TG EAPCET 2025 (Online) 3rd May Evening Shift

There is a possible error of 0.03 cm in a scale of length 1 foot with which the height of a closed right circular cylinder and the diameter of a sphere are measured as 3.5 feet each. If the radii of both cylinder and sphere are same, then t...

If the tangent and the normal drawn to the curve $x y^2+x^2 y=12$ at the point $(1,3)$ meet the X -axis in $T$ and $N$ respectively, then $T N=$

A man of 5 feet height is walking away from a light fixed at a height of 15 feet at the rate of of $K$ miles/hour. If the rate of increase of his shadow is $\frac{11}{5}$ feet $/ \mathrm{sec}$, then $K=($ Take 1 mile $=5280$ feet $)$

If the point $P\left(x_1, y_1\right)$ lying on the curve $y=x^2-x+1$ is the closest point to the line $y=x-3$, then the perpendicular distance from $P$ to the line $3 x+4 y-2=0$ is

If $C_0, C_1, C_2, \ldots, C_n$ are the binomial coefficients in the expansion of $(1+x)^n$ then the value of $\Sigma r^3 \cdot C_r$ when $n=5$ is

The coefficient of $x^{12}$ in the expansion of $\left(x^2+2 x+2\right)^8$ is

The centre of the circle touching the circles $x^2+y^2-4 x-6 y-12=0$
$x^2+y^2+6 x+18 y+26=0$ at their point of contact and passing through the point $(1,-1)$ is

If $(\alpha, \beta)$ is the centre of the circle which passes through the point $(1,-1)$ and cuts the circles
\(x^2+y^2+2 x-3 y-5=0, x^2+y^2-3 x+2 y+1=0\)
orthogonally, then $\alpha-5 \beta=$

If $2 x-3 y+5=0$ and $4 x-5 y+7=0$ are the equations of the normals drawn to a circle and $(2,5)$ is a point on the given circle, then the radius of the circle is

If $(3,-2)$ is the centre of the circle $S \equiv x^2+y^2+2 g x+2 f y-23=0$ and $A$ is a point on the circle $S=0$ such that its distance from a point $P(-1,-5)$ is least, then $A=$

The lines $4 x-3 y+2=0$ intersects the circle $x^2+y^2-2 x+6 y+c=0$ at two points $A, B$ and $A B=8$. If $(1, k)$ is a point on the given circle and $k>0$, then $k=$

Two circles which touch both the coordinate axes intersect at the points $A$ and $B$. If $A=(1,2)$, then $A B=$

The amplitude of the complex number $\frac{(\sqrt{3}+i)(1-\sqrt{3} i)}{(-1+i)(-1-i)}$ is

\((\sqrt{3}+i)^{10}+(\sqrt{3}-i)^{10}=\)

If a complex number $z=x+i y$ represents a point $p(x, y)$ in the argand plane and $z$ satisfies the condition that the imaginary part of $\frac{z-3}{z+3 i}$ is zero, then the locus of the point $P$ is

Number of real values of $(-1-\sqrt{3 i})^{3 / 4}$ is

\(\int_{-1}^1 \frac{\log 2-\log (1+x)}{\sqrt{1-x^2}} d x=\)

\(\int_{-2}^4\left|2-x^2\right| d x=\)

\(\int_0^{\frac{\pi}{4}} \frac{\sec x}{3 \cos x+4 \sin x} d x=\)

If the differential equation having $y=A e^x+B \sin x$ as its general solution is $f(x) \frac{d^2 y}{d x^2}+g(x) \frac{d y}{d x}+h(x) y=0$, then $f(x)+g(x)+h(x)=$

The general solution of the differential equation $\frac{d y}{d x}+(\sec x \operatorname{cosec} x) y=\cos ^2 x$

If $x=\sqrt{1-\tan y}$, then $\frac{d y}{d x}=$

If $x=\sin 2 \theta \cos 3 \theta, y=\sin 3 \theta \cos 2 \theta$, then $\frac{d y}{d x}=$

If $y=\sqrt{\log \left(x^2+1\right)+\sqrt{\log \left(x^2+1\right)+\sqrt{\log \left(x^2+1\right)+\ldots+\infty}}, \text {, } 100.00}$, $|x|<1$, then $\frac{d y}{d x}=$

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

If the perpendicular distance from the focus of an ellipse $\frac{x^2}{9}+\frac{y^2}{b^2}=1(b<3)$ to its corresponding directrix is $\frac{4}{\sqrt{5}}$, then the slope of the tangent to this ellipse drawn at $\left(\frac{3}{\sqrt{2}}, \fra...

Let $f: R \rightarrow R$ be defined by $f(x)=5^{-|x|}+\operatorname{sgn}\left(5^{-x}\right)$, where sgn $x$ denotes signum function of $x$. Then $f$ is

For a real number ' $a$ ', if a real valued function $f(x)=4 x^3+a x^2+3 x-2$ is monotonic in its domain, then the range of ' $a$ ' is

If the range of the real valued function $f(x)=\frac{x^2+x+k}{x^2-x+k}$ is $\left[\frac{1}{3}, 3\right]$, then $k=$

Let $P, Q, R, S$ be the points of intersection of the circle $x^2+y^2=4$ and the hyperbola $x y=\sqrt{3}$. If $P=(\alpha, \beta)$ and $\alpha>\beta>0$, then the equation of the tangent drawn at $P$ to the hyperbola is

If $\frac{5 \pi}{4} < x < \frac{7 \pi}{4}$, then $\int \sqrt{\frac{1-\sin 2 x}{1+\sin 2 x}} d x=$

If $\frac{x^2+1}{\left(x^2+2\right)\left(x^2+3\right)}=\frac{A x+B}{x^2+2}+\frac{C x+D}{x^2+3}$, then $A+B+C+D=$

\(\int \frac{3^x(x \log 3-1)}{x^2} d x=\)

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

\(\int \frac{1}{(2 \cos x+\sin x)^2} d x=\)

The number of values of $x$ satisfying the equation, $\tan ^{-1}\left(x+\frac{\sqrt{2}}{x}\right)+\tan ^{-1}\left(x-\frac{\sqrt{2}}{x}\right)=\tan ^{-1}(x)$ is

If $y=\sec ^{-1} x$, then $\frac{d^2 y}{d x^2}=$

If the real valued function
$$ f(x)=\left\{\begin{array}{ccc} \frac{\cos 3 x-\cos x}{x \sin x}, & \text { if } & x<0 \\ p, & \text { if } & x=0 \\ \frac{\log (1+q \sin x)}{x}, & \text { if } & x>0 \end{array}\right. $$
is continuous at $x=0...

If $[t]$ represents the greatest integer $\leq t$, then the value of $\lim\limits_{x \rightarrow 3} \frac{11-[2-x]}{[x+10]}$ is

If $A=\left[\begin{array}{lll}1 & 5 & 2 \\ 4 & 1 & 3 \\ 2 & 6 & 3\end{array}\right]$, then $\left|(\operatorname{adj} A)^{-1}\right|=$

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

If the system of linear equations $(\sin \theta) x-y+z=0$, $x-(\cos \theta) y+z=0, x+y+(\sin \theta) z=0$ has non-trivial solution, then the least positive value of $\theta$ is

If the system of simultaneous linear equations $x-2 y+z=0,2 x+3 y+z=6$ and $x+2 y+p z=q$ has infinitely many solutions, then

If $m_1$ and $m_2$ are the slopes of the tangents drawn from the point $(1,4)$ to the parabola $y^2=11 x$, then $2\left(m_1^2+m_2^2\right)=$

The number of normals that can be drawn through the point $(2,0)$ to the parabola $y^2=7 x$ is

The number of ways in which 6 boys and 4 girls can be arranged in a row such that between any two girls there must be exactly 2 boys is

The number of all possible three letter words that can be formed by choosing three letters from the letters of the word FEBRUARY so that a vowel always occupies the middle place is

If all the letters of the word ACADEMICIAN are permuted in all possible ways, then the number of permutations in which no two $A^{\prime} s$ are together and all the consonants are together is

If the mean and variance of a binomial distribution are $\frac{4}{3}$ and $\frac{10}{9}$ respectively, then $P(X \geq 6)=$

Out of the given 25 consecutive position integers, three integers are drawn. If the least integer among given 25 integers is an odd number, then the probability that the sum of the three integers drawn is an even number is