List of top Questions asked in TS EAMCET- 2024

If $ f(x) $ is given as: \[ f(x) = \begin{cases} \frac{8}{x^3} - 6x, & \text{if } 0 < x \leq 1 \\ \frac{x - 1}{\sqrt{x}-1}, & \text{if } x > 1 \end{cases} \] is a real-valued function, then at $ x = 1 $, $ f $ is:

$\lim_{x \to 0}$ $\frac{3^{\sin x} - 2^{\tan x}}{\sin x}$ =

If the function

$ f(x) = \begin{cases} \frac{\cos ax - \cos 9x}{x^2}, & \text{if } x \neq 0 \\ 16, & \text{if } x = 0 \end{cases} $

is continuous at $ x = 0 $, then $ a = ? $

$ \lim_{x \to -\frac{3}{2}} \frac{(4x^2 - 6x)(4x^2 + 6x + 9)}{\sqrt{2x - \sqrt{3}}} $

\[ f(x) = \begin{cases} \frac{(4^x - 1)^4 \cot(x \log 4)}{\sin(x \log 4) \log(1 + x^2 \log 4)}, & \text{if } x \neq 0 \\ k, & \text{if } x = 0 \end{cases} \]

Find $ e^k $ if $ f(x) $ is continuous at $ x = 0 $.

A function $ f : \mathbb{R} \rightarrow \mathbb{R} $ is such that $ yf(x+y) + \cos mxy = 1 + yf(x) $. If $ m = 2 $, find $ f'(x) $.

For a given function $ y = f(x) $, $ \delta y $ denotes the actual error in $ y $ corresponding to actual error $ \delta x $ in $ x $ and $ dy $ denotes the approximate value of $ \delta y $. If $ y = f(x) = 2x^2 - 3x + 4 $ and $ \delta x = 0.02 $, then the value of $ \delta y - dy $ when $ x = 5 $ is: \vspace{0.5cm}

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

If

\[ A = \{ P(\alpha, \beta) \mid \text{the tangent drawn at P to the curve } y^3 - 3xy + 2 = 0 \text{ is a horizontal line} \} \]

and

\[ B = \{ Q(a, b) \mid \text{the tangent drawn at Q to the curve } y^3 - 3xy + 2 = 0 \text{ is a vertical line} \} \]

then $ n(A) + n(B) = $

Evaluate the integral: \[ \int \frac{\sec x}{3(\sec x + \tan x) + 2} \,dx \]

If \[ 2x^2 - 3xy + 4y^2 + 2x - 3y + 4 = 0 \] then \[ \left( \frac{dy}{dx} \right)_{(3,2)} = \]

The general solution of the differential equation $(3x^2 - 2xy)dy + (y^2 - 2xy)dx = 0$ is

If the slope of the tangent drawn at any point $(x, y)$ on the curve $y = f(x)$ is $y = 6x^2 + 10x - 9$ and $f(2) = 0$, then $f(-2) = $?

If $y = \sin ax + \cos bx$ then $y'' + b^2y =$

If $ f(x) = (2x-1)(3x+2)(4x-3) $ is a real-valued function defined on $\left[\frac{1}{2}, \frac{3}{4}\right]$, then the value(s) of 'c' as defined in the statement of Rolle's theorem is:

Evaluate $ \int_{-2}^{2} x^4 (4 - x^2)^{\frac{7}{2}} \, dx $:

If \[ x = \frac{9t^2}{1 + t^4}, \quad y = \frac{16t^2}{1 - t^4} \] then \[ \frac{dy}{dx} = \]

Area of the region enclosed between the curves $y^{2}=4(x+7)$ and $y^{2}=5(2-x)$ is

\[ y = \sqrt{\sin(\log(2x)) + \sqrt{\sin(\log(2x)) + \sqrt{\sin(\log(2x))} + \dots \infty}} \]

The area of the region (in square units) enclosed by the curves $ y = 8x^3 - 1 $, $ y = 0 $, $ x = -1 $, and $ x = 1 $ is:

The general solution of the differential equation \[ (9x - 3y + 5) dy = (3x - y + 1) dx. \]

The approximate value of $ \sqrt[3]{730} $ obtained by the application of derivatives is:

If $ y = \frac{\tan x \cos^{-1}x}{\sqrt{1 - x^2}} $, then the value of $ \frac{dy}{dx} $ when $ x = 0 $ is:

If $AX = D$ represents the system of linear equations \[ 3x - 4y + 7z + 6=0,\quad 5x + 2y - 4z + 9=0,\quad 8x - 6y - z + 5=0, \] then AX = D ?

The system of equations $ x + 3y + 7 = 0 $, $ 3x + 10y - 3z + 18 = 0 $, and $ 3y - 9z + 2 = 0 $ has: