Prepare a cumulative frequency table and draw the cumulative frequency curve for the distribution. Use your curve to estimate the: (i) Median mark (ii) Upper and lower quartiles (iii) Semi-inter quartile range

Here's the solution for question 7.

7. (a) A function $y = 2x^4 - Px^3 + 2x$, where P is a constant, has a stationary value at $x = -1$. Find the value of P.

Step 1: Find the first derivative of the function with respect to $x$.
For a stationary value, the derivative $\frac{dy}{dx}$ must be equal to zero.

$$y = 2x^4 - Px^3 + 2x$$ $$\frac{dy}{dx} = \frac{d}{dx}(2x^4) - \frac{d}{dx}(Px^3) + \frac{d}{dx}(2x)$$ $$\frac{dy}{dx} = 2(4x^{4-1}) - P(3x^{3-1}) + 2(1)$$ $$\frac{dy}{dx} = 8x^3 - 3Px^2 + 2$$

Step 2: Set the derivative to zero and substitute $x = -1$.
Since there is a stationary value at $x = -1$, we have $\frac{dy}{dx} = 0$ when $x = -1$.

$$0 = 8(-1)^3 - 3P(-1)^2 + 2$$ $$0 = 8(-1) - 3P(1) + 2$$ $$0 = -8 - 3P + 2$$

Step 3: Solve for P.

$$0 = -6 - 3P$$ $$3P = -6$$ $$P = \frac{-6}{3}$$ $$P = \boxed{-2}$$

7. (b) The motion of a particle along a straight line is defined by the equation $x = 5t^3 - 4t^2$. Find after 4 seconds, the: