The uncertainty in the position of electron $(\Delta x)$ is approximately 100 pm . The uncertainty in momentum (in $\mathrm{kg} \mathrm{ms}^{-1}$ ) of an electron $\left[h=6.626 \times 10^{-34} \mathrm{Js}\right]$

Which of the following statements are correct?

I. The energy of hydrogen atom in its ground state is -13.6 eV .

II. On the basis of Bohr's model, the radius of the 3rd orbit of hydrogen atom is 158.7 pm .

III. The order of radius of the first orbit of $\mathrm{H}, \mathrm{He}^{+}, \mathrm{Li}^{2+}$ and $\mathrm{Be}^{3+}$ is $\mathrm{H}>\mathrm{He}^{+}>\mathrm{Li}^{2+}>\mathrm{Be}^{3+}$.

Which of the following orders is not correct about the property shown against it?

Consider the following changes I and II

$$ \mathrm{O}_2^{-} \underset{\text { II }}{\longleftarrow} \mathrm{O}_2 \xrightarrow[\text { I }]{ } \mathrm{O}_2^{+} $$

The correct statements about these changes (I) and (II) in accordance with MO theory are

(A) In (I) bond order increases by 0.5 from the existing value

(B) In (II) bond order decreases by 1.0 from the existing value

(C) In both (I) and (II) magnetic property is not changed

(D) In both (I) and (II) magnetic property is changed

The increasing order of number of lone pair of electrons on the central atom of the following molecules is

(I) $\mathrm{ClF}_3$

(II) $\mathrm{XeF}_2$

(III) $\mathrm{SF}_4$

(IV) $\mathrm{SiH}_4$

$$ \text { Which of the following is correct for an ideal gas? } $$

At 256 K , rms speed of $\mathrm{SO}_2$ gas molecules is $3.16 \times 10^2 \mathrm{~ms}^{-1}$. What is the most probable velocity (in $\mathrm{ms}^{-1}$ ) of same gas at same temperature?

209 g of an element reacts with chlorine to form 315.5 g of its chloride. What is the weight (in g ) of oxygen that reacts with 418 g of same element ?

$$ (\mathrm{Cl}=35.5 \mathrm{u} ; \mathrm{O}=16 \mathrm{u}) $$

Consider the following :

Statement I : During isothermal expansion of an ideal gas its enthalpy decreases.

Statement II : When 2.0 L of an ideal gas expands isothermally into vaccum, $\Delta U=0$.

The correct answer is :

The energy required to increase the temperature of 180 g of liquid water from $10^{\circ} \mathrm{C}$ to $15^{\circ} \mathrm{C}$ is 3765 J . What is $C_p$ of water in $\mathrm{J} \mathrm{mol}^{-1} \mathrm{~K}^{-1} ?\left(\mathrm{H}_2 \mathrm{O}=18 \mathrm{u}\right)$

At $25^{\circ} \mathrm{C}$, the percentage of ionisation of $x \mathrm{M}$ acetic acid is 4.242 . What is the pH of the acetic acid solution?

$$ \begin{aligned} & (\log 4.242=0.6275) ;(\log 0.04242=-1.372) \\ & \left(K_{\mathrm{a}}=1.8 \times 10^{-5}\right) \end{aligned} $$

At 298 K , the value of $K_c$ for the following reaction is $x \mathrm{~mol} \mathrm{~L}^{-1}$.

What is the approximate $K_{\mathrm{P}}$ value for this reaction?

$$ \begin{array}{r} \left(R=0.082 \mathrm{~L} \mathrm{~atm} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}\right) \\ \mathrm{A}_2 \mathrm{O}_4(\mathrm{~g}) \rightleftharpoons 2 \mathrm{AO}_2(\mathrm{~g}) \end{array} $$

$\mathrm{H}_2 \mathrm{O}_2$ with $\mathrm{KMnO}_4$ in acidic medium gives a manganese compound ' $X$ ' and in basic medium gives another manganese compound ' $Y$ '. The oxidation state of manganese in $X$ and $Y$ respectively are

Which of the following orders are correct against the stated property?

I. $\mathrm{NaO}_2<\mathrm{KO}_2<\mathrm{RbO}_2<\mathrm{CsO}_2$ - stability

II. $\mathrm{Mg}(\mathrm{OH})_2<\mathrm{Ca}(\mathrm{OH})_2<\mathrm{Sr}(\mathrm{OH})_2$ - basic strength

III. $\mathrm{MgCO}_3<\mathrm{CaCO}_3<\mathrm{SrCO}_3$ - thermal stability

In the structure of diborane, the number of 2-centre-2-electron bonds is $X$ and 3-centre-2-electron bonds is $Y$. The value of $(X+Y)$ is

$$ \text { Match the following } $$

$$ \begin{array}{clll} \hline & \text { List-I (Compound) } & & \text { List-II (Use) } \\ \hline \text { (A) } & \text { Kieselghur } & \text { (I) } & \text { Chromatographic material } \\ \hline \text { (B) } & \text { Silica gel } & \text { (II) } & \text { Softening of hard Water } \\ \hline \text { (C) } & \text { ZSM-5 } & \text { (III) } & \text { Filtration plants } \\ \hline \text { (D) } & \text { Hydrated zeolites } & \text { (IV) } & \begin{array}{l} \text { To convert alcohol directly } \\ \text { into gasoline } \end{array} \\ \hline \end{array} $$

The correct answer is

Identify the air pollutant which in high concentration leads to stiffness of flower buds?

The number of primary $\left(1^{\circ}\right)$, secondary $\left(2^{\circ}\right)$ and tertiary $\left(3^{\circ}\right)$ alcohols possible for the formula $\mathrm{C}_5 \mathrm{H}_{12} \mathrm{O}$ respectively are

The catalyst used for the isomerisation of $n$-alkanes to branched chain alkanes is

An element crystallises in bcc lattice. The atomic radius of the element is $2.598 \mathop {\rm{A}}\limits^{\rm{o}}$. What is the volume (in $\mathrm{cm}^3$ ) of one unit cell?

A centi molar solution of acetic acid is $50 \%$ dissociated at $27^{\circ} \mathrm{C}$. The osmotic pressure of the solution (in atm ) is $\left(R=0.083 \mathrm{~L}\right.$ atm $\left.\mathrm{K}^{-1} \mathrm{~mol}^{-1}\right)$

At 300 K vapour pressure of a pure liquid. ' $A$ ' is 70 mm Hg . It forms an ideal solution with another liquid ' $B$ '. The mole fraction of $B$ in the solution is 0.2 and total vapour pressure of solution is 84 mm Hg at same temperature. What is the vapour pressure (in mm ) of pure liquid $B$ at 300 K ?

The specific conductance of 0.05 M NaOH solution is $0.0115 \mathrm{~S} \mathrm{~cm}^{-1}$ What is its molar conductance ( $\wedge_{\mathrm{m}}$ ) in $\mathrm{Scm}^2 \mathrm{~mol}^{-1}$ ?

Consider the reaction given below

$$ A+2 B \longrightarrow 3 C+2 D $$

If rate of disappearance of $B$ is $x \times 10^{-2} \mathrm{~mol} \mathrm{~L}^{-1} \mathrm{~s}^{-1}$, the ratio of rate of reaction and rate of appearance of $C$ is

Identify the catalytic reaction in which both reactants are in different phases.

Consider the following.

Statement-I : Gold sol is prepared by Bredig's arc method.

Statement-II : Bredig's arc method involves only dispersion but not condensation.

The correct answer is

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

$$ \begin{array}{cll} \hline & \text { Metal } & \text { Refining Process } \\ \hline \text { I. } & \mathrm{Hg} & \text { Distillation } \\ \hline \text { II. } & \mathrm{Cu} & \text { Poling } \\ \hline \text { III. } & \mathrm{B} & \text { Zone refining } \\ \hline \text { IV. } & \mathrm{Ti} & \text { Liquation } \\ \hline \end{array} $$

The oxides of nitrogen obtained by the reaction of nitric acid with (i) $\mathrm{P}_4 \mathrm{O}_{10'}$ (ii) $\mathrm{P}_4$ respectively are

$$ \text { Match the following } $$

Correct answer is

The ion with $4 f^7$ configuration is

Which of the following is the common monomer for the polymers bakelite and melamine?

Activation energy for the hydrolysis of sucrose by acid is $X \mathrm{~kJ} \mathrm{~mol}^{-1}$ whereas activation energy for the hydrolysis of sucrose by sucrase is $Y \mathrm{~kJ} \mathrm{~mol}^{-1} . X$ and $Y$ respectively are

The structure of the nitrogen containing heterocyclic base given below represents

What is the drug used to control depression and hypertension?

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

In the following sequence of reactions, what is the end product $(D)$ ?

$$ \mathrm{C}_2 \mathrm{H}_5 \mathrm{Br} \xrightarrow{\mathrm{KCN}} A \xrightarrow{\mathrm{H}_3 \mathrm{O}^{+}} B \xrightarrow{\mathrm{LiAlH}_4} C \xrightarrow[573 \mathrm{~K}]{\mathrm{Cu}} D $$

$$ \text { The most acidic carboxylic acid is } $$

A carbonyl compound $X\left(\mathrm{C}_8 \mathrm{H}_8 \mathrm{O}\right)$ gives yellow precipitate with NaOI .

Hemiacetal of $X$ with methanol/dry HCl is

Which of the following does not involve in Friedel-Craft reaction?

Consider the following

Statement I : $\mathrm{CH}_3 \mathrm{NH}_2$ is more basic than $\mathrm{NH}_3$ but $\mathrm{C}_6 \mathrm{H}_5 \mathrm{NH}_2$ is less basic than $\mathrm{NH}_3$.

Statement II : The order of basic strength of amines in aqueous phase follows the order $\left(\mathrm{C}_2 \mathrm{H}_5\right)_3 \mathrm{~N}>\left(\mathrm{C}_2 \mathrm{H}_5\right)_2 \mathrm{NH} >\mathrm{C}_2 \mathrm{H}_5 \mathrm{NH}_2$

The correct answer is

The set of real values of $x$ such that $f(x)=\sqrt{\frac{[x]-1}{\left.[x]^2-[x]-6\right]}}$ is a real valued function is

If a function $f: Z \rightarrow Z$ is defined by $f(x)=x-(-1)^x$, then $f(x)$ is

If $2 \cdot 5+5 \cdot 9+8 \cdot 13+11 \cdot 17+\ldots$ to $n$ terms $=a n^3+b n^2+c n+d$, then $a-b+c-d=$

If $A=\left[\begin{array}{ccc}1 & 2 & -2 \\ 2 & -1 & 2 \\ -1 & 1 & -2\end{array}\right]$, then $A+2 A^{-1}=$

If $A=\left[\begin{array}{ccc}a & b & c \\ d & e & f \\ l & m & n\end{array}\right]$ is a matrix such that $|A|>0$ and $\operatorname{adj}(A)=\left[\begin{array}{ccc}0 & 4 & -6 \\ 10 & 8 & 0 \\ 2 & 4 & -4\end{array}\right]$, then $\frac{c d}{f b}+\frac{\ln }{e m}=$

In solving a system of linear equations $A X=B$ by Cramer's rule, in the usual notation, if $\Delta_1=\left|\begin{array}{ccc}-11 & 1 & -7 \\ -4 & 1 & -2 \\ 5 & 1 & 1\end{array}\right|$ and $\Delta_3=\left|\begin{array}{ccc}4 & 1 & -11 \\ 1 & 1 & -4 \\ 4 & 1 & 5\end{array}\right|$, then $X=$

If $a=\operatorname{Im}\left(\frac{1+z^2}{2 i z}\right)$ and $z$ is any non-zero complex number such that $|z|=1$, then $a=$

If $(3+4 i)^{2025}=5^{2023}(x+i y)$, then $\sqrt{x^2+y^2}=$

If $\left(\frac{\cos \theta+i \sin \theta}{\sin \theta+i \cos \theta}\right)^{2024}+\left(\frac{1+\cos \theta+i \sin \theta}{1-\cos \theta+i \sin \theta}\right)^{2025}=x+i y$ then the value of $x+y$ at $\theta=\frac{\pi}{2}$ is

The roots $\alpha, \beta$ of the equation $x^2-6(k-1) x+4(k-2)=0$ are equal in magnitude but opposite in sign, if $\alpha>\beta$, then the product of the roots of the equation $2 x^2-\alpha x+6 \beta(\alpha+1)=0$

If $a x^2+b x+c<0 \forall x \in R$ and the expressions $c x^2+a x+b$ and $a x^2+b x+c$ have their extreme values at the same point $x$, then for the expression $c x^2+a x+b$

If $a \pm i b$ and $b \pm a i$ are the roots of $x^4-10 x^3+50 x^2-130 x+169=0$, then $\frac{a}{b}+\frac{b}{a}=$

If $x^2-5 x+6$ is a factor of $f(x)=x^4-17 x^3+k x^2-247 x+210$, then the other quadratic factor of $f(x)$ is

If all the letters of the word COMBINATION are arranged in all possible ways to form 11 letter words (with or without meaning), then the number of words among them in which $C$ and $N$ occupy the end positions and no vowel appears exactly in the middle position is

The number of ways of distributing 3 dozen fruits (no two fruits are identical) to 9 persons such that each gets the same number of fruits is

If $\binom{p}{q}={ }^p C_q$ and $\sum\limits_{i=0}^m\binom{10}{i}\binom{20}{m-i}$ is maximum, then $m=$

Coefficient of $x^2$ in the expansion of $\left(x^2+x-2\right)^5$ is

If $P_n$ denotes the product of the binomial coefficients in the expansion of $(1+x)^n$, then $\frac{P_{n+1}}{P_n}=$

The coefficient of $x^3$ in the expansion of $\frac{x^4+1}{\left(x^2+1\right)(x-1)}$ when it is expressed in terms of positive integral powers of $x$, is

$$ \begin{aligned} \frac{1}{\sin 1^{\circ} \sin 2^{\circ}}+\frac{1}{\sin 2^{\circ} \sin 3^{\circ}}+\frac{1}{\sin 3^{\circ} \sin 4^{\circ}} & +\frac{1}{\sin 89^{\circ} \sin 90^{\circ}}= \end{aligned} $$

$$ \cos ^3 \frac{\pi}{8} \cos \frac{3 \pi}{8}+\sin ^3 \frac{\pi}{8} \sin \frac{3 \pi}{8}= $$

If $A+B+C=\frac{\pi}{4}$, then $\sin 4 A+\sin 4 B+\sin 4 C=$

Number of solutions of the equation $\cos \theta+\cos 2 \theta-\sqrt{3}(\sin \theta+\sin 2 \theta)+1=0$ lying in the interval $(0,2 \pi)$ is

If $x$ is a real number, then the number of solutions of $\tan ^{-1}(\sqrt{x(x+1)})+\sin ^{-1}\left(\sqrt{x^2+x+1}\right)=\frac{\pi}{2}$ is

Domain of the real valued function $f(x)=\log \left(x^2-1\right)+x \operatorname{coth}^{-1} x$ is

In a $\triangle A B C$, if $\sin \frac{A}{2}=\frac{1}{4} \sqrt{\frac{3}{5}}, a=2, c=5$ and $b$ is an integer, then the area (in sq. units) of $\triangle A B C$ is

In a $\triangle A B C$ if $a+c=5 b$, then $\cot \frac{A}{2} \cot \frac{C}{2}=$

In a $\triangle A B C$, if $r_1=3, r_2=4, r_3=6$, then $b=$

Let the position vectors of the vertices of a $\triangle A B C$ be $\mathbf{a , b}, \mathbf{c}$. If on the plane of the triangle, $P$ is a point having position vector $\mathbf{x}$ such that $\mathbf{x} \cdot(\mathbf{c}-\mathbf{b})=\mathbf{a} \cdot \mathbf{c}-\mathbf{a} \cdot \mathbf{b}$ and $\mathbf{x} \cdot(\mathbf{a}-\mathbf{c})=\mathbf{a b}-\mathbf{b} \mathbf{c}$, then for the $\triangle A B C, P$ is the

The point of intersection of the lines represented by $\mathbf{r}=(\hat{\mathbf{i}}-6 \hat{\mathbf{j}}+2 \hat{\mathbf{k}})+\mathbf{t}(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}})$ and $\mathbf{r}=(4 \hat{\mathbf{j}}+\hat{\mathbf{k}})+\mathbf{s}(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}})$ is

$\mathbf{a}, \mathbf{b}, \mathbf{c}$ are three vectors such that $|\mathbf{a}|=2,|\mathbf{b}|=3$, $|\mathbf{c}|=5,|\mathbf{a}+\mathbf{b}+\mathbf{c}|=\sqrt{69}$. If $(\mathbf{a} \cdot \mathbf{b})=(\mathbf{b} \cdot \mathbf{c})=\frac{\pi}{3}$, then $(\mathbf{c}, \mathbf{a})=$

If the points $A, B, C, D$ with positions vectors $\hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}, \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}, \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{\mathbf{k}}, 2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$ respectively form a tetrahedron, then the angle between the faces $A B C$ and $A B D$ of the tetrahedron is

$\mathbf{a}, \mathbf{b}, \mathbf{c}$ are unit vectors. If $\mathbf{a}, \mathbf{b}$ are perpendicular vectors, $(\mathbf{a}-\mathbf{c}) \cdot(\mathbf{b}+\mathbf{c})=0$ and $\mathbf{c}=l \mathbf{a}+m \mathbf{b}+n(\mathbf{a} \times \mathbf{b}) ;$ ( $l, m, n$ are scalars), then $n^2=$

If the variance of the first $n$ natural numbers is 10 and the variance of the first $m$ even natural numbers is 16 , then $n: m=$

Given $f(x)=x^2-5 x+4$. Out of first 20 natural numbers, if a number $x$ is chosen at random, then the probability that the chosen $x$ satisfies the inequality $f(x)>10$ is

A problem in Algebra is given to two students $A$ and $B$ whose chances of solving it are $\frac{2}{5}$ and $\frac{3}{4}$ respectively.

The probability that the problem is solved if both of them try independently is

Three dice are thrown simultaneously and the sum of the numbers appeared on them is noted. If $A$ is the event of getting a sum greater than 14 and $B$ is the event of getting a sum which is a multiple of 3 , then $P(A \cap \bar{B})+P(\bar{A} \cap B)=$

A manufacturing company of bulbs has 3 units $A, B$ and $C$ which produce $25 \%, 35 \%$ and $40 \%$ of the bulbs respectively. Out of the bulbs produced by $A, B, C$ units, $5 \%, 4 \%$ and $2 \%$ are defective, respectively. If a bulb is chosen at random and found to be defective, then the probability that it is produced by unit $B$ is

The probability distribution of a random variable $X$ is given below

$$ \begin{array}{ccccccc} \hline X & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline P\left(X=x_i\right) & \alpha & \alpha & \alpha & \beta & \beta & 0.3 \\ \hline \end{array} $$

If $\mu$ and $\sigma^2$ represent the mean and variance of $X$ and $\mu=4.2$, then $\sigma^2+\mu^2=$

The probability that a student gets distinction in a Mathematics test is $\frac{2}{3}$. If five such tests are conducted over a certain period of time, then the probability that he gets distinction in atleast 3 tests is

If $P$ is a variable point which is at a distance of 2 units. from the line $2 x-3 y+1=0$ and $\sqrt{13}$ units from the point $(5,6)$, then the equation of the locus of $P$ is

If the equation $3 x^2+4 y^2-x y+k=0$ is the transformed equation of $3 x^2+4 y^2-x y-5 x-7 y+2=0$ after shifting the origin to the point $(\alpha, \beta)$ by the translation of axes, then $\alpha+\beta-k=$

If the intercept of a straight line $L$ made between the straight lines $5 x-y-4=0$ and $3 x+4 y-4=0$ is bisected at the point $(1,5)$, then the equation of $L$ is

$A$ line $L$ passes through the point $P(1,2)$ and makes an angle of $60^{\circ}$ with $O X$ in the positive direction. $A$ and $B$ are two points lying on $L$ at a distance of 4 units from $P$. If $O$ is the origin, then the area of $\triangle O A B$ is

The equation $(2 p-3) x^2+2 p x y-y^2=0$ represents a pair of distinct lines

The equation of a chord $A B$ of an ellipse $2 x^2+y^2=1$ is $x-y+1=0$. If $O$ is the origin, then $\sqrt{A O B}=$

If a circle $S$ passes through the origin and makes an intercept of length 4 units on the line $x=2$, then the equation of the curve on which the centre of $S$ lies is

A circle touches the line $2 x+y-10=0$ at $(3,4)$ and passes through the point $(1,-2)$. Then, a point that lies on the circle is

If $(a, b)$ is the common point for the circles $x^2+y^2-4 x+4 y-1=0$ and $x^2+y^2+2 x-4 y+1=0$, then $a^2+b^2=$

The angle between the tangents drawn from the point $(2,2)$ to the circle $x^2+y^2+4 x+4 y+c=0$ is $\cos ^{-1}\left(\frac{7}{16}\right)$. If two such circles exist, then sum of the values of $c$ is

If the circle $S=x^2+y^2+2 g x+4 y+1=0$ bisects the circumference of the circle $x^2+y^2-2 x-3=0$, then the radius of circle $S=0$ is

The angle between the tangents drawn from the point $(1,4)$ to the parabola $y^2=4 x$ is

The square of the slope of a common tangent drawn to the circle $4 x^2+4 y^2=25$ and the ellipse $4 x^2+9 y^2=36$ is

The tangents drawn to the hyperbola $5 x^2-9 y^2=90$ through a variable point $P$ make the angles $\alpha$ and $\beta$ with its transverse axis. If $\alpha, \beta$ are the complementary angles then the locus of $P$ is

If $\theta$ is the acute angle between the asymptotes of a hyperbola $7 x^2-9 y^2=63$, then $\cos \theta=$

If $O(0,0,0), A(1,2,1), B(2,1,3)$ and $C(-1,1,2)$ are the vertices of a tetrahedron, then the acute angle between its face $O A B$ and edge $B C$ is

If the angles between the sides of the $\triangle A B C$ formed by $A(2,3,5), B(-1,3,2)$ and $C(3,5,-2)$ are $\alpha, \beta$ and $\gamma$, then $\sin ^2 \alpha+\sin ^2 \beta+\sin ^2 \gamma=$

If the four points $(6,2,4),(1,3,5),(1,-2,3)$ and $(6, k, 2)$ are coplanar, then $k=$

$$ \mathop {\lim }\limits_{x \to - \infty } \frac{5 x^3-x^2 \sin 5 x}{x \cos 4 x+7|x|^3-4|x|+3}= $$

If $\mathop {\lim }\limits_{x \to {a^ + }} f(x)=p, \mathop {\lim }\limits_{x \to {a^ - }} f(x)=m$ and $f(a)=k$, then which one of the following is true?

If a function $f$ defined by

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

is continuous at $x=0$, then $a=$

If $y=\tanh ^{-1} \sqrt{\frac{1-x}{1+x}}$, then $\frac{d y}{d x}=$

If $x^2+y^2=t-\frac{1}{t}$ and $x^4+y^4=t^2+\frac{1}{t^2}$, then $\frac{d y}{d x}=$

If $y=(a x+b) \cos x$, then

$$ y_2+y_1 \sin 2 x+y\left(1+\sin ^2 x\right)= $$

If the normal drawn at the point $P$ on the curve $y=x \log x$ is parallel to the line $2 x-2 y=3$, then $P=$

If the curves $y^2=16 x$ and $9 x^2+\alpha y^2=25$ intersect at right angles, then $\alpha=$

If the function $y=\sin x(1+\cos x)$ is defined in the interval $[-\pi, \pi]$, then $y$ is strictly increasing in the interval

If the velocity of a particle moving on a straight line is proportional to the cube root of its displacement, then its acceleration is

If $\int e^{\sin x}(1+\sec x \tan x) d x=e^{\sin x} f(x)+c$, then in $0 \leq x \leq 2 \pi$, then number of solutions of $f(x)=1$ is

If $\int \frac{d x}{(x-1)^{\frac{3}{2}}(x-3)^{\frac{1}{2}}}=\sqrt{f(x)}+C$, then $f(-1)-f(0)=$

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

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

If $\int x^2 \cos ^2 x d x=\frac{1}{6} f(x)+g(x) \sin 2 x +h(x) \cos 2 x+c$, then $f(1)+g(2)+h\left(\frac{1}{2}\right)=$

$$ \int_0^{\frac{\pi}{2}} \log |\tan x+\cot x| d x= $$

$$ \int_0^\pi x \cdot \sin ^5 x \cdot \cos ^6 x d x= $$

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

The area of the region (in sq. units) enclosed between the curves $y=|x|, y=[x]$ and the ordinates $x=-1$, $x=0, x=1$ is

The general solution of the differential equation $\frac{d y}{d x}+x y=4 x-2 y+8$ is

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

The general solution of the differential equation $\frac{d y}{d x}+\frac{x+y+1}{x-3 y+5}=0$ is

If the maximum and minimum temperatures at a place on a day are measured as $44^{\circ} \mathrm{C} \pm 0.5^{\circ} \mathrm{C}$ and $22^{\circ} \mathrm{C} \pm 0.5^{\circ} \mathrm{C}$ respectively, then the temperature difference is

If a ball projected vertically upwards with certain initial velocity from the ground crosses a point at a height of 25 m twice in a time interval of 4 s , then the initial velocity of the ball is

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

If a particle of mass ' $m$ ' covers half of the horizontal circle with constant speed ' $v$ ', then the change in its kinetic energy is

A car is moving with a velocity of $4 \mathrm{~ms}^{-1}$ towards east. After a time of 4 s , if it is heading north-east with a velocity of $4 \sqrt{2} \mathrm{~ms}^{-1}$, then the average velocity of the car is

A body of mass 5 kg starts from the origin with an initial velocity $(30 \hat{\mathbf{i}}+40 \hat{\mathbf{j}}) \mathrm{ms}^{-1}$. If a constant force $-(\hat{\mathbf{i}}+5 \hat{\mathbf{j}}) \mathrm{N}$ acts on the body, then the time in which the $y$-component of its velocity becomes zero is

A block of mass 10 kg moving with a speed of $5 \hat{\mathrm{i}} \mathrm{ms}^{-1}$ on a frictionless horizontal surface suddenly explodes into two pieces. If one piece with mass 4 kg moves with a speed of $10 \hat{\mathbf{i}} \mathrm{~ms}^{-1}$, then the velocity of the second piece is

The bob of a simple pendulum of length 200 cm is released from horizontal position. If $10 \%$ of its initial energy is lost due to air resistance, then the speed of bob at the mean position is

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

A steel sphere of radius 1.2 cm collides a second steel sphere at rest. If the collision is elastic and after the collision the first sphere continues to move in its initial direction with a velocity of $\frac{7}{9}$ times its initial velocity, then the radius of the second sphere is

Ratio of angular velocity of hour hand of a watch and the angular velocity of rotation of Earth is

If two bodies of masses 2 kg and 3 kg are moving at right angles with velocities $20 \mathrm{~ms}^{-1}$ and $10 \mathrm{~ms}^{-1}$ respectively, then the velocity of the centre of mass of the system of the two bodies is

The kinetic energy of a particle executing simple harmonic motion at a displacement of 3 cm from the mean position is 4 mJ . If the amplitude of the particle is 5 cm , then the maximum force acting on the particle is

A body of mass 1 kg is attached to the lower end of a vertically suspended spring of force constant $600 \mathrm{~N}-\mathrm{m}^{-1}$. If another body of mass 0.5 kg moving vertically upward hits the suspended body with a velocity $3 \mathrm{~ms}^{-1}$ and embedded in it, then the frequency of the oscillation is

If the angular velocity of a planet about its axis is halved, the distance of the stationary satellite of this planet from the centre of the planet becomes $2^n$ times the initial distance. Then, the value of ' $n$ ' is

When a wire of length ' $L$ ' clamped at one end is pulled by a force ' $F$ ' from the other end, its length increases by ' $L$ '. If the radius of the wire and the applied force were halved, then the increase in its length is

A liquid drop of diameter $D$ splits into 3375 small identical drops. If $S$ is the surface tension of the liquid, then the change in the surface energy in the process is

When a sphere is taken to the bottom of a sea of depth 1 km , it contracts in volume by $0.01 \%$, then the Bulk modulus of the material of the sphere is

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

If a gas of volume 400 cc at an initial pressure $p$ is suddenly compressed to 100 cc , then its final pressure is

(The ratio of the specific heat capacities of the gas at constant pressure and constant volume is 1.5 )

A Carnot engine having efficiency $60 \%$ receives heat from a source at a temperature 600 K . For the same sink temperature, to increase its efficiency to $80 \%$, then the temperature of the source is

A gaseous mixture consists of 2 moles of oxygen and 4 moles of argon at an absolute temperature $T$. Neglecting all vibrational modes, the total internal energy of the mixture of the gases is

The average translational kinetic energy of the oxygen molecules at a temperature of $127^{\circ} \mathrm{C}$ is

(Boltzmann constant $=1.38 \times 10^{-23} \mathrm{JK}^{-1}$ )

The speed of a stationary wave represented by the equation

$$ y=0.7 \sin \left(\frac{7 \pi}{4} x\right) \cos (350 \pi t) \text { is } $$

(In the given equation $x$ and $y$ are in metre and $t$ is in second)

Two thin convex lenses are kept in contact coaxially. If the focal length of the combination of the lenses is 4 cm and sum of the focal lengths of the two lenses is 18 cm , then the focal length of the lens of low power is

For an observer on the Earth, if a spectral line of wavelength $6600\mathop {\rm{A}}\limits^{\rm{o}}$ emitted by a star is found to be red shifted by $22 \mathop {\rm{A}}\limits^{\rm{o}}$, then the star is

Three particles of each charge $q$ are placed at the vertices of an equilateral triangle of side $L$. The work to be done to decrease the side of the triangle to $\frac{L}{2}$ is

The radii of the inner and outer spheres of a spherical capacitor are 8 cm and 9 cm respectively. The outer sphere is earthed and the inner sphere is charged. If the space between the concentric spheres is filled with a liquid of dielectric constant 5 , the capacitance of the capacitor is

If 27 charged water droplets, each of radius $10^{-3} \mathrm{~m}$ and charge $10^{-12} \mathrm{C}$ coalesce to form a single big spherical drop, then the potential of the big drop is

A straight wire of resistance $18 \Omega$ is bent in the form of an equilateral triangular loop. The effective resistance between any two vertices of the triangle is

The power dissipated by a uniform wire of resistance $100 \Omega$ when a potential difference of 120 V is applied across its ends is

If a straight current carrying wire of linear density $0.12 \mathrm{~kg} \mathrm{~m}^{-1}$ is suspended in mid air by a uniform horizontal magnetic field of 0.5 T normal to the length of the wire, then the current through the wire is (Acceleration due to gravity $=10 \mathrm{~ms}^{-2}$, Neglect Earth's magnetic field)

Two concentric loops $A$ and $B$ of same radius $2 \pi \mathrm{~cm}$ are placed at right angles to each other. If the currents flowing through $A$ and $B$ are 3 A and 4 A respectively, then the net magnetic field at their common centre is

A short bar magnet is placed in a uniform magnetic field of 2 T such that the axis of the magnet makes an angle of $45^{\circ}$ with the direction of the magnetic field. If the torque acting on the magnet is $0.36 \sqrt{2} \mathrm{~N}-\mathrm{m}$, then the moment of the magnet is

A horizontal telegraph wire of length 30 m spread east to west fell down freely from a height of 20 m . If the resistance of the wire is $40 \Omega$ and the horizontal component of the Earth's magnetic field at the place is $2 \times 10^{-5} \mathrm{~T}$, then the induced current when the wire reaches the ground is

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

In an LCR series circuit, if the potential differences across inductor, capacitor and resistor are $60 \mathrm{~V}, 30 \mathrm{~V}$ and 40 V respectively, then the AC voltage applied to the circuit is

A plane electromagnetic wave of frequency 25 MHz propagates in vacuum along positive $x$-direction. At a particular point in space and time, if the electric field is $63 \hat{\mathrm{j}} \mathrm{Vm}^{-1}$, then the magnitude of the magnetic field of the wave at this point at the same time is

A particle of mass $8 \mu \mathrm{~g}$ in motion collides with another stationary particle of mass $4 \mu \mathrm{~g}$. If the collision is perfectly elastic and one dimensional, the ratio of their de-Broglie wavelengths after collision is

The difference between the frequencies of the first and second Lyman lines of hydrogen atom is ( $R=$ Rydberg constant and $c=$ speed of light in vacuum)

If the half-life of a radioactive element is 12.5 hours, then the time taken to disintegrate 256 g of the substance into 1 g is (in hours)

A transistor works as an amplifier when

If five logic gates are connected as shown in the figure, then the values of $y_1, y_2$ and $y_3$, are respectively

In amplitude modulation of waves, the maximum amplitude is 30 mV and minimum amplitude is 5 mV , then the modulation index is