Chemistry
In the two elements, ${ }_{Z_1} A^{M_1}$ and ${ }_{Z_2} B^{M_2}$, the following relations are true. $M_1 \neq M_2$ and $Z_1 \neq Z_2$ but $M_1-Z_1=M_2-Z_2$. These elements are ( $M$ is atomic weight, $Z$ is atomic number)
The correct order of decreasing energy for the electrons whose quantum numbers $n$ and $l$ are given below, is
A. $n=5$ and $l=2$
B. $n=5$ and $l=0$
C. $n=4$ and $l=3$
D. $n=4$ and $l=1$
Find the correct set with isoelectronic species?
Among the following, the pair of elements having same electronegativity values are
I. H and $\mathrm{P} \quad$ II. Be and Al III. N and $\mathrm{Cl} \quad$ IV. C and P
The correct order of $\mathrm{H}-\mathrm{N}-\mathrm{H}$ bond angles of ammonia, ammonium ion and amide are
Find out the correct hybridisation of the central atom in $\mathrm{BCl}_3, \mathrm{PCl}_5, \mathrm{NH}_3$ and $\mathrm{SF}_6$.
Which among the following graphs, correctly represents the Boyle's Law?
A vessel of volume 24.6 L contains 1.5 moles of $\mathrm{H}_2$ and 2.5 moles of $\mathrm{N}_2$ at 300 K . Calculate the partial pressure of $\mathrm{N}_2$ in the vessel.
For a given unbalanced reaction, $\mathrm{MnO}_2+\mathrm{HCl} \longrightarrow \mathrm{MnCl}_2+\mathrm{H}_2 \mathrm{O}$, which is the limiting reagent, if the initial amount for each of the reactant is 100 grams?
[Molar masses : $\mathrm{MnO}_2=86.9 ; \mathrm{HCl}=36.5 ; \mathrm{MnCl}_2 \left.=125.8 ; \mathrm{Cl}_2=70.9 ; \mathrm{H}_2 \mathrm{O}=18\right]$
The emperical formula of a compound is $\mathrm{C}_2 \mathrm{H}_5 \mathrm{O}$ and its vapour density is 45 . What is the molecular formula of the compound?
Find the value of the equilibrium constant $(K)$ of a reaction at 300 K , when standard Gibbs free energy change is $-25 \mathrm{~kJ} \mathrm{~mol}^{-1}$ ? (Consider $R=8.33 \mathrm{Jmol}^{-1} \mathrm{~K}^{-1}$ )
What is the pH of 10 L of a buffer solution containing $0.01 \mathrm{M} \mathrm{NH}_4 \mathrm{Cl}$ and $0.1 \mathrm{M} \mathrm{NH}_4 \mathrm{OH}$ having $\mathrm{pK}_b$ of 5 ?
Decreasing order of number of ionisable hydrogen atoms in the following molecules is
I. $\mathrm{H}_3 \mathrm{PO}_4$
II. $\mathrm{H}_3 \mathrm{PO}_3$
III. $\mathrm{H}_3 \mathrm{PO}_2$
IV. $\mathrm{H}_4 \mathrm{P}_2 \mathrm{O}_6$
What will be the organic compound formed, when aluminium carbide react with deuterated water?
What is the increasing order of hydration enthalpies of alkali metal ions?
When orthoboric acid $\left(\mathrm{H}_3 \mathrm{BO}_3\right)$ is subjected to strong heating, the residue left is
When graphite is heated at $300^{\circ} \mathrm{C}$ with potassium vapour, it forms $\mathrm{C}_8 \mathrm{~K}$ compound that shows one of the following property.
Which of the following molecules is present in photochemical smog?
The correct order of the relative stability of the following carbanions is
I. $\mathrm{HC} \equiv \mathrm{C}^{-}$
II. $\mathrm{CH}_2=\mathrm{CH}-\mathrm{CH}_2^{-}$
III. $\mathrm{H}_2 \mathrm{C}=\mathrm{CH}^{-}$
IV. $\left(\mathrm{C}_6 \mathrm{H}_5\right)_2 \mathrm{CH}^{-}$
Which of the following sets of functional groups is meta-directing group?
The correct order of reactivity of hydrogen halides with propene is
The angle between (100) and (110) planes of FCC lattice is
$15 \%$ aqueous solution of glucose (molecular weight $=180 \mathrm{~g} / \mathrm{mol}$ ) is isotonic with $8 \%$ aqueous solution containing an unknown non-dissociable solute. What is the molecular weight of the unknown solute?
The vapour pressure of pure water is 23 mmHg . The vapour pressure of an aqueous solution, which contains 10 mass per cent of solute ' $A$ ' having molecular weight 50 is
What is the standard cell potential for the reaction with $K=1$ (equilibrium constant)
For a first order reaction ( $A \rightarrow B$ ), the temperature ( $T$ ) dependent rate constant $(k)$ in $\mathrm{s}^{-1}$ was found to follow the equation $: \log k=\left(-\frac{20}{T}\right)+4$. The activation energy ( $E_a$ ) and pre-exponential factor ( $A$ ) respectively, are
The method, which is based on the principle that "different components of a mixture are differently adsorbed on an adsorbent" is called
The solubilities of the impurities in the melt and solid states is not same. This principle is applied during the extraction of boron. This method is known as
How many bridging oxygen atoms are present in $\mathrm{P}_4 \mathrm{O}_{10}$ ?
Ozone is obtained from oxygen
What is the oxidation state of Cr , when the pH of the aqueous solution of potassium dichromate changes from acidic to basic?
Coordination number of Fe in the complexes $\left[\mathrm{Fe}(\mathrm{CN})_6\right]^{4-},\left(\mathrm{Fe}(\mathrm{CN})_6\right]^{3-}$ and $\left[\mathrm{FeCl}_4\right]^{-}$would be respectively
Type of reaction involved in the initial step in the formation of bakelite
Which of the following are reducing sugars?
I. Sucrose
III. Lactose
II. Ribose
IV. Fructose
The number of six and five membered rings present in norethindrone (a synthetic progesterone) are respectively
$$ \text { What are } X \text { and } Y \text { in the following reaction? } $$
The major products $P$ and $Q$ in the following reaction sequences are
$$ \text { Identify the correct decreasing order of acidic strength. } $$
$$ \text { Identify } X \text { in the following. } $$
Mathematics
If $f:[-3,2] \rightarrow[0, \sqrt[3]{x}]$ is an onto function defined by $f(n)=\left\{\begin{array}{cc}2+\sqrt[3]{n}, & -3 \leq n \leq-1 \\ n^{2 / 3}, & -1 \leq n \leq 2\end{array}\right.$, then $x=$
Let $[x]$ denote the greatest integer not more than $x$. If $A$ and $B$ are the domains of the functions $f(x)=\frac{x-[x]}{\sqrt{|x|-x}}$ and $g(x)=\frac{x-[x]}{\sqrt{|x|+x}}$ respectively, then
$n^5-5 n^3+4 n$ is divisible by 120 is true for
A value of $\theta$ in $\left(0, \frac{\pi}{2}\right)$ and satisfying $\left|\begin{array}{ccc}1+\sin ^2 \theta & \cos ^2 \theta & 4 \sin 4 \theta \\ \sin ^2 \theta & 1+\cos ^2 \theta & 4 \sin 4 \theta \\ \sin ^2 \theta & \cos ^2 \theta & 1+4 \sin 4 \theta\end{array}\right|=0$ is
Let $[A]_{3 \times 3}$ be a non-singular matrix such that
$$ A^{-1}=\frac{1}{3}\left(A^2-5 A+7 I\right) . $$
Then $17 A^8-85 A^7+119 A^6-51 A^5-19 A^4+95 A^3-133 A^2+58 A+I=$
If $\left[\begin{array}{ccc}2 & 1 & 1 \\ 0 & 3 & -1 \\ 1 & -1 & 1\end{array}\right]\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{l}1 \\ 1 \\ 0\end{array}\right]$, then $\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=$
For some $a, b, c \in \mathbf{R}$, if $\sin 5 \theta=a \cos ^4 \theta \sin \theta+b \cos ^2 \theta \sin ^3 \theta+c \sin ^5 \theta$, then $a b c=$
$A\left(z_1=2+2 i\right), B\left(z_2\right), C\left(z_3\right)$ are three points on the Argand plane satisfying $\left|z_k-2 i\right|=2,(k=1,2,3)$. If $\triangle A B C$ encloses the maximum area, then the sum of the imaginary parts of $z_2$ and $z_3$ is
For $n \in \mathbf{N}$, If $A_n=\cos \left(\frac{\pi}{2^n}\right)+i \sin \left(\frac{\pi}{2^n}\right)$, then $\left(A_1 A_2 A_3 A_4\right)^4=$
Let $A_r=\left(x+\frac{1}{x}\right)^3 \cdot\left(x^2+\frac{1}{x^2}\right)^3 \cdot\left(x^3+\frac{1}{x^3}\right)^3 \cdots\left(x^r+\frac{1}{x^r}\right)^3$. If $x^2+x+1=0$, then $\frac{1}{A_3}+\frac{1}{A_6}+\frac{1}{A_9}+\frac{1}{A_{12}}+\ldots . \infty=$
$p$ and $q$ are two roots of the equation $x^2+7 x+3=0$. If $\frac{3 p}{1-2 p}, \frac{3 q}{1-2 q}$ are the roots of $l x^2+m x+n=0$ and the greatest common divisor of $l, m, n$ is 1 , then $l-m+n=$
If the quadratic equations $3 x^2-7 x+2=0$ and $k x^2+7 x-3=0$ have a common root then the positive value of $k$ is
If the roots of $x^3+a x^2+b x+c=0$ are in arithmetic progression with common difference 1 , then
If $\alpha, \beta, \gamma$ are the roots of the equation $x^3+3 x^2-x-3=0$, then $\left(1+\alpha^2\right)\left(1+\beta^2\right)\left(1+\gamma^2\right)=$
The number of integers $x, y, z, w$ satisfying $x+y+z+w=25$ and $x, y, z \geq-1, w \geq 1$, is
If 3 sisters and 8 other girls are together playing a game, then the number of ways in which all the girls are seated around a circle such that the three sisters are not seated together, is
Suppose $1, m, n$ respectively represent the coefficient of $x^{10}$, the constant term and the coefficient of $x^{-10}$ in the expansion of $\left(a x^2+\frac{b}{x^3}\right)^{15}$. If $\frac{l}{m}+\frac{m}{n}=\frac{26}{11}$, then $a^2: b^2=$
For $z \in \mathbf{C}$, if $(1+z)^n=1+{ }^n C_1 z+{ }^n C_2 z^2+\ldots{ }^n C_n z^n$ and $\sum_{r=0}^{100} 100 c_r(\sin r x)=\left(2 \cos \frac{x}{2}\right)^{100} \sin k x$, then $k=$
If $\frac{1}{x^4+x^2+1}=\frac{A x+B}{x^2+x+1}+\frac{C x+D}{x^2-x+1}$, then $\cos ^{-1}(A+B+C+D)=$
The period of $\frac{\sin x}{\cos 3 x}+\frac{\sin 3 x}{\cos 9 x}+\frac{\sin 9 x}{\cos 27 x}+\frac{\sin 27 x}{\cos 81 x}$ is
If $\alpha=\frac{\sin ^3 x}{\cos ^2 x}, \beta=\frac{\cos ^3 x}{\sin ^2 x}$ and $\sin x+\cos x=k$, then $\alpha \sin x+\beta \cos x+3=$
If $A+B+C=60^{\circ}$, then $\cos \left(30^{\circ}-A\right)+\cos \left(30^{\circ}-B\right)+\cos \left(30^{\circ}-C\right)+\sin (A+B+C)=$
If $\left|\sin x-\cos ^2 x\right| \geq\left|3-3 \sin x+\sin ^2 x\right|+4|\sin x-1|$, then $x=$
The number of real roots of the equation $\sin \left[2 \cos ^{-1}\left\{\cot \left(2 \tan ^{-1} x\right)\right\}\right]=0$ that are greater than or equal to one are
If $\operatorname{sech}^{-1}(1 / 2)-\operatorname{cosech}^{-1}(3 / 4)=\log _e k$, then
The perimeter of a $\triangle A B C$ is 6 times the arithmetic mean of the sine of its angles. If its side $B C$ is of unit length, then $\angle A=$
In a $\triangle A B C$, with usual notation, if $r=r_1-r_2-r_3$, then $2 R=$
In a $\triangle A B C$, let $a, b, c, s, r, R, I, S, r_1, r_2, r_3$ stand for their usual meaning. Then Match the items of List-I with those of the items of List-II.
| List-I | List-II |
| A. |
I. |
| B. |
II. |
| C. |
III. |
| D. |
IV. |
| V. |
$$ \text { The correct match is } $$
If $\mathbf{a}, \mathbf{b}, \mathbf{c}$ are the position vectors of the points $A, B, C$ respectively, then match the items of list-I with those of list-II.
| $$ \text { List-I } $$ |
$$ \text { List-II } $$ |
||
|---|---|---|---|
| A. | $$ \text { } \begin{aligned} \mathbf{a} & =2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}, \\ \mathbf{b} & =3 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}, \\ \mathbf{c} & =4 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}} \end{aligned} $$ |
I. | $A, B, C$ are collinear |
| B. | $$ \text { } \begin{aligned} \mathbf{a} & =\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}, \\ \mathbf{b} & =3 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}+7 \hat{\mathbf{k}}, \\ \mathbf{c} & =-3 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}-5 \hat{\mathbf{k}} \end{aligned} $$ |
II. | $\triangle A B C$ is an isosceles triangle |
| C. | $$ \begin{aligned} &\text { }\\ &\begin{aligned} & a=2 \hat{i}-\hat{j}+\hat{k}, \\ & b=\hat{i}-3 \hat{j}-5 \hat{k}, \\ & c=-3 \hat{i}-4 \hat{j}-4 \hat{k} \end{aligned} \end{aligned} $$ |
III. | $\triangle A B C$ is a right-angled triangle |
| D. | $$ \begin{aligned} & a=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \\ & b=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}, \\ & c=2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}, \end{aligned} $$ |
IV. | $\triangle A B C$ is a right-angled isosceles triangle |
| V. | $$ \triangle A B C \text { is an equilateral triangle } $$ |
||
$$ \text { The correct match is } $$
If the point of intersection of the lines $\mathbf{r}=\hat{\mathbf{i}}-6 \hat{\mathbf{j}}+(p \sec \alpha) \hat{\mathbf{k}}+t(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}})$ and $\mathbf{r}=4 \hat{\mathbf{j}}+\hat{\mathbf{k}}+\lambda(2 \hat{\mathbf{i}}+(p \tan \alpha) \hat{\mathbf{j}}+2 \hat{\mathbf{k}})$ is $8 \hat{\mathbf{i}}+8 \hat{\mathbf{j}}+9 \hat{\mathbf{k}}$, (where $\left.0<\alpha<\frac{\pi}{2}\right)$, then $p=$
$\mathbf{1}, \mathbf{m}, \mathbf{n}$ are three unit vectors in a right handed system and $L$ is a line through the points $A, B, C$ whose position vectors are $p \mathbf{1}+7 \mathbf{m}-6 \mathbf{n}, 2 \mathbf{1}+5 \mathbf{m}-4 \mathbf{n}$ and $1+4 \mathbf{m}-3 \mathbf{n}$ respectively. If the equation of the plane containing $L$ and the points ( $-p, p, p+1$ ) is $a x+b y+c z=1$, then $p(a+b+c)=$
Let $\mathbf{a}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ and $\mathbf{b}=2 \hat{\mathbf{i}}-\hat{\mathbf{j}}-2 \hat{\mathbf{k}}$. If the orthogonal projection vector of $\mathbf{a}$ on $\mathbf{b}$ be $\mathbf{x}$ and the orthogonal projection vector of $\mathbf{b}$ on $\mathbf{a}$ be $\mathbf{y}$, then $|\mathbf{x}-\mathbf{y}|=$
Let $\mathbf{p}, \mathbf{q}, \mathbf{r}$ be three non-coplanar vectors and $\left[\begin{array}{lll}\mathbf{p} & \mathbf{q} & \mathbf{r}] \mathbf{a}=\mathbf{q} \times \mathbf{r},\left[\begin{array}{lll}\mathbf{p} & \mathbf{q} & \mathbf{r}] \mathbf{b}=\mathbf{r} \times \mathbf{p},[ \end{array}[\mathbf{p}\right. \\ \mathbf{q} & \mathbf{r}] \mathbf{c}=\mathbf{p} \times \mathbf{q} \text {. If }\end{array}\right. \mathbf{a}, \mathbf{b}, \mathbf{c}$ denote the coterminous edges of a parallelopiped, then its height with the base having a and $\mathbf{c}$ is
If $\mathbf{b}, \mathbf{c}$ are non collinear vectors, $|\mathbf{c}| \neq 0$, $\mathbf{a} \times(\mathbf{b} \times \mathbf{c})+(\mathbf{a} \cdot \mathbf{b}) \mathbf{b}=(4-2 \beta-\sin \alpha) \mathbf{b}+\left(\beta^2-1\right) \mathbf{c}$ and $(\mathbf{c} \cdot \mathbf{c}) \mathbf{a}=\mathbf{c}$, then the scalars $\alpha$ and $\beta$ are
$$ \text { The variance of the following frequency distribution is } $$
$$ \begin{array}{ccccccc} \hline \text { Classes } & 0-10 & 10-20 & 20-30 & 30-40 & 40-50 & 50-60 \\ \hline \text { Frequency } & 11 & 29 & 18 & 4 & 5 & 3 \\ \hline \end{array} $$
The mean deviation about the mean of the following data is nearly
$$ \begin{array}{ccccccccc} \hline \text { Size }(x) & 1 & 3 & 5 & 7 & 9 & 11 & 13 & 15 \\ \hline \text { Frequency }(f) & 3 & 3 & 4 & 14 & 7 & 4 & 3 & 4 \\ \hline \end{array} $$
If the roots of each of the equations $2 x^2+x-1=0$, $3 x^2-10 x+3=0$ and $6 x^2+11 x-2=0$ corresponds to probabilities of three events of a random experiment, then those events are
Cards are drawn one after the other without replacement from a well shuffled pack of cards until and ace card appears. If the probability that exactly 5 cards are drawn before the first ace card appears is $\frac{4}{49}\left(\frac{p_1 \cdot p_2 \cdot p_3}{p_4 \cdot p_5 \cdot p_6}\right),\left(p_i\right.$ is prime, $\left.i=1,2,3,4,5,6\right)$ then $\left(\max \left\{p_i\right\}-\min \left\{p_i\right\}\right)=$
A number is selected at random from the set $\{1,2, \ldots \ldots ., 100\}$. Given that the number selected is divisible 2 , the probability that it is also divisible by 3 or 5 , is
A target is to be destroyed in a bombing exercise and there is a $75 \%$ chance that a bomb will hit the target. Assuming that two direct hits are required to destroy the target completely, the minimum number of bombs to be dropped in order that the probability of destroying the target is not less than $99 \%$, is
Let $X \sim B(n, p)$ with mean $\mu$ and variance $\sigma^2$. If $\mu=2 \sigma^2$ and $\mu+\sigma^2=3$, then $P(X \leq 3)=$
If $A=(1,2), B=(2,1)$ and $P$ is any point satisfying the condition $P A+P B=3$, then the equation of the locus of $P$ is
Let $C$ be a curvea $x^2+2 h x y+b y^2+2 g x+2 f y+c=0$ in a cartesian plane. By rotating the coordinate axes through an angle $\frac{\pi}{4}$ in the positive direction, if the transformed equation of $C$ is $Y^2+X Y-X=0$, then $\left(h^2-a b\right)-2 g f=$
If the straight line passing through the point $P(3,4)$ makes an angle $\frac{\pi}{6}$ with the positive direction of $X$-axis and meets the line $12 x+5 y+10=0$ at $Q$, then the length of $P Q$ is
If the equation of the straight line passing through the point of intersection of $x+2 y-19=0, x-2 y-3=0$ and which is at a perpendicular distance of 5 units from the point $(-2,4)$ is $5 x+b y+c=0$, then a possible value of $5+b+c$ is
If two equal sides of an isosceles triangle are given by the equations $7 x-y+3=0$ and $x+y-3=0$, then the equation of its third side passing through the point $(2,-5)$ is
Suppose $O(0,0)$ is the origin and the line $L=x+y-\lambda=0$ meets the curve $x^2+y^2-2 x-4 y+2=0$ at $A$ and $B$. If $\angle A O B=90^{\circ}$, then the distance between such lines $L=0$ is
Let $P$ be the point of intersection of the lines $L_1 \equiv x-y-7=0$ and $L_2 \equiv x+y-5=0 . A\left(x_1, y_1\right)$ and $B\left(x_2, y_2\right)$ are points on the lines $L_1=0$ and $L_2=0$ respectively such that $P A=3 \sqrt{2}$, $P B=\sqrt{2}, x_1, y_1 \geq 0, x_2, y_2 \geq 0$, then the angle made by the line segment $A B$ at the origin is
If the poles of the line $x-y=0$ with respect to the circles $x^2+y^2-2 g_i x+c_i^2=0(i=1,2,3)$ are ( $\alpha_i, \beta_i$ ), then $\sum_{i=1}^3 \frac{\alpha_i+\beta_i}{g_i}=$
If the circles $x^2+y^2-4 x+6 y+13-a^2=0$ and $x^2+y^2-10 x-2 y+17=0$ intersect in two distinct points, then ' $a$ ' is
If a circle of radius $r$ touches the positive coordinate axes and also the circle $x^2+y^2-12 x-10 y+52=0$ externally, then the distance between the centres of the two circles is
If the circles $x^2+y^2-2 x-2 y+k=0$ and $x^2+y^2+4 x+6 y+4=0$ touch each other externally, then the point of contact of the two circles is
The centre of the circle passing through the points of intersection of the circles $(x+3)^2+(y+2)^2=25$ and $(x-2)^2+(y-3)^2=25$ and cutting the circle $(x+1)^2+(y-2)^2=16$ orthogonally is
If a circle with its centre at the focus of the parabola $y^2=2 p x$ is such that it touches the directrix of the parabola, then a point of intersection of the circle and the parabola is
If the tangent drawn at the point $P(4,8)$ to the parabola $y^2=16 x$ meets the parabola $y^2=16 x+80$ at $A$ and $B$, then the mid-point of $A B$ is
If the sum of the distances from the foci to the centre $O(0,0)$ of an ellipse is $8 \sqrt{6}$ units and the area of the smallest rectangle in which that ellipse is inscribed is 80 sq. units, then the equation of such an ellipse is
The equation of the ellipse with directrix $3 x+4 y-5=0$, focus $(1,2)$ and eccentricity $1 / 2$, is
A rectangular hyperbola passing through $(3,2)$ has its asymptotes parallel to the coordinate axes. If $(1,1)$ is the point of intersection of the two perpendicular tangents of that hyperbola, then its equation is
$E(1,0,0), F(0,2,0), G(0,0,3)$ are respectively the mid-points of the sides $A B, B C, C A$ of $\triangle A B C$. If $a_1, b_1, c_1$ and $a_2, b_2, c_2$ are respectively the direction ratios of $A F$ and $B G$, then $\frac{a_1^2+b_1^2+c_1^2}{a_2^2+b_2^2+c_2^2}=$
If the direction ratios $a, b, c$ of a line $L$ satisfy the relations $a b+b c+c a=0$ and $6 a b+9 b c+8 c a=0$, then the direction cosines of the line $L$ are
The equation of the plane passing through the line of intersection of planes $\pi_1=2 x+6 y+4 z-7=0$, $\pi_2=x-y-2 z-2=03$ and perpendicular to the plane $x+y+2 z-5=0$ is
$$\mathop {\lim }\limits_{x \to 0} \frac{x \tan 4 x-2 x \tan 2 x}{(1-\cos 4 x)^2}= $$
Assertion (A) $f(x)=|x-a|+|x-b|$, is continuous on $\mathbf{R}$
Reason (R) $\frac{|x-\alpha|}{x-\alpha}$ is continuous at $x \in \mathbf{R}-\{\alpha\}$.
The correct option among the following is
If $x \sqrt{1+y}+y \sqrt{1+x}=0$, then $\frac{d y}{d x}=$
If $p(x)$ be a polynomial satisfying $p(2 x)=p^{\prime}(x) \cdot p^{\prime \prime}(x)$, then $\sum_{x=1}^5 p(x)=$
If $2^x+2^y=2^{x+y}$, then $\frac{d y}{d x}=$
If the tangent and normal drawn to the curve $x=a(\theta+\sin \theta), y=a(1-\cos \theta)$ at $P\left(\theta=\frac{\pi}{2}\right)$ cuts the $X$-axis at $A$ and $B$ respectively, then the area (in sq. units) of $\triangle P A B$ is
$x_1, x_2 \in \mathbf{N}$. If a line having slope 2 is a tangent to the curve $y=x^4-6 x^3+13 x^2-10 x+5$ at points $P\left(x_1, y_1\right)$ and $Q\left(x_2, y_2\right)$, then $x_1 x_2+y_1 y_2=$
Consider the following statements
Statement I If $a_0+\frac{a_1}{2}+\frac{a_2}{3}+\ldots .+\frac{a_n}{n+1}=0$, where $a_0, a_1, \ldots, a_n$ are real numbers, then the polynomial $a_0+a_1 x+a_2 x^2+\ldots .+a_n x^n$ has a zero in the interval $(0,1)$.
Statement II If $f:[a, b] \rightarrow \mathbf{R}$ is continuous on $[a, b]$ and $f$ is differentiable in $(a, b)$, where $a>0$ and if $\frac{f(a)}{a}=\frac{f(b)}{b}$, then there exists $c \in(a, b)$, such that $c f^{\prime}(c)=f(c)$.
Which one of the following options is true?
A function $y=f(x)$ with $f(-1)=-249$ has no maximum and has only one minimum at $x=5$ with $f(5)=75$. Which one of the following is true?
If $\int e^{\sin ^2 x}\left(\sin x \cos x+\cos ^3 x \sin x\right) d x=e^{\sin ^2 x}(1+f(x))+c$, then $f^{\prime}(x)=$
$$ \int \frac{25 x^2+8}{\sqrt{25 x^2+9}} d x= $$
$$ I_{m, n}=\int x^m(\log x)^n d x= $$
The positive integer $n \leq 5$ for which $\int_0^1 e^x(x-1)^n d x=16-6 e$ is
If $f(x)=\sin ^6 x+\cos ^6 x+2 \sin ^3 x \cos ^3 x$, then $\int_0^{\pi / 4} \frac{\sin ^2 2 x}{f(x)} d x=$
$$ \int_3^5(x-3)^3(5-x)^5 d x= $$
The area (in sq. units) of the portion lying above the $X$-axis and enclosed between the curves $y^2=2 a x-x^2$ and $y^2=a x$ is
The differential equation for which $l x^2+m y^2=x+y$ is the general solution is
The general solution of the differential equation $(x-2 y+1) d y-(3 x-6 y+2) d x=0$ is
The general solution of the differential equation $\left(1+y^2\right) d x=\left(\tan ^{-1} y-x\right) d y$ is
Physics
Let $G, W, E$ and $S$ be relative strength of gravitational, weak-nuclear, electromagnetic and strong-nuclear forces, respectively. Which of the correct statement?
What is the dimension of $\frac{1}{\mu_0 \varepsilon_0}$ ? ( $\mu_0=$ magnetic permeability and $\varepsilon_0=$ permittivity of free space)
A person moves 30 m North and then 20 m towards East and finally $30 \sqrt{2} \mathrm{~m}$ in South-West direction. The displacement of the person from the origin will be
A particle moving along $X$-axis has acceleration $f$ at time $t$ given by $f=f_0\left(1-\frac{t}{T}\right)$, where $f_0$ and $T$ are constants. The particle at $t=0$ has zero velocity. In the time interval between $t=0$ and the instant when $f=0$, the particle's velocity is
Consider the following vectors.
Choose the correct statement.
A river 200 m wide is flowing at a rate of $3.0 \mathrm{~m} / \mathrm{s}$. A boat is sailing at a velocity of $15 \mathrm{~m} / \mathrm{s}$ with respect to the water in a direction perpendicular to the river. How far from the point directly opposite to the starting point does the boat reach on the opposite bank?
An infinite number of masses are placed on a frictionless table and they are connected via massless strings. Their masses follow the sequence, $m, \frac{m}{2}, \frac{m}{6}, \ldots \ldots \ldots . . \frac{m}{n!}, \ldots \ldots$. and they are further connected to a mass $m$ that hangs over a massless pulley. The acceleration of the hanging mass is
A block of mass $m=2 \mathrm{~kg}$ is initially at rest on a horizontal surface. A horizontal force $\mathbf{F}_1=(6 \mathrm{~N}) \hat{\mathbf{i}}$ and a vertical force $\mathbf{F}_2=(10 \mathrm{~N}) \hat{\mathbf{j}}$ are then applied to the block. The coefficients of static friction and kinetic friction for the block and the surfaces are 0.4 and 0.25 , respectively. The magnitude of the frictional force acting on the block is (assume, $g=10 \mathrm{~m} / \mathrm{s}^2$ )
A force of 4 N acts on a 10 kg body initially at rest. Let $W_1$ is work done by force during $0 \leq t \leq \mathrm{ls}$. Likewise $W_2$ is the work done by force during $\mathrm{l} \mathrm{s} \leq t \leq 2 \mathrm{~s}$, where $t$ is time in second. The ratio $\frac{W_2}{W_1}$ is
An elevator of mass 500 kg is ascending upwards with a constant acceleration $a=2 \mathrm{~m} / \mathrm{s}^2$. What is the work done by the tension in the elevator cable during its climb by 12 m ? (Take, $g=10 \mathrm{~m} / \mathrm{s}^2$ )
A solid sphere of mass 2 kg rolls on a smooth horizontal surface at $10 \mathrm{~m} / \mathrm{s}$. It then rolls up a smooth inclined plane of inclination $30^{\circ}$ with the horizontal. The height attained by the sphere before it stops is [take $g=10 \mathrm{~m} / \mathrm{s}^2$ ]
A rod of length $L$ revolves in a horizontal plane about the axis passing through its centre and perpendicular to its length. The angular velocity of the rod is $\omega$. If $A$ is the area of cross-section of the rod and $\rho$ is its density, then the rotational kinetic energy of the rod is
Consider a simple harmonic motion (SHM). Let $K$ and $U$ be kinetic energy and potential energy when the displacement in SHM is one-half $\left(\frac{1}{2}\right)$ the amplitude.
Which of the correct statement?
A planet is moving in an elliptical orbit around the sun. The work done on the planet by the gravitational force of the sun
(i) is zero in no part of the motion.
(ii) is zero in some parts of the orbit.
(iii) is zero in one complete revolution.
(iv) is zero in any small part of the orbit.
Which of the following is true?
A steel rod has a radius of 10 mm and a length of 1 m . A 80 kN force stretches it along its length. If the Young's modulus of the rod is $2 \times 10^{11} \mathrm{~N} / \mathrm{m}^2$, then the change in length is
A block of iron contains a hollow cavity as shown below. The block weighs 6000 N in air and 4000 N in water. If the density of iron and water are $6 \mathrm{~g} / \mathrm{cm}^3$ and $1 \mathrm{~g} / \mathrm{cm}^3$, then the volume of the cavity is (assume, $g=10 \mathrm{~m} / \mathrm{s}^2$ )
A heating element of mass 100 g and having specific heat of $1 \mathrm{~J} /\left(\mathrm{g}^{\circ} \mathrm{C}\right)$ is exposed to surrounding air at $27^{\circ} \mathrm{C}$. The element attains a steady state temperature of $127^{\circ} \mathrm{C}$, while absorbing 100W of electric power. If the power is switched Off, then approximate time taken by the element to cool down to $126^{\circ} \mathrm{C}$ will be (neglect radiation)
An ideal gas at temperature $T$, pressure $p$ occupies a volume $V$. If its temperature is halved and pressure doubled, what is its new volume?
A Carnot engine whose efficiency is $40 \%$, receives heat at 500 K . If the efficiency is to be $50 \%$, the source temperature for the same exhaust temperature is
A system goes from $A$ and $B$ via two processes I and II as shown in figure. If $\Delta U_1$ and $\Delta U_2$ are the changes in internal energies in the processes I and II respectively, then the relation between $\Delta U_1$ and $\Delta U_2$ is
An organ pipe with both ends open has a length $L=25$ cm . An extra hole is created at position $L / 2$. The lowest frequency of sound produced is (assume, speed of sound $=340 \mathrm{~m} / \mathrm{s}$ )
The magnifications produced by a convex lens for two position of an object are 4 and 3, respectively. If the distance of separation between the two positions of the object is 2 cm , then the focal length of the lens is
What minimum separation between two objects a human eye would be able to resolve, if the eye pupil diameter is 2 mm and the two objects are 20 m away from the eye?
(Assume, human eye to be equivalent to a convex lens and consider the average wave length of light as 600 nm )
The shape of wavefront of light diverging from point source
A cube of side $L$ has point charges $+q$ located at its seven vertices and $-q$ at remaining one vertex. The electric field at its centre is found to be $|\mathbf{E}|=\alpha\left(\frac{q}{4 \pi \varepsilon_0 L^2}\right)$.
The magnitude of constant $\alpha$ is
A capacitor is fully charged with a battery and then disconnected. A dielectric is then inserted into the capacitor. How do charges on surface of the dielectric and outer surface of the plates of the capacitor would change respectively?
A conducting wire of cross-sectional area $1 \mathrm{~cm}^2$ has $3 \times 10^{23}$ charge carriers per $\mathrm{m}^3$. If wire carries a current of 24 mA , then drift velocity of carriers is
The total current supplied to the following circuit by the battery is
A conducting wire is in the shape of a regular hexagon which is inscribed inside an imaginary circle of radius $R$. If a current $I$ flows the wire, the magnitude of magnetic field at the centre of the circle is
A coil having 100 turns is wound tightly in the form of a spiral with inner and outer radii 1 cm and 2 cm , respectively. When a current 1 A passes through the coil, the magnetic field at the centre of the coil is
Let $m$ and $r$ are the dipole moment and radius of earth respectively. Then, the earth's magnetic field at the equator is
A circular coil consists 70 closely wound turns and has a radius of 10 cm . An externally produced magnetic field of magnitude $2 \times 10^{-3} \mathrm{~T}$ is applied perpendicular to the coil. The net flux through the coil is found to vanish when the current in the coil is 2.2 A . The inductance of the coil is
The $L-C-R$ circuit shown below is driven by an ideal AC voltage source.
At frequency $f=\frac{1}{2 \pi \sqrt{L C}}$, choose the correct statement.
At an instant, a plane electromagnetic wave has its magnetic field in the direction of the vector $\hat{\mathbf{i}}-\hat{\mathbf{j}}$ and its electric field is in the direction of $\hat{\mathbf{i}}+\hat{\mathbf{j}}$. The wave is travelling along which direction?
When the energy of incident radiation is increased by $20 \%$, the kinetic energy of the photoelectrons emitted from a metal surface increased from 0.5 eV to 0.8 eV . The work function of the metal is
In the Bohr model an electron of mass $m$ moves in a circular orbit around the proton. Considering the orbiting electron to be a circular current loop, the magnetic moment of the hydrogen atom, when the electron is in $n$th excited state. (Assume, $h=$ Planck's constant)
A radioactive element which can decay by two processes, has half-life $t_1$ for first process and half-life $t_2$ for second process. Let $\langle t\rangle$ be the effective average-life of this element. Which of the following is correct?
A $p-n$ junction diode can withstand upto 20 mA current under forward bias. The diode has a potential difference of 0.5 V across it, which is assumed to be independent of current. What is the maximum voltage of the battery used to forward bias the diode when a resistance of $125 \Omega$ is connected in series with it?
In a Zener diode,
A message signal is super-imposed with a carrier signal. The resulting modulating signal $C_m(t)$ is given by $C_m(t)=A_1 \sin \left(\omega_1 t\right)+A_2 \sin \left(\omega_2 t\right)-A_2 \sin \left(\omega_3 t\right)$, where $\omega_2<\omega_1<\omega_3$. The modulation index and the angular frequency fo the message signal respectively, are