Chemistry
The maximum number of orbitals present in $n=4$ energy level of an atom and the maximum number of electrons with spin value $+\frac{1}{2}$ in the same orbitals are respectively
The approximate ratio of the speed of light in vacuum to that of an electron in the first Bohr orbit of hydrogen atom is
Which of the following species are isoelectronic species?
(A) $\mathrm{O}^{2-}$
(B) $\mathrm{F}^{-}$
(C) $\mathrm{Na}^{+}$
(D) $\mathrm{Mg}^{2+}$
Arrange the following in increasing order of ionic radii
$$ \mathrm{O}^{2-}, \mathrm{Na}^{+}, \mathrm{F}^{-}, \mathrm{Mg}^{2+} $$
The set of molecules among the following with zero dipole moment is $\mathrm{CCl}_4, \mathrm{BF}_3, \mathrm{CHCl}_3, \mathrm{CS}_2, \mathrm{NH}_3$,
1, 4-dichlorobenzene, $\mathrm{CO}_2$
The correct pair of species which are not isostructural is
The rate of diffusion of methane at 1.0 atm pressure is twice than that of another gas ' $X$ ' kept at 1.45 atm . The molecular mass of the gas ' $X$ ' is
Which of the following gases has the maximum van der Waals' constant ' $a$ '?
Electrolysis of aqueous $\mathrm{Na}_2 \mathrm{SO}_4$ was carried out by passing a current of 3 ampere for 10 min . The volume of the gas (in litre) at STP at the anode of the cell is approximately
Magnetite can be reduced with CO to yield iron metal and carbon dioxide. Calculate the mass of magnetite (in kg ) needed to obtain 4 kg of iron if the process is $80 \%$ efficient. [Atomic weight of Fe and O are 56 g and 16 g , respectively]
In the reaction, $\mathrm{H}_2 \mathrm{O}(l) E \longrightarrow \mathrm{H}_2 \mathrm{O}(s)$ at $0^{\circ} \mathrm{C}$ and 1 atom, the internal energy change is $-41 \mathrm{~kJ} / \mathrm{mol}$. What will be the value of molar enthalpy change?
For the formation of ammonia from its constituent elements ( 1 mole of $\mathrm{N}_2$ and 3 moles of $\mathrm{H}_2$ ) in a closed vessel of volume $V(\mathrm{~L})$, the value of $K_C$ is [units of $K_C=\mathrm{mol}^{-2} \mathrm{~L}^2$ ]
How many of the following are diprotic acids? Citric acid, chromic acid, oxalic acid, pyrosulphuric acid, sulphurous acid
Deionised water is obtained by passing hard water through
Unknown inorganic compound ' $A$ ' is used for water softening. ' $A$ ' reacts with $\mathrm{Na}_2 \mathrm{CO}_3$ to generate an alkaline compound ' $B$ ' whose $\mathrm{pH}=14$. ' $A$ ' on reaction with $\mathrm{CO}_2$ gives cloudy ppt. ' $B$ ' +CaO reacts with unknown organic compound ' $C$ ' to give $\mathrm{C}_6 \mathrm{H}_6$. $A$, $B$ and $C$, respectively, are
Assertion $(\mathrm{A})\left[\mathrm{B}\left(\mathrm{OH}_2\right)_6\right]^{3+}$ and $\left[\mathrm{B}(\mathrm{OH})_4\right]^{-}$form octahedral and tetrahedral structures.
Reason (R) Being electron deficient, boron readily reacts with Lewis bases like $\mathrm{H}_2 \mathrm{O}, \mathrm{OH}^{-}$ The correct option among the following is
The correct order of " $\Delta H_f^{\circ}$ " values of diamond (I), graphite (II) and fullerene (III) is
$$ \text { Which of the following lacks hyperconjugative stability? } $$
The organic compound 5-allylcyclohex-3-ene-1-ol is reacted with cold, dilute, aqueous solution of $\mathrm{KMnO}_4$. The total number of hydroxyl group(s) $(-\mathrm{OH})$ present in the product is
The molecular formula of the product formed when benzene is reacted with excess of chlorine molecules under ultra-violet light is
KBr has rock salt type structural arrangements and has a density of
$3.70 \mathrm{~g} / \mathrm{cm}^3$. The edge length of the unit cell is approximately [molecular weight of $\mathrm{KBr}=120 \mathrm{~g} / \mathrm{mol}$ ]
A liquid mixture is an ideal solution, if
(A) it obeys ideal gas equation
(B) it obeys Raoult's law at all concentrations
(C) solute - solute, solute - solvent and solvent solvent interactions are similar
The freezing point of equimolal aqueous solution will be highest for
The variation of $\lambda_{\mathrm{m}}$ of acetic acid with concentration is correctly represented as
- Rate constants in the following reaction are Reaction 1 :
$$ A \xrightarrow{\text { Catalyst } 1} P_1, k_1=1 \mathrm{~s}^{-1} $$
Reaction 2 :
$$ A \xrightarrow{\text { Catalyst } 2} P_2, k_2=0.1 \mathrm{~L} \mathrm{~mol}^{-1} \mathrm{~s}^{-1} $$
Reaction 3 :
$$ A \xrightarrow{\text { Catalyst } 3} P_3, k_3=0.01 \mathrm{~L} \mathrm{~mol}^{-1} \mathrm{~s}^{-1} $$
The correct relations between the rate of the reactions at 1 M of $A$ are
10 g of a gas is adsorbed on 500 g of solid at 10 bar. If the pressure is increased at 20 bar, 14 g of the gas is adsorbed by the same solid at the same temperature. What is the slope of Freundlich adsorption isotherm?
The minimum temperature required for a non-catalytic reaction between $\mathrm{N}_2$ and $\mathrm{O}_2$ is
Assertion (A) Both rhombic and monoclinic sulphur have $\mathrm{S}_8$ molecules.
Reason (R) They have planar structure.
The correct option among the following is
Assertion (A) Hydrogen fluoride has higher boiling point than other hydrogen halides.
Reason (R) Hydrogen fluoride exhibits strong hydrogen bonding.
The correct option among the following is
The chemical structures of $\mathrm{XeO}_3$ and $\mathrm{XeOF}_4$, respectively, are
The elements with full $d^{10}$ electronic configuration in their " +2 " oxidation state are
The pair in which both the species have same magnetic moment (spin only) is
$$ \text { Consider the following structures } $$
Which of the pairs represent $D$ and $L$-fructose, respectively?
$$ \text { The major product in the following reaction is } $$
The intramolecular hydrogen bonding is present in
Assertion (A) Tertiary alcohols produce turbidity immediately with Lucas reagent.
Reason ( $\mathbf{R}$ ) Lucas reagent is a $1: 1$ mixture of conc. $\mathrm{HNO}_3$ and anhydrous $\mathrm{ZnCl}_2$.
The correct option among the following is
$$ \text { The major product of the following reaction is } $$
Assertion (A) Ammonia and its derivatives form $\mathrm{H}_2 \mathrm{~N}-\mathrm{Z}$ undergo condensation reaction with carbonyl compounds (aldehydes and ketones).
Reason (R) This reaction is an example of irreversible reaction.
The correct option among the following is
The correct order of $\mathrm{p} K_a$ of the following is
$$ \underset{\mathrm{I}}{\mathrm{CCl}_3 \mathrm{COOH}} \quad \underset{\mathrm{II}}{\mathrm{CF}_3 \mathrm{COOH}} $$
$$ \underset{\text { III }}{\mathrm{NO}_2 \mathrm{CH}_2 \mathrm{COOH}} \underset{\text { IV }}{\mathrm{NCCH}_2 \mathrm{COOH}} $$
$$ \text { Match the following. } $$
| List-I | List-II |
| (A) Amide | (I) Carbylamine reaction |
| (B) Nitrile | (II) Hinsberg's reagent |
| (C) |
(III) Hoffmann's bromamide |
| (D) |
(IV) |
$$ \text { The correct match is } $$
Mathematics
The domain of the real valued function $f(x)=\sqrt{\frac{2 x^2-7 x+5}{3 x^2-5 x-2}}$ is
The range of the real valued function $f(x)=|x-2|+|x-3|$ is
$A=\left[\begin{array}{lll}1 & 2 & 3 \\ 4 & 3 & 2\end{array}\right]$, then $\left(A+A^T\right)\left(A-A^T\right)=$
If $f(x)=\left|\begin{array}{ccc}x & x+1 & x+3 \\ x+2 & x+4 & x+7 \\ x+6 & x+9 & x+13\end{array}\right|$, then $f(5)=$
Let $A=\left[\begin{array}{lll}2 & 1 & 1 \\ 0 & 1 & 0 \\ 1 & 1 & 2\end{array}\right]$. If $A^{-1}=\alpha A^2+\beta A+\gamma I$, where $\alpha, \beta$ and $\gamma$ are real numbers and $I$ is a $3 \times 3$ identity matrix, then $17 \alpha+5 \beta+\gamma=$
For a system of simultaneous linear equations, if $A X=\left[\begin{array}{l}1 \\ 1 \\ 2\end{array}\right], \operatorname{Adj} A=\left[\begin{array}{ccc}1 & -1 & -1 \\ 1 & 1 & -1 \\ 1 & 1 & 1\end{array}\right]$ and $\operatorname{det} A>0$, then $X=$
$\{x \in[0,2 \pi] / \sin x+i \cos 2 x$ and $\cos x-i \sin 2 x$ are conjugate to each other} $=$
If $|x+i y|=\sqrt{x^2+y^2}$, then $\left|(1-\sqrt{3} i)^9+(\sqrt{3}+i)^9\right|=$
If $1, \omega, \omega^2$ are the cube roots of unity and $1, \alpha, \alpha^2, \alpha^3$ are the fourth roots of unity in usual notation, then $\alpha+\alpha \omega-\alpha^3 \omega^2=$
If $\alpha$ and $\beta$ are the roots of a quadratic equation $x^2+b x+c=0$ such that $\alpha^2+\beta^2=5$ and $\alpha^3+\beta^3=9$, then $b+c=$
The set of all real values of the expression $\frac{x^2-x+2}{x^2+x-2} \forall x \in R-\{-2,1\}$ is
If $\alpha, \beta$ and $\gamma$ are the roots of the equation $x^3-9 x^2+23 x-15=0$, then $\alpha^3+\beta^3+\gamma^3=$
If $\alpha, \beta$ and $2 \beta$ are the real roots of the equation $x^3-9 x^2+k=0$ and $k \in R-\{0\}$, then $14 \beta=$
The sum of all distinct roots of the equation $x^5-3 x^4+5 x^3-5 x^2+3 x-1=0$ is
$\left(x^4+1\right)=\frac{1}{a}(x+1)^4$ is a reciprocal equation
Let $a, b, c \in N$ and $a+b+c=5$. Let $L, M$ be the least and greatest values of $2^a 3^b 5^c$, respectively. Then $M-L=$
The number of positive divisors of 360 which are multiples of 3 is
$\frac{1}{8}-\frac{7}{8 \cdot 12}+\frac{7 \cdot 10}{8 \cdot 12 \cdot 16}-\ldots=$
If $\frac{2 x^2-3 x+5}{(x-7)^3}=\frac{A}{x-7}+\frac{B}{(x-7)^2}+\frac{C}{(x-7)^3}$, then $2 A-3 B+C=$
If $\frac{3 x^2+a x+3}{(2 x+3)\left(x^2+2\right)}=\frac{3}{2 x+3}+\frac{B x+C}{x^2+2}$, then $a(B+C)=$
If $\sin (A+B) \sin (A-B)+\cos (A+B) \cos (A-B) =\frac{1}{2}$ and $0
$$ \frac{1}{\sin 250^{\circ}}+\frac{\sqrt{3}}{\cos 290^{\circ}}= $$
If $A+B+C=\frac{\pi}{2}$, then $\sqrt{2} \cos \left(\frac{\pi}{4}-A\right)$
$$ +\sqrt{2} \cos \left(\frac{\pi}{4}-B\right)+\sqrt{2} \cos \left(\frac{\pi}{4}-C\right)+1= $$
If $\sinh x=\tan A$, then $|\tanh x|=$
$$ \frac{\sinh (x+y)+\sinh (x-y)}{\cosh (x+y)-\cosh (x-y)}= $$
If $a, b$ and $c$ are the sides of $a \triangle A B C$ and $\left|\begin{array}{lll}b & 1 & a \\ a & 1 & c \\ c & 1 & b\end{array}\right|=0$, then $2(\cos A+\cos B+\cos C)=$
In $\triangle A B C$, if $A=\frac{\pi}{3}$ and $B=\frac{\pi}{4}$, then $\frac{a^2-b^2}{c^2}=$
In a $\triangle A B C$, if $a=3, b=7$ and $c=8$, then $\sin \frac{B}{2} \tan \frac{C-A}{2}=$
Let $A B C$ be a triangle and $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ be the position vectors of $A, B$ and $C$ respectively. If $D$ divides $B C$ in the ratio $2: 3$ internally and $E$ divides $C A$ in the ratio $2: 1$ internally, then the position vector of the point $P$ which divides $D E$ in the ratio $3: 5$ internally is
If $2 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}-5 \hat{\mathbf{k}}, \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ are the position vectors of the vertices $A, B, C$ of a triangle respectively, then a unit vector along the median drawn through the vertex $A$ is
Let $L$ be a line passing through the points $2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+8 \hat{\mathbf{k}}$ and $\hat{\mathbf{i}}+6 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}$. Let $P$ be a plane passing through $-5 \hat{\mathbf{i}}+19 \hat{\mathbf{j}}-14 \hat{\mathbf{k}}$ and parallel to the vectors $\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$. If $L$ meets the plane $P$ at a point $A$, then the position vector of $A$, is
Let $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ be three unit vectors satisfying $|\mathbf{a}-\mathbf{b}|^2+|\mathbf{a}-\mathbf{c}|^2=10$. Then,
Statement (I): $|\mathbf{a}+2 \mathbf{b}|^2+|2 \mathbf{a}+\mathbf{c}|^2=2$
Statement (II) : $|2 a+3 b|^2+|3 a+2 c|^2=10$
Which of the above statements is (are) true?
If $\mathbf{r} \cdot(2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}})=5, \mathbf{r} \cdot(\hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}})=7$ are two planes and $(16,-9,0)$ is a point common to both the planes, then the vector equation of the line of intersection of the planes is $\mathbf{r}=$
Let $A B C$ be a triangle. Let a point $P$ divide $A B$ in the ratio $1: 2$ internally and a point $Q$ divide $B C$ in the ratio $1: 2$ internally. Let $D$ be the point of intersection of $A Q$ and $C P$. If the area of the $\triangle A B C$ is $k$ square units, then the area of the $\triangle B C D$ in (sq. units) is
If 10 is the mean deviation of ' $n$ ' observations $x_1, x_2, x_3, \ldots, x_n$, then the mean deviation of the observations $\frac{2 x_1+5}{3}, \frac{2 x_2+5}{3}, \frac{2 x_3+5}{3}, \ldots . \frac{2 x_n+5}{3}$ is
A bag contains 9 identical black balls numbered 1 to 9 . and 4 identical white balls numbered 1 to 4 . If 3 balls are drawn at a time randomly from that bag, then the probability of getting atleast one white ball is
The probabilities of two persons to hit a target are $1 / 4$ and $1 / 5$ respectively. The probability that the target is being hit when both of them attempt independently is
When 3 dice are thrown at a time, the sum of the numbers appeared on 3 dice were found to be 15 . Then, the probability that the number 5 does not appear on any one of the dice is
If the probability distribution of a random variable $X$ is given by
$$ \begin{array}{|c|c|c|c|c|c|c|} \hline X=x & 0 & 2 & 4 & 6 & 8 & 10 \\ \hline P(X=x) & 0 & k & 2 k & 5 k^2 & 2 k^2 & 3 k \\ \hline \end{array} $$
then the mean of $X$ is
The probability of getting a success in a trail is five times that of a failure. The probability of getting atmost one success in 5 trails, is
$B(2,3), C(5,-2), D(1,-1)$ are three points. If $A$ is a variable point such that the area of the quadrilateral $A B C D$ is 10 sq. units, then the locus of $A$ is
A line makes intercepts 5 and 7 on the coordinate axes. The axes are rotated through an angle $\theta$ in the positive direction about the origin so that the line makes equal intercepts on the new axes, then $|\tan \theta|=$
$L \equiv 7 x-y+8=0$ is one of the diagonals of a square for which $(-4,5)$ and $(3,4)$ are two vertices. Then, the coordinates of the two vertices lying on the diagonal $L=0$ are
The locus of the image of a variable point $(\alpha, 2 \alpha-1)$ with respect to the line $3 x-2 y+4=0$, is
Let $M$ be the foot of the perpendicular drawn from the point $(5,-7)$ to the line $3 x-5 y+1=0$. Then, the perpendicular distance from $M$ to the line $2 x+5 y-3=0$ is
If $P$ is a point equidistant from all the vertices $A(-1,3), B(3,5), C(5,7)$ of a $\triangle A B C$, then $P A=$
4 different pairs of lines are given in List I and the cosine of the angle between every pair of lines is given in List II. Match the following :
| List-I | List-II |
| (A) |
(I) |
| (B) |
(II) |
| (C) |
(III) |
| (D) |
(IV) |
| (V) |
If $a x^2+6 x y-2 y^2=0$ represents a pair of perpendicular lines and $9 x^2+2 h x y+4 y^2=0(h>0)$ represents a pair of coincident lines, then $h=$
The line $x+2 y=k$ meets the curve $2 x^2-2 x y+3 y^2+2 x-y-1=0$ at two points $A$ and $B$. Let $O$ be the origin. If the line segments $O A$ and $O B$ are perpendicular to each other, then $k=$
Let the centre of the circle $S=0$ lie on the line $x+y-5=0$ and also lie in the first quadrant. If this circle touches both the lines $x-2=0$ and $y-5=0$, then the area of the circle is
The straight line $x+2 y=1$ cuts the $X$-axis at $A$ and $Y$-axis at $B, A$ circle is drawn through $A, B$ and the origin. The sum of the perpendicular distances from $A$ and $B$ on to the tangent drawn at origin to the circle $S$ is
Let $P$ and $Q$ be two external points of the circle $S=x^2+y^2-a^2=0$. Let the chord of contact of the point $P$ with respect to the circle $S=0$ passes through $Q$. If $l_1$ and $l_2$ are the lengths of the tangents drawn from $P$ and $Q$ to the circle $S=0$, then $P Q=$
$A\left(x_1, y_1\right)$ is the internal centre of similitude and $B\left(x_2, y_2\right)$ is the external centre of similitude of two circles $C_1$ and $C_2$ whose centes are $P(\alpha, \beta)$ and $Q(\gamma, \delta)$, respectively. If $P A=3, A B=5, Q B=2$, then ratio of the radii of the two circles is
The equation of the direct common tangent of the circles $x^2+y^2-6 x-4 y-23=0$ and $x^2+y^2+2 x+2 y+1=0$ is
The length of the common chord of the two circles $x^2+y^2-4 x-8 y+4=0$ and $x^2+y^2-8 x-12 y+16=0$ is
If the focal chord drawn through the point $(1,2)$ to the parabola $y^2=8 x$ meets this parabola in $\left(x_1, y_1\right)$ and $\left(x_2, y_2\right)$, then $x_1+x_2=$
If $\left(2 t^2, 4 t\right)$ is a point on the parabola $y^2=8 x$ such that its focal distance is 3 , then $t=$
The length of the latusrectum of an ellipse is 6 units and the distance between a focus and its nearest vertex on the major-axis is $5 / 3$ units. If $e$ is the eccentricity of this ellipse, then $e$ satisfies the equation
If the line $2 x-3 y+4=0$ cuts the ellipse $x=3 \cos \theta, y=5 \sin \theta$ in $A$ and $B$ and $(\alpha, \beta)$ is the mid-point of $A B$, then $3 \beta-2 \alpha=$
Let $e_1$ be the eccentricity of a hyperbola for which distance between its focii is 2 times the distance between its directrices and $e_2$ be the eccentricity of another hyperbola for which the length of its transverse axis is twice the length of its conjugate axis. Then, $e_1 e_2=$
- Assertion (A) The distance between the points $p\left(\frac{\pi}{4}\right)$ and $p\left(\frac{\pi}{3}\right)$ on the hyperbola $9 x^2+16 y^2=9$ is
$$ \frac{1}{2 \sqrt{2}} \sqrt{66-33 \sqrt{2}-9 \sqrt{3}} $$
Reason (R) $x=a \cosh t, y=b \sinh t$ are the parametric equations of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$
The correct option among the following is
$A(1,1,1), B(1,-4,3), C(2,-2,0)$ and $D(8,1,4)$ are the vertices of a tetrahedron. $G_1, G_2, G_3$ and $G_4$ are the centroids of the faces $A B C, B C D, C D A$ and $D A B$. Then, the centroid of the tetrahedron having $G_1, G_2, G_3$ and $G_4$ as its vertices is
Let $A(2,3,-1), B(4,1,0), C(-1,-1,1)$ be the vertices of a $\triangle A B C$. Let $D$ be the point where the bisector of $B A C$ meet the side $B C$. Then, the direction ratios of $A D$ are
If a plane passing through the points $(2,3,0),(0,-5,2)$ and ( $-2,0,3$ ) meets the $X, Y$ and $Z$-axes in $A, B$ and $C$ respectively, then $A=$
Let [ $x$ ] denote the greatest integer less than or equal to $x$ and $f(x)=2 x-[2 x]$. If $\mathop {\lim }\limits_{x \to {2^ - }} f(x)=l_1$ and $\mathop {\lim }\limits_{x \to {2^ + }} f(x)=l_2$, then $l_1+l_2=$
$$ \mathop {\lim }\limits_{x \to 0} \frac{\left(2^x-1\right)(1+\sin x)^{\frac{2}{\sin x}}}{\log (1+2 x)}= $$
Let $f(x)$ be a differentiable function such that $f(0)=0$ and $f^{\prime}(0)=20$. For $x \in\left(0, \frac{\pi}{2}\right]$, if
$A(x)=2 f(x) \operatorname{cosec} 4 x+4 f(x)\left(\cos ^2 x+1\right)-4 \cos ^2 x$, then $\mathop {\lim }\limits_{x \to 0} A(x)=$
If $y=\frac{e^{\sin x}+\sinh ^3 x}{\cosh x-\tan x}$, then $y^{\prime}(0)=$
The approximate value of $\sqrt[3]{28}$ rounded up to 3 decimal places is
$y=x^2$ is the given curve. Imagine that this curve is dragged along the positive $X$-axis to a distance of ' $a$ ' units. If the acute angle between the curves at two positions is $\theta$, then
If $x$ and $y$ are two positive integers such that $x+2 y=10$ and $x^2 y^3$ is maximum, then $x^2+2 y^3=$
If $\frac{3 \pi}{4}
If $\tan \alpha=\frac{4}{3}$, then $\int \frac{1}{3 \cos x-4 \sin x} d x=$
If $x \neq(2 n+1) \frac{\pi}{2}$, then $\int \frac{\cos ^3 x}{(1+\sin x)^4} d x=$
- Given that $\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{r=1}^{n p} f\left(\frac{r}{n}\right)=\int_0^p f(x) d x$. If $f: R \rightarrow R$ is defined by $f(x)=x^2+2$, then
$$ \lim _{n \rightarrow \infty} \frac{3}{n}\left[f\left(\frac{7}{n}\right)+f\left(\frac{14}{n}\right)+f\left(\frac{21}{n}\right)+\ldots+f(7)\right]= $$
If $f(x)=\left|\begin{array}{ccc}2 \cos ^2 x & \sin 2 x & \sin x \\ \sin 2 x & 2 \sin ^2 x & -\cos x \\ \sin x & -\cos x & 0\end{array}\right|$, then
$$ \left.\int_0^{\pi / 4}|2| f(x) \mid+5 f^{\prime}(x)\right) d x= $$
The number of arbitrary constants that appear in the general solution of the differential equation $\left(\frac{d^4 y}{d x^4}+\frac{d^2 y}{d x^2}\right)^{3 / 2}=5 \frac{d^3 y}{d x^3}$ is
Assertion (A) The degree of the differential equation $y^{\prime \prime}+2 x y^{\prime}+\log _e\left(\frac{d y}{d x}\right)=0$ is 2 .
Reason (R) The degree of a differential equation is the highest degree of the highest order derivative occurring in the equation, after the equation is expressed in the form of a polynomial in differential coefficients. The correct option among the following
Let $S$ be the family of curves given by the general solution of the differential equation $\frac{y^2 e^{-1 / y}}{\sqrt{x}} d x-2 \sec \sqrt{x} d y=0$. Then, the equation of the curve belonging to $S$ and passing through $\left(\pi^2, 1\right)$ is
Physics
Which of the following statements is true?
If the velocity of light $c$, the gravitational constant $G$ and Planck's constant $h$ are chosen as the fundamental units, the dimension of density in the new system is
A ball projected up passes the same height $H$ at 2 s and 10 s . The value of $H$ is (use, $g=9.8 \mathrm{~m} / \mathrm{s}^2$ )
Two towns $X$ and $Y$ are connected by a regular bus service. A bus leaves in either direction at every $t=T$ minutes. A man moving with same speed in the direction $X$ to $Y$ find that a bus goes past him every $t=t_1$ minutes in the direction of his motion, and every $t=t_2$ minutes in the opposite direction. Then, $T$ is given by
Statement I An object subjected to velocities $\mathbf{v}_1$ and $\mathbf{v}_2$ has a resultant velocity with magnitude $|\mathbf{v}|=\left|\mathbf{v}_1\right|+\left|\mathbf{v}_2\right|$.
Statement II The magnitude of displacement is either less or equal to the path length of an object between two points.
Statement III The instantaeous acceleration is the limiting value of the average acceleration as the time interval approaches zero.
Which of the following is correct?
For a projectile, if $\alpha$ is the angle of projection, $R$ is the range, $h$ is the maximum height, $t$ is the time of flight then
At time $t=0$, a force $F=\alpha t$, where $t$ is time in seconds, is applied to a body of mass 1 kg , resting on a smooth horizontal plane. If the direction of the force makes an angle of $45^{\circ}$ with the horizontal, then the velocity of the body at the moment of its breaking off the plane is
Statement I The slope of kinetic energy-displacement curve of a body in motion will be directly proportional to its acceleration.
Statement II From a height of 15 m , a ball is projected vertically upwards with a velocity of $30 \mathrm{~m} / \mathrm{s}$. If the ball rises to the same height after hitting the ground, the loss of its energy on hitting the ground is $30 \%$.
Statement III The velocity acquired by a body of mass $m$ after travelling a fixed distance from rest under the action of a constant force is directly proportional to mass $m$.
Which of the following is correct?
An object is moving in a straight line under the influence of a source of constant power. If $v$ and $t$ are velocity and time respectively, then
A solid spherical ball is rolled up an inclined plane of angle of inclination $30^{\circ}$ with an initial speed of $4 \mathrm{~m} / \mathrm{s}$ at the bottom of the inclination. How far will the ball go up the plane?
(Use, $g=10 \mathrm{~m} / \mathrm{s}^2$ )
A particle performs simple harmonic motion with a time period of 16 s . At a time $t=2 \mathrm{~s}$, the particle passes through the origin and at $t=4 \mathrm{~s}$ its velocity is $4 \mathrm{~m} / \mathrm{s}$. The amplitude of the motion is
Let the escape speed of an object on the earth's surface be $v_0$. The object is projected out with speed $5 v_0$. The speed of the object far away from the earth will be
One end of a steel rod of radius 10.0 mm and length 50.0 cm is clamped on a horizontal table. The other end of the rod is pulled with a force of magnitude 10.0 $\times \pi \mathrm{kN}$. This force is uniform across the flat surface of the rod and is perpendicular to it. The change in the length of the rod due to this applied force is (use Young's modulus, $Y=2.0 \times 10^{11} \mathrm{~N} / \mathrm{m}^2$ )
A venturimeter has a pipe diameter of 4 cm and a throat diameter of 2 cm . Velocity of water in the pipe section is $10 \mathrm{~m} / \mathrm{s}$. The pressure drop, between pipe section and the throat section is (use, density of water $=1000 \mathrm{~kg} / \mathrm{m}^3$ )
A soap bubble of initial radius $R$ is to be blown up. The surface tension of the soap film is $T$. The surface energy needed to double the diameter of the bubble is
Two metal rods $A$ and $B$ each of length 50 cm can diameter 4.0 mm are joined together at temperature $30^{\circ} \mathrm{C}$. What is the change in length of the combined rod at $230^{\circ} \mathrm{C}$ ? (Given, linear expansion coefficients of rods $A$ and $B$ are respectively, $2.0 \times 10^{-5} /{ }^{\circ} \mathrm{C}$ and $1.0 \times 10^{-5} /{ }^{\circ} \mathrm{C}$ )
Find the difference in temperature between the water at the top and the bottom of 20 m high waterfall assuming 10\% of the energy of fall is spent in heating the water (use, specific heat capacity of water $=4000 \mathrm{~J} \mathrm{kg}^{-1} \mathrm{~K}^{-1}$ and $g=10 \mathrm{~m} / \mathrm{s}^2$ )
Assertion (A) The zeroth law of thermodynamics leads to the concept of temperature.
Reason (R) The zeroth law states that two systems in thermal equilibrium with a third system are in thermal equilibrium with each other.
The correct option among the following is
When a gas expands adiabatically, its volume is doubled while its absolute temperature is decreased by a factor of 2 . The value of the adiabatic constant is
An amount of 700 J of heat is transferred to a diatomic gas allowing it to expand with the pressure held constant. The work done on the gas is
Which of the following wave has the largest wave speed?
What is the refractive index of the material of a double convex lens having radii of curvature of 5 cm and 10 cm and focal length of $\frac{20}{3} \mathrm{~cm}$
In an interference pattern of Young's double slit experiment, at a point we observe the 12 th order maximum for a monochromatic light source with wavelength $6000 \mathop {\rm{A}}\limits^{\rm{o}} $. What order will be visible here, if the source is replaced by a light of wavelength $4800 \mathop {\rm{A}}\limits^{\rm{o}} $ ?
Two charges are $+10 \mu \mathrm{C}$ and $-10 \mu \mathrm{C}$ are separated by 10 cm . The magnitude of force acting on another charge $5 \mu \mathrm{C}$ placed at the mid-point of the line joining the two charges will be (use, $\frac{1}{4 \pi \varepsilon_0}=9 \times 10^9$ in SI unit)
A sphere-1 with redius $R$ has charge $q$. Sphere-2 with radius $3 R$ is far from sphere-1 and is initially uncharged. If the two spheres are now connected with a thin conducting wire, then the ratio $\frac{\sigma_1}{\sigma_2}$ of the surface charge densities is
Statement I Specific resistance depends on nature of material and independent of temperature of the material. Statement II A wire of resistance $6 \Omega$ is drawn out, so that its new length is four times its original length. The resistance of the new wire is $48 \Omega$.
Statement III Drift velocity is the average constant velocity acquired by free electrons inside a metal by the application of an electric field which results in current. Which of the following is correct?
Find the mobility of electron in a wire, if its average collision time is $9.1 \times 10^{-15} \mathrm{~s}$. (Charge of electron $=1.6 \times 10^{-19} \mathrm{C}$ and mass of electron $=9.1 \times 10^{-31} \mathrm{~kg}$ )
A current carrying loop $A B C D$ has two circular arcs $A D$ and $B C$ with radius 1 cm and 2 cm respectively, as shown in the figure. The two arcs $A D$ and $B C$ subtend a common angle $30^{\circ}$ at the centre $O$. If the current flowing in the loop is $\frac{12}{\pi} \mathrm{~A}$. Then, the magnitude of net magnetic field at $O$ is (given, $\mu_0=4 \pi \times 10^{-7}$ )
Three parallel wires $a, b$ and $c$ carrying currents $i_a, i_b$ and $i_c$ as shown in the figure are placed next to each other.
The magnitude force on a length $l$ of the wire $a$, if $d_2=2 d_1, i_b=i_a$ and $i_c=4 i_a$ is
An iron bar having a cross-sectional area of $2 \times 10^{-5} \mathrm{~m}^2$ and magnetising field of $2400 \mathrm{~A} / \mathrm{m}$ produce a magnetic flux $2.4 \pi \times 10^{-5} \mathrm{~Wb}$. What will be the value of permeability $\mu$ and susceptibility $\chi$ of the bar (given, $\mu_0=4 \pi \times 10^{-7}$ )
A metal disc of radius 30 cm rotates with a constant angular velocity $\omega=100 \mathrm{rad} / \mathrm{s}$ about its axis. Find the magnitude of potential difference between the centre and the rim of the disc of the external uniform magnetic field on induction $B=4 \mathrm{mT}$ is directed perpendicular to the disc.
A capacitor of capacitance $100 \mu \mathrm{~F}$ and a coil of resistance $20 \Omega$ and inductance 12.5 mH are connected in series with a $220 \mathrm{~V}, \frac{200}{\pi} \mathrm{~Hz}, \mathrm{AC}$ source. The maximum value of instantaneous current in the circuit is
On a particular day, the sun delivers an average power of $\left(\frac{6}{\pi} \times 10^3\right) \frac{\mathrm{W}}{\mathrm{m}^2}$ to the top of earth's atmosphere. Find the amplitude of magnetic field for the electromagnetic waves above atmosphere.
(Take, $\mu_0=4 \pi \times 10^{-7}$ SI unit)
Statement I By increasing the potential difference between cathode and anode continuously in a photoelectric experiment, the photocurrent always increases continuously.
Statement II If two photons $A$ and $B$ of energies 2.5 eV and 3.5 eV respectively, fall on a metal surface of work function 2.0 eV , then the ratio of maximum kinetic energies emitted between $A$ and $B$ is 3 .
Statement III The maximum energy needed by an electron to come out from a metal surface is called the work function of the metal.
Which of the following is correct?
Which of the following has the largest de-Broglie wavelength?
The energy of an electron in the fourth excited state of the hydrogen atom is
Estimate the approximate volume of aluminium nucleus $(A=27)$, use $\binom{R_0 \simeq 1.0 \times 10^{-15} \mathrm{~m}}{\pi \simeq 3}$
A $p-n$ junction is fabricated from a semiconductor with band gap of 2.8 eV . what approximate wavelength it cannot detect? [Use, $h=6 \times 10^{-34} \mathrm{~m}^2 \mathrm{~kg} / \mathrm{s}$ ]
Identify the logic operation performed by the following circuit.
A carrier wave of peak voltage 10 V is used to transmit a message signal. What should be the peak voltage of the modulating signal in order to have a modulation index of $80 \%$ ?