List of top Mathematics Questions asked in AP EAPCET

Let \( a>1 \) and \( 0<b<1 \). -∞\( f : \mathbb{R} \to [0, 1] \) is defined by \( f(x) = \begin{cases} a^x & \text{if } x<0 b^x & \text{if } 0 \leq x \leq 1 \end{cases} \), then \( f(x) \) is:

If \[ A = \begin{bmatrix} 1 & 0 & 2\\ 2 & 1 & 3 \\3 & 2 & 4 \end{bmatrix}, \] then evaluate \( A^2 - 5A + 6I \)=

Sum of the positive roots of the equation: \[ \begin{vmatrix} x^2 + 2x + 2 & x + 2 & 1 \\ 2x + 1 & x - 1 & 1 \\ x + 2 & -1 & 1 \end{vmatrix} = is \; 0. \]

If the solution of the system of simultaneous linear equations: $$ x + y - z = 6, $$ $$ 3x + 2y - z = 5, $$ $$ 2x - y - 2z + 3 = 0 $$ is \( x = \alpha, y = \beta, z = \gamma \), then \( \alpha + \beta = ?\) 
If the point \( P \) represents the complex number \( z = x + iy \) in the Argand plane and if \[ \frac{z\bar{z} + 1}{z - 1} \] is a purely imaginary number, then the locus of \( P \) is:
The set \( S = \{ z \in \mathbb{C} : |z + 1 - i| = 1 \} \) represents:

If \(\cos \alpha + \cos \beta + \cos \gamma = \sin \alpha + \sin \beta + \sin \gamma = 0,\) then evaluate \((\cos^3 \alpha + \cos^3 \beta + \cos^3 \gamma)^2 + (\sin^3 \alpha + \sin^3 \beta + \sin^3 \gamma)^2 =\)

If $\alpha$ and $\beta$ are two double roots of the equation: $$ x^2 + 3(a + 3)x - 9a = 0 $$ for different values of $a$ (where $\alpha > \beta$), then the minimum value of the equation: $$ x^2 + \alpha x - \beta = 0 $$ is:
If \( 2x^2 + 3x - 2 = 0 \) and \( 3x^2 + \alpha x - 2 = 0 \) have one common root, then the sum of all possible values of \( \alpha \) is:

If the sum of two roots of \( x^3 + px^2 + qx - 5 = 0 \) is equal to its third root, then \( p(q^2 - 4q) = \) ?

If \( P(x) = x^5 + ax^4 + bx^3 + cx^2 + dx + e \) is a polynomial such that: \[ P(0) = 1, \quad P(1) = 2, \quad P(2) = 5, \] \[ P(3) = 10, \quad P(4) = 17, \] then find the value of \( P(5) \)=
If a polygon of \( n \) sides has 275 diagonals, then \( n \) is:
The number of positive divisors of 1080 is:
If \( a_n = \sum_{r=0}^{n} \frac{1}{\binom{n}{r}} \), then \( \sum_{r=0}^{n} r \binom{n}{r} = \):
The coefficient of \( x^5 \) in the expansion of \( \left( 2x^3 - \frac{1}{3x^2} \right)^5 \) is:
Evaluate the infinite series: \[ 1 + \frac{1}{3} + \frac{1.3}{3.6} + \frac{1.3.5}{3.6.9} + \dots \text{ to } \infty = \]
Given the equation: $$ \frac{A}{x-a} + \frac{Bx + C}{x^2 + b^2} = \frac{1}{(x-a)(x^2 + b^2)} $$ then $ C $=
If \[ \cos \frac{ \pi }{8} + \cos \frac{3 \pi }{8} + \cos \frac{5 \pi }{8} + \cos \frac{7 \pi }{8} = k, \] then evaluate \[ \sin^{-1} \left( \frac{k}{\sqrt{2}} \right) + \cos^{-1} \left( \frac{k}{3} \right)= \]
Evaluate the expression: \[ \frac{\cos 10^\circ + \cos 80^\circ}{\sin 80^\circ - \sin 10^\circ}. \]
Evaluate the expression: \[ \frac{\sin 1^\circ + \sin 2^\circ + \dots + \sin 89^\circ}{2(\cos 1^\circ + \cos 2^\circ + \dots + \cos 44^\circ) + 1} =\]
The number of ordered pairs \( (x, y) \) satisfying the equations: \[ \sin x + \sin y = \sin (x + y) \quad \text{and} \quad |x| + |y| = 1. \]
Evaluate: \[ 4 \tan^{-1} \frac{1}{5} - \tan^{-1} \frac{1}{70} + \tan^{-1} \frac{1}{99}= \]
If \( 5 \sinh x - \cosh x = 5 \), then one of the values of \( \tanh x \) is:
In \( \triangle ABC \), if \( r_1 = 4 \), \( r_2 = 8 \), \( r_3 = 24 \), then find \( a \)=
If a circle is inscribed in an equilateral triangle of side \( a \), then the area of any square inscribed in this circle (in square units) is: