PUC Science 2nd PUC Class 12 - Karnataka Board PUC Question Bank Solutions for Mathematics

If f(x) = (cos x + i sin x) (cos 2x + i sin 2x) (cos 3x + i sin 3x) ...... (cos nx + i sin nx) and f(1) = 1, then f'' (1) is equal to

If y = a sin mx + b cos mx, then \[\frac{d^2 y}{d x^2}\]   is equal to

If \[f\left( x \right) = \frac{\sin^{- 1} x}{\sqrt{1 - x^2}}\] then (1 − x)² f '' (x) − xf(x) =

If \[y = \tan^{- 1} \left\{ \frac{\log_e \left( e/ x^2 \right)}{\log_e \left( e x^2 \right)} \right\} + \tan^{- 1} \left( \frac{3 + 2 \log_e x}{1 - 6 \log_e x} \right)\], then \[\frac{d^2 y}{d x^2} =\]

Let f(x) be a polynomial. Then, the second order derivative of f(e^x) is

If y = a cos (log_e x) + b sin (log_e x), then x² y₂ + xy₁ =

If x = 2 at, y = at², where a is a constant, then \[\frac{d^2 y}{d x^2} \text { at x } = \frac{1}{2}\] is 

If x = f(t) and y = g(t), then \[\frac{d^2 y}{d x^2}\] is equal to

If y = sin (m sin⁻¹ x), then (1 − x²) y₂ − xy₁ is equal to

If y = (sin⁻¹ x)², then (1 − x²)y₂ is equal to

If y = e^tan x, then (cos² x)y₂ =

If \[y = \frac{ax + b}{x^2 + c}\] then (2xy₁ + y)y₃ = 

If \[y = \log_e \left( \frac{x}{a + bx} \right)^x\] then x³ y₂ =

If x = f(t) cos t − f' (t) sin t and y = f(t) sin t + f'(t) cos t, then\[\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2 =\]

If \[y^\frac{1}{n} + y^{- \frac{1}{n}} = 2x, \text { then find } \left( x^2 - 1 \right) y_2 + x y_1 =\] ?

If \[\frac{d}{dx}\left[ x^n - a_1 x^{n - 1} + a_2 x^{n - 2} + . . . + \left( - 1 \right)^n a_n \right] e^x = x^n e^x\] then the value of a_r, 0 < r ≤ n, is equal to 

If y = xⁿ⁻¹ log x then x² y₂ + (3 − 2n) xy₁ is equal to

If xy − log_e y = 1 satisfies the equation \[x\left( y y_2 + y_1^2 \right) - y_2 + \lambda y y_1 = 0\]

If y² = ax² + bx + c, then \[y^3 \frac{d^2 y}{d x^2}\] is