TG EAPCET 2024 (Online) 11th May Morning Shift

Paper was held on Sat, May 11, 2024 3:30 AM

Chemistry

If $n, l$ represent the principal and azimuthal quantum numbers respectively, the formula used to know the number of radial nodes possible for a given orbital is

If the radius of first orbit of hydrogen like ion is $1.763 \times 10^{-2} \mathrm{~nm}$, the energy associated with that orbit (in J ) is

If first ionisation enthalpy $\left(\Delta_{i} H\right)$ values of $\mathrm{Na}, \mathrm{Mg}$ and Si are respectively 496,737 and $786 \mathrm{~kJ} \mathrm{~mol}^{-1}$, the first ionisation enthalpy value of $\mathrm{Al}\left(\mathrm{in} \mathrm{kJ} \mathrm{mol}^{-1}\right.$ ) will be

Among the oxides $\mathrm{SiO}_{2}, \mathrm{SO}_{2}, \mathrm{Al}_{2} \mathrm{O}_{3}$ and $\mathrm{P}_{2} \mathrm{O}_{3}$, the correct order of acidic strength is

According to molecular orbital theory, which of the following statements is not correct?

The melting point of $o$-hydroxybenzaldehyde $(A)$ is lower than that of $p$-hydroxybenzaldehyde $(B)$. This is because

At what temperature will the RMS velocity of sulphur dioxide molecules at 400 K be the same as the most probable velocity of oxygen molecules?

0.43 g of a metal of valence 2 was dissolved in 50 mL of $0.5 \mathrm{M} \mathrm{H}_{2} \mathrm{SO}_{4}$ solution. The unreacted acid required 14.2 mL of 1 M NaOH solution for neutralisation. The atomic weight of the metal is

At $300 \mathrm{~K}, 3.0$ moles of an ideal gas at 3.0 atm pressure is compressed isothermally to one half of its volume by an external pressure of 6.0 atm . The work done (in kJ ) is (Given, $\left.R=0.082 \mathrm{~L} \mathrm{~atm} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\right)(1 \mathrm{~L} \mathrm{~atm}=1013 \mathrm{~J})$

At $T(K)$ the equilibrium constants for the following two reactions are given below

$ 2 A(g) \rightleftharpoons B(g)+C(g) ; K_{1}=16 $

$ 2 B(g)+C(g) \rightleftharpoons 2 D(g) ; K_{2}=25 $

What is the value of equilibrium constant $(K)$ for the reaction given below at $T(K)$ ?

$ A(g)+\frac{1}{2} B(g) \rightleftharpoons D(g) $

Identify the pair of hydrides which have polymeric structure

Match the following

List I (Alloy)	List II (Use)
A Li-Pb	I. In aircraft construction
B Be-Cu	II. To make bearings for motor engines
C Mg-Al	III. To make tetraethyl lead
D Na-Pb	IV. To make high strength springs

The correct answer is

The hydroxide of which of the following metal reacts with both acid and alkali?

The correct formula of borax is $\mathrm{Na}_{2}\left[\mathrm{~B}_{4} \mathrm{O}_{5}(\mathrm{OH})_{x}\right] \cdot y \mathrm{H}_{2} \mathrm{O}$. The sum of $x$ and $y$ is

Formic acid on heating with concentrated $\mathrm{H}_{2} \mathrm{SO}_{4}$ at 373 K gives $X$, a colourless substance and $Y$, a good reducing agent. The number of $\sigma$ and $\pi$ bonds in $X, Y$ are respectively

Eutrophication can lead to

In which of the following options, the IUPAC name is not correctly matched with the structure of the compound?

Consider the following carbocations.

Arrange the above carbocations in the order of decreasing stability

Consider the following reaction sequence

$ \text { 2-methylpropane } \xrightarrow{\mathrm{KMnO}_{4}} X \xrightarrow[358 \mathrm{~K}]{20 \% \mathrm{H}_3 \mathrm{PO}_4} Y \xrightarrow[\text { (ii) } \mathrm{Zn} / \mathrm{H}_{2} \mathrm{O}]{\text { (i) } \mathrm{O}_{3}} A+B \text {. } $

What are $A$ and $B$ ?

Identify the end product $(Z)$ in the sequence of the following reactions.

In bcc lattice containing $X$ and $Y$ type of atoms, $X$ type of atoms are present at the corners and $Y$ type of atoms are present at the centers. In its unit cell, if three atoms are missing in the corners, the formula of the compound is

At 300 K , the vapour pressure of toluene and benzene are 3.63 kPa and 9.7 kPa respectively. What is the composition of vapour in equilibrium with the solution containing 0.4 mole fraction of toluene?

(Assume the solution is ideal)

0.592 g of copper is deposited in 60 minutes by passing

0.5 A current through a solution of copper (II) sulphate. The electro chemical equivalent of copper (II) (in $\mathrm{gC}^{-1}$ ) is

( $F=96500 \mathrm{C} \mathrm{mol}^{-1}$ )

For the gaseous reaction, $\mathrm{N}_{2} \mathrm{O}_{5} \longrightarrow 2 \mathrm{NO}_{2}+\frac{1}{2} \mathrm{O}_{2}$

the rate can be expressed as

$ \begin{array}{l} -\frac{d\left[\mathrm{~N}_{2} \mathrm{O}_{5}\right]}{d t}=K_{1}\left[\mathrm{~N}_{2} \mathrm{O}_{5}\right] \\\\ +\frac{d\left[\mathrm{NO}_{2}\right]}{d t}=K_{2}\left[\mathrm{~N}_{2} \mathrm{O}_{5}\right] \\\\ +\frac{d\left[\mathrm{O}_{2}\right]}{d t}=K_{3}\left[\mathrm{~N}_{2} \mathrm{O}_{5}\right] \end{array} $

The correct relation between $K_{1}, K_{2}$ and $K_{3}$

Match the following

List I (Industrial process)	List II (Catalyst used)
A Ostwald's process	I $\mathrm{CuCl}_2$
B Haber's process	II Zeolites
C Deacon's process	III Pt gauge
D Cracking of hydrocarbons	IV Fe

The correct answer is

Copper matte is mixture of

$\mathrm{C}+$ Conc. $\mathrm{H}_{2} \mathrm{SO}_{4} \xrightarrow{\Delta} X+Y+\mathrm{H}_{2} \mathrm{O}$

$X$ and $Y$ in the above reaction are

Which among the following oxoacids of phosphorous will have $\mathrm{P}-\mathrm{O}-\mathrm{P}$ bonds?

I. $\mathrm{H}_{4} \mathrm{P}_{2} \mathrm{O}_{5}$

II. $\mathrm{H}_{4} \mathrm{P}_{2} \mathrm{O}_{6}$

III. $\mathrm{H}_{4} \mathrm{P}_{2} \mathrm{O}_{7}$

IV. $\left(\mathrm{HPO}_{3}\right)_{3}$

The bond angles $\mathrm{H}-\mathrm{O}-\mathrm{N}$ and $\mathrm{O}-\mathrm{N}-\mathrm{O}$ in the planar structure of nitric acid molecule are respectively

Observe the following $f$-block elements

$\mathrm{Eu}(Z=63) ; \mathrm{Pu}(Z=94) ; \mathrm{Cf}(Z=98)$;

$\operatorname{Sm}(Z=62) ; \mathrm{Gd}(Z=64) ; \mathrm{Cm}(Z=96)$

How many of the above have half-filled $f$-orbitals in their ground state?

Which one of the following complex ions has geometrical isomers?

Which one of the following is not an example of condensation polymer?

What is the IUPAC name of the product $Y$ in the given reaction sequence?

What is the value of ' $n$ ' in ' $Z$ ' of the following sequence?

Lauryl alcohol $\xrightarrow{\mathrm{H}_{2} \mathrm{SO}_{4}}$

( $X$ )

Lauryl hydrogen sulphate $\xrightarrow{\mathrm{NaOH}(a q)}$

( $Y$ )

$ \mathrm{CH}_{3}-\left(\mathrm{CH}_{2}\right)_{n}-\mathrm{CH}_{2} \mathrm{OSO}_{3} \mathrm{Na} $

(Z)

Sodium lauryl sulphate

The organic halide, which does not undergo hydrolysis by $\mathrm{S}_{\mathrm{N}} 1$ mechanism is

What is ' $Z$ ' in the given sequence of reactions?

What is the % carbon in the product ' $Z$ ' formed in the reaction?

Match the following

The correct answer is

What are $Y$ and $Z$ respectively in the given reaction sequence?

What is $C$ in the given sequence of reactions?

TG EAPCET 2024 (Online) 11th May Morning Shift Chemistry - Carboxylic Acids and Its Derivatives Question 1 English

Mathematics

If $f(x)$ is a quadratic function such that $f(x) f\left(\frac{1}{x}\right)=f(x)+f\left(\frac{1}{x}\right)$, then $\sqrt{f\left(\frac{2}{3}\right)+f\left(\frac{3}{2}\right)}=$

$f(x)=a x^{2}+b x+c$ is an even function and

$g(x)=p x^{3}+q x^{2}+r x$ is an odd function.

If $h(x)=f(x)+g(x)$ and $h(-2)=0$, then $8 p+4 q+2 r=$

If $1 \cdot 3 \cdot 5+3 \cdot 5 \cdot 7+5 \cdot 7 \cdot 9 \ldots$ to $n$ terms $=n(n+1) f(n)$, then $f(2)=$

$A=\left[\begin{array}{ll}1 & 2 \\\\ 2 & 1\end{array}\right]$ and $B=\left[\begin{array}{ll}x & y \\\\ 1 & 2\end{array}\right]$ are two matrices such that $(A+B)(A-B)=A^{2}-B^{2}$ If $C=\left[\begin{array}{ll}x & 2 \\\\ 1 & y\end{array}\right]$, then trace $(C)=$

If $x=k$ satisfies the equation $\left|\begin{array}{ccc}x-2 & 3 x-3 & 5 x-5 \\\\ x-4 & 3 x-9 & 5 x-25 \\\\ x-8 & 3 x-27 & 5 x-125\end{array}\right|=0$, then $x=k$ also satisfies the equation

If $A$ is a non-singular matrix, then $\operatorname{adj}\left(A^{-1}\right)=$

If the homogeneous system of linear equations $x-2 y+3 z=0,2 x+4 y-5 z=0,3 x+\lambda y+\mu z=0$ has non-trivial solution, then $8 \mu+11 \lambda=$

If $z=\frac{(2-i)(1+i)^{3}}{(1-i)^{2}}$, then $\arg (z)=$

$z=x+i y$ and the point $P$ represents $z$ in the argand plane. If the amplitude of $\left(\frac{2 z-i}{z+2 i}\right)$ is $\frac{\pi}{4}$, then the equation of the locus of $P$ is

$\alpha, \beta$ are the roots of the equation $x^{2}+2 x+4=0$. If the point representing $\alpha$ in the argand diagram lies in the 2nd quadrant and $\alpha^{2024}-\beta^{2024}=i k,(i=\sqrt{-1})$, then $k=$

If $\alpha$ is a root of the equation $x^{2}-x+1=0$, then $\left(\alpha+\frac{1}{\alpha}\right)^{3}+\left(\alpha^{2}+\frac{1}{\alpha^{2}}\right)^{3}+\left(\alpha^{3}+\frac{1}{\alpha^{3}}\right)^{3}+\left(\alpha^{4}+\frac{1}{\alpha^{4}}\right)^{3}=$

$\alpha, \beta$ are the real roots of the equation $x^{2}+a x+b=0$. If $\alpha+\beta=\frac{1}{2}$ and $\alpha^{3}+\beta^{3}=\frac{37}{8}$, then $a-\frac{1}{b}=$

The solution set of the inequation $\sqrt{x^{2}+x-2} > (1-x)$ is

If $\alpha, \beta, \gamma$ are the roots of the equation $4 x^{3}-3 x^{2}+2 x-1=0$, then $\alpha^{3}+\beta^{3}+\gamma^{3}=$

The equation $16 x^{4}+16 x^{3}-4 x-1=0$ has a multiple root. If $\alpha, \beta, \gamma, \delta$ are the roots of this equation, then $\frac{1}{\alpha^{4}}+\frac{1}{\beta^{4}}+\frac{1}{\gamma^{4}}+\frac{1}{\delta^{4}}=$

The sum of all the 4-digit numbers formed by taking all the digits from $0,3,6,9$ without repetition is

The number of ways in which 6 distinct things can be distributed into 2 boxes so that no box is empty is

Number of ways in which the number 831600 can be split into two factors which are relatively prime is

The coefficient of $x y^{2} z^{3}$ in the expansion of $(x-2 y+3 z)^{3}$ is

The set of all real values of $x$ for which the expansion of $\left(125 x^{2}-\frac{27}{x}\right)^{\frac{-2}{3}}$ is valid, is

If $\frac{x^{2}}{2 x^{4}+7 x^{2}+6}=\frac{A x+B}{x^{2}+a}+\frac{C x+D}{a x^{2}+3}$, then $A+B+C-2 D=$

If $(\sin \theta-\operatorname{cosec} \theta)^{2}+(\cos \theta+\sec \theta)^{2}=5$ and $\theta$ lies in the third quadrant, then $(\sin \theta+\cos \theta)^{3}=$

If $0 < B < A < \frac{\pi}{4}, \cos ^{2} B-\sin ^{2} A=\frac{\sqrt{3}+1}{4 \sqrt{2}}$ and $2 \cos A \cos B=\frac{1+\sqrt{2}+\sqrt{3}}{2 \sqrt{2}}$, then $\cos ^{2} \frac{4 B}{3}-\sin ^{2} \frac{4 A}{5}=$

If $\theta$ is an acute angle and $2 \sin ^{2} \theta=\cos ^{4} \frac{\pi}{8}+\sin ^{4} \frac{3 \pi}{8}+\cos ^{4} \frac{5 \pi}{8}+\sin ^{4} \frac{7 \pi}{8}$, then $\theta=$

If $2 \tan ^{2} \theta-4 \sec \theta+3=0$, then $2 \sec \theta=$

If $\sin ^{-1} x-\cos ^{-1} 2 x=\sin ^{-1}\left(\frac{\sqrt{3}}{2}\right)-\cos ^{-1}\left(\frac{\sqrt{3}}{2}\right)$, then $\tan ^{-1} x+\tan ^{-1}\left(\frac{x}{x+1}\right)=$

$\operatorname{sech}^{-1}\left(\frac{3}{5}\right)-\tanh ^{-1}\left(\frac{3}{5}\right)=$

In a $\triangle A B C$, if $a=5, b=3, c=7$, then $\sqrt{\frac{\sin (A-B)}{\sin (A+B)}}=$

In a $\triangle A B C$, if $r_{1}=6, r_{2}=9, r_{3}=18$, then $\cos A=$

$2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}$ are the position vectores of two points $A$ and $B$ respectively and $C$ divides $A B$ in the ratio $3: 2$ : If $3 \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ is the position of vector of a point $D$, then the unit vector in the direction of $C D$ is

A plane $\pi$ passing through the points $2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}, 3 \hat{\mathbf{i}}+4 \hat{\mathbf{k}}$ is parallel to the vector $2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}-4 \hat{\mathbf{k}}$. If a line joining the points $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}$ and $\hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ intersects the plane $\pi$ at the point $a \hat{\mathbf{i}}+b \hat{\mathbf{j}}+c \hat{\mathbf{k}}$, then $a+b+2 c=$

A unit vector $\hat{\mathbf{e}}=a \hat{\mathbf{i}}+b \hat{\mathbf{j}}+c \hat{\mathbf{k}}$ is coplanar with the vectors $\hat{\mathbf{i}}-3 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}$, and $3 \hat{\mathbf{i}}+\hat{\mathbf{j}}-5 \hat{\mathbf{k}}$. If $\hat{\mathbf{e}}$ is perpendicular to the vector $\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$, then $2 a^{2}+3 b^{2}+4 c^{2}=$

$\mathbf{a}=\hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}}, \hat{\mathbf{b}}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\hat{\mathbf{c}}=2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}$ are three vectors. If $\hat{\mathbf{d}}$ is a normal to the plane of $\hat{\mathbf{a}}$ and $\hat{\mathbf{b}}$ and d. $\hat{\mathbf{c}}=2$, then $|\hat{\mathbf{d}}|=$

$\hat{\mathbf{r}} .(\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}})=5$ and $\hat{\mathbf{r}} .(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}})=3$ are two planes. A plane $\pi$ passing through the line of intersection of these two planes, passes through the point $(0,1,2)$. If the equation of $\pi$ is $\hat{\mathbf{r}} .(a \hat{\mathbf{i}}+b \hat{\mathbf{j}}+c \hat{\mathbf{k}})=m$, then $\frac{b c}{a^{2}}=$

The variance of the data: $1,2,3,5,8,13,17$ is approximately

The numbers $2,3,5,7,11,13$ are written on six distinct paper chits. If 3 of them are chosen at random, then the probability that the sum of the numbers on the obtained chits is divisible by 3 , is

If 4 letters are selected at random from the letters of the word PROBABILITY, then the probability of getting a combination of letters in which atleast one letter is repeated is

If two dice are rolled, then the probability of getting a multiple of 3 as the sum of the numbers appeared on the top faces of the dice, if it is known that their sum is an odd number, is

If a random variable $X$ has the following probability distribution, then its variance is

X = x	1	3	5	2
P(X = x)	$3 K^2$	K	$K^2$	2K

The mean and variance of a binomial variate $X$ are $\frac{16}{5}$ and $\frac{48}{25}$ respectively. IfP $(X > 1)=1-K\left(\frac{3}{5}\right)^{7}$, then $5 K=$

$P$ and $Q$ are the points of trisection of the line segment joining the points $(3,-7)$ and $(-5,3)$. If $P Q$ subtends right angle at a variable point $R$, then the locus of $R$ is

$(a, b)$ is the point to which the origin has to be shifted by translation of axes so as to remove the first-degree terms from the equation $2 x^{2}-3 x y+4 y^{2}+5 y-6=0$. If the angle by which the axes are to be rotated in positive direction about the origin to remove the $x y$-term from the equation $a x^{2}+23 a b x y+b y^{2}=0$ is $\theta$, then $\tan 2 \theta=$

$A(1,-2), B(-2,3), C(-1,-3)$ are the vertices of a $\triangle A B C . L_{1}$ is the perpendicular drawn from $A$ to $B C$ and $L_{2}$ is the perpendicular bisector of $A B$. If $(l, m)$ is the point of intersection of $L_{1}$ and $L_{2}$, then $26 m-3=$

The area of the parallelogram formed by the lines $L_{1} \equiv \lambda x+4 y+2=0, L_{2} \equiv 3 x+4 y-3=0$, $L_{3} \equiv 2 x+\mu y+6=0, L_{4} \equiv 2 x+y+3=0$, where $L_{1}$ is parallel to $L_{2}$ and $L_{3}$ is parallel to $L_{4}$ is

If $A(1,2), B(2,1)$ are two vertices of an acute angled triangle and $S(0,0)$ is its circumcenter, then the angle subtended by $A B$ at the third vertex is

If the angle between the pair of lines given by the equation $a x^{2}+4 x y+2 y^{2}=0$ is $45^{\circ}$, then the possible values of $a$

A circle passing through the points $(1,1)$ and $(2,0)$ touches the line $3 x-y-1=0$. If the equation of this circle is $x^{2}+y^{2}+2 g x+2 f y+c=0$, then a possible value of $g$ is

A circle passes through the points $(2,0)$ and $(1,2)$. If the power of the point $(0,2)$ with respect to this circle is 4 , then the radius of the circle is

$x-2 y-6=0$ is a normal to the circle $x^{2}+y^{2}+2 g x+2 f y-8=0$. If the line $y=2$ touches this circle, then the radius of the circle can be

The line $x+y+1=0$ intersects the circle $x^{2}+y^{2}-4 x+2 y-4=0$ at the points $A$ and $B$. If $M(a, b)$ is the mid-point of $A B$, then $a-b=$

A circle $S$ passes through the points of intersection of the circles $x^{2}+y^{2}-2 x-3=0$ and $x^{2}+y^{2}-2 y=0$. If $x+y+1=0$ is a tangent to the circle $S$, then equation of $S$ is

If the common chord of the circles $x^{2}+y^{2}-2 x+2 y+1=0$ and $x^{2}+y^{2}-2 x-2 y-2=0$ is the diameter of a circle $S$, then the center of the circles is

$(1,1)$ is the vertex and $x+y+1=0$ is the directrix of a parabola. If $(a, b)$ is its focus and $(c, d)$ is the point of intersection of the directrix and the axis of the parabola, then $a+b+c+d=$

The axis of a parabola is parallel to $Y$-axis. If this parabola passes through the points $(1,0),(0,2),(-1,-1)$ and its equation is $a x^{2}+b x+c y+d=0$, then $\frac{a d}{b c}=$

If the focus of an ellipse is $(-1,-1)$, equation of its directrix corresponding to this focus is $x+y+1=0$ and its eccentricity is $\frac{1}{\sqrt{2}}$, then the length of its major axis is

If the normal drawn at the point $(2,-1)$ to the ellipse $x^{2}+4 y^{i}=8$ meets the ellipse again at $(a, b)$, then $17 a=$

$P(\theta)$ is a point on the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{9}=1, S$ is its $\mathrm{fOO}_{4 /}$ lying on the positive $X$-axis and $Q=(0,1)$. If $S Q=\sqrt{26}$ and $S P=6$, then $\theta=$

If $A(-2,4, a), B(1, b, 3), C(c, 0,4)$ and $D(-5,6,1)$ are collinear points, then $a+b+c=$

$A(1,-2,1)$ and $B(2,-1,2)$ are the end points of a line segment. If $D(\alpha, \beta, \gamma)$ is the foot of the perpendicular drawn from $C(1,2,3)$ to $A B$, then $\alpha^{2}+\beta^{2}+\gamma^{2}=$

The foot of the perpendicular drawn from the point $(-2,-1,3)$ to a plane $\pi$ is $(1,0,-2)$. If $a, b, c$ are the intercepts made by the plane $\pi$ on $X, Y, Z$-axis respectively, then $3 a+b+5 c=$

$\lim\limits_{x \rightarrow \frac{3}{2}} \frac{\left(4 x^{2}-6 x\right)\left(4 x^{2}+6 x+9\right)}{\sqrt[3]{2 x}-\sqrt[3]{3}}=$

If the real valued function $f(x)=\int \frac{\left(4^{x}-1\right)^{4} \cot (x \log 4)}{\sin (x \log 4) \log \left(1+x^{2} \log 4\right)}, \quad$ if $x \neq 0$ is continuous at $x=0$, then $e^{k}=$

A function $f: R \rightarrow R$ is such that $y f(x+y)+\cos m x y=1+y f(x)$. If $m=2$, then $f^{\prime}(x)=$

If $y=\sqrt{\sin (\log 2 x)+\sqrt{\sin (\log 2 x)+\sqrt{\sin (\log 2 x)+\ldots \infty,}}}$ then $\frac{d y}{d x}=$

If $y=\tan ^{-1}\left[\frac{\sin ^{3}(2 x)-3 x^{2} \sin (2 x)}{3 x \sin ^{2}(2 x)-x^{3}}\right]$, then $\frac{d y}{d x}=$

Derivative of $(\sin x)^{x}$ with respect to $x^{(\sin x)}$ is

For a given function $y=f(x), \delta y$ denote the actual error in $y$ corresponding to actual error $\delta x$ in $x$ and $d y$ denotes the approximately value of $\delta y$. If $y=f(x)=2 x^{2}-3 x+4$ and $\delta x=0.02$, then the value of $\delta y-d y$ when $x=5$ is

The length of the normal drawn at $t=\frac{\pi}{4}$ on the curve $x=2(\cos 2 t+t \sin 2 t), y=4(\sin 2 t+t \cos 2 t)$ is

If Water is poured into a cylindrical tank of radius 3.5 ft at the rate of $1 \mathrm{cu} \mathrm{ft} / \mathrm{min}$, then the rate at which the level of the water in the tank increases (in $\mathrm{ft} / \mathrm{min}$ ) is

$y=2 x^{3}-8 x^{2}+10 x-4$ is a function defined on [1,2]. If the tangent drawn at a point $(a, b)$ on the graph of this function is parallel to X-axis $a \in(1,2)$, then $a=$

If $m$ and $M$ are respectively the absolute minimum and absolute maximum values of a function $f(x)=2 x^{3}+9 x^{2}+12 x+1$ defined on $[-3,0]$, then $m+M=$

$\int \frac{\sec x}{3(\sec x+\tan x)+2} d x=$

$\int \frac{d x}{4+3 \cot x} d x=$

$\int \frac{d x}{(x+1) \sqrt{x^{2}+4}}=$

If $\int e^{x}\left(x^{3}+x^{2}-x+4\right) d x=e^{x} f(x)+c$, then $f(1)=$

$\int_{\frac{\pi}{5}}^{\frac{3 \pi}{10}} \frac{d x}{\sec ^{2} x+\left(\tan ^{2024} x-1\right)\left(\sec ^{2} x-1\right)}=$

$\int_{-\pi / 15}^{\pi / 5} \frac{\cos 5 x}{1+e^{5 x}} d x=$

The area of the region (in sq units) enclosed by the curves $y=8 x^{3}-1, y=0, x=-1$ and $x=1$ is

If the equation of the curve which passes through the point $(1,1)$ satisfies the differential equation $\frac{d y}{d x}=\frac{2 x-5 y+3}{5 x+2 y-3}$, then the equation of that curve is

The general solution of the differential equation $\left(6 x^{2}-2 x y-18 x+3 y\right) d x-\left(x^{2}-3 x\right) d y=0$ is

Physics

The range of gravitational forces is

In a simple pendulum experiment for the determination of acceleration due to gravity, the error in the measurement of the length of the pendulum is $1 \%$ and the error in the measurement of the time period is $2 \%$. The error in the estimation of acceleration due to gravity is

The position $x$ (in metre) of a particle moving along a straight line is given by $x=t^{3}-12 t+3$, where $t$ is time (in second). The acceleration of the particle when its velocity becomes $15 \mathrm{~ms}^{-1}$ is

The maximum horizontal range of a ball projected from the ground is 32 m . If the ball is thrown with the same speed horizontally from the top of a tower of . height 25 m , the maximum horizontal distance covered by the ball is

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

A block of mass 5 kg is kept on a smooth horizontal surface. A horizontal stream of water coming out of a pipe of area of cross-section $5 \mathrm{~cm}^{2}$ hits the block with a velocity of $5 \mathrm{~ms}^{-1}$ and rebounds back with the same velocity. The initial acceleration of the block is (density of water is $1 \mathrm{~g} / \mathrm{cc}$ )

A constant force of $(8 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+6 \hat{\mathbf{k}}) \mathrm{N}$ acting on a body of mass 2 kg displaces the body from $(2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}-4 \hat{\mathbf{k}}) \mathrm{m}$ to $(4 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+6 \hat{\mathbf{k}}) \mathrm{m}$. The work done in the process is

A ball $A$ of mass 1.2 kg moving with a velocity of $8.4 \mathrm{~ms}^{-1}$ makes one-dimensional elastic collision with a ball $B$ of mass 3.6 kg at rest. The percentage of kinetic energy transferred by ball $A$ to ball $B$ is

A meter scale is balanced on a knife edge at its centre. When two coins, each of mass 9 g are kept one above the other at the 10 cm mark, the scale is found to be balanced at 35 cm . The mass of the meter scale is

A body of mass $m$ and radius $r$ rolling horizontally $m$ an inclined plane to a vertical

a velocity $v$ rolls up an height $\frac{v^{2}}{g}$. The body is

A massless spring of length $l$ and spring constant $k$ oscillates with a time period $T$ when loaded with a mass $m$. The spring is now cut into three equal parts and are connected in parallel. The frequency of oscillation of the combination when it is loaded with ${ }_{3}$ mass 4 m is

An object of mass $m$ at a distance of $20 R$ from the centre of a planet of mass $M$ and radius $R$ has an initity velocity $u$. The velocity with which the object hits the surface of the planet is

( $G$-Universal gravitational constant)

A simple pendulum is made of a metal wire of length $L$, area of cross-section $A$, material of Young's modulus $Y$ and a bob of mass $m$. This pendulum is hung in abus moving with a uniform speed $v$ on a horizontal circular road of radius $R$. The elongation in the wire is

If the excess pressures inside two soap bubbles are in the ratio $2: 3$, then the ratio of the volumes of the somp bubbles is

The velocities of air above and below the surfaces of a flying aeroplane wing are $50 \mathrm{~ms}^{-1}$ and $40 \mathrm{~ms}^{-1}$, respectively. If the area of the wing is $10 \mathrm{~m}^{2}$ and the mass of the aeroplane is 500 kg , then as time passes of (density of air $=13 \mathrm{~kg} \mathrm{~m}^{-3}$ )

A pendulum clock loses 10.8 s a day when the temperature is $38^{\circ} \mathrm{C}$ and gains 108 s a day when the temperature is $18^{\circ} \mathrm{C}$. The coefficient of linear expansion of the metal of the pendulum clock is

A liquid cools from a temperature of 368 K to 358 K in 22 min . In the same room, the same liquid takes 12.5 min to cool from 358 K to 353 K . The room temperature is

For a gas in a thermodynamic process, the relation between internal energy $U$, the pressure $p$ and the volume $V$ is $U=3+15 p V$. The ratio of the specific heat capacities of the gas at constant volume and constant pressure is

At a pressure $p$ and temperature $127^{\circ} \mathrm{C}$, a vessel contains 21 g of a gas. A small hole is made into the vessel, so that the gas in it leaks out. At a pressure of $\frac{2 p}{3}$ and a temperature of $t^{\circ} \mathrm{C}$, the mass of the gas leaked out is 5 g . Then, $t=$

The tension applied to a metal wire of one metre length produces an elastic strain of $1 \%$. The density of the metal is $8000 \mathrm{kgm}^{-3}$ and Young's modulus of the metal is $2 \times 10^{11} \mathrm{Nm}^{-2}$. The fundamental frequency of the transverse waves in the metal wire is

Two closed pipes when sounded simultaneously in their fundamental modes produce 6 beats per second. If the length of the shorter pipe is 150 cm , then the length of the longer pipe is

(Speed of sound in air $=336 \mathrm{~ms}^{-1}$ )

An object placed at a distance of 24 cm from a concave mirror forms an image at a distance of 12 cm from the mirror. If the object is moved with a speed of $12 \mathrm{~ms}^{-1}$, then the speed of the image is

When the object and the screen are 90 cm apart, it is observed that a clear image is formed on the screen when a convex lens is placed at two positions separated by 30 cm between the object and the screen. The focal length of the lens is

When a monochromatic light is incident on a surface separating two media, both the reflected and refracted lights have the same

The electric flux due to an electric field $\mathbf{E}=(8 \hat{\mathbf{i}}+13 \hat{\mathbf{j}}) \mathrm{NC}^{-1}$ through an area $3 \mathrm{~m}^{2}$ lying in the $X Z$-plane is

A capacitor of capacitance $C$ is charged to a potential $V$ and disconnected from the battery. Now, if the space between the plates is completely filled with a substance of dielectric constant $K$, the final charge and the final potential on the capacitor are respectively.

A voltmeter of resistance $400 \Omega$ is used to measure the emf of a cell with an internal resistance of $4 \Omega$. The error in the measurement of emf of the cell is

When two wires are connected in the two gaps of a meter bridge, the balancing length is 50 cm . When the wire in the right gap is stretched to double its lengt ${ }^{-}$ and again connected in the same gap, then the new balancing length from the left. end of the bridge wiu :

A magnetic field is applied in $y$-direction on an $\alpha$-particle travelling along $x$-direction. The motion of the $\alpha$-particle will be

A straight wire carrying a current of $2 \sqrt{2} \mathrm{~A}$ is making an angle of $45^{\circ}$ with the direction of uniform magnetic field of 3 T . The force per unit length of the wire due to the magnetic field is

The magnetising field which produces a magnetic flux of $22 \times 10^{-6} \mathrm{~Wb}$ in a metal bar of area of cross-section $2 \times 10^{-5} \mathrm{~m}^{2}$ is (susceptibility of the metal $=699$ )

The energy stored in a coil of inductance 80 mH carrying a current of 2.5 A is