The derivative of x³ + 2x² - 5x + 1 is:

The value of the definite integral $I = \int_{1}^{2} \frac{1}{x(1 + x^2)}dx$ is:

If $y = ax^2 + bx$ has minima at $x = 2$ and the minimum value is -12, then which of the following are correct? (A) $a = 3$ (B) $a = -3$ (C) $b = 12$ (D) $b = -12$ Choose the correct answer from the options given below:

A balloon which always remains spherical, has a variable diameter $\frac{3}{2}(5x+7)$. Then the rate of change of its volume with respect to x is

Match List-I with List-II | List-I | List-II | |---|---| | **Differential equation** | **Degree** | | (A) $\frac{d^2y}{dx^2} + \sqrt{\frac{dy}{dx}} - y = 0$ | (I) 6 | | (B) $\sqrt{\frac{d^3y}{dx^3}} - \sqrt[12]{\frac{d^2y}{dx^2}} = 0$ | (II) Not defined | | (C) $\left(\frac{d^2y}{dx^2}\right)^2 + \frac{dy}{dx} + e^{\frac{dx}{dx}} = x^2$ | (III) 3 | | (D) $\sqrt[3]{\frac{dy}{dx}} - \frac{d^2y}{dx^2} = e^x$ | (IV) 2 | Choose the correct answer from the options given below:

Which of the following are NOT correct regarding the equation of tangent and normal to the curve $y = \frac{x-11}{(x-2)(x-3)}$ at the point, where it cuts the $x$-axis? (A) The point of contact is (11, 0). (B) The equation of tangent is $x - 72y - 11 = 0$ (C) The equation of normal is $72x + y - 11 = 0$ (D) The slope of the tangent at the given point of contact is $\frac{1}{88}$ Choose the **correct** answer from the options given below: