List of top Mathematics Questions asked in TS EAMCET

If the real-valued function

\[ f(x) = \sin^{-1}(x^2 - 1) - 3\log_3(3^x - 2) \]

is not defined for all $ x \in (-\infty, a] \cup (b, \infty) $, then what is $ 3^a + b^2 $?

$$ x^5 + 4x^4 - 13x^3 - 52x^2 + 36x + 144 = 0 $$

The point of intersection of two tangents drawn to the hyperbola \[ \frac{x^2}{a^2} - \frac{y^2}{4} = 1 \] lie on the circle \[ x^2 + y^2 = 5. \] If these tangents are perpendicular to each other, then $ a $ is:

A circle $ S $ passes through the points of intersection of the circles $ x^2 + y^2 - 2x - 3 = 0 $ and $ x^2 + y^2 - 2y = 0 $. If $ x + y + 1 = 0 $ is a tangent to the circle $ S $, then the equation of $ S $ is:

If $ (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 $ is:

Among the 5 married couples, if the names of 5 men are matched with the names of their wives randomly, then the probability that no man is matched with the name of his own wife is ?

If all the numbers which are greater than 6000 and less than 10000 are formed with the digits $ 3,5,6,7,8 $ without repetition of the digits, then the difference between the number of odd numbers and the number of even numbers among them is:

If $\theta$ is the angle between the vectors $4\mathbf{i} - \mathbf{j} + 2\mathbf{k}$ and $ \mathbf{i} + 3\mathbf{j} - 2\mathbf{k}$, then $\sin 2\theta$ is:

Given the position vectors of points $A$ and $B$ as $ \mathbf{A} = 2\mathbf{i} - 3\mathbf{j} + \mathbf{k} $ and $ \mathbf{B} = \mathbf{i} + 2\mathbf{j} - 3\mathbf{k} $, and $C$ divides $AB$ in the ratio 3:2. If $ \mathbf{D} = 3\mathbf{i} - \mathbf{j} + 2\mathbf{k} $ is the position vector of point $D$, find the unit vector in the direction of $ \mathbf{CD} $:

In a triangle $ABC$, if

\[ (a - b)^2 \cos^2 \frac{C}{2} + (a + b)^2 \sin^2 \frac{C}{2} = a^2 + b^2, \]

then $ \cos A $ is:

If \[ \frac{1}{x^4 + x^2 + 1} = \frac{Ax + B}{x^2 + ax + 1} + \frac{Cx + D}{x^2 - ax + 1} \] then $ A + B - C + D = ? $

If $ \theta $ is the acute angle between the curves $ y^2 = x $ and $ x^2 + y^2 = 2 $, then $ \tan \theta $ is:

The length of the latus rectum of the ellipse $ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $ $ (a>b) $ is $ \frac{8}{3} $. If the distance from the center of the ellipse to its focus is $ \sqrt{5} $, then $ \sqrt{a^2 + 6ab + b^2} = \text{?} $

The slope of the tangent drawn from the point $ (1,1) $ to the hyperbola $ 2x^2 - y^2 = 4 $ is:

If $ 0 \leq x \leq \frac{\pi}{2} $, then \[ \lim\limits_{x \to a} \frac{2\cos x - 1}{2\cos x - 1} \] Options:

If the function

\[ f(x) = \begin{cases} \frac{(e^x - 1) \sin kx}{4 \tan x}, & x \neq 0 \\ P, & x = 0 \end{cases} \]

is differentiable at $ x = 0 $, then:

Evaluate the integral: \[ \int e^{-2x} \left( \tan 2x - 2\sec^2 2x \tan 2x \right) dx. \]

The number of ways in which 6 distinct things can be distributed into 2 boxes so that no box is empty is:

The variance of the first 10 natural numbers which are multiples of 3 is:

The domain of the real-valued function $$ f(x) = \sin^{-1} \left( \log_2 \left( \frac{x^2}{2} \right) \right) $$ is:

The domain of the real valued function: \[ f(x) \;=\; \sqrt[3]{\frac{x \,-\, 2}{2x^2 \,-\, 7x \,+\,5}} \;+\; \log\bigl(x^2 \,-\, x \,-\, 2\bigr). \]

If $ Z_1 = \sqrt{3} + i\sqrt{3} $ and $ Z_2 = \sqrt{3} + i $, and \[ \left( \frac{Z_1}{Z_2} \right)^{50} = x + iy, \] then the point $ (x,y) $ lies in:

If $ f $ is a real valued function from $ A $ onto $ B $ defined by $ f(x) = \frac{1}{\sqrt{|x| - |x|}} $, then $ A \cap B $ is:}

{If $f(x)$ is a quadratic function such that $f\left(\frac{1}{x}\right) = f(x) + f\left(\frac{1}{1-x}\right)$, then $\sqrt{f\left(\frac{2}{3}\right) + f\left(\frac{3}{2}\right)} =$}

If $ \alpha,\,\beta,\,\gamma $ are the roots of the equation \[ \begin{vmatrix} x & 2 & 2\\ 2 & x & 2\\ 2 & 2 & x \end{vmatrix} = 0 \]