first 5 prime numbers is

A person tossing a biased coin indefinitely wins the game by getting head for the first time. The probability that he wins the game in odd number of tosses is $3 / 4$. If 5 such coins are tossed at a time then the probability that head appears on all the coins is

Let $B(\alpha, \beta, \gamma)$ represents that a bag $B$ contains $\alpha$ red balls, $\beta$ green balls and $\gamma$ blue balls. Given $B_1(2,3,2), B_2(3,2,2), B_3(2,2,3)$. A die is rolled. If the die shows up 2 or 3 or 5 , then a ball will be drawn at random from bag $B_1$. If the die shows up 4 or 6 , then a ball will be drawn at random from bag $B_2$. If the die shows up 1 , then from bag $B_3$ a ball will be drawn at random. Then the probability of drawing a green ball from a bag thus chosen is

If the coefficients $a$ and $b$ of a quadratic expression $x^2+a x+b$ are chosen from the sets $A=\{3,4,5\}$ and $B=\{1,2,3,4\}$ respectively, then the probability that the equation $x^2+a x+b=0$ has real roots is

A random variable $X$ has the following probability distribution

$$ \begin{array}{|c|c|c|c|c|c|c|c|c|} \hline X=x & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline P(X=x) & 0.15 & 0.23 & K & 0.10 & 0.20 & 0.08 & 0.07 & 0.05 \\ \hline \end{array} $$

For the event $E=\{X / X$ is a prime number $\}$ and the event $F=\{X / X<4\}$, the probability $P(E \cup F)=$

If a variable line is moving such that the intercepts made by it on the coordinate axes are reciprocal to each other, then the points $P(x, y)$ on such lines satisfy

If a variable line is moving such that the intercepts made by it on the coordinate axes are reciprocal to each other, then the points $P(x, y)$ on such lines satisfy

Suppose the axes $X$ and $Y$ are obtained by rotating the axes $x$ and $y$ an angle $\theta$. If the equation $x^2+2 \sqrt{3} x y-y^2=4 a^2$ is transformed to $X^2-Y^2=2 a^2$ with respect to the $X Y$-axes, then $\theta$ is equal to

If the lines drawn along the diagonals of the two squares formed by two pairs of lines $x^2-3|x|+2=0$ and $y^2-3 y+2=0$ form a square $A B C D$, then the equations of two adjacent sides of the square $A B C D$ are

If $\pi / 3$ is the angle between the straight lines $p x+q y+r=0$ and $x \sin \alpha+y \cos \alpha=r(r \neq 0)$ which meet at a point $A$ and the straight line $x \cos \alpha-y \sin \alpha=0$ also passes through the point $A$, then

The distance between the point $(2,1)$ and the image of the point $(3,-1)$ with respect to the line $2 x+y-1=0$ is

Let $O A B C$ be a parallelogram. The equation of one diagonal $A C$ is $x+y-1=0$ and the combined equation of the sides $O A, O C$ is $2 x^2-y^2=0$. If $G$ is centroid of the triangle $O A C$, then $B G=$

The acute angle between the pair of straight lines joining the origin to the points of intersection of the line $x+y-1=0$ with the pair of straight lines $k x^2+8 x y-3 y^2+2 x-4 y-1=0$ is

The centre and radius of the circumcircle of the triangle formed by the lines $2 x+3 y=10, y=x$ and the $X$-axis are respectively.

If the straight lines $3 x-4 y+4=0$ and $6 x-8 y-7=0$ are the tangents to the same circle, then the area of that circle (in square units) is

In the List-I each item contains equations of two circles, List-II contains the number of common tangents for each pair of circles given in List-I. Match the items of List-I with those of the items of List-II

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

$\left(0, \frac{3}{4}\right)$ is the radical centre of the circles $S_1: x^2+y^2-2 x+6 y=0, S_2: x^2+y^2+2 g x-2 y+6=0$ and $S_3: x^2+y^2-12 x+2 f y+3=0$. If $S_2$ and $S_3$ intersect orthogonally, then $(g, f)=$

For the circles $(x-a)^2+y^2=a^2$ and $x^2+(y-a)^2=a^2$, where $a>0$, which one of the following is not true?

If $S(a, b)$ is a fixed point and $P(\alpha, \beta)$ is such a variable point that $4\left[(x-a)^2+(y-b)^2\right]=(\alpha x+\beta y+7)^2$ represents a parabola, then the locus of $P(\alpha, \beta)$ is

If $P(-3,2)$ is an end point of the focal chord $P Q$ of the parabola $y^2+4 x+4 y=0$, then the slope of the normal drawn at $Q$ is

Equation of a common tangent to the circle $x^2+y^2=4$ and to the ellipse $2 x^2+25 y^2=50$ is

The $\theta$ is the angle made by the common tangent to the circle $x^2+y^2=16$ and the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$ with positive $X$-axis, then $\cos 2 \theta=$

For the hyperbola $x^2-y^2-4 x+2 y+c=0$, if the focus is $S(2+2 \sqrt{2}, k)$ and the directrix that is adjacent to $S$ is $x=2+\sqrt{2}$, then $c=$

The quadrilateral formed by the points $A(1,2,5), B(-1,6,1), C(3,4,-3)$ and $D(5,0,1)$ is a

A line with direction cosines proportional to $2,1,2$ meets the line $L_1$ passing through $(0,-1,0)$ with direction ratios $1,1,1$ at $A(x, y, z)$ and another line $L_2$ at $B(1,1,1)$ then $x+y+z=$

If a plane $\pi$ passes through the point $(-1,6,2)$ is perpendicular to the planes $x+2 y+2 z-5=0$ and $3 x+3 y+2 z-8=0$, then, the perpendicular distance from the point $(1,-1,1)$ to the plane $\pi$ is

If $f: \mathbf{R} \rightarrow \mathbf{R}$ and $g: \mathbf{R} \rightarrow \mathbf{R}$ be defined by $f(x)=\left\{\begin{array}{cc}x+2, & x>0 \\ 2-x, & x \leq 0\end{array}\right.$ and $g(x)=\left\{\begin{array}{cc}x^2-2 x-2, & 1 \leq x<2 \\ x-7 & x \geq 2 \\ x+5, & x<1\end{array}\right.$ then $\lim _{x \rightarrow 0} g \circ f(x)$

Define $f: R \rightarrow R$ by $f(x)= \begin{cases}(x-a) \frac{e^{\frac{1}{(x-a)}}-1}{\frac{1}{(x-a)}}+1 & \text { for } x \neq a \\ 0, \quad \text { at } x=a\end{cases}$

Then which one of the following is true?

If $\operatorname{Lt}_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}=e^x(x+1)$ and $f(0)=0$, then $\frac{d}{d x}\left(f(x) e^{-x}\right)+\frac{d}{d x}\left(\frac{f(x)}{x}\right)=$

Let $f, g: \mathbf{R} \rightarrow \mathbf{R}$ be functions defined by

$$ f(x)=\left\{\begin{array}{cc} x \sin \left(\frac{1}{x}\right), & \text { for } x \neq 0 \\ 0, & \text { for } x=0 \end{array}\right. $$

and $g(x)=x f(x)$

Consider the following statements

(i) $f(x)$ is continuous at $x=0$ but not differentiable at $x=0$

(ii) $g(x)$ is differentiable at $x=0$, but $g^1(x)$ is not continuous at $x=0$

Then, which one of the following is true?

If $y(x)=\tan ^{-1}\left(\frac{\sqrt{1+a^2 x^2}-1}{a x}\right)$ and $\left(1+a^2 x^2\right) y^{\prime \prime}+g(x) y^{\prime}=0$ then, the sum of the roots of the equation $1+a^2 x^2+g(x)=0$ is

A vessel in the shape of an inverted cone of height 10 ft and semi vertical angle $30^{\circ}$ is full of water. Due to a hole at the vertex, the slant height of the water in the vessel is decreasing at a constant rate of $\frac{1}{\sqrt{3}}$ feet per minute. The rate (in cu. feet/min) at which the volume of water in the vessel is decreasing, when the volume of water is $\frac{8 \pi}{\sqrt{3}}$ cubic feet, is

The area (in sq. units) of the triangle formed by the tangent and normal drawn to the curve $\left(\frac{x}{3}\right)^n+\left(\frac{y}{4}\right)^n=2$ at $(3,4)$ and $x$-axis is

If the curves $a x^2+b y^2=1$ and $c x^2+d y^2=1$ intersect orthogonally, then $\frac{b-a}{d-c}=$

The ratio of the maximum and minimum values attained by the function $f(x)=1+2 \sin x+3 \cos ^2 x, 0 \leq x \leq \frac{2 \pi}{3}$ is

If $5(f(x))^2=x f(x)+30$ and

$$ \begin{aligned} & \int \frac{\left(3 x^3+\left(1-30 x^2\right) f(x)\right)}{(10 f(x)-x)\left(x^3-f(x)\right)^2} d x \\ & =\frac{A}{B x^3+D f(x)}+C \text { then } A+B+D= \end{aligned} $$

If $\int x[\log (1+x)]^3 d x=\frac{(1+x)^2}{16}(f(x))+(1+x)(g(x))$, then

$$ f(x)+g(x)= $$

$$ \int \frac{\left(x+\sqrt{1+x^2}\right)^2}{\sqrt{1+x^2}} d x= $$

$$ \int\left[\frac{x^4-x}{x^{20}}\right]^{1 / 4} d x= $$

$$ \begin{array}{r}\mathop {\lim }\limits_{n \to \infty }\left[\frac{n^{3 / 2}}{n^{5 / 2}}-\frac{n^{1 / 2}}{n^{3 / 2}}+\frac{n^{3 / 2}}{(n+2)^{5 / 2}}-\frac{n^{1 / 2}}{(n+3)^{3 / 2}}\right. \\ +\frac{n^{3 / 2}}{(n+4)^{5 / 2}}-\frac{n^{1 / 2}}{(n+6)^{3 / 2}}+\ldots+\frac{n^{3 / 2}}{(n+2(n-1))^{5 / 2}} \\ \left.-\frac{n^{1 / 2}}{(n+3(n-1))^{3 / 2}}\right]= \end{array} $$

$$ \lim _{n \rightarrow \infty}\left[\frac{n+3}{n^2+1^2}+\frac{n+6}{n^2+2^2}+\frac{n+9}{n^2+3^2}+\ldots+\frac{2}{n}\right]= $$

If the area lying in the first quadrant and bounded by the circle $x^2+y^2-4 x=0$, the parabola $y^2=x$ and the $X$-axis is $A$, then $6 A-9 \sqrt{3}=$

If the order and degree of the differential equation corresponding to the family of curves $(x-2)^2+(y-a)^2=b^2$, (where $a$ and $b$ are parameters) are $m$ and $n$ respectively, then $m^2+n=$

Consider the differential equation $\frac{d y}{d x}=\frac{1}{a x+4 y+7}$ and the following statements

A. The given differential equation is linear in $x$.

B. The given differential equation is not linear in $y$.

C. The given differential equation is linear in $y$.

D. $e^{a x}$ is the integrating factor of the given differential equation.

Which one of the following options is true?

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

Choose the correct statement.

The dimension of $\frac{E^2}{\mu_0}$ in mass $(M)$, length $(L)$ and time

$(T)$ is $(E=$ electric field,

$\mu_0=$ permeability of free space)

A body moves in a straight line with speed $v_1$ and $v_2$ for distance which are in ratio $1: 2$. Find the average speed.

A block of mass $m$ placed on a rough horizontal plane is pulled by a constant power $P$. The coefficient of friction between the block and the surface is $\mu$. The maximum velocity of the block will be

Consider a particle is moving with a minimum speed $v$ at highest point of vertical circle of radius $R$. If the radius of the circle doubled the corresponding minimum speed will be

A ball is projected with a velocity $5 \mathrm{~m} / \mathrm{s}$, so that its horizontal range is twice the greatest height attained. The value of range is

A block rests on a fixed wedge inclined at an angle $\theta$. The coefficient of friction between the block and plane is $\mu$. The maximum value of $\theta$ for the block to remain motionless on the

wedge is

A block of mass 4 kg at rest on a rough inclined plane making an angle of $\theta$ with the horizontal. The coefficient of static friction between the block and plane is 0.5 and the frictional force on the block is 14.14 N , find the value of $\theta$ ?

Two blocks of equal masses are connected with a massless spring of spring constant $2500 \mathrm{~N} / \mathrm{m}^2$ and length 10 cm at rest on the frictionless horizontal plane. If a constant horizontal force 10 N is applied as shown in the figure, find the maximum distance between the blocks.

The graph of potential energy $U(x)$ versus distance $x$ is shown in the following figure. The force $F$ versus distance $x$ graph will be represented by (assume, that the force is conservative)

A point moves in the $x y$-plane according to the following equation, $x=a \sin \omega t, y=a(1-\cos \omega t)$, where $a$ and $\omega$ are positive constants. Find the angle between the point's velocity and acceleration vectors.

Assume proton is rotating along a circular path of radius 1 m under a centrifugal force of $4 \times 10^{-12} \mathrm{~N}$. If the mass of proton is $1.6 \times 10^{-27} \mathrm{~kg}$, then its angular velocity of rotation is

A particle is exhibiting simple harmonic motion has its displacement $x$ and velocity $v$ related as $4 v^2=25-x^2$. The time period of SHM is

If the escape velocity on earth is $11.2 \mathrm{~km} / \mathrm{s}$, its value for a planet having double the radius and 8 time the mass of earth is

If the bulk modulus of water is $2 \times 10^9 \mathrm{Nm}^{-2}$, then the required pressure to reduce the given volume of water by $2 \%$ is

The surface tension of 70 dynes $/ \mathrm{cm}$ is equal to

Assertion A thermos bottle consists of a double walled glass vessel with the space between the two walls evacuated, so that the heat transfer between the contents of the bottle and outside is minimised.

Reason The vacuum between the two walls inhibits the heat transfer by radiation mechanism.

Which of the following is correct?

How much heat energy is supplied when 5 kg of water at $20^{\circ} \mathrm{C}$ is brought to its boiling point? (Assume, specific heat of water $=4.2 \mathrm{~J} / \mathrm{g}^{\circ} \mathrm{C}$ )

What is the name of ideal-gas process in which no heat is transferred?

Mean free path of molecules in a polyatomic gas is independent of

Three masses $700 \mathrm{~g}, 500 \mathrm{~g}$ and 400 g are suspended at the end of a spring shown in figure and are in equilibrium. When the 700 g mass is removed, the system oscillates with a time period of 3 s . If 500 g mass is further removed, then it will oscillate with a period of

A thin prism of angle $6^{\circ}$ made of glass of refractive index 1.5 is combined with another prism made of refractive index 1.75 to produce dispersion without deviation. Then, the angle of the second prism is

A small object is placed in the air, at a distance 45 cm from a convex refracting surface of radius of curvature 15 cm . If the surface separates air from glass of refractive index 1.5, then the position of image is

Which of the following phenomena produces the colour in soap bubble?

Choose the incorrect statement.

In a regular polygon of 10 sides, each corner is at a distance $R$ from the centre. Identical charges are placed at 9 corners. At the centre, the magnitude of electric field is $E$ and the potential is $V$. The ratio $\frac{V}{E}$ is

The resistance of a wire is $20 \Omega$. It is stretched, so that the length becomes three times, then the new resistance of the wire will be

In a meter bridge the resistances $R$ and $S$ are such that the null point is found at a distance of 40 cm from one end. If a resistance of $10 \Omega$ is connected in parallel with shunt $S$, then the null point occurs at 90 cm from the same end. The values of the two resistances $R$ and $S$ respectively are

Two similar coils which are separated by 2 m having radius of 1 m and number of turns 80 have a common axis. Calculate the strength of magnetic field at a point midway between them on their common axis when a current is 0.2 A .

A particle of mass $m$ and charge $Q$ moving with a speed $v$ in a circular path of radius $R$ has magnetic moment $\mu$. If the mass of the particle is doubled and maintains the same speed while revolving in the same circular path, then the magnetic moment will be

A short bar magnet placed with its axis at $30^{\circ}$ with an external field of 800 G experiences a torque of 0.016 Nm . The magnetic moment of the bar magnet is

Two concentric circular coils, one of small radius $r$ and the other of large radius $R$ are placed co-axially with centres coinciding. If the radius $r$ is changed by $2 \%$, then the change in mutual inductance of the arrangement is (assume, $r \ll R$ )

A $L-C-R$ series circuit is connected to a source of alternating current. At resonance, the applied voltage and the current flowing through the circuit will have a phase difference of

A radiation of energy $E$ falls on a perfectly reflecting surface. The momentum transferred to the surface is (let $c \equiv$ speed of light)

According to the photoelectric effect, the plot of kinetic energy of the emitted photo-electrons from a metal versus the frequency of the incident radiation gives a straight line whose slope

Half-life of radioactive sample is 24 h . If a newly prepared radioactive sample shows 4 times the allowed and safe value of radio activity, the minimum time after which one can work safely with the source is

The current gain of a transistor is 0.98 . If the transistor is used in a common emitter arrangement what would be the change in collector current corresponding to a change of 0.5 mA in the base current?

In a $p-n-p$ transistor, the current carriers are

The function of a detector is to demodulate the modulated carrier wave and the steps for this process are