TG EAPCET 2025 (Online) 2nd May Evening Shift

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3-P x^2+Q x-R=0$ and $(\alpha-2)^2,(\beta-2)^2,(\gamma-2)^2$ are the roots of the equation $x^3-5 x^2+4 x=0$, then the possible least value of $P+Q+R$ is

The value of the greatest positive integer $k$, such that $49^k+1$ is a factor of $48\left(49^{125}+49^{124}+\ldots+49^2+49+1\right)$ is

If the variance of the numbers $9,15,21, \ldots,(6 n+3)$ is $P$, then the variance of the first $n$ even numbers is

The value of ' $a$ ' for which the equation $\left(a^2-3\right) x^2+16 x y -2 a y^2+4 x-8 y-2=0$ represents a pair of perpendicular lines is

If the line passing through the point $(4,-3)$ and having negative slope makes an angle of $45^{\circ}$ with the line joining the points $(1,1),(2,3)$, then the sum of intercepts of that line is

The point $P(\alpha, \beta)(\alpha>0, \beta>0)$ undergoes the following transformations successively.
(a) Translation to a distance of 3 units in positive direction of $X$-axis.
(b) Reflection about the line $y=-x$.
(c) Rotation of axes thr...

$O(0,0), B(-3,-1)$ and $C(-1,-3)$ are vertices of a $\triangle O B C$. $D$ is a point on $O C$ and $E$ is a point on $O B$. If the equation of $D E$ is $2 x+2 y+\sqrt{2}=0$, then the ratio in which the line $D E$ divides the altitude of the...

Every point on the curve $3 x+2 y-3 x y=0$ is the centroid of a triangle formed by the coordinate axes and a line $(L)$ intersecting both the coordinates axes. Then, all such lines $(L)$

If $L$ is a line common to the planes $3 x+4 y+7 z=1$, $x-y+z=5$, then the direction ratios of the line $L$ are

Let $A$ be a point having position vector $\hat{\mathbf{i}}-3 \hat{\mathbf{j}}$ and $\mathbf{r}=(\hat{\mathbf{i}}-3 \hat{\mathbf{j}})+t(\hat{\mathbf{j}}-2 \hat{\mathbf{k}})$ be a line. If $P$ is a point on this line and is at a minimum dist...

If the points $(1,1, \lambda)$ and $(-3,0,1)$ are equidistant from the plane $3 x+4 y-12 z+13=0$, then the values of $\lambda$ are

$1+\cos x+\cos ^2 x+\cos ^3 x+\ldots$ to $\infty=4+2 \sqrt{3}$, then $x=$

If $3 \sin (\alpha-\beta)=5 \cos (\alpha+\beta)$ and $\alpha+\beta \neq \frac{\pi}{2}$, then $\frac{\tan \left(\frac{\pi}{4}-\alpha\right)}{\tan \left(\frac{\pi}{4}-\beta\right)}=$

\(\frac{\sin 1^{\circ}+\sin 2^{\circ}+\ldots \sin 89^{\circ}}{2\left(\cos 1^{\circ}+\cos 2^{\circ}+\ldots+\cos 44^{\circ}\right)+1}=\)

If $5 \sin \theta+3 \cos \left(\theta+\frac{\pi}{3}\right)+3$ lies between $\alpha$ and $\beta$ (including $\alpha, \beta$ also), then $(\alpha-\beta)(\alpha+\beta-6)=$

The four points whose position vectors are given by $2 a+3 b-c, a-2 b+3 c, 3 a+4 b-2 c$ and $a-6 b+6 c$ are

If $a=|\mathbf{a}| ; b=|\mathbf{b}|$, then $\left(\frac{\mathbf{a}}{a^2}-\frac{\mathbf{b}}{b^2}\right)^2$

$\mathbf{a}, \mathbf{b}, \mathbf{c}$ are three unit vectors such that $x \mathbf{a}+y \mathbf{b}+z \mathbf{c}= p(\mathbf{b} \times \mathbf{c})+q(\mathbf{c} \times \mathbf{a})+r(\mathbf{a} \times \mathbf{b})$. If $(\mathbf{a}, \mathbf{b})=(\...

$O(0,0,0), A(3,1,4), B(1,3,2)$ and $C(0,4,-2)$ are the vertices of a tetrahedron. If $G$ is the centroid of the tetrahedron and $G_1$ is the centroid of its face $A B C$, then the point which divides $G G_1$ in the ratio $1: 2$ is

$A, B, C$ and $D$, are any four points. If $E$ and $F$ are mid-points of $A C$ and $B D$ respectively, then $\mathbf{A B}+\mathbf{C B}+\mathbf{C D}+\mathbf{A D}=$

An inductor and a resistor are connected in series to an AC supply. If the potential difference across the inductor and the resistor are 180 V and 240 V respectively, then the voltage of the AC supply is

In $\beta^{-}$decay, a neutron transforms into a proton within the nucleus according to the equation :
neutron $\rightarrow$ proton $+\beta^{-}+x$
In this equation the particle represented by ' $x$ ' is

Bose-Einstein statistics is applicable to particles with

Two radioactive substances $A$ and $B$ have same number of initial nuclei. If the half lives of $A$ and $B$ are 1.5 days and 4.5 days respectively, then the ratio of the number of nuclei remaining in $A$ and $B$ after 9 days is

The ratio of the kinetic energies of the electrons in the third and fourth excited states of hydrogen atom is

Coaxial cable, a widely used wire medium offers and approximate frequency bandwidth of

In a potentiometer experiment, a wire of length 10 m and resistance $5 \Omega$ is connected to a cell of emf 2.2 V . If the potential difference between two points separated by a distance of 660 cm on potentiometer wire is 1.1 V , then the ...

When the right gap of a metre bridge consists of two equal resistors in series, the balancing point is at 50 cm . When one of the resisters in the right gap is removed and is connected in parallel to the resistor in the left gap, the balanc...

When a photosensitive material is illuminated by photons of energy 3.1 eV , the stopping potential of the photoelectrons is 1.7 V . When the same photosensitive material is illuminated by photons of energy 2.5 eV , the stopping potential of...

Two wires $A$ and $B$ made of same material and areas of cross-section in the ratio $1: 2$ are stretched by same force. If the masses of the wires $A$ and $B$ are in the ratio $2: 3$, then the ratio of the elongations of the wires $A$ and $...

A coil of resistance $8 \Omega$, number of turns 250 and area $120 \mathrm{~cm}^2$ is placed in a uniform magnetic field of 2 T such that the plane of the coil makes and angle of $\frac{\pi}{6}$ with the direction of the magnetic field. In ...

If electromagnetic waves of power 600 W incident on a non-reflecting surface, then the total force acting on the surface is

An electron and a positron enter a uniform electric field $E$ perpendicular to it with equal speeds at the same time. The distance of separation between them in the direction of the field after a time ' $t$ ' is
( $\frac{e}{m}$ is specific ...

A charge $q$ is placed at the centre ' $O$ ' of a circle of radius $R$ and two other charges $q$ and $q$ are placed at the ends of the diameter $A B$ of the circle. The work done to move the charge at point $B$ along the circumference of th...

For which of the following Reynold's number, a flow is streamlined?

Water is filled in a tank up to a height of 20 cm from the bottom of the tank. Water flows through a hole of area $1 \mathrm{~mm}^2$ at its bottom. The mass of the water coming out from the hole in a time of 0.6 s is

The ratio of the time periods of a simple pendulum at heights $2 R_E$ and $3 R_E$ from the surface of the Earth is ( $R_E$ is radius of the Earth)

A body cools from a temperature of $60^{\circ} \mathrm{C}$ to $50^{\circ} \mathrm{C}$ in 10 minutes and $50^{\circ} \mathrm{C}$ to $40^{\circ} \mathrm{C}$ in 15 minutes. The time taken in minutes for the body to cool from $40^{\circ} \mathr...

If the heat required to increase the rms speed of 4 moles of a diatomic gas from $v$ to $\sqrt{3} v$ is 83.1 kJ , then the initial temperature of the gas is
(universal gas constant $=8.31 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}$ )

When the temperature of a gas in a closed vessel is increased by $2.4^{\circ} \mathrm{C}$, its pressure increases by $0.5 \%$. The initial temperature of the gas is

A gas is suddenly compressed such that its absolute temperature is doubled. If the ratio of the specific heat capacities of the gas is 1.5 , then the percentage decrease in the volume of the gas is

If the system of blocks shown in the figure is released from rest, the ratio of the tensions $T_1$ and $T_2$ is (Neglect the mass of the string shown in the figure)

If $B_V$ and $B_H$ are respectively the vertical and horizontal components of the Earth's magnetic field at a place where the angle of $\operatorname{dip}$ is $60^{\circ}$, then the total magnetic field at that place is

Two bodies are projected from the same point with the same initial velocity ' $u$ ' making angles ' $\theta^{\prime}$ and $\left(90^{\circ}-\theta\right)$ with the horizontal in opposite directions. The horizontal distance between their pos...

The ratio of times taken by a freely falling body to travel first 5 m , second 5 m , third 5 m distances is

A body projected vertically up with an initial speed of $10 \mathrm{~ms}^{-1}$ reaches the point of projection after sometime with a speed of $8 \mathrm{~ms}^{-1}$. The maximum height reached by the body is
(Acceleration due to gravity $=10...

Two identical wires, carrying equal currents are bent into circular coils $A$ and $B$ with 2 and 3 turns respectively. The ratio of the magnetic fields at the centres of the coils $A$ and $B$ is

A current of 4 A is passed through a square loop of side 5 cm made of a uniform manganin wire as shown in the figure. The magnetic field at the centre of the loop is

A compound microscope has an objective of focal length 1.25 cm and an eyepiece of focal length 5 cm separated by a distance of 7.5 cm . The total magnification produced by the microscope when the final image forms at infinity is

The property of light that explains the formation of coloured images due to thick lenses is