List of top Mathematics Questions on Some Properties of a Triangle asked in AP EAMCET

If \( O(0, 0, 0), A(3, 0, 0), B(0, 4, 0) \) form a triangle then the incenter of triangle OAB is:
If \(f(x) = \begin{cases} 2x + 3, & \text{if } x \leq 1 \\ 2ax + bx, & \text{if } x > 1 \end{cases}\) is differentiable \(\forall x \in \mathbb{R}\), then \(f(2) =\)
Evaluate the limit: \[ \lim_{x \to 0} \frac{e^{x} - a - \log(1+x)}{\sin x} = 0, { then find } a. \]
If \[ \frac{x^2+3}{x^4+2x^2+9} = \frac{Ax+B}{x^2+ax+b} + \frac{Cx+D}{x^2+cx+d} \] then \( aA + bB + cC + dD = \)
If the area of the triangle formed by the straight lines \(-15x^2 + 4xy + 4y^2 = 0\) and \(x = a\) is \(200 \, {sq. units}\), then \(|a| =\)

Evaluate the integral \( \int \frac{dx}{\sqrt{\sin^3 x \cdot \cos (x - \alpha)}} \):

In a triangle, if the angles are in the ratio \( 3:2:1 \), then the ratio of its sides is:
If \( \theta = \sec^{-1}(\cosh u) \), then \( u = \)
Evaluate the integral \( \int \frac{\sin^{-1} \left(\frac{x}{\sqrt{a + x}}\right) \sqrt{a + x}}{dx} \):
In \(\triangle ABC\), \(bc - r_2 r_3 = \)
In \(\triangle ABC\), if \(4r_1 = 5r_2 = 6r_3\), then \(\sin^2 \frac{A}{2} + \sin^2 \frac{B}{2} + \sin^2 \frac{C}{2} =\)
In \(\triangle ABC\), \(r r_1 \cot^ \frac{A}{2} + r r_2 \cot^ \frac{B}{2} + r r_3 \cot^ \frac{C}{2} = \)
In \( \triangle ABC \), evaluate: \[ \frac{r_2 (r_1 + r_3)}{\sqrt{r_1 r_2 + r_2 r_3 + r_3 r_1}}. \]
Evaluate the integral \( \int \frac{e^{2x}}{\sqrt{e^x + 1}} \, dx \):
Evaluate the integral \( \int \frac{dx}{x(x^4 + 1)} \):
Evaluate the integral \( \int_{-\frac{1}{24}}^{\frac{1}{24}} \sec(x) \log\left(\frac{1-x}{1+x}\right) \, dx \):
In \( \triangle ABC \), if \( a = 13 \), \( b = 14 \), and \( \cos \frac{C}{2} = \frac{3}{\sqrt{13}} \), then \( 2r_1 = \):
If \(\sec \theta + \tan \theta = \frac{1}{3}\), then the quadrant in which \(2\theta\) lies is:
If \(2.4^{2n+1} + 3^{3n+1}\) is divisible by \(k\) for all \(n \in \mathbb{N}\), then \(k\) is:
If \([x]\) is the greatest integer function, then evaluate the integral \( \int_{0}^{5} [x] \, dx \):
If \( (a, \beta) \) is the orthocenter of the triangle with the vertices \( A(2, 5), B(1, 5), C(1, 4) \), then \( a + \beta = \):
If \( \frac{3 - 2i \sin \theta}{1 + 2i \sin \theta}\) is a purely imaginary number, then \( \theta \) is:
In \( \triangle ABC \), if \( AB:BC:CA = 6:4.5 \), then \( R : r = \)
In \( \triangle ABC \), if \( (r_2 - r_1)(r_3 - r_1) = 2r_2r_3 \), then \( 2(r + R) = \):
If 7 and 8 are the lengths of two sides of a triangle and \( a \) is the length of its smallest side. The angles of the triangle are in AP and \( a \) has two values \( a_1 \) and \( a_2 \) satisfying this condition. If \( a_1<a_2 \), then \( 2a_1 + 3a_2 = \):