List of top Questions asked in TS EAMCET- 2024

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

The sum of two roots of the equation $x^4 - x^3 - 16x^2 + 4x + 48 = 0$ is zero. If $\alpha, \beta, \gamma, \delta$ are the roots of this equation, then $\alpha^4 + \beta^4 + \gamma^4 + \delta^4$ is:

A man has 7 relatives, 4 of them are ladies and 3 gents; his wife has 7 other relatives, 3 of them are ladies and 4 gents. The number of ways they can invite them to a party of 3 ladies and 3 gents so that there are 3 of man's relatives and 3 of wife's relatives, is

If $ \vec{b} = \mathbf{i}-\mathbf{j} +2\mathbf{k},\;\vec{c} = \mathbf{i} +2\mathbf{j} -\mathbf{k} $ are two vectors and $ \vec{a} $ is a vector such that $ \cos\angle(\vec{a},\vec{b}\times \vec{c}) = \frac{2}{\sqrt{3}} $. If $ \vec{a} $ is a unit vector, then $ \lvert \vec{a}\times(\vec{b}\times \vec{c})\rvert =\,? $

A point $ P $ is moving on the curve $ x^3 y^4 = 27 $. The x-coordinate of $ P $ is decreasing at the rate of 8 units per second. When the point $ P $ is at $ (2, 2) $, the y-coordinate of $ P $ is:

Given the function:

\[ f(x) = \frac{2x - 3}{3x - 2} \]

and if $ f_n(x) = (f \circ f \circ \ldots \circ f)(x) $ is applied $ n $ times, find $ f_{32}(x) $.

If 4 letters are selected at random from the letters of the word PROBABILITY, compute the probability that at least one letter is repeated in the selection.

If $\cos x + \cos y = \tfrac{2}{3}$ and $\sin x - \sin y = \tfrac{3}{4}$, then \[ \sin(x - y) \;+\; \cos(x - y) \;=\;? \]

In a triangle $ABC$, if $\tan\tfrac{A}{2} : \tan\tfrac{B}{2} : \tan\tfrac{C}{2} = 15 : 10 : 6$, then $\frac{a}{b - c} = \,?$

In triangle ABC, if $ a = 4, b = 3, c = 2 $, then $ 2(a - b \cos C)(a - c \sec B) = $:

\[ D = \begin{vmatrix} -\frac{bc}{a^2} & \frac{c}{a} & \frac{b}{a} \\ \frac{c}{b} & -\frac{ac}{b^2} & \frac{a}{b} \\ \frac{b}{c} & \frac{a}{c} & -\frac{ab}{c^2} \end{vmatrix} \]

The equation of the straight line whose slope is $ -\frac{2}{3} $ and which divides the line segment joining $ (1,2) $ and $ (-3,5) $ in the ratio 4:3 externally is:

If the function $ f(x) $ is given by \[ f(x) = \begin{cases} \frac{\tan(a(x-1))}{\frac{x-1}{x}}, & tif0<x<1
\frac{x^3-125}{x^2 - 25} , & \text{if } 1 \leq x \leq 4
\frac{b^x - 1}{x}, & \text{if } x>4 \end{cases} \] is continuous in its domain, then find $ 6a + 9b^4 $.

If $ f(x) = kx^3 - 3x^2 - 12x + 8 $ is strictly decreasing for all $ x \in \mathbb{R} $, then:

Given $(\sin \theta - \csc \theta)^2 + (\cos \theta + \sec \theta)^2 = 5$ and $\theta$ lies in the third quadrant, find $(\sin \theta + \cos \theta)^3$.

If \[ \frac{3x^4 - 2x^2 +1}{(x-2)^4} = A + \frac{B}{x-2} + \frac{C}{(x-2)^2} + \frac{D}{(x-2)^3} + \frac{E}{(x-2)^4}, \] then $2A + 3B - C - D + E =\;?$

If $ A = \begin{pmatrix} 9 & 3 & 0 \\ 1 & 5 & 8 \\ 7 & 6 & 2 \end{pmatrix} $ and

\[ A^T A^{-2} = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix} \]

then

\[ \sum_{1 \leq i \leq 3} \sum_{1 \leq j \leq 3} a_{ij} \]

is:

If $ T_4 $ represents the 4th term in the expansion of $ \left( 5x + \frac{7}{x} \right)^{-\frac{3}{2}} $ and $ x \not\in \left[ \frac{\sqrt{7}}{5}, \frac{\sqrt{7}}{5} \right] $, then $ \left( x^3 \cdot \sqrt{5x} \right) T_4 = $:}

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

If the line \[ 2x + by + 5 = 0 \] forms an equilateral triangle with \[ ax^2 - 96bxy + ky^2 = 0, \] then $ a + 3k $ is: \

The variance of the following continuous frequency distribution is: