List of top Questions asked in TS EAMCET

If \( \alpha, \beta \) are the roots of the equation \( x + \frac{4}{x} = 2\sqrt{3} \), then \( \frac{2}{\sqrt{3}}\left| \alpha^{2024} - \beta^{2024} \right| \) is:
If \( \lim_{x \to 4} \frac{2x^2 + (3+2a)x + 3a{x^3 - 2x^2 - 23x + 60} = \frac{11}{9} \), then find \( \lim_{x \to a} \frac{x^2 + 9x + 20}{x^2 - x - 20} \).}
If \( \mathbf{a} = 2\overline{i} - \overline{j}, \overline{b} = 2\overline{j} - \overline{k}, \overline{c} = 2\overline{k} - \overline{i} \) are three vectors and \( \overline{d} \) is a unit vector perpendicular to \( \overline{c} \), If \( \overline{a}, \overline{b}, \overline{d} \) are coplanar vectors, then \( |\overline{d} \cdot \overline{b}| = \):
The vertices of triangle \( \Delta ABC \) are \( A(2, 3, k) \), \( B(-1, k, -1) \), and \( C(4, -3, 2) \). If \( AB = AC \) and \( k>0 \), then the triangle \( ABC \) is:
Define \( f: \mathbb{R} \to \mathbb{R} \) by \[ f(x) = \begin{cases} \frac{1 - \cos 4x}{x^2}, & x < 0 \\ a, & x = 0 \\ \frac{\sqrt{x}}{\sqrt{16 + \sqrt{x}} - 4}, & x > 0 \end{cases} \] Find the value of \( a \) such that \( f \) is continuous at \( x = 0 \).
If two dice are thrown, then the probability of getting co-prime numbers on the dice is:

If \( x \) and \( y \) are two positive real numbers such that \( x + iy = \frac{13\sqrt{5} + 12i}{(2 - 3i)(3 + 2i)} \), then \( 13y - 26x = \):

If \( y (\cos x)^{\sin x} = (\sin x)^{\sin x} \), then the value of \( \frac{dy}{dx} \) at \( x = \frac{\pi}{4} \) 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 \( z = x + iy \) and the point \( P \) represents \( z \) in the Argand plane, then the locus of \( z \) satisfying the equation \( |z-1| + |z+i| = 2 \) is:
The system of equations \( x + 3y + 7 = 0 \), \( 3x + 10y - 3z + 18 = 0 \), and \( 3y - 9z + 2 = 0 \) has:
The slope of the tangent drawn from the point \( (1,1) \) to the hyperbola \( 2x^2 - y^2 = 4 \) is:
The position vectors of the vertices \( A, B, C \) of a triangle are given as: \[ \vec{A} = \hat{i} - 2\hat{j} + k, \quad \vec{B} = 2\hat{i} + \hat{j} - k, \quad \vec{C} = \hat{i} - \hat{j} - 2k \] If \( D \) and \( E \) are the midpoints of \( BC \) and \( CA \) respectively, then the unit vector along \( \overrightarrow{DE} \) is:
The equation of the common tangent to the parabola \(y^2 = 8x\) and the circle \(x^2 + y^2 = 2\) is \(ax + by + 2 = 0\). If \(-\frac{a}{b}>0\), then \(3a^2 + 2b + 1 =\)

Matrix Inverse Sum Calculation

Given the matrix:

    A =    | 1  2  2 |    | 3  2  3 |    | 1  1  2 |    

The inverse matrix is represented as:

    A-1 =    | a11  a12  a13 |    | a21  a22  a23 |    | a31  a32  a33 |    

The sum of all elements in A-1 is:

∑ aij = 3

Solving the System of Linear Equations

If (x,y,z) = (α,β,γ) is the unique solution of the system of simultaneous linear equations:

    3x - 4y + 2z + 7 = 0,    2x + 3y - z = 10,    x - 2y - 3z = 3,    

then α = ?

The number of common roots among the 12th and 30th roots of unity is ?

The product of all the values of \(\bigl(\sqrt{3} - i\bigr)^{25}\) is ?

The domain of the real valued function: \[ f(x) \;=\; \sqrt[3]{\frac{x \,-\, 2}{2x^2 \,-\, 7x \,+\,5}} \;+\; \log\bigl(x^2 \,-\, x \,-\, 2\bigr). \]
The general solution of the differential equation \[ \frac{dy}{dx} + \frac{\sin(2x + y)}{\cos x} + 2 = 0 \] is:
If \(2\,\tan^{-1}x = 3\,\sin^{-1}x\) and \(x \neq 0\), then \(8x^2 +1 =\;?\)
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:
If \(A + B + C = 2S\), then \[ \sin(S - A)\,\cos(S - B)\;-\;\sin(S - C)\,\cos S \;=\;? \]

The maximum value of the function \[ f(x) = 3\sin^{12}x + 4\cos^{16}x \] is ?

The equation of the normal drawn to the curve \( y^3 = 4x^5 \) at the point \( (4,16) \) is: