Chemistry
What is the approximate angular momentum (in J s ) of electron in hydrogen atom in its ground state?
$$ \left(h=6.625 \times 10^{-34} \mathrm{~J} \mathrm{~s}\right) $$
The energy of electron in hydrogen atom when present in $n=1, n=2$ and $n=3$ will be in the ratio of
What is the correct order with respect to metallic property of Zr, Cd, Sn, Sr?
Identify the number of molecules in which the central atom has one lone pair of electrons from the following list.
$$ \mathrm{PbCl}_2, \mathrm{PH}_3, \mathrm{ClF}_3, \mathrm{SF}_4, \mathrm{BF}_3, \mathrm{SnCl}_2 $$
In which of the following molecules, the number of lone pairs of electrons on central atom and the number of $d$-orbitals involved in the hybridisation of central atom, is same?
4 g of an ideal gas $A$ (molar mass $=M_A$ ) present in a vessel of volume $V$ litre exerted a pressure of 5 atm at 300 K . When 16 g of another ideal gas $B$(molar mass $=M_B$ ) was introduced into this vessel at the same temperature, its pressure increased to 10 atm . What is the correct relationship between $M_A$ and $M_B$ ?
The sum of three values $12.0,19.034$ and 2.0143 is equal to $X$. The number of significant figures in $X$ is
In. Observe the following properties.
- Molar volume
II. Mass
III. Internal energy
IV. Volume
v. Enthalpy
VI. Temperature
VII. Density
The intensive properties in the above list are
229, $\mathrm{At} T(\mathrm{~K}), K_C$ value for the reaction, $\frac{1}{3} \mathrm{~N}_2(g)+\mathrm{H}_2(g) \rightleftharpoons \frac{2}{3} \mathrm{NH}_3(g)$ is 50 . The $K_C$ value for the reaction, $2 \mathrm{NH}_3(g) \rightleftharpoons \mathrm{N}_2(g)+3 \mathrm{H}_2(g)$ at the same temperature is
Identify the correct statements from the following.
A. In photosynthesis reaction, water is oxidised to oxygen.
B. An example for interstitial hydride is $\mathrm{MgH}_2$.
C. Sodium hexametaphosphate is used in the removal of permanent hardness of water.
Which one of the following statements is not correct?
Thermal decomposition of lithium nitrate gives
Consider the following statements about group 13 elements.
A. $\mathrm{AlCl}_3$ gets stability by forming a dimer.
B. $\mathrm{BCl}_3$ is an electron deficient molecule.
C. $E_{M^{3+} / M}^{\circ}(\mathrm{V})$ is +1.26 for aluminium.
D. In +1 oxidation state thallium is unstable.
The incorrect statements are
What is the correct order of melting temperature of $\mathrm{C}, \mathrm{Si}, \mathrm{Ge}$ ?
$$ \text { The IUPAC name of the following compound is } $$
The delocalisation of $\sigma$ electrons of C-H bond of an alkyl group with the $\pi$ electrons of benzene is observed in
An alkene $X\left(\mathrm{C}_6 \mathrm{H}_{12}\right)$ on ozonolysis gave acetaldehyde and ethyl methyl ketone. What is the product formed when $X$ reacts with HBr ?
$$ \text { What is } X \text { in the following reaction sequence? } $$
A compound is formed by elements $A, B$ and O . Atoms of oxygen form ccp lattice. Atoms of $A$ (cation) occupy $\frac{1}{8}$ th of tetrahedral voids and atoms of $B$ (cation) occupy half of octahedral voids. What is the molecular formula of the compound?
Liquids $A$ and $B$ form an ideal solution. The vapour pressures of $A$ and $B$ are 50 and 32 mm Hg respectively at 300 K . One mole of liquid $A$ is mixed with 1 mole of liquid $B$. What is the approximate mole fraction of $A$ in vapour phase?
$A$ and $B$ are two metals. The standard reduction potential of $A^{+}(a q) / A(s)$ and $B^{+}(a q) / B(s)$ are -0.5 V and +0.5 V respectively. What is the $\log K_c$ value for the following reaction at 298 K ?
$$ \begin{aligned} & A(s)+B^{+}(a q) \rightleftharpoons A^{+}(a q)+B(s) \\ & \left(\text { Given }: \frac{2.303 R T}{F}=0.06 \mathrm{~V}\right) \end{aligned} $$
For a zero order reaction $A \rightarrow$ product, a plot of $[A]$ (on $y$-axis) and time (on $x$-axis) gave a straight line with slope equal to $-3 \times 10^{-3} \mathrm{M} \mathrm{min}^{-1}$ and intercept equal to $2 \times 10^{-2} \mathrm{M}$ (on y -axis). What is the rate constant (in M $\mathrm{min}^{-1}$ ) of this reaction?
$$ \text { Match the following. } $$
$$ \begin{array}{llll} \hline & \text { List I } & & \text { List II } \\ \hline \text { A. } & \text { Negatively charged sol } & \text { I. } & \text { Emulsion } \\ \hline \text { B. } & \text { Milk } & \text { II. } & \text { Kala azar } \\ \hline \text { C. } & \text { Gold number } & \text { III. } & \begin{array}{l} \text { FeCl} \mathbf{C}_3 \text { solution is added to } \\ \text { excess NaOH solution } \end{array} \\ \hline \text { D. } & \text { Colloidal antimony } & \text { IV. } & \text { Protection of colloids } \\ \hline \end{array} $$
The correct answer is
$$ \begin{aligned} 4 \mathrm{Ag}(s)+8 \mathrm{CN}^{-}(a q) & +2 \mathrm{H}_2 \mathrm{O}(a q)+\mathrm{O}_2(g) \longrightarrow 4\left[\mathrm{Ag}(\mathrm{CN})_2\right]^{-}(a q)+4 \mathrm{OH}^{-}(a q) \end{aligned} $$
The above reaction represents the process of concentration of ore in the extraction of silver. The process is
Which one of the following statements is correct?
Which oxo-acid of sulphur contains $\mathrm{S}-\mathrm{O}-\mathrm{S}$ bond?
Which of the following reaction gives nitrogen (II) oxides as one of the products?
Identify the correct pairs in which the chemical substance given is correctly matched its use
| Chemical substance | Use |
| A. |
Preparation of phosgene |
| B. |
Estimation of CO |
| C. |
Disinfectant |
Observe the following ions
$$ \mathrm{V}^{2+}, \mathrm{Zn}^{2+}, \mathrm{Cu}^{2+}, \mathrm{Fe}^{2+}, \mathrm{Fe}^{3+}, \mathrm{Ti}^{3+}, \mathrm{Sc}^{3+}, \mathrm{Ti}^{4+}, \mathrm{Ni}^{3+}, \mathrm{Co}^{3+}, \mathrm{Cu}^{+} $$
How many ions in the above list have zero magnetic moment?
Identify the correct set for $\left[\mathrm{Co}\left(\mathrm{NH}_3\right)_6\right]^{3+}$ ion.
(Hybridisation of $\mathrm{Co}^{3+}$, type of complex, number of unpaired electrons in the complex ion respectively.)
Identify the correct statement from the following.
A. Glyptal is made from the monomers ethylene glycol and phthalic acid.
B. Bakelite is used in making electrical switches.
C. Nylon 2-nylon-6 is a biodegradable polymer.
A vitamin $X$ is soluble in fat and its source is egg yolk. Deficiency of $X$ causes the disease
Identify the pair of drugs which act as tranquilizers
An alkyl halide $X\left(\mathrm{C}_4 \mathrm{H}_9 \mathrm{Br}\right)$ undergoes nucleophilic substitution by $\mathrm{S}_{\mathrm{N}} 2$ reaction. The product of $X$ on reaction with $\mathrm{Mg} /$ dry ether followed by $\mathrm{D}_2 \mathrm{O}$ is
Assertion (A) $\mathrm{p} K_a$ of phenol is 4.19 and that of benzoic acid is 10 .
Reason (R) Phenoxide ion is stabilised by non-equivalent resonance structures whereas benzoate ion by two equivalent resonance structures.
The correct option among the following is
In which of the following pairs reactant is correctly matched with reagent that would form benzaldehyde as product?
An alkene $X$ with formula $\mathrm{C}_4 \mathrm{H}_8$ does not exhibit geometrical isomerism. In the conversion of $X$ to $Y$, the correct sequence of reagents $A$ and $B$ used are ( $Y$ gives iodoform test)
$$ \text { Which of the following reaction is feasible? } $$
The sequence of reagents which convert $p$-methyl aniline to $p$-methyl benzoic acid are
An amine $(X)$ reacts with $p$-toluene sulphonyl chloride to give the product $Y$, which is insoluble in alkali. The product of $X$ with benzoyl chloride is
Mathematics
Which one of the following functions is a bijection?
The domain of the real valued function $f(x)=\frac{\sqrt{|x|-x}}{\sqrt{x-[x]}}$ is
The range of the function defined by
$$ f(x)=\left\{\begin{array}{lc} 2 x-3, & \text { if } x<-1 \\ 1-x^2, & \text { if }-1 \leq x \leq 1 \text { is } \\ 3 x^2+2, & \text { if } x>1 \end{array}\right. $$
- If $A=\left[\begin{array}{lll}b & a & 0 \\ c & 0 & b \\ a & a & b\end{array}\right]$ and $B=\left[\begin{array}{lll}0 & a & b \\ b & 0 & c \\ b & a & a\end{array}\right]$ are two matrices such that $A B=\left[\begin{array}{ccc}2 & 2 & 7 \\ 1 & 8 & 5 \\ 3 & 6 & 10\end{array}\right]$, then $a^2+b^2+c^2=$
If $A=\left[\begin{array}{lll}1 & a & 3 \\ b & 2 & c \\ 3 & d & 4\end{array}\right]$ is a symmetric matrix and $B=\left[\begin{array}{ccc}0 & 5 & b \\ -5 & 0 & -7 \\ 6 & c & 0\end{array}\right]$ is a skew-symmetric matrix, then $A B=$
If the inverse of the matrix $A=\left[\begin{array}{ccc}-1 & -3 & -2 \\ 0 & 1 & 2 \\ 3 & 4 & 5\end{array}\right]$ is $A^{-1}=\left[\begin{array}{lll}a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3\end{array}\right]$, then $a_1+c_2+b_3=$
If $x=\alpha, y=\beta, z=\gamma$ is the unique solution of the system of linear equations $2 x-3 y+5 z=12,5 x+2 y+3 z=11$ and $x+2 y-3 z=-3$, then $2 \alpha+5 \beta+3 \gamma=$
If $i^2=-1$, then $(1+\sqrt{3} i)^{2022}-(\sqrt{3}-i)^{2022}=$
If $\left(\frac{\sqrt{3}+i}{\sqrt{3}-i}\right)^4+\left(\frac{\sqrt{3}-i}{\sqrt{3}+i}\right)^4=r$ cis $\theta$, then one of the values of $\sqrt{r \operatorname{cis} \theta}$ is
If $z=x+i y$ and the point $P$ in the argand plane represents $z$, then the locus of $z$ satisfying the equation $|z-2|+|z-2 i|=4$ is
One of the values of $(\sqrt{3}-i)^{2 / 5}$ is
If $\alpha, \beta, \gamma$ and $\delta$ are the roots of the equation $x^4+x^2+1=0$ such that $\alpha+\beta=-1, \gamma+\delta=1, \alpha^2=\beta$ and $\gamma^2=-\delta$, then $\alpha^{2023}+\beta^{2023}+\gamma^{2022}+\delta^{2022}=$
Let the equations $a x^2-7 x+c=0$ and $a x^2+5 x-c=0$ have a common root and $a c \neq 0$. If 3 is a root of $a x^2-7 x+c=0$ other than the common root, then the common root of the given equations is
The set of all values of $x$ for which the inequalities $x^2-7 x+10 \geq 0$ and $2 x+3-x^2>0$ hold simultaneously is
If $\alpha, \beta, \gamma$ are the roots of the equation $2 x^3+x^2-13 x+6=0$, then $\alpha^3+\beta^3+\gamma^3=$
If $\alpha, \beta, \gamma$ are the real roots of the equation $18 x^3-15 x^2-4 x+4=0$ such that $\alpha=\beta$ and $\alpha>\gamma$, then $\alpha+\beta^2+\gamma^3=$
If $\alpha$ is a multiple root of the equation $x^5-6 x^4+11 x^3-2 x^2-12 x+8=0$, then $3 \alpha^2-2 \alpha+1=$
All the letters of the word 'INDEED' are taken and permuted in all possible ways to form distinct 6 letter strings (words with or without meaning). If they are listed in dictionary order, then the rank position of the string 'NIDDEE' is
All possible 5-digit numbers each having 5 distinct digits are formed using the digits $1,2,3,5,6,8$. Among them, the number of numbers which are divisible by 3 but not by 6 is
The total number of ways of forming a committee of 5 members out of 7 Indians, 6 Americans, 5 Russians and 4 Australians, so that every committee contains atleast one member from each country is
The numerically greatest term in the binomial expansion of $(2 x-3 y)^5$, when $x=\frac{3}{2}$ and $y=\frac{2}{3}$ is
When $3^{2023}$ is divided by 16 , the remainder obtained is
If $3 x=1+\frac{5}{8}+\frac{5}{8} \cdot \frac{9}{13}+\frac{5}{16}+\ldots$, then $x^4+4 x^3+6 x^2+4 x=$
If $\frac{2 x^3+3 x^2+3 x+5}{\left(x^2+1\right)\left(x^2+2\right)}$ is expanded in terms of the powers of $x$, then the coefficient of $x^5$ is
$$ \sin 6^{\circ}+\sin 54^{\circ}+\sin 126^{\circ}+\cos 156^{\circ}= $$
If $\tan \alpha=\frac{-12}{5}, \cot \beta=\frac{7}{24}, \alpha$ does not belong to second quadrant and $\beta$ does not belong to first quadrant, then $\sqrt{13} \sin \frac{\alpha}{2}+\cos \frac{\beta}{2}+\tan \frac{\alpha}{2} \cot \frac{\beta}{2}=$
$\cos \frac{\pi}{7} \cos \frac{2 \pi}{7} \cos \frac{3 \pi}{7} \cos \frac{\pi}{14} \cos \frac{3 \pi}{14} \cos \frac{5 \pi}{14}=$
If $\sinh x=-\frac{4}{3}$, then $\sinh 2 x+\cosh 2 x=$
In $\triangle A B C$, if $b=6, c=7$ and $\tan \frac{A}{2}=\frac{1}{\sqrt{6}}$, then the inradius of $\triangle A B C$ is
In $\triangle A B C$, if $a=7, b=8$ and $c=9$, then $\frac{1}{r_1^2}+\frac{1}{r_2^2}+\frac{1}{r_3^2}=$
II. If the points with position vectors $\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}, 2 \hat{\mathbf{i}}-\hat{\mathbf{k}}$, $\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ and $\hat{\mathbf{i}}+\hat{\mathbf{j}}+\lambda \hat{\mathbf{k}}$ are coplanar, then the magnitude of the vector $6 \lambda \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+6 \hat{\mathbf{k}}$ is
Let $\mathbf{a}, \mathbf{b}, \mathbf{c}$ be three non-coplanar vectors and $L$ be the line passing through the points $\mathbf{a}-\mathbf{b}+\mathbf{c}$ and $\mathbf{b}-\mathbf{c}$. If $\pi$ is a plane passing through the points $2 \mathbf{a}-\mathbf{b}, 2 \mathbf{b}-\mathbf{c}$ and $2 c-\mathbf{a}$, then the point of intersection of $L$ and $\pi$ is
Let $\mathbf{a}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}, \mathbf{b}=6 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}$ and $\mathbf{c}=3 \hat{\mathbf{i}}-4 \hat{\mathbf{j}}-12 \hat{\mathbf{k}}$ be three vectors. If $\mathbf{p}$ is the projection of $\mathbf{b}$ on $\mathbf{a}$ and $\mathbf{q}$ is the projection of $\mathbf{c}$ on $\mathbf{a}$, then $13 \mathbf{p}=$
Let $\mathbf{a}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}, \mathbf{b}=3 \hat{\mathbf{i}}-\hat{\mathbf{j}}+5 \hat{\mathbf{k}}$ and $\mathbf{c}=\hat{\mathbf{i}}-4 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ be three vectors. Let $\mathbf{r}$ be a vector perpendicular to both $\mathbf{b}$, $c$ and $\mathbf{r} \cdot \mathbf{a}=11$. Then, the vector among the following that is perpendicular to $\mathbf{r}$ is
The volume of the tetrahedron with $\hat{\mathbf{i}}-\lambda \hat{\mathbf{j}}+\hat{\mathbf{k}}$, $\lambda \hat{\mathbf{i}}-\hat{\mathbf{j}}-\hat{\mathbf{k}}$ and $\hat{\mathbf{i}}+\hat{\mathbf{j}}+\lambda \hat{\mathbf{k}}$ as coterminous edges is 2 . If $\lambda$ is an integer, then $|\lambda \hat{\mathbf{i}}-3 \lambda \hat{\mathbf{j}}+3 \hat{\mathbf{k}}|=$
If $M$ and $\sigma^2$ represent respectively the mean deviation from the mean and the variance for the data $1,3,5,7$, $11,13,17,19,23$, then $3\left(\sigma^2-M\right)=$
A bag contains 3 red, 5 black and 7 blue balls. If three balls are drawn at random simultaneously from the bag, then the probability of getting at least two blue balls is
In a game, two dice are thrown simultaneously by a person $A$ and two cards are drawn at random simultaneously from a pack of 52 playing cards by a person $B$. They win the game, if $A$ gets a prime score as the sum of the numbers appear on both the dice and $B$ gets a face card and a card having a prime number. Then, the probability that both $A$ and $B$ win is
Two players $A$ and $B$ alternatively toss 3 coins simultaneously. The player who gets 2 heads and 1 tail first, wins the game. If game continues until someone wins and if $A$ begins the game, the probability that B wins the game is
If $X$ is a Poisson variate satisfying the condition $3 P(X=2)=P(X=4)$, then $P(X=6)=$
Let $A=(1,2), B=(2,1), C=(-1,-1)$ be three points. If $P$ is a point such that the area of the quadrilateral $P A B C$ is twice the area of the $\triangle P A B$, then the equation of the locus of $P$ is
When the origin is shifted to the point $(h, k)$ by translating the coordinates axes, the equation $S \equiv 2 x^2-x y+y^2+2 x+3 y+1=0$ is changed to $S \equiv a x^2+2 h x y+b y^2-3=0$. Again by rotating the coordinate axes about the new origin through the angle $\theta$ in the positive direction, $S^{\prime}=0$ is changed to $A x^2+B y^2+C=0$. Then, $h+k+\tan 2 \theta=$
Two points $P(a, 2)$ and $Q(1, b)$ lie on either side of the line $2 x-3 y+1=0$. If $P$ is the point of intersection of the lines $4 x+3 y+k=0$ and $3 x+4 y+k=0$, then the range of $b$ is
Let the angle between the lines $x-2 y+3=0$ and $k x-y+2=0$ be $45^{\circ}$. If $k_1, k_2\left(k_1>k_2\right)$ are two distinct real values of $k$, then $k_1-2=$
If the lines $4 x+3 y-k=0,2 x+y+3=0$ and $3 x+2 y+k=0$ are concurrent, then the perpendicular distance from the point of concurrency of these lines to the line $3 x+4 y+2=0$ is
Let $A(1,3)$ and $B(2,5)$ be two points and $C(h, k)$ be a point such that $B C$ is perpendicular to $A C$. If $\angle C A B=\angle C B A$, then $h=$
Let the line $2 x-3 y-1=0$ intersect the curve $x^2+2 x y+5 y^2+2 x+3 y-1=0$ in distinct points $A$ and $B$. If ' $O$ ' is the origin, then $\cos \angle A O B=$
The equation of the circle inscribed in a square formed by the lines $x+y-2=0, x+y-6=0, x-y+1=0$ and $x-y+5=0$ is
Let the circle $S \equiv x^2+y^2+2 g x+2 f y+c=0$ touch the positive $X$-axis and the positive $Y$-axis. Let $(2,4)$ be a point on the circle $S=0$. If two such circles exist, then the difference of their areas is
If the equation $2 x-3 y+3=0,2 x+y+1=0$ and $6 x+4 y+1=0$ represent the sides of a triangle, then the equation of the circle passing through the vertices of this triangle is
If $T_1 T^{\prime}{ }_1$ and $T_2 T_2^{\prime}$ are the common tangents of the circles $S \equiv x^2+y^2-2 x-4 y-4=0$ and $S \equiv x^2+y^2+4 x+4=0$, where $T_1, T^{\prime}{ }_1, T_2, T^{\prime}{ }_2$ are the points of contact, then the distance between $T_1$ and $T_1^{\prime}$ is
A circle $S \equiv x^2+y^2+2 g x+2 f y+4=0$ cuts the circle $x^2+y^2-4 x-4 y-4=0$ orthogonally and makes an angle of $60^{\circ}$ with the circle $x^2+y^2+4 x+4 y+4=0$. Then, the radius of the circle $S=0$ is
If the circle $S \equiv x^2+y^2+2 g x+2 f y+c=0$ cuts each of the three circles $x^2+y^2+4 x+4 y+7=0$, $x^2+y^2-4 x+4 y+7=0$ and $x^2+y^2-4 x-4 y+7=0$ orthogonally, then the equation of the tangent drawn at the point $(\sqrt{3}, 2)$ to the circle $S=0$ is
If the line $2 x+3 y+n=0$ is a tangent to the parabola $y^2=8 x$, then the equation of the normal drawn at the point $(2 n, 4 \sqrt{n})$ to the parabola $y^2=8 x$ is
$a x-y+c=0$ is the equation of the common tangent to the parabola $y^2=8 \sqrt{5} x$ and the circle $x^2+y^2=1$. If this tangent makes an acute angle with the positive $X$-axis in the positive direction, then $a^2 c^2=$
In an ellipse, the distance from one of the foci to its corresponding end of the major axis is $4-\sqrt{7}$ and the distance from same focus to one end of the minor axis is 4 . Then, the cosine of the angle subtended by the line segment joining its foci at one end of its minor axis is
If the equations $x=1+2 \cos \theta, y=2+\sin \theta, 0 \leq \theta<2 \pi$ represent an ellipse, then the point of intersection of the normal drawn at $P\left(\frac{\pi}{4}\right)$ to this ellipse and its major axis is
If the equation $x+y+n=0$ represents a normal to the hyperbola $\frac{x^2}{6}-\frac{y^2}{2}=1$, then $n=$
$A(1,2,3), B(2,3,1)$ and $C(3,1,2)$ are three points. If the point $P$ divides $A B$ in the ratio $1: 2$ and the point $Q$ divides $B C$ in the ratio $-2: 3$, then the distance between $P$ and $Q$ is
If the image of the point $(1,-2,1)$ with respect to the line passing through the points $B(1,1,2)$ and $C(2,2,1)$ is $(l, m, n)$, then $l^2+m^2+n^2=$
A plane $\pi$ passing through the point $(1,1,1)$ is perpendicular to the line joining the points $(6,3,2)$ and $(1,-4,-9)$. If $a x+b y+c z-23=0$ is the equation of the plane $\pi$, then $a+b-c=$
$$ \mathop {\lim }\limits_{x \to 2} \frac{\sqrt[3]{6+x}-\sqrt[3]{10-x}}{x-2}= $$
$\mathop {\lim }\limits_{x \to 0} \frac{\tan ^4 x-\sin ^4 x}{x^6}=$
If $f(x)=\sqrt{\log \left(x^2+x+1\right)+\sqrt{\cosh (2 x-3)}}$, then $f^{\prime}(0)=$
- If $x=\cos ^3 \theta-\sin ^3 \theta$ and $y=\sqrt[3]{\cos \theta}-\sqrt[3]{\sin \theta}$, then the value of $\frac{d y}{d x}$ at $\theta=\frac{\pi}{4}$ is
If $2 x^2+3 x y-y^2+4 x-5 y+6=0$, then the value of $\frac{d y}{d x}$ at $(x, y)=(1,-2)$ is
The diameter of a sphere is measured as 42 cm . If there is an error of $1 / 77 \mathrm{~cm}$ in measuring it, then the error involved in the volume of that sphere (in cubic centimeters) is
For $h, k \in N$, let $P(h, k)$ be the point of intersection of the curves $x^2 y-x^3=8$ and $y^3-x y^2=32$. If $\theta$ is the acute angle between these two curves at $P$, then $\tan \theta=$
If the absolute maximum and absolute minimum values of the function $f(x)=x^3-2 x^2+x-3$ defined on $[0,2]$ are $M$ and $m$ respectively, then $M+m=$
$\int \frac{1}{\left(x+\frac{2}{x}\right) \sqrt{x^4+4 x^2+3}} d x=$
If $\frac{3 \pi}{2} < x < \frac{5 \pi}{2}$ and $\int(\sqrt{1-\sin x}+\sqrt{1+\sin x}) d x=f(x)+C$, where $C$ is the constant of integration, then $f\left(\frac{\pi}{3}\right)-f(0)=$
If $\int \frac{2 \sin 2 x-3 \cos x}{2 \sin ^2 x-3 \sin x+4} d x=f(x)+C$, where $C$ is the constant of integration, then $f\left(\frac{\pi}{2}\right)-f(0)=$
$\int \frac{2 x+3}{\sqrt{3 x^2-2 x+1}} d x=$
$$ \int_0^\pi \frac{x \cos ^2 x}{1+\sin x} d x= $$
If $[x]$ represents greatest integer function, then
$$ \int_{-2}^2[2-x] d x= $$
$$ \int_0^2 \frac{x}{(2-x)^{\frac{3}{4}}} d x= $$
$$ \int_0^2 x^3(2-x)^4 d x= $$
If the slope of the tangent drawn at any point $(x, y)$ to the curve $y=f(x)$ is $3 x^2-5$ and $f(1)=2$, then the tangent at $(1,2)$ to the curve $y=f(x)$ intersects the curve at the point
The general solution of the differential equation $(3 x-4 y)(d x-3 d y)+(6 d x-4 d y)=0$ is
The general solution of the differential equation $(\sec x+\tan x) \frac{d y}{d x}+\left(\sec ^2 x+\sec x \tan x\right) y=1$ is
Physics
The ratio of relative strengths of the gravitational force and the electromagnetic force between two charged particles is
The efficiency of an engine is given by $\eta=\frac{\alpha \beta}{\sin \theta} \cdot \log _e \frac{\beta x}{k T}$, where $\alpha$ and $\beta$ are constants. If $T$ is the absolute temperature, $k$ is Boltzmann constant, $\theta$ is angular displacement and $x$ is distance, then the incorrect statement is
A bird flies with a velocity $(t-2) \mathrm{ms}^{-1}$ along a straight line, where $t$ is the time in seconds. The distance covered by it in a time of 4 seconds is
A car is travelling with linear velocity $v$ on a circular road of radius $r$. If its velocity is increasing at a rate of $a \mathrm{~ms}^{-2}$, then the resultant acceleration will be
Two blocks of masses $w_1$ and $w_2$ are suspended from the ends of a light string passing over a smooth fixed pulley. If the pulley is pulled up with an acceleration $g$, then the tension in the string will be
A body is moved along a straight line by an engine which delivers a constant power. The distance moved by the body in time $t$ is proportional to
A body of mass 3 kg is moving under the action of a force which causes a displacement of $\left(t^3 / 3\right) \mathrm{m}$, where $t$ is time in seconds. The work done by the force in first 2 sec is
Two blocks of masses 2 kg and 1 kg are tied to the ends of a string which passes over a light frictionless pulley. The blocks are held at the same horizontal level and then released suddenly. The distance traversed by their centre of mass in 2 sec is
(acceleration due to gravity $=10 \mathrm{~ms}^{-2}$ )
A particle of mass $m$ is moving along a line $y=x+a$ with a constant velocity $v$. The angular momentum of the particle about the origin is
A force of 6.4 N stretches a vertical spring by 0.1 m . If it were to oscillate with a period of $\pi / 4$, then the mass that is to be suspended from the spring is
The ratio of orbital velocity of a body near to the surface of a planet and escape velocity of a body from the surface of the same planet is
The length of a metal rod at $30^{\circ} \mathrm{C}$ is 30 cm . If its temperature is raised to $105^{\circ} \mathrm{C}$, its length is increased by 0.027 cm . Then, the coefficient of linear expansion of the metal is
The angle of contact is $120^{\circ}$ when a cylindrical rod is vertically placed in a liquid. If the same rod is placed horizontally in the liquid, then the angle of contact is
In a well the pressure at a point 10 m below the surface of water is $\left(g=10 \mathrm{~ms}^{-2}\right)$
The heat energy required to convert 10 kg of ice at $-10^{\circ} \mathrm{C}$ into water at $0^{\circ} \mathrm{C}$ is (specific heat capacity of ice $=0.5 \mathrm{calg}^{-1}$ and latent heat of fusion of ice $=80 \mathrm{calg}^{-1}$ )
If the reading in Fahrenheit scale is twice the reading in Celsius scale, then the reading in Fahrenheit scale is
When some amount of heat energy is supplied to a monatomic gas, the percentage of heat energy used for increasing the internal energy of the gas $(\gamma=5 / 3)$ is
The average energy possessed by an oscillator at a temperature 300 K is (Boltzmann constant $=1.38 \times 10^{-23} \mathrm{JK}^{-1}$ )
A wave is given by $y=5 \times 10^{-3} \sin \left(12.5 \pi x-\frac{\pi}{2} t\right)$. Then its wavelength and time period are respectively ( $y$ and $x$ are in metres and $t$ is in seconds)
A tuning fork $A$ of frequency 250 Hz and another tuning fork $B$ of frequency $x$ produced 5 beats per second when vibrated together. If the fork $B$ is waxed and vibrated together with $A$, then 3 beats per second are produced. Then, $x=$
A convex lens forms a real image of a point object placed on its principal axis. If the upper half of the lens is painted black, then
Two slits separated by a distance of 1 mm are illuminated with light of wavelength $6.5 \times 10^{-7} \mathrm{~m}$. The interference fringes are observed on a screen placed at 1 m from the slits. The distance between the third dark fringe and the fifth bright fringe is equal to
Two conducting spheres of radii $r_1$ and $r_2$ are charged to the same surface charge density. The ratio of electric fields near their surfaces is
Two electric charges $+2 \mu \mathrm{C}$ and $-4 \mu \mathrm{C}$ are separated by a distance 3 m in air. At a point $P$ located on the line joining the two charges and in between them, the electric potential is zero. Then the electric field at a point $P$ (in $\mathrm{NC}^{-1}$ ) is
If the masses of three wires of same material are in the ratio of $1: 2: 3$ and their lengths are in the ratio of $3: 2: 1$, then electrical resistances of these wires are in the ratio
As shown in the figure, in a Wheatstone's bridge, three resistances $P, Q$ and $R$ are connected in the three arms and the fourth arm is formed by two resistances $S_1$ and $S_2$ connected in parallel. The condition for the bridge to be balanced is
A current $i$ flows in an infinitely long, straight and thin walled pipe, then
A closely wound solenoid of 80 cm long has 5 layers of windings of 400 turns each. The diameter of the solenoid is 1.8 cm . If the current carried is 8 A , then the magnitude of the magnetic field inside the solenoid near its centre is approximately
The period of oscillation of a bar magnet at a place is 2 s . At the same place, the period of oscillation of another identical bar magnet whose magnetic moment is 4 times so that of first magnet is
The self inductance of a coil depends on
A conducting circular coil is place in a uniform magnetic field with the magnetic field initially directed perpendicular to the plane of the coil. In step $A$, the coil is rotated from its initial position by $60^{\circ}$ about its diameter in time $t$. In step $B$, the coil is further rotated about the same axis in the same sense by another $120^{\circ}$ in time $2 t$. Ratio of emf induced in the coil in step $A$ to that in step $B$ is
An alternating emf given by the equation $E=200 \sin (50 \pi t)$ (where, $E$ is in volts and $t$ is in seconds) is applied across a series combination of an inductor and a resistor having inductive reactance $40 \Omega$ and resistance $30 \Omega$ respectively. At time $t=1 \mathrm{~s}$, the power dissipated by the resistor is close to $\left(\cos 53^{\circ}=0.6\right)$
The speed of electromagnetic waves in a medium is $1.5 \times 10^8 \mathrm{~ms}^{-1}$. If relative permittivity of that medium is 2 , then its magnetic susceptibility is (speed of light in vacuum is $3 \times 10^8 \mathrm{~ms}^{-1}$ ).
Consider two black bodies $A$ and $B$ having equal surface area. On the surface of $A, n$ photons of frequency $f$ are incident perpendicularly in a time $t$. On the surface of $B$, $2 n$ photons of frequency $3 f$ are incident perpendicularly in a time $4 t$. The ratio of average intensity of radiation on surface $A$ to that on surface $B$ is
A photon released by the transition of an electron from the second excited state to the ground state of Hydrogen atom is incident on the surface of a metal of work function 3.1 eV . The de-Broglie wavelength of the most energetic electron emitted from that metal surface is nearly
The radius of a nucleus of mass number 27 is $R$. Which of the following is true about a nucleus whose radius is $2 R$ ?
The nucleus ${ }_{50}^{120} X$ undergoes the series of reactions given below:
$$ { }_Z^A X \xrightarrow{\alpha \text {-decay }} P \xrightarrow{\beta^{-} \text {-decay }} Q \xrightarrow{\alpha \text {-decay }} R $$
The number of neutrons in the nucleus $R$ is
In the logic circuit given below, if $X=1$ and $Y=1$, then the values of $P, Q$ and $R$ are
The symbol given below represents
A telephonic communication service is working at a carrier frequency of 20 GHz . Only $20 \%$ of it is utilised for transmission. If each channel requires a bandwidth of 5 kHz , then the number of telephonic channels that can be transmitted simultaneously are