TG EAPCET 2025 (Online) 4th May Morning Shift

If two smallest squares are chosen at random on a chess board, then the probability of getting these squares such that they do not have a side in common is

An urn contains 7 red, 5 white and 3 black balls. Three balls are drawn randomly one after the other without replacement. If it is known that first ball drawn is red and the second ball drawn is white, then the probability that the third ba...

Let $A$ and $B$ be two events in a random experiment . If $P(A \cap \bar{B})=0.1, P(\bar{A} \cap B)=0.2$ and $P(B)=0.5$, then $P(\bar{A} \cap \bar{B})=$

If $a=3, b=5, c=7$ are the sides of a $\triangle A B C$, then $\cot A+\cot B+\cot C=$

Let $p_1, p_2$ and $p_3$ be the altitudes of a $\triangle A B C$ drawn through the vertices $A, B$ and $C$ respectively. If $r_1=4$, $r_2=6, r_3=12$ are the ex-radii of $\triangle A B C$, then $\frac{1}{p_1^2}+\frac{1}{p_2^2}+\frac{1}{p_3^2...

If $\alpha, \beta, \gamma, \delta$ and $\varepsilon$ are the roots of the equation $x^5+x^4-13 x^3-13 x^2+36 x+36=0$ and $\alpha<\beta<\gamma<\delta<\varepsilon$ then $\frac{\varepsilon}{\alpha}+\frac{\delta}{\beta}+\frac{1}{\gamma}=$

If one of the roots of the equation $6 x^3-25 x^2+2 x+8=0$ is an integer and $\alpha>0, \beta<0$ are the other two roots, then $\frac{4}{\alpha}+\frac{1}{\beta}=$

If $\alpha, \beta$ are the roots of the equation $x^2+3 x+k=0$ and $\alpha+\frac{1}{\alpha}, \beta+\frac{1}{\beta}$ are the roots of the equation $4 x^2+p x+18=0$, then $k$ satisfies the equation

If $f(x)=x^2+b x+c$ and $f(1+k)=f(1-k) \forall k \in R$, for two real numbers $b$ and $c$ then

If $f(x)$ is a second degree polynomial such that $f(x) \geq 0 \forall x \in R, f(-3)=0$ and $f(0)=18$, then $f(3)=$

$t_1, t_2, t_3, \ldots, t_n$ are positive integers, $S_n=t_1+t_2+t_3+\ldots+t_n$, $S_1=1^2, S_2=3^2, S_3=6^2, S_4=10^2, S_5=15^2$ and similarly other terms are there. Following this pattern, if $S_{10}=k^2$ then $k=$

$K=\left|\begin{array}{cc}3 & 4 \\ 5 & 4\end{array}\right|+\left|\begin{array}{cc}1 & -1 \\ 5 & 4\end{array}\right|+\left|\begin{array}{cc}\frac{1}{3} & \frac{1}{4} \\ 5 & 4\end{array}\right|+\left|\begin{array}{cc}\frac{1}{9} & -\frac{1}{1...

\(\text { The coefficient of variation for the following data is }\)
$$ \begin{array}{llllll} \hline \text { Class interval } & 0-2 & 2-4 & 4-6 & 6-8 & 8-10 \\ \hline \text { Frequency } & 2 & 3 & 5 & 3 & 2 \\ \hline \end{array} $$

If $(-1,-1)$ is the point of intersection of the pair of lines $2 x^2+5 x y-3 y^2+2 g x+2 f y+c=0$. Then $g+f$

$(a, b)$ are the new coordinates of the point $(2,3)$ after shifting the origin to the point $(3,2)$ by translation of axes. If $(c, d)$ are the new coordinates of the point $(a, b)$ after rotating the axes through an angle $\frac{\pi}{4}$ ...

The area of the quadrilateral formed by the lines $x+2 y+3=0,2 x+4 y+9=0, x-2 y+3=0$ and $3 x-6 y+11=0$

The lines $x+y+4=0, x-2 y-4=0$ and $3 x+4 y-2=0$

The area of the triangle formed by the line $L$ with the coordinate axes is 12 sq. units. If $L$ passes through the point $(12,4)$ and the product $P$ of $X$ - intercept of $L$ and square of the $Y$-intercept of $L$ is negative, then $P=$

The equation of the locus of a point whose distance from $X Y$-plane is twice its distance from $Z$-axis is

If the angle between the planes $a x-y+3 z=2 a$ and $3 x+a y+z=3 a$ is $\frac{\pi}{3}$, then the direction ratio of the line perpendicular to the plane $(a+2) x+(a-4) y+2 a z=a$ are

If $\alpha$ is the angle between any two diagonals of a cube and $\beta$ is the angle between a diagonal of a cube and a diagonal of its face, which intersects this diagonal of the cube, then $\cos \alpha+\cos ^2 \beta=$

Number of solutions of the equation $\tan ^2 x+3 \cot ^2 x=2 \sec ^2 x$ lying in the interval $[0,2 \pi]$ is

If $0 \leq A, B \leq \frac{\pi}{4}$ and $\cot A+\cot B+\tan A+ \tan B=\cot A \cot B-\tan A \tan B$, then $\sin (A+B)=$

If $x=\log _e 3$, then $\tanh 2 x+\operatorname{sech} 2 x=$

If $\cos \theta+\sin \theta=\sqrt{2} \cos \theta$ and $0<\theta<\frac{\pi}{2}$, then $\sec 2 \theta+\tan 2 \theta=$

If $\mathbf{a}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}, \mathbf{b}=6 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}, \mathbf{c}=-4 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+12 \hat{\mathbf{k}}$ are three vectors, then $\sqr...

If $\mathbf{a}$ and $\mathbf{b}$ are two vectors such that $|\mathbf{a}|=|\mathbf{b}|=\sqrt{6}$ and $\mathbf{a} \cdot \mathbf{b}=-1$, then $|\mathbf{a} \times \mathbf{b}| \sin (\mathbf{a}, \mathbf{b})=$

Let $\mathbf{a}$ and $\mathbf{b}$ be two vectors such that $|\mathbf{a}|=|\mathbf{b}|$ and $|\mathbf{a}+2 \mathbf{b}|=|2 \mathbf{a}-\mathbf{b}|$. If $\mathbf{c}$ is a vector parallel to $\mathbf{a}$, then the angle between $\mathbf{b}$ and ...

$A B C D$ is a tetrahedron, $\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}},-2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+3 \hat{\mathbf{k}}$, $3 \bar{i}+2 \bar{j}-\bar{k}$ are the the position vectors of the points $A, B$ and $C$ respective...

If the volume of a tetrahedron having $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}, 2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-3 \hat{\mathbf{k}}$ and $3 \hat{\mathbf{i}}-\hat{\mathbf{j}}+p \hat{\mathbf{k}}$ as its coterminous edges is ...