Correct Answer: D. 96cm²

Correct Answer: D. \(y = x^{3} - 2x^{2} + 16\)

Explanation

The gradient of a curve is gotten by differentiating the equation of the curve. Therefore, given the gradient, integrate to get the equation of the curve back.

\(\frac{\mathrm d y}{\mathrm d x} = 3x^{2} - 4x\)

\(y = \int {(3x^{2} - 4x)} \mathrm {d} x = \frac{3x^{2+1}}{2+1} - \frac{4x^{1+1}}{1+1} + c\)

= \(x^{3} - 2x^{2} + c \)

To find c (the constant of integration), when x = -2, y = 0

\(0 = (-2^{3}) - 2(-2^{2}) + c\)

\(0 = -8 - 8 + c \implies c = 16\)

\(\therefore y = x^{3} - 2x^{2} + 16\)

Correct Answer: C. 11

Correct Answer: D. 1 - 2log 2

Correct Answer: A. 3

Explanation

4y - 9 > y + 3x; 4y - y > 3x + 9

3y > 3(x + 3); y = > \(\frac{3(x + 3)}{3}\)

y > x + 3; y - 3 > x

y is greater than x

Correct Answer: B. 1/3

Correct Answer: B. \(\frac{65}{12}\)

Explanation

\({1_0^∫} x^2(x^3+2)^3\)dx

let \( u = x^3 + 2, du = 3x^2dx\)

when x = 1, u = 3

when x = 0, u = 2

dx = \(\frac{du}{3x^2}\)

\({3_2^∫}\) \(\frac{x^2[u]^3}{3x^2}\)

\({3_2^∫}\) \(\frac{u^3}{3}\) du

= \(\frac{u^4}{3*4}\)\(_2\)3

\(\frac{1}{12} [u^4]\)\(_2\)3

\(\frac{1}{12} [3^4 - 2^4]\)

\(\frac{1}{12}[81 - 16]\)

\(\frac{65}{12}\)

Explanation

(a)

(b)

An histogram to represent the distribution of 40 students in an examination.

(c) Modal mark : \(\frac{74.5 + 79.5}{2} = \frac{154}{2} = 77\)

(d) Number of students who obtained marks greater than 79 : 8 + 7 + 2 + 1 = 18.

Total number of students = 40

\(\therefore\) P(mark greater than 79) = \(\frac{18}{40} = \frac{9}{20}\)

(a)

(b)

An histogram to represent the distribution of 40 students in an examination.

(c) Modal mark : \(\frac{74.5 + 79.5}{2} = \frac{154}{2} = 77\)

(d) Number of students who obtained marks greater than 79 : 8 + 7 + 2 + 1 = 18.

Total number of students = 40

\(\therefore\) P(mark greater than 79) = \(\frac{18}{40} = \frac{9}{20}\)

Correct Answer: B. 4.00

Explanation

\(\log_{2}(12x - 10) = 1 + \log_{2}(4x + 3)\)

Recall, \( 1 = \log_{2}2\), so

\(\log_{2}(12x - 10) = \log_{2}2 + \log_{2}(4x + 3)\)

= \(\log_{2}(12x - 10) = \log_{2}(2(4x + 3))\)

\(\implies (12x - 10) = 2(4x + 3) \therefore 12x - 10 = 8x + 6\)

\(12x - 8x = 4x = 6 + 10 = 16 \implies x = 4\)

Correct Answer: B. \((\frac{2}{5}, 5\frac{2}{5})\)

Explanation

\((x = \frac{nx_{1} + mx_{2}}{n + m}, y = \frac{ny_{1} + my_{2}}{n + m})\)

Given P(-2, 3) and Q(4, 9),

\((\frac{2(4) + 3(-2)}{2 + 3}, \frac{2(9) + 3(3)}{2 + 3})\)

= \((\frac{2}{5}, \frac{27}{5})\)

= \((\frac{2}{5}, 5\frac{2}{5})\)