List of top Mathematics Questions on Locus of Normals asked in AP EAMCET

Evaluate the limit: \[ \lim\limits_{x \to 2} \frac{\sqrt{1 + 4x} - \sqrt{3} + 3x}{x^3 - 8}. \]

If \[ 3f(x) - 2f\left(\frac{1}{x}\right) = x, \] then \( f'(2) \) is:

If \[ \frac{x + 2}{(x^2 + 3)(x^4 + x^2)(x^2 + 2)} = \frac{Ax + B}{x^2 + 3} + \frac{Cx + D}{x^2 + 2} + \frac{Ex^3 + Fx^2 + Gx + H}{x^4 + x^2}, \] then \[ (E + F)(C + D)(A) = \]

Assertion (A): If \( A = 10^\circ, B = 16^\circ, C = 19^\circ \), then: \[ \tan(2A) \tan(2B) + \tan(2B) \tan(2C) + \tan(2C) \tan(2A) = 1. \] Reason (R): If \( A + B + C = 180^\circ \), then: \[ \cot\left(\frac{A}{2}\right) + \cot\left(\frac{B}{2}\right) + \cot\left(\frac{C}{2}\right) = \cot\left(\frac{A}{2}\right) \cot\left(\frac{B}{2}\right) \cot\left(\frac{C}{2}\right). \]

If $$ \lim\limits_{x \to \infty} \frac{\left(\sqrt{2x+1} + \sqrt{2x-1}\right) + \left(\sqrt{2x+1} - \sqrt{2x-1}\right) P x^4 - 16} {(x+\sqrt{x^2 - 2}) + (x - \sqrt{x^2 - 2})} = 1, $$ then P = ? 

If the function \[ f(x) = \frac{\sqrt{1+x} - 1}{x} \] is continuous at \( x = 0 \), then \( f(0) \) is:

If \[ 729 \int_1^3 \frac{1}{x^3 (x^2 + 9)^2} \, dx = a + \log b, \] then \( a - b = \) ?

Evaluate the limit: $$ \lim_{n \to \infty} \frac{17^7 + 27^7 + \dots + n^{77}}{n^{78}}. $$ 

If \[ \int_{1}^{n} f(x) \,dx = 120, \] then \( n \) is:

Evaluate the expression \[ 2 \cot h^{-1}(4) + \sec h^{-1}\left( \frac{3}{5} \right). \]

Evaluate the limit: $$ \lim_{x \to 1} \frac{x + x^2 + x^3 + \dots + x^n - n}{x - 1}. $$ 

If \( P + Q + R = \frac{\pi}{4} \), then \[ \cos \left( \frac{\pi}{8} - P \right) + \cos \left( \frac{\pi}{8} - Q \right) + \cos \left( \frac{\pi}{8} - R \right) = P + Q + R = \frac{\pi}{4}. \]

If \( \sin x + \sin y = \alpha \), \( \cos x + \cos y = \beta \), then \( \csc(x + y) \) =

If the amplitude of \( (Z - 2) \) is \( \frac{\pi}{2} \), then the locus of \( Z \) is:

If \[ \frac{d}{dx} \left(\frac{1 + x^2 + x^4}{1 + x + x^2}\right) = ax + b, \] then \( (a,b) \) is:

If \[ y = \sin^{-1} x, \] then \[ (1 - x^2)y_2 - xy_1 = 0. \]

Evaluate the limit: \[ \lim_{x \to 0} \frac{\sin(\pi \cos^2 x)}{x^2}. \]