TG EAPCET 2024 (Online) 11th May Morning Shift

$\alpha, \beta$ 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 $a-\frac{1}{b}=$

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 $f(2)=$

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

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

$(a, b)$ 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 direct...

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 $a$

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

$A(1,-2), B(-2,3), C(-1,-3)$ 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 $L_{2}$, t...

$\hat{\mathbf{r}} .(\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}})=5$ 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 ...

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 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...

$A(1,-2,1)$ 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 $3 a+b+5 c=$

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

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 $\theta=$

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 $3 \hat{\mathbf{i}}+\hat{\mathbf{j}}-5 \hat{\mathbf{k}}$. I...

$\mathbf{a}=\hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}}, \hat{\mathbf{b}}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\hat{\mathbf{c}}=2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}$ are three vectors. If $\hat...

$2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+\hat{\mathbf{k}}$ 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{\m...

A capacitor and a resistor are connected in series to an AC source. If the ratio of the capacitive reactance of the capacitor and the resistance of the resistor is $4: 3$, then the power factor of the circuit is

Half-life periods of two nuclei $A$ and $B$ are $T$ and $2 T$ respectively. Initially $A$ and $B$ have same number of nuclei. After a time of $4 T$, the ratio of the remaining number of nuclei of $A$ and $B$ is

In hydrogen atom, the frequency of the photon emitted when an electron jumps from second orbit to first orbit is $f$. The frequency of the photon emitted when an electron jumps from third excited state to first excited state is

If the ratio of the radii of nuclei ${ }_{52} X^{A}$ and ${ }_{13} \mathrm{AI}^{27}$ is $5: 3$, then the number of neutrons in the nucleus $X$ is

For the displacement current through the plates of a parallel plate capacitor of capacitance $30 \mu \mathrm{~F}$ to be $150 \mu \mathrm{~A}$, the potential difference across the plates of the capacitor has to vary at the rate of

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 ...

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

The maximum distance between the transmitting and receiving antennas is $D$. If the heights of both transmitting and receiving antennas are doubled, then the maximum distance between the two antennas 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 fro...

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

The work functions of two photosensitive metal surfaces $A$ and $B$ are in the ratio $2: 3$. If $x$ and $y$ are the slopes of the graphs drawn between the stopping potential and frequency of incident light for the surfaces $A$ and $B$ respe...

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...

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

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

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 , the...

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

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)

The range of gravitational forces 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 $t^{\circ} \mathrm{C}$...

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

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

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

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 ...

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 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...

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

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

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

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