Chemistry
The degeneracy of the level of hydrogen atoms that contain the energy of $\left(\frac{-R_{\mathrm{H}}}{16}\right)$ is
Wavelength of $\mathrm{H}^{+}$ion with kinetic energy 1.65 eV is (mass of proton $=1.6726 \times 10^{-27} \mathrm{~kg}$ )
Assertion (A) $\mathrm{Mg}^{2+}$ and $\mathrm{Al}^{3+}$ are isoelectronic but the magnitude of ionic radius of $\mathrm{Al}^{3+}$ is less than that in $\mathrm{Mg}^{2+}$.
Reason (R) The effective nuclear charge on the outermost electrons in $\mathrm{Al}^{3+}$ is greater than that in $\mathrm{Mg}^{2+}$.The correct option among the following is
The successive ionisation energy values for an element ' $X$ ' are given below :
(i) 1st ionisation energy $=410 \mathrm{~kJ} \mathrm{~mol}^{-1}$
(ii) 2nd ionisation energy $=820 \mathrm{~kJ} \mathrm{~mol}^{-1}$
(iii) 3rd ionisation energy $=1100 \mathrm{~kJ} \mathrm{~mol}^{-1}$
(iv) 4th ionisation energy $=1500 \mathrm{~kJ} \mathrm{~mol}^{-1}$
(v) 5th ionisation energy $=3200 \mathrm{~kJ} \mathrm{~mol}^{-1}$
$$ \text { Match the following : } $$
| List-I | List-II | ||
| A. | I. | Tetrahedral | |
| B. | II. | Trigonal planar | |
| C. | III. | T-shape | |
| D. | IV. | Trigonal pyramidal | |
$$ \text { The correct match is } $$
Which of the following molecules does not exist according to molecular orbital theory?
Root mean square ( rms ) speed of $\mathrm{O}_2$ is $500 \mathrm{~m} / \mathrm{s}$ at a constant temperature. Calculate the rms speed and the average kinetic energy of $\mathrm{H}_2$ at the same temperature. (Consider, $R=8.33 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}$ )
Which of the following describes an ideal gas?
(i) The volume occupied by a gas molecule is negligible.
(ii) The collision between ideal gases are elastic.
(iii) Particles are very small compared to the distance between each other.
What is the $\%$ strength of 22.4 volume of $\mathrm{H}_2 \mathrm{O}_2$ solution?
$\mathrm{KMnO}_4$ oxidises $\mathrm{C}_2 \mathrm{H}_2 \mathrm{O}_4$ to form $\mathrm{CO}_2$. In which of the following, the reaction will be faster?
$\Delta H$ and $\Delta S$ for a reaction are $+30.0 \mathrm{~kJ} \mathrm{~mol}^{-1}$ and 0.06 $\mathrm{kJK}^{-1} \mathrm{~mol}^{-1}$ at 1 atm pressure. The temperature at which free energy change is equal to zero and nature of the reaction below this temperature are
The vapour density of $\mathrm{N}_2 \mathrm{O}_4$ in $\mathrm{N}_2 \mathrm{O}_4 \rightleftharpoons 2 \mathrm{NO}_2$ is 40 . The degree of dissociation is
What is the equilibrium constant $\left(K_C\right)$ for the given reaction?
$$ \mathrm{N}_2+\mathrm{O}_2 \rightleftharpoons 2 \mathrm{NO} $$
Where the equilibrium concentration of $\mathrm{N}_2, \mathrm{O}_2$ and NO are found to be $4 \times 10^{-3}, 3 \times 10^{-3}$ and $3 \times 10^{-3} \mathrm{M}$ respectively.
Hard water contains ion of
Predict the feasibility of the given reactions in aqueous solution
(i) $\mathrm{Be}(\mathrm{OH})_2+2 \mathrm{OH}^{-} \longrightarrow\left[\mathrm{Be}(\mathrm{OH})_4\right]^{2-}$
(ii) $\mathrm{Be}(\mathrm{OH})_2+2 \mathrm{H}^{+} \longrightarrow\left[\mathrm{Be}(\mathrm{OH})_4\right]^{2+}$
What is the nature of the bonding in anhydrous $\mathrm{AlCl}_3$ and hydrated $\mathrm{AlCl}_3$ respectively?
Which of the following elements reacts with water?
Biochemical oxygen demand (BOD) is a measure of organic materials present in water, BOD value less than 5 ppm indicates a water sample to be
Sodium fusion extract of aniline when heated with ferrous sulphate solution and then acidified with concentrated $\mathrm{H}_2 \mathrm{SO}_4$ from which of the following complexes?
Ethyl phenyl acetylene (1-phenyl-1-butyne) on reduction with partially deactiveated palladised charcoal (Lindlar's catalyst) gives
Which one of the following methods is suitable to generate aromatic compound(s) from linear aliphatic saturated hydrocarbons with at least six carbon atoms?
Copper crystallises in ccp arrangement and accepted value of metal ion radius was found to be $1.14 \mathop {\rm{A}}\limits^{\rm{o}}$. Calculate the density of copper in grams per cubic centimetre. (Atomic weight of copper is 64 , $N_A=6 \times 10^{23}$ )
$A^{2+}, B^{2+}$ and $C^{-}$form an ionic complex like $A_{x-2}\left[B(C)_x\right]_2$. If the complex is $75 \%$ dissociated in a solvent with $i=4$, the coordination number of $B$ is
The freezing point of equimolal aqueous solution will be highest for
The standard electrode potentials of $\mathrm{Ag}^{+} / \mathrm{Ag}$ is +0.80 V and $\mathrm{Cu}^{+} / \mathrm{Cu}$ is +0.34 V . If these electrodes are connected through a salt-bridge, which of the following statements is correct?
For a zero order reaction, the plot of concentration of reactant vs time is (Hint: Consider the intercept on the concentration axis)
The gold numbers of gelatin, haemoglobin and sodium acetate are $5 \times 10^{-3}, 5 \times 10^{-2}$ and $7 \times 10^{-1}$, respectively. The protective actions will be in the order
Which one of the following ores does not contain iron?
When copper metal is treated with cold and dilute nitric acid, it forms
Which one of the following is not a colourless compound?
Which of the following statement is not true about interstitial complexes?
Which of the following molecules is colourless?
Which of the following vinyl derivatives is the most reactive towards anionic polymerisation?
Amino acids containing heterocyclic ring are
(i) Histidine
(ii) Valine
(iii) Arginine
(iv) Proline
Which among the following is an arsenic based antibiotic drug, for which Paul Ehrlich was awarded Noble prize in 1908.
$$ \text { The major product in the following reaction sequence is } $$
$$ \text { Which of the following gives alcohol/phenol products? } $$
$$ \text { The major products } P \text { and } Q \text { in the following reactions are } $$
$$ \text { In the following reactions, } P, Q \text { and } R \text { are } $$
$$ \text { Following transformation can be accomplished by } $$
Mathematics
Let $f:[0,10] \rightarrow[1,20]$ be a function defined as
$$ f(x)=\left\{\begin{array}{ll} \frac{60-5 x}{3}, & 0 \leq x \leq 6 \\ 10, & 6 \leq x \leq 7 \\ 31-3 x, & 7 \leq x \leq 10 \end{array} \text { then } f\right. \text { is } $$
The domain of the function, $f(x)=\sqrt{\log _{10}\left(\frac{5 x-x^2}{4}\right)}$ is
Let the greatest common divisor of $m, n$ be 1 . If $\frac{1}{1 \cdot 7}+\frac{1}{7 \cdot 13}+\frac{1}{13 \cdot 19}+\ldots \ldots$. upto 20 terms $=\frac{m}{n}$, then $5 m+2 n=$
If $A, B$ are two non singular matrices of order $3,|B|=k$, a positive integer, then match the items of list-I with the items of list-II.
| $$ \text { List-I } $$ |
$$ \text { List-II } $$ |
||
|---|---|---|---|
| A. | $\quad\left|k^{-1} A^{-1}\right|$ | I. | $$ B A^k+A^k B $$ |
| B. | $\left|\operatorname{Adj}\left(A^{-1}\right)\right|$ | II. | $$ \frac{B \operatorname{Adj}(B)}{|B|} $$ |
| C. | $B A B^{-1}=I, \Rightarrow B A^k B^{-1}=$ | III. | $$ \frac{1}{|B|^3|A|} $$ |
| D. | $\quad \operatorname{Adj}\left(\operatorname{Adj}\left(A^{-1}\right)\right)=$ | IV. | $$ \frac{1}{|A|}\left(A^{-1}\right) $$ |
| V. | $$ \frac{1}{|A|^2} $$ |
||
$$ \text { The correct match is } $$
All the real values of $p, q$ so that the system of equations
$$ 2 x+p y+6 z=8, x+2 y+q z=5 $$
and $\quad x+y+3 z=4$
may have no solution are
If $p$ and $q$ are two distinct real values of $\lambda$ for which the system of equations
$$ \begin{array}{r} (\lambda-1) x+(3 \lambda+1) y+2 \lambda z=0 \\ (\lambda-1) x+(4 \lambda-2) y+(\lambda+3) z=0 \\ 2 x+(3 \lambda+1) y+3(\lambda-1) z=0 \end{array} $$
has non-zero solution, then $p^2+q^2-p q=$
Let $z=x+i y$ be a complex number, $A=\{z /|z| \leq 2\}$ and $B=\{z /(1-i) z+(1+i) \bar{z} \geq 4\}$ Then which one of the following options belongs to $A \cap B$ ?
The solutions of the equation $z^2\left(1-z^2\right)=16, z \in \mathbf{C}$, lie on the curve
If $z, \bar{z},-z,-\bar{z}$ forms a rectangle of area $2 \sqrt{3}$ square units, then one such $z$ is
$$ \left(\frac{\cos \theta+i \sin \theta}{\sin \theta+i \cos \theta}\right)^8+\left(\frac{1+\cos \theta-i \sin \theta}{1+\cos \theta+i \sin \theta}\right)^{16}= $$
Let $S$ be the set of all possible integral values of $\lambda$ in the interval $(-3,7)$ for which the roots of the quadratic equation $\lambda x^2+13 x+7=0$ are all rational numbers. Then the sum of the elements in $S$ is
$\alpha$ is the maximum value of $1-2 x-5 x^2$ and $\beta$ is the minimum value of $x^2-2 x+r$. If $5 \alpha x^2+\beta x+6>0$ for all real values $x$, then the interval in which $r$ lies is
For the equation $x^4+x^3-4 x^2+x-1=0$ the ratio of the sum of the squares of all the roots to the product of the distinct roots is
If $\alpha_1, \beta_1, \gamma_1, \delta_1$ are the roots of the equation $a x^4+b x^3+c x^2+d x+e=0$ and $\alpha_2, \beta_2, \gamma_2, \delta_2$ are the roots of the equation $e x^4+d x^3+c x^2+b x+a=0$ such that $0<\alpha_1<\beta_1<\gamma_1<\delta_1, 0<\alpha_2<\beta_2<\gamma_2<\delta_2$, $\alpha_1-\delta_2=2=\beta_1-\gamma_2 ; \gamma_1-\beta_2=\delta_1-\alpha_2=4$, then $a+b+c+d+e=$
The total number of three digit and five digit integers which can be formed by using the digits $0,1,2,3,4,5$ but using each digit not more than once in each number is
At an election a voter may vote for any number of candidates not exceeding the number to be elected. If 4 candidates are to be elected out of the 12 contested in the election and voter votes for at least one candidate, then the number of ways in which a voter can vote is
Let $x \in \mathbf{R}$ be so small that the powers of $x$ beyond two are insignificant and negligibly small. For such $x$, if $(1-x)^3(2+x)^6$ is approximated by $a+b x+c x^2$, then $a+b+c=$
For $0 < x < 1$, the expansion of $\left(1+\frac{1}{x}\right)^{\frac{1}{2}}$ is
If $\frac{4 x^2+5 x^4+7}{\left(x^2+1\right)\left(x^4+x^2+1\right)}=\frac{A x+B}{x^2+1} +\frac{C x^3+D x^2+E x+F}{x^4+x^2+1}$, then $B+2(D+F+E)-C \cdot A=$
$$ \begin{aligned} \sin ^4 \frac{\pi}{8}+\cos ^4 \frac{3 \pi}{8}-\sin ^4 \frac{3 \pi}{8} & +\sin ^4 \frac{5 \pi}{8} +\cos ^4 \frac{7 \pi}{8}-\sin ^4 \frac{7 \pi}{8}= \end{aligned} $$
$$ \operatorname{cosec}^{-1}\left[\left(\frac{\tan ^2\left(\frac{\alpha-\pi}{4}\right)-1}{\tan ^2\left(\frac{\alpha-\pi}{4}\right)+1}+\cos \frac{\alpha}{2} \cdot \cot 5 \alpha\right) \sec \frac{11 \alpha}{2}\right] $$
Assertion (A) If $A=15^{\circ}, B=17^{\circ}$ and $C=13^{\circ}$, then $\cot 2 A+\cot 2 B+\cot 2 C=\cot 2 A \cot 2 B \cot 2 C$
Reason (R) In a $\triangle P Q R$,
$$ \tan \frac{P}{2} \tan \frac{Q}{2}+\tan \frac{Q}{2} \tan \frac{R}{2}+\tan \frac{P}{2} \tan \frac{R}{2}=1 $$
The correct option among the following is
The solution set of the trigonometric equation $\tan \theta+5 \cot \theta=\sec \theta$ is
If $\tan ^{-1} \frac{1}{5}+\frac{1}{2} \sec ^{-1} x+\tan ^{-1} \frac{1}{8}=\frac{\pi}{8}$, then $x^2=$
Assertion $(\mathrm{A}) \operatorname{cosech}^{-1}(3)=\log \left(\frac{1+\sqrt{10}}{3}\right)$
Reason (R) $e^{\operatorname{cosech}^{-1} x}$ is a root of the quadratic equation $x p^2-2 p-x=0$
The correct option among the following is
In a $\triangle A B C$ if $\angle A=3 \angle B, C A=9$ and $B C=16$, then the length of $A B$ is
In $\triangle A B C, \frac{1+\cos C}{r_1+r_2}+\frac{1+\cos A}{r_2+r_3}+\frac{1+\cos B}{r_1+r_3}=$
In a triangle $A B C$, if $\cos A \cos B+\sin A \sin B \sin C=1$, then $a: b: c=$
Let $\mathbf{O A}=\mathbf{a}, \mathbf{O B}=\mathbf{b}$ be two non collinear vectors,
$\mathbf{O P}=x_1 \mathbf{a}+y_1 \mathbf{b}, \mathbf{O Q}=x_2 \mathbf{a}+y_2 \mathbf{b}$ and $\mathbf{A}^{\prime} \mathbf{O}=\mathbf{O A}$,
$\mathbf{B}^{\prime} \mathbf{O}=\mathbf{O B}$. If $x_1=\frac{-3}{4}, x_2=\frac{1}{3}, y_1=\frac{7}{4}, y_2=\frac{5}{3}$, then
The position vector of a point $P$ is $2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ and $\mathbf{a}=-\hat{\mathbf{i}}-2 \hat{\mathbf{k}}, \mathbf{b}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ are two vectors which determine a plane $\pi$. The equation of a line through $P$ normal to $\mathbf{b}$ and lying on the plane $\pi$ is
In a quadrilateral $A B C D$, the point $P$ divides $D C$ in the ratio $1: 3$ internally and $Q$ is the mid-point of $A C$. If $\mathbf{A B}+\mathbf{A D}+\mathbf{B C}-2 \mathbf{D C}=\lambda \mathbf{P Q}$, then the value of $\lambda$ is
$\mathbf{p}=2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{q}=\hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}$. If the vectors $\mathbf{a}$ and $\mathbf{b}$ are the orthogonal projections of $\mathbf{p}$ on $\mathbf{q}$ and $\mathbf{q}$ on $\mathbf{p}$ respectively, then $\frac{\mathbf{a} \times \mathbf{b}}{\mathbf{a} \cdot \mathbf{b}}=$
Let $\mathbf{a}=2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}, \mathbf{b}=7 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}, \mathbf{c}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$. The vector $\mathbf{x}$ such that $\mathbf{x} \cdot \mathbf{c}=60$ and perpendicular to both $\mathbf{a}, \mathbf{b}$ is
The shortest distance between the line $\mathbf{r}=2 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}+\lambda(\hat{\mathbf{i}}-\hat{\mathbf{j}}+4 \hat{\mathbf{k}})$ and the plane $\mathbf{r} \cdot(\hat{\mathbf{i}}+5 \hat{\mathbf{j}}+\hat{\mathbf{k}})=5$ is
For the following frequency distribution, the variance is approximately equal to
$$ \begin{array}{cccccc} \hline \begin{array}{c} \text { Class } \\ \text { Interval } \end{array} & 0-5 & 5-10 & 10-15 & 15-20 & 20-25 \\ \hline \text { Frequency } & 4 & 1 & 10 & 3 & 2 \\ \hline \end{array} $$
If the mean of the discrete distribution $8,9,6,5, x, 4$, 6, 5 is 6 , then its standard deviation (nearest to two decimal places) is
If $A$ and $B$ are events of a sample space such that $P(A \cup B)=\frac{3}{4}, P(A \cap B)=\frac{1}{4}$ and $P(\bar{A})=\frac{2}{3}$, then $P(\bar{A} \cap B)$ is
Let $X$ and $Y$ be two events of a sample space such that $P(X)=\frac{1}{3}, P(X / Y)=\frac{1}{2}$ and $P(Y / X)=\frac{2}{5}$ then
Let $A$ and $B$ be not mutually exclusive events. If $P(A)=\frac{4}{9}, P(A \cap \bar{B})=\frac{3}{7}$ then $P\left(\frac{B}{A}\right)=$
If $20 \%$ of the bolts produced by a machine are defective then the probability that out of 4 bolts chosen at random, less than 2 bolts will be defective, is
In a book consisting of 600 pages, there are 60 typographical errors. The probability that a randomly chosen page will contain at most two errors, is
If $M$ is the foot of the perpendicular drawn from the origin $O$ on to the variable line $L$, passing through a fixed point $(a, b)$, then the locus of the mid-point of $O M$ is
When the origin is shifted to the point $\left(\frac{3}{2}, \frac{3}{2}\right)$ by the translation of coordinate axes, then the transformed equation of $32 x^2+8 x y+32 y^2-108 x-108 y+99=0$ is
A line $L_1$ passing through $A(3,4)$ and having slope 1 cuts another line $L_2$ passing through $C$ at $B$, such that $A B=A C$. If the equation of line $B C$ is $2 x-y+4=0$, then the equation of $A C$ is
Angles made with the $X$-axis by the two lines passing through the point $P(1,2)$ and cutting the line $x+y=4$ at a distance $\frac{\sqrt{6}}{3}$ units from the point $P$ are
The straight lines $x+3 y-9=0,4 x+5 y-1=0$, $p x+q y+10=0$ are concurrent, if the line $5 x+6 y+10=0$ passes through the point
The straight lines $x+3 y-9=0,4 x+5 y-1=0$, $p x+q y+10=0$ are concurrent, if the line $5 x+6 y+10=0$ passes through the point
The centroid of the triangle formed by the lines $x+y=1$ and $2 y^2-x y-6 x^2=0$ is
If the polar of a point $P$ with respect to a circle of radius $r$ which touches the coordinate axes and lies in the first quadrant is $x+2 y=4 r$, then the point $P$ is
If the circles $x^2+y^2-2 x-2(3+\sqrt{7}) y+8+6 \sqrt{7}=0$ and $x^2+y^2-8 x-6 y+k^2=0, k \in \mathbf{Z}$, have exactly two common tangents, then the number of possible values of $k$ is
The circle $S=0$ cuts the circles
$C_1=x^2+y^2-8 x-2 y+16=0$ and $C_2=x^2+y^2-4 x-4 y-1=0$ orthogonally. If the common chord of $S=0$ and $C_1=0$ is $2 x+13 y-15=0$, then the centre of $S=0$ is
The equation of the circle passing through the points of intersection of the two orthogonal circles $S_1=x^2+y^2+k x-4 y-1=0$, $S_2=3 x^2+3 y^2-14 x+23 y-15=0$ and passing through the point $(-1,-1)$ is
Consider the parabola $y^2+2 x+2 y-3=0$ and match the items of List-I with those of the List-II.
$$ \begin{array}{llll} \hline & \text { List-I } & & \text { List-II } \\ \hline \text { A. } & 2 x-5=0 & \text { I. } & \text { Vertex } \\ \hline \text { B. } & \left(\frac{3}{2},-1\right) & \text { II. } & \text { Focus } \\ \hline \text { C. } & y+1=0 & \text { III. } & \text { Equation of directrix } \\ \hline \text { D. } & (2,-1) & \text { IV. } & \text { Equation of the axis } \\ \hline & & \text { V. } & \text { Equation of the Latus rectum } \\ \hline \end{array} $$
$$ \text { The correct match is } $$The normal at a point on the parabola $y^2=4 x$ passes through $(5,0)$. If there are two more normals to this parabola which pass through $(5,0)$, the centroid of the triangle formed by the feet of these three normals is
The eccentricity of an ellipse passing through $(3 \sqrt{2}, \sqrt{10})$ with foci at $(-4,0)$ and $(4,0)$ is
If the product of the lengths of the perpendiculars drawn from the foci to the tangent $y=\frac{-3}{4} x+3 \sqrt{2}$ of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is 9 , then the eccentricity of that ellipse is
The equation of the hyperbola, whose eccentricity is $\sqrt{2}$ and whose foci are 16 units apart, is
If the points $A(-1,0,7), B(3,2, t), C(5, k,-2)$ are collinear, then the ratio in which the point $P(t, k-2 t, t+k)$ divides the line segment $B C$ is
The direction cosines $l, m, n$ of two lines are satisfying $3 l+m+5 n=0$ and $6 m n-2 n l+5 l m=0$. If $\theta$ is the angle between those lines then $|\cos \theta|=$
A tetrahedron has vertices $O(0,0,0), A(1,2,1)$, $B(2,1,3), C(-1,1,2)$. If $\theta$ is the angle between the faces $O A B$ and $A B C$, then $\cos \theta=$
If $\log (1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\ldots \ldots \infty$ and $\mathop {\lim }\limits_{x \to 0} \frac{\log (1+x)^{1+x}}{x^2}-\frac{1}{x}=k$, then $12 k=$
If $f(x)=\left\{\begin{array}{ll}k, & \text { for } x=1 \\ \frac{(9 x-1)(\sqrt{x}-1)}{3 x^2+2 x-5}, & \text { for } x \neq 1\end{array}\right.$ is continuous on $[0, \infty)$, then $k=$
let $g(x) \neq 0, g^{\prime}(x) \neq 0, f(x) \neq 0, f^{\prime}(x) \neq 0$. If
$F(x)=f(x) g(x), G(x)=f^{\prime}(x) g^{\prime}(x)$ and
$F^{\prime}(x)=G(x) H(x)=F(x) K(x)$, then $H(x)+K(x)=$
If $y=\frac{x \sin ^{-1} x}{\sqrt{1-x^2}}+\log \sqrt{1-x^2}$, then $\frac{d y}{d x}=$
Let $f(x)$ and $g(x)$ be twice differentiable functions such that $f(x)=x^2+g^{\prime}(1) x+g^{\prime \prime}(2)$ and $g(x)=f(1) x^2+x f^{\prime}(x)+f^{\prime \prime}(x)$. Then $f(x)-g(x)=$
If the area of a circle increases at the rate of $\frac{1}{\sqrt{\pi}}$ sq. units/sec, then the rate (in units/sec) at which the perimeter of the circle changes, when perimeter is $\sqrt{\pi}$ units, is
Let $a$ be a fixed positive real number and $n$ be an arbitrary constant. For the curve $y=\frac{x^n}{a^{n-1}}$, if the length of the subnormal at any point $(\alpha, \beta)$ is proportional to $a^2$, then $n=$
In each of the choices given below, a function and an interval are given. The correct choice having a function and the associated interval for which the Lagrange's mean value theorem is not valid is
Let $P(x)$ be a polynomial of degree 3 having extreme value at $x=1$. If $\mathop {\lim }\limits_{x \to 0}\left(\frac{P(x)+4}{x^2}+2\right)=6$, then $\left(\frac{d P}{d x}\right)_{x=\frac{1}{2}}=$
$$ \int \frac{y^2+\sqrt[3]{y^4}+\sqrt[6]{y^2}}{y\left(1+\sqrt[3]{y^2}\right)} d y= $$
For $k \in(1, \infty), \int \frac{1}{1+k \cos x} d x=$
$$ \int e^{-3 x}\left(x^2+\sin 4 x\right) d x= $$
If $\int \frac{2 x^{12}+5 x^9}{\left(1+x^3+x^5\right)^3} d x=\frac{x^m}{l\left(1+x^3+x^5\right)^r}+C$, then $\frac{m-l}{r}=$
$$ \mathop {\lim }\limits_{x \to \infty }\left[\left(1+\frac{1}{n^2}\right)\left(1+\frac{2^2}{n^2}\right) \ldots \ldots\left(1+\frac{n^2}{n^2}\right)\right]^{1 / n}= $$
$$ \int_{\pi / 4}^{\pi / 2} \frac{3 d x}{1+e^{\sqrt{8} \sin \left(x-\frac{3 \pi}{8}\right)}}= $$
If the area of the region bounded by $y=\cos x, y=\sin x$, $x=\pi / 4$ and $x=\pi$ is bisected by the line $x=a$, then $\sin \left(a+\frac{\pi}{4}\right)=$
If the family of curves $y=a e^{4 x}+b e^{-x}$, where $a, b$ are arbitrary constants represents the general solution of the differential equation
$$ f\left(x, y \frac{d y}{d x}, \frac{d^2 y}{d x^2}\right)=0, \text { then } \frac{d f}{d x}= $$
If the length of the sub tangent at any point $p(x, y)$ on a curve $f(x, y)=0$ is $x+7 y^2$, then $f(x, y)=$
If the general solution of the differential equation $(y-x+1) d y-(y+x+2) d x=0$ is $f(x, y, c)=0$, then the value of $c$ such that $f(1,1, c)=0$ is
Physics
The nuclear forces are
Due to an explosion underneath water, a bubble started oscillating. If this oscillation has time period $T$,which is proportional to $p^\alpha S^\beta E^\gamma$, where $p$ is static pressure, $S$ is density of water and $E$ is total energy of explosion. Determine $\alpha, \beta$ and $\gamma$.
A car travelling at $15 \mathrm{~m} / \mathrm{s}$ overtake another car travelling at $10 \mathrm{~m} / \mathrm{s}$. Assuming, each car is 4 m long. What is the time taken during the overtake?
If a body moving in a circular path maintains constant speed of $10 \mathrm{~ms}^{-1}$, then which of the following correctly describes the relation between acceleration (a) and radius $(r)$ ?
If $0.5 \hat{\mathbf{i}}+0.8 \hat{\mathbf{j}}+c \hat{\mathbf{k}}$ is a unit vector, then $c$ is
A projectile is launched from point $A$ of the given landscape with a water body as shown in the diagram. The launching angle is $15^{\circ}$. From the following, identify the right initial velocity of the projectile with which it will fall somewhere in between the points $C$ and $D$. [Assume, $g=10 \mathrm{~m} / \mathrm{s}^2$ ]
When a bullet is fired from a rifle its momentum becomes $20 \mathrm{~kg}-\mathrm{ms}^{-1}$. If the velocity of the bullet is $1000 \mathrm{~ms}^{-1}$, then what is its mass?
A block is between two surfaces as shown in the figure. Find the normal reaction at both surfaces. [Assume, $g=10 \mathrm{~m} / \mathrm{s}^2$ ]
When a body is acted upon by a resultant force, then the work done by the resultant force is equal to
A force acts on a body of mass 10 kg , resulting in its displacement given as $x=\left(\frac{t^3}{25}\right) \mathrm{m}$, where $t$ is the time in seconds. The work done by the force in 5 s is
A bullet of mass 25 g moves horizontally at a speed of $250 \mathrm{~m} / \mathrm{s}$ is fired into a wooden block of mass 1 kg suspended by a long string. The bullet crosses the block and emerges on the other side. If the centre of the mass of the block rises through a height of 20 cm . The speed of the bullet as it emerges from the block is (take, $g=10 \mathrm{~m} / \mathrm{s}^2$ )
A circular hole of radius 3 cm is cut out from a uniform circular disc of radius 6 cm . The centre of the hole is at 3 cm , from the centre of the original disc. The distance of centre of gravity of the resulting flat body from the centre of the original disc is
For a particle executing SHM, determine the ratio of average acceleration of the particle between extreme position and equilibrium position w.r.t. the maximum acceleration.
Choose the correct statement.
A slab of side 50 cm and thickness 10 cm is subjected to a shearing force of $10^5 \mathrm{~N}$ on its narrow edge. If the lower edge is riveted to the floor and upper edge is displaced by 0.2 mm , then shear modulus of the material of the slab is
A meniscus drop of radius 1 cm is sprayed into $10^6$ droplets of equal size. Calculate the energy expended if surface tension of mercury is $435 \times 10^{-3} \mathrm{~N} / \mathrm{m}$.
The specific heat of helium at constant volume is 12.6 J $\mathrm{mol}^{-1} \mathrm{~K}^{-1}$. The specific heat of helium at constant pressure in $\mathrm{J} \mathrm{mol}^{-1} \mathrm{~K}^{-1}$ is approximately (assume, the universal gas constant, $R=8.314 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}$ )
A composite slab is prepared with two different materials $A$ and $B$. The relation between their coefficient of thermal conductivity and thickness is given as $K_A=\frac{K_B}{2}$ and $X_A=2 X_B$, respectively. If the temperature of faces of $A$ and $B$ are $75^{\circ} \mathrm{C}$ and $50^{\circ} \mathrm{C}$ respectively, what will be the temperature of common surface?
Work done on heating one mole of monoatomic gas adiabatically through $20^{\circ} \mathrm{C}$ is $W$. Then, the work done on heating 6 moles of rigid diatomic gas through the same change in temperature
If a gas has $n$ degrees of freedom, then the ratio of $\frac{C_p}{C_V}$ is
A bus moving with an uniform speed of $72 \mathrm{~km} / \mathrm{h}$ towards a building blows a horn of frequency 1.7 kHz . If speed of sound in air is $340 \mathrm{~m} / \mathrm{s}$, what will be the frequency of echo heard by bus driver?
If the image of an object is at the focal point $f$ to the right side of a convex lens, the position of the object on the left of the lens is at
On using red light $(\lambda=6600 \mathop {\rm{A}}\limits^{\rm{o}})$ in Young's double slit experiment, 60 fringes are observed in the field of view. If violet light ( $\lambda=4400 \mathop {\rm{A}}\limits^{\rm{o}}$ ) is used, the number of fringes observed will be
Young's double slit experiment is carried out by using green, red and blue light, one colour at a time. The fringe width recorded are $\beta_{G^{\prime}} \beta_{R^{\prime}} \beta_B$ respectively, then
If a proton is moved against the coulomb force of an electric field, then
Assume each oil drop consists of a capacitance of $C$. If combine $n$ drops to form a bigger drop, then the capacitance of bigger drop $C^{\prime}$ would be
A conductor of length 100 cm and area of cross-section $1 \mathrm{~mm}^2$ carries a current of 5 A . If the resistivity of the material of the conductor is $3.0 \times 10^{-8} \Omega-\mathrm{m}$, then the electric field across the conductor is
If the Wheatstone's bridge with four resistors $R_1, R_2$ and $R_3, R_4$ is balanced, then the correct expression is
A circular coil of 10 turns and radius 10 cm is placed in a uniform magnetic field of 0.1 T normal to the plane of the coil. If the current in the coil is 5 A , then the magnitude of the torque on the coil is
A 50 cm long solenoid has winding of 400 turns. What current must pass through it to produce a magnetic field of induction $4 \pi \times 10^{-3} \mathrm{~T}$ at the centre?
If relative permeability of iron is 5500 , then its susceptibility is
A moving coil galvanometer of resistance $100 \Omega$ is used as an ammeter using a resistance $0.1 \Omega$. The maximum deflection current in the galvanometer is $100 \mu \mathrm{~A}$. Find the minimum current in the circuit, so that ammeter shows maximum deflection?
In $C R$-circuit the growth of charge on the capacitor is
What is the amplitude of the electric field in a parallel beam of light intensity $\left(\frac{15}{\pi}\right) \frac{\mathrm{W}}{\mathrm{m}^2}$ ?
$$ \left[\text { Assume }, \frac{1}{4 \pi \varepsilon_0}=9 \times 10^9 \frac{\mathrm{Nm}^2}{\mathrm{C}^2}\right] $$
Photons of energy 2.4 eV and wavelength $\lambda$ fall on a metal plate and release photoelectrons with a maximum velocity $v$. By decreasing $\lambda$ by $50 \%$, the maximum velocity of photoelectrons becomes $3 v$. The work function of the material of the metal plate is
The ratio of maximum to minimum wavelength in Balmer series of an hydrogenic atom is
Alpha rays emitted from a radioactive substance are
Which of the following depicts the output of the full wave rectifier with capacitor filter for the following AC input?
The Boolean expression of the circuit given in figure is
A message signal of frequency 10 kHz and peak voltage of 15 V is used to modulate a carrier frequency of 1 MHz and peak voltage of 30 V . Determine the modulation index.