TG EAPCET 2024 (Online) 11th May Morning Shift

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 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

$x-2 y-6=0$ 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

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

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

$P$ 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

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

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

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

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

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 $\frac{d y}{d x}=$

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 $f^{\prime}(x)=$

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

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 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 $17 a=$

$f(x)=a x^{2}+b x+c$ 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 $8 p+4 q+2 r=$

$P(\theta)$ 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 $\theta=$

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

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

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

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

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 $\tan ^{-1} x+\tan ^{-1}\left(\frac{x}{x+1}\right)=$

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

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

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 $e^{k}=$

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

If $x=k$ 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

$A=\left[\begin{array}{ll}1 & 2 \\\\ 2 & 1\end{array}\right]$ 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}\rig...

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 $\frac{a d}{b c}=$

$(1,1)$ 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 $a+b+c+d=$

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

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

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

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 $5 K=$

If a random variable $X$ has the following probability distribution, then its variance is .tg {border-collapse:collapse;border-spacing:0;} .tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-si...

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 $\cos A=$

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}}\righ...

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 $\frac{1}{\alpha^{4}}+\frac{1}{\beta^{4}}+\frac{1}{\gamma^{4}}+\frac{1}{\delta^{4}}=$

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

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 $\sqrt{f\left(\frac{2}{3}\right)+f\left(\frac{3}{2}\right)}=$