List of top Mathematics Questions asked in TS EAMCET

The equations \[ 7x + y - 24 = 0 \quad \text{and} \quad x + 7y - 24 = 0 \] represent the equal sides of an isosceles triangle. If the third side passes through \( (-1,1) \), then a possible equation for the third side is: \

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

By shifting the origin to the point \( (h,5) \) by the translation of coordinate axes, if the equation \[ y = x^2 - 9x^2 + cx - d \] transforms to \( Y = X^2 \), then \( \left( \frac{d - c}{h} \right) \) is: \

If \( X \sim B(6, p) \) is a binomial variate and \[ \frac{P(X=4)}{P(X=2)} = \frac{1}{9}, \] then the value of \( p \) is:

If the locus of the centroid of the triangle with vertices \( A(a,0) \), \( B(\cos t, a\sin t) \) and \( C(\sin t, -b\cos t) \) (\( t \) is a parameter) is given by \[ 9x^2 + 9y^2 - 6x = 49, \] then the area of the triangle formed by the line \[ \frac{x}{a} + \frac{y}{b} = 1 \] with the coordinate axes is: \

If \( z = x + iy \) satisfies the equation \[ z^2 + az + a^2 = 0, \quad a \in \mathbb{R}, \] then:

If \( \omega \) is the complex cube root of unity and \[ \left( \frac{a + b\omega + c\omega^2}{c + a\omega + b\omega^2} \right)^k + \left( \frac{a + b\omega + c\omega^2}{b + a\omega^2 + c\omega} \right)^2 = 2, \] then \( 2k + 1 \) is always:

A, B, C, D are square matrices such that A + B is symmetric, A - B is skew-symmetric, and D is the transpose of C.

If

\[ A = \begin{bmatrix} -1 & 2 & 3 \\ 4 & 3 & -2 \\ 3 & -4 & 5 \end{bmatrix} \]

and

\[ C = \begin{bmatrix} 0 & 1 & -2 \\ 2 & -1 & 0 \\ 0 & 2 & 1 \end{bmatrix} \]

then the matrix \( B + D \) is:

If

\[ \cosh^{-1}\left(\frac{5}{3}\right) + \sinh^{-1}\left(\frac{3}{4}\right) = k, \]

then \( e^k \) is:

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:

\( \vec{a}, \vec{b}, \vec{c} \) are three vectors such that \(|\vec{a}| = 3\), \(|\vec{b}| = 2\sqrt{2}\), \(|\vec{c}| = 5\), and \( \vec{c} \) is perpendicular to the plane of \( \vec{a} \) and \( \vec{b} \).

If the angle between the vectors \( \vec{a} \) and \( \vec{b} \) is \( \frac{\pi}{4} \), then

\[ |\vec{a} + \vec{b} + \vec{c}| = \ ? \]

If \( \vec{a}, \vec{b}, \vec{c} \) are non-coplanar vectors and the points \[ \lambda \vec{a} + 3\vec{b} - \vec{c}, \quad \vec{a} - \lambda \vec{b} + 3\vec{c}, \quad 3\vec{a} + 4\vec{b} - \lambda \vec{c}, \quad \vec{a} - 6\vec{b} + 6\vec{c} \] are coplanar, then one of the values of \( \lambda \) is:

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 \( a = ? \)

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:

\[ \lim_{n\to\infty}\left[\left(1+\frac{1}{n^2}\right)\left(1+\frac{4}{n^2}\right)\left(1+\frac{9}{n^2}\right)\cdots\left(1+\frac{n^2}{n^2}\right)\right]^{\frac{1}{n}} = \]

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