TG EAPCET 2025 (Online) 4th May Evening Shift

If the angular bisector of the angle $A$ of the $\triangle A B C$ meets its circumcircle at $E$ and the opposite side $B C$ at $D$, then $D E \cos \frac{A}{2}=$

$y-x=0$ is the equation of a side of a $\triangle A B C$. The orthocentre and circumcentre of the $\triangle A B C$ are respectively $(5,8)$ and $(2,3)$. The reflection of orthocentre with respect to any side of the triangle lies on its cir...

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3+\frac{a}{2} x+b=0$ and $(\alpha-\beta)(\alpha-\gamma),(\beta-\alpha)(\beta-\gamma),(\gamma-\alpha),(\gamma-\beta)$ are the roots of the equation
$(y+a)^3+K(y+a)^2+L=0$, then $\fr...

The mean deviation about median of the numbers $3 x, 6 x, 9 x, \ldots .81 x$ is 91 , then $|x|=$

If the equations $3 x^2+2 h x y-3 y^2=0$ and $3 x^2+2 h x y-3 y^2+2 x-4 y+c=0$ represent the four sides of a square, then $\frac{h}{c}=$

Two families of lines are given by $a x+b y+c=0$ and $4 a^2+9 b^2-c^2-12 a b=0$. Then, the line common to both the families is

$A(2,0), B(0,2), C(-2,0)$ are three points. Let $a, b, c$ be the perpendicular distances from a variable point $P$ on to the lines $A B, B C$ and $C A$ respectively. If $a, b, c$ are in arithmetic progression, then the locus of $P$ is

Two non-parallel sides of a rhombus are parallel to the lines $x+y-1=0$ and $7 x-y-5=0$. If $(1,3)$ is the centre of the rhombus and one of its vertices $A(\alpha, \beta)$ lies on $15 x-5 y=6$, then one of the possible values of $(\alpha+\b...

The direction ratios of the line bisecting the angle between the $X$-axis and the line having direction ratios $(3,-1,5)$ are

If the plane $-4 x-2 y+2 z+\alpha=0$ is at a distance of two units from the plane $2 x+y-z+1=0$, then the product of all the possible values of $\alpha$ is

Let $A(\alpha, 4,7)$ and $B(3, \beta, 8)$ be two points in space. If $Y Z$ plane and $Z X$-plane respectively divide the line segment joining the points $A$ and $B$ in the ratio $2: 3$ and $4: 5$, then the point $C$ which divides $A B$ in t...

If $a, b$ are real numbers and $\alpha$ is a real roots of $x^2+12+3 \sin (a+b x)+6 x=0$, then the value of $\cos (a+b \alpha)$ for the least positive value of $a+b \alpha$ is

If $\tan \left(\frac{\pi}{4}+\frac{\alpha}{2}\right)=\tan ^3\left(\frac{\pi}{4}+\frac{\beta}{2}\right)$, then $\frac{3+\sin ^2 \beta}{1+3 \sin ^2 \beta}=$

If $P=\sin \frac{2 \pi}{7}+\sin \frac{4 \pi}{7}+\sin \frac{8 \pi}{7}$ and $Q=\cos \frac{2 \pi}{7}+\frac{4 \pi}{7}+\cos \frac{8 \pi}{7}$, then the point $(P, Q)$ lies on the circle of radius

If $\cos \alpha=\frac{l \cos \beta+m}{l+m \cos \beta}$, then $\left(\frac{\tan \frac{\alpha}{2}}{\tan \frac{\beta}{2}}\right)^2=$

A plane $\pi_1$ contains the vectors $\hat{\mathbf{i}}+\hat{\mathbf{j}}$ and $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}$. Another plane $\pi_2$ contains the vectors $2 \hat{\mathbf{i}}-\hat{\mathbf{j}}$ and $3 \hat{\mathbf{i}}+2 \hat{\mathbf{k}}$...

$\mathbf{a}, \mathbf{b}, \mathbf{c}$ are three non- coplanar and mutually perpendicular vectors of same magnitude $K . r$ is any vectors satisfying $\mathbf{a} \times((\mathbf{r}-\mathbf{b}) \times \mathbf{a})+\mathbf{b} \times((\mathbf{r}-...

Consider the following
Assertion (A) The two lines $\mathbf{r}=\mathbf{a}+t(\mathbf{b})$ and $\mathbf{r}=\mathbf{b}+s(\mathbf{a})$ intersect each other.
Reason (R) The shortest distance between the lines $\mathbf{r}=\mathbf{p}+t(\mathbf{q})...

In a quadrilateral $A B C D, \mathbf{A}=\frac{2 \pi}{3}$ and $A C$ is the bisector of angle $\mathbf{A}$. If $15|\mathbf{A C}|=5|\mathbf{A D}|=3|\mathbf{A B}|$, then angle between $\mathbf{A B}$ and $\mathbf{B C}$ is

Two adjacent sides of a triangle are represented by the vectors $2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ and $2 \sqrt{3} \hat{\mathbf{i}}-2 \sqrt{3} \hat{\mathbf{j}}+\sqrt{3} \hat{\mathbf{k}}$. Then, the least angle of the t...

When an element ${ }_{90}^{232} \mathrm{Th}$ decays into ${ }_{82}^{208} \mathrm{~Pb}$, the number of $\alpha$ and $\beta^{-}$particles emitted respectively are

The maximum wavelength of incident radiation required to ionize a hydrogen atom in its ground state is nearly

During the disintegration of a radioactive nucleus of mass number 208 at rest, two alpha particles each with kinetic energy $E$ are emitted. The total kinetic energy of the emitted alpha particles and the daughter nucleus after the disinteg...

If the frequencies of the carrier wave and message signal are 1 MHz and 28 kHz respectively, then the frequencies of the side bands are

The area of cross-section of a potentiometer wire is $6 \times 10^{-7} \mathrm{~m}^2$. The potential difference per unit length of the potentiometer wire when it is connected to a cell of negligible internal resistance and a resistor in ser...

For the circuit shown in the figure, the current through $6 \Omega$ resistor connected between the junctions $A$ and $B$ is

In a photoelectric experiment, the slope of the graph drawn between stopping potential along $Y$-axis and frequency of incident radiation along $X$-axis is (Planck's constant $=6.6 \times 10^{-34} \mathrm{Js}$ )

A steel rod with a circular cross-section of diameter 1 cm and another steel rod with a square cross-section of side 1 cm have equal mass. If the two rods are subjected to same tension, the ratio of the elongations of the two rods is

The small energy losses in transformers due to eddy currents can be reduced by

A coil of resistance $16 \Omega$ is placed with its plane perpendicular to a uniform magnetic field whose flux ( $\phi$ in $10^{-3}$ weber) changes with time ( $t$ in second) as $\phi=5 t^2+4 t+2$. The induced current at time $t=6$ second i...

If the electric field of a plane electromagnetic wave is $E_z=60 \sin \left(0.5 \times 10^3 x+1.5 \times 10^{11} t\right) \mathrm{Vm}^{-1}$, then the magnetic field of the wave is

The electrostatic force between two charges kept in air is $F$. If $30 \%$ of the space between the charges is filled with a medium, then the electrostatic force between the charges becomes $\frac{F}{2.56}$. The dielectric constant of the m...

729 small identical spheres each charged to an electric potential 3V combine to form a bigger sphere. The electric potential of the bigger sphere is

A cube of side 40 cm is floating with $\frac{1}{4}$ th of its volume immersed in water. When a circular disc is placed on the cube, it floats with $\frac{2}{5}$ th of its volume immersed in water. The mass of the disc is

The maximum length of water column that can stay without falling in a vertically held capillary tube of diameter 1 mm and open at both the ends is (Acceleration due to gravity $=10 \mathrm{~ms}^{-2}$ and surface tension of water $=0.07 \mat...

The force of mutual attraction between any two objects by virtue of their masses is

Which of the following is incorrect about the gravitational force between two bodies?

A steel pendulum clock manufactured at $32^{\circ} \mathrm{C}$ and working at $47^{\circ} \mathrm{C}$ is nearly
(Coefficient of linear expansion of steel $=12 \times 10^{-6} /{ }^{\circ} \mathrm{C}$ )

The ratio of the average translational kinetic energies of hydrogen and oxygen at the same temperature is

A Carnot engine uses diatomic gas as a working substance. During the adiabatic expansion part of the cycle, if the volume of the gas becomes 32 times its initial volume, then the efficiency of the engine is

A metal metre scale that is accurate up to 0.5 mm is made at a temperature of $25^{\circ} \mathrm{C}$. The range of temperatures within which it can be used is (Coefficient of linear expansion of the metal $=10^{-5} /{ }^{\circ} \mathrm{C}$...

A block of mass $\sqrt{2} \mathrm{~kg}$ is placed on a rough horizontal surface. A force ' $F$ ' acting upwards at an angle of $45^{\circ}$ with the horizontal causes the block to start motion. If the coefficient of static friction between ...

At a certain place in the magnetic meridian the Earth's magnetic field is twice its vertical component. The ratio of horizontal component of Earth's magnetic field and the total magnetic field of the Earth at the place is

The vertical displacement ( $y$ in metre) of a projectile in term of its horizontal displacement ( $x$ in metre) is given by $y=\left(\sqrt{3} x-0.2 x^2\right)$. The time of flight of the projectile is
(Acceleration due to gravity $=10 \mat...

For a particle moving along a straight line path, the displacements in third and fifth seconds of its motion are 10 m and 18 m respectively. The speed of the particle at time $t=4 \mathrm{~s}$ is

A proton moving with a velocity of $8 \times 10^5 \mathrm{~ms}^{-1}$ enters a uniform magnetic fleld normal to the direction of the magnetic field. If the radius of the circular path of the proton in the magnetic field is 8.3 cm , then the ...

As shown in the figure, a uniform straight wire of length $30 \sqrt{3} \mathrm{~cm}$ is bent in the form of an equilateral triangle $A B C$. A uniform magnetic field $2 T$ is applied parallel to the side $B C$. If the current through the wi...

A ray of light incidents at an angle of $9.3^{\circ}$ on one face of a small angle prism of refracting angle $6^{\circ}$. If the ray of light emerges normally from the second face, the refractive index of the material of the prism is

A straight metal rod of length 6 cm is placed along the principal axis of a concave mirror of focal length 9 cm such that the end of the rod closer to the mirror is at a distance of 15 cm from the pole of the mirror. The length of the image...

If the length of a thin uniform rod is ' $L$ ' and the radius of gyration of the rod about an axis perpendicular to its length and passing through one end is $K$, then $K: L=$