List of top Mathematics Questions asked in TS EAMCET

If \(\vec{a},\,\vec{b},\,\vec{c}\) are non-coplanar vectors. If the three points \[ \lambda \vec{a} - 2\vec{b} + \vec{c},\quad 2 \vec{a} + \lambda\vec{b} - 2\vec{c},\quad 4\vec{a} + 7\vec{b} - 8\vec{c} \] are collinear, then \(\lambda = \,?\)
The point \((p, q)\) is the point of intersection of a latus rectum and an asymptote of the hyperbola \(9x^2 - 16y^2 = 144\). If \(p>0\) and \(q>0\), then \(q = \ldots\)
\(\vec{r}\) is a vector perpendicular to the plane determined by \(\vec{r_1}=2\mathbf{i}-\mathbf{j}\) and \(\vec{r_2}=\mathbf{j}+2\mathbf{k}\). If the magnitude of the projection of \(\vec{r}\) on the vector \(2\mathbf{i}+\mathbf{j}+2\mathbf{k}\) is 1, then \(\lvert \vec{r}\rvert =\,?\)
The equations of the directrices of the ellipse \(9x^2 + 4y^2 - 18x - 16y - 11 = 0\) are:
If \[ f(x) = 3x^{15} - 5x^{10} + 7x^5 + 50\cos(x - 1), \] then \[ \lim_{h \to 0} \frac{f(1 - h) - f(1)}{h^2 + 3h} \] is:
(a,b) is the point of concurrency of the lines \(x-3y+3=0\), \(Kx+y+k=0\) and \(2x+y-8=0\). If the perpendicular distance from the origin to the line \(L: ax-by+2k=0\) is \(p\), then the perpendicular distance from the point \((2,3)\) to \(L=0\) is:
If \[ \int \sqrt{\csc x + 1} \, dx = k \tan^{-1} (f(x)) + c, \] then \[ \frac{1}{k} f\left(\frac{\pi}{6}\right) = ? \]
A rhombus is inscribed in the region common to the two circles \[ x^2 + y^2 - 4x - 12 = 0 \] and \[ x^2 + y^2 + 4x - 12 = 0. \] If the line joining the centres of these circles and the common chord of them are the diagonals of this rhombus, then the area (in Sq. units) of the rhombus is:
If \( 3^{2n+2} - 8n - 9 \) is divisible by \( 2^p \) \(\forall n \in \mathbb{N}\), then the maximum value of \( p \) is

\[ \textbf{If } | \text{Adj} \ A | = x \text{ and } | \text{Adj} \ B | = y, \text{ then } \left( | \text{Adj}(AB) | \right)^{-1} \text{ is } \]

If the expression \( 7 + 6x - 3x^2 \) attains its extreme value \( \beta \) at \( x = \alpha \), then the sum of the squares of the roots of the equation \( x^2 + ax - \beta = 0 \) is:

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

The area of the region enclosed by the curves \( y^2 = 4(x+1) \) and \( y^2 = 5(x-4) \) is:

For \( n \in \mathbb{N} \), the largest positive integer that divides \( 81^n + 20n - 1 \) is \( k \). If \( S \) is the sum of all positive divisors of \( k \), then find \( S - k \).

If \(1\cdot3\cdot5 + 3\cdot5\cdot7 + 5\cdot7\cdot9 + \ldots\) to \(n\) terms is \(n(n+1)f(n)\), then \(f(2) =\)
\(\frac{1}{3.6} + \frac{1}{6.9} + \frac{1}{9.12} + \dots\) up to 9 terms \(=\)
If \(A\) is a non singular matrix, then \(\text{Adj}(A^{-1}) =\)
If \[ \frac{x^4}{(x^2+1)(x-2)} = f(x) + \frac{Ax+B}{x^2+1} + \frac{C}{x-2} \] then \( f(14) + 2A - B = \) ?
If the normal drawn at the point \((2, -1)\) to the ellipse \(x^2 + 4y^2 = 8\) meets the ellipse again at \((a, b)\), then \(17a\) is:
The position vector of the point of intersection of the line joining the points \( \mathbf{i} - \mathbf{j} + \mathbf{k} \) and the line joining the points \( 2\mathbf{i} + \mathbf{j} - 6\mathbf{k} \), \( 3\mathbf{i} - \mathbf{j} - 7\mathbf{k} \) is:

If the function \( f(x) = x^3 + ax^2 + bx + 40 \) satisfies the conditions of Rolle's theorem on the interval \( [-5, 4] \) and \( -5, 4 \) are two roots of the equation \( f(x) = 0 \), then one of the values of \( c \) as stated in that theorem is:

In a triangle ABC, if \( r_1 = 6 \), \( r_2 = 9 \), \( r_3 = 18 \), then \( \cos A \) is:
If the quadratic equation \( 3x^2 + (2k + 1)x - 5k = 0 \) has real and equal roots, then the value of \( k \) such that \( -\frac{1}{2}<k<0 \) is:
If \( y = \log \left[ \tan \left( \sqrt\frac{2x - 1}{2x + 1} \right) \right] \) for \( x>0 \), then find \[ \left( \frac{dy}{dx} \right)_{x = 1}. \]

If \( \sqrt{5} - i\sqrt{15} = r(\cos\theta + i\sin\theta), -\pi < \theta < \pi, \) then

\[ r^2(\sec\theta + 3\csc^2\theta) = \]