TG EAPCET 2025 (Online) 3rd May Morning Shift

If $m_1, m_2$ are the slopes of the tangents drawn through the point $(-1,-2)$ to the circle $(x-3)^2+(y-4)^2=4$, then $\sqrt{3}\left|m_1-m_2\right|=$

The equation of the locus of a point, which is at a distance of 5 units from a fixed point $(1,4)$ and also from a fixed line $2 x+3 y-1=0$ is

One of the values of $\sqrt{24-70 i}+\sqrt{-24+70 i}$ is

The set of all values of $\theta$ such that $\frac{1-i \cos \theta}{1+2 i \sin \theta}$ is purely imaginary is

If $\alpha$ is a root of the equation $x^2-x+1=0$, then
$\left(\alpha+\frac{1}{\alpha}\right)^3+\left(\alpha^2+\frac{1}{\alpha^2}\right)^3+\left(\alpha^3+\frac{1}{\alpha^3}\right)^3+\left(\alpha^4+\frac{1}{\alpha^4}\right)^3+\ldots$ to 12 ...

\(\int_0^{\pi / 4} \frac{1}{5 \cos ^2 x+16 \sin ^2 x+8 \sin x \cos x} d x=\)

\(\int_8^{18} \frac{1}{(x+2) \sqrt{x-3}} d x=\)

If [.] denotes the greatest integer function, then $\int_1^2\left[x^2\right] d x=$

The differential equation of a family of hyperbolas whose axes are parallel to coordinate axes, centres lie on the line $y=2 x$ and eccentricity is $\sqrt{3}$ is

The general solution of the differential equation $\left(x^3-y^3\right) d x=\left(x^2 y-x y^2\right) d y$ is

If $3^x y^x=x^{3 y}$, then the value of $\frac{d y}{d x}$ at $x=1$ is

If $y=\left(1-x^2\right) \tanh ^{-1} x$, then $\frac{d^2 y}{d x^2}=$

If $f(x)=\log _{\left(x^2-2 x+1\right)}\left(x^2-3 x+2\right), x \in R-[1,2]$ and $x \neq 0$, then $f^{\prime}(3)=$

The mid-point of the chord of the ellipse $x^2+\frac{y^2}{4}=1$ formed on the line $y=x+1$ is

If a normal is drawn at a variable point $P(x, y)$ on the curve $9 x^2+16 y^2-144=0$, then the maximum distance from the centre of the curve to the normal is

Let $P$ be a point on the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ and let the perpendicular drawn through $P$ to the major axis meet its auxiliary circle at $Q$. If the normals drawn at $P$ and $Q$ to the ellipse and the auxiliary circle re...

If $D \subseteq R$ and $f: D \rightarrow R$ defined by $f(x)=\frac{x^2+x+a}{x^2-x+a}$ is a surjection, then ' $a$ ' lies in the interval.

A real valued function $f:[4, \infty) \rightarrow R$ is defined as $f(x)=\left(x^2+x+1\right)^{\left(x^2-3 x-4\right)}$, then $f$ is

If the domain of the real valued function $f(x)=\frac{1}{\sqrt{\log _{\frac{1}{3}}\left(\frac{x-1}{2-x}\right)}}$ is $(a, b)$, then $2 b=$

If the tangent drawn at the point $P(3 \sqrt{2}, 4)$ on the hyperbola $\frac{x^2}{9}-\frac{y^2}{16}=1$ meets its directrix at $Q(\alpha, \beta)$ in fourth quadrant, then $\beta=$

If $\frac{x+3}{(x+1)\left(x^2+2\right)}=\frac{a}{x+1}+\frac{b x+c}{x^2+2}$, then $a-b+c=$

\(\int e^{-x}\left(x^3-2 x^2+3 x-4\right) d x=\)

\(\int\left(1+\tan ^2 x\right)(1+2 x \tan x) d x=\)

\(\int \frac{x^2 \tan ^{-1} x}{\left(1+x^2\right)^2} d x=\)

\(\int \frac{\log x}{(1+x)^3} d x=\)

If $0 \leq x<\frac{3}{4}$, then the number of values of $x$ satisfying the equation $\tan ^{-1}(2 x-1)+\tan ^{-1} 2 x= \tan ^{-1} 4 x-\tan ^{-1}(2 x+1)$ is

If $\sinh ^{-1} x=\cosh ^{-1} y=\log (1+\sqrt{2})$, then $\tan ^{-1}(x+y)$

If $\{x\}=x-[x]$, where $[x]$ is the greatest integer $\leq x$ and $\mathop {\lim }\limits_{x \to {0^ - }} \frac{\cos ^{-1}\left(1-\{x\}^2\right) \sin ^{-1}(1-\{x\})}{\{x\}-\{x\}^4}=\theta$, then $\tan \theta$

For $a \neq 0$ and $b \neq 0$, if the real valued function $f(x)=\frac{\sqrt[5]{a(625+x)}-5}{\sqrt[4]{625+b x}-5}$ is continuous at $x=0$, then $f(0)=$

The value of $x$ at which the real valued function $f(x)=7|2 x+1|-19|3 x-5|$ is not differentiable is

The system of linear equation $(\sin \theta) x+y-2 z=0$, $2 x-y+(\cos \theta) z=0$ and $-3 x+(\sec \theta) y+3 z=0$, where $\theta \neq(2 n+1) \frac{\pi}{2}$, has non-trivial solution for

If $A=\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right]$, then $\operatorname{adj}(\operatorname{adj}(\operatorname{adj} A))$

The sum of all the roots of the equation
$\left|\begin{array}{ccc}x & -3 & 2 \\ -1 & -2 & (x-1) \\ 1 & (x-2) & 3\end{array}\right|=0$ is

If the system of simultaneous linear equations $x+\lambda y-2 z=1, x-y+\lambda z=2$ and $x-2 y+3 z=3$ is inconsistent for $\lambda=\lambda_1$ and $\lambda_2$, then $\lambda_1+\lambda_2=$

If the normals drawn at the points $P\left(\frac{3}{4}, \frac{3}{2}\right)$ and $Q(3,3)$ on the parabola $y^2=3 x$ intersect again on $y^2=3 x$ at $R$, then $R=$

If $\theta$ is the acute angle between the tangents drawn from the point $(1,5)$ to the parabola $y^2=9 x$, then

Number of all possible ways of distributing eight identical apples among three persons is

Number of all possible words (with or without meaning) that can be formed using all the letters of the word CABINET in which neither the word CAB nor the word NET appear is

There are 15 stations on a train route and the train has to be stopped at exactly 5 stations among these 15 stations. If it stops at atleast two consecutive stations, then the number of ways in which the train can be stopped is

If a number $x$ is drawn randomly from the set of numbers $\{1,2,3, \ldots ., 50\}$, then the probability that number $x$ that is drawn satisfies the inequation $x+\frac{10}{x} \leq 11$ is

Two cards are drawn randomly from a pack of 52 playing cards one after the other with replacement. If $A$ is the event of drawing a face card in first draw and $B$ is the event of drawing a clubs card in second draw, then $P\left(\frac{\bar...

If $X$ is a random variable with probability distribution $P(X=k)=\frac{(2 k+3) c}{3^k}, k=0,1,2, \ldots .$. to $\infty$, then $P(X=3)=$

If a coin is tossed seven times, then the probability of getting exactly three heads such that number two heads occur consecutively is

In a $\triangle A B C$, if $c^2-a^2=b(\sqrt{3} c-b)$ and $b^2-a^2=c(c-a)$ then, $\angle A B C$

Let $A B C$ be a triangle right angled at $B$. If $a=13$ and $c=84$, then $r+R=$

If the equations $x^2+p x+2=0$ and $x^2+x+2 p=0$ have a common root, then the sum of the roots of the equation $x^2+2 p x+8=0$ is

If both roots of the equation $x^2-5 a x+6 a=0$ exceed 1 , then the range of ' $a$ ' is

The equation having the multiple root of the equation $x^4+4 x^3-16 x-16=0$ as its roots is

If $\alpha, \beta, \gamma$ and $\delta$ are the roots of the equation $x^4-4 x^3+3 x^2+2 x-2=0$ such that $\alpha$ and $\beta$ are integers and $\gamma, \delta$ are irrational numbers, then $\alpha+2 \beta+\gamma^2+\delta^2=$

If $\frac{1}{2 \cdot 7}+\frac{1}{7 \cdot 12}+\frac{1}{12 \cdot 17}+\frac{1}{17 \cdot 22}+\ldots$ to 10 terms $=k$, then $k=$