List of top Mathematics Questions on Algebra asked in UP Board XII

(e) If \( y = x \cos (a + y) \) and \( \cos a \neq \pm 1 \), prove that \[ \frac{dy}{dx} = \frac{\cos^2 (a + y)}{\sin a}. \]
(b) Show that \[ \begin{vmatrix} 1 + a & 1 & 1 \\ 1 & 1 + b & 1 \\ 1 & 1 & 1 + c \end{vmatrix} = abc \left( 1 + \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \right) \] :
(c) Solve the differential equation \( (\tan^{-1} y - x) dy = (1 + y^2) dx \):
(d) If \( (\cos x)^y = (\cos y)^x \), find \( \frac{dy}{dx} \):
(a) Prove that for the two vectors \( \vec{a} \) and \( \vec{b} \), \( |\vec{a} \cdot \vec{b}| \leq |\vec{a}| |\vec{b}| \):
If \( f(x) = x + \frac{1}{x} \), prove that \( \left[ f(x) \right]^3 = f(x^3) + 3f\left( \frac{1}{x} \right) \).
Solve the inequality \( 8x + 4<7x + 8 \).