Correct Answer: C. \(f(x) = x^{3} - 3x^{2} + x + 2\)

Explanation

\(f ' (x) = 3x^{2} - 6x + 1\)

\(f(x) = \int (3x^{2} - 6x + 1) \mathrm {d} x\)

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

\(f(3) = 5 = 3^{3} - 3(3^{2}) + 3 + c\)

\(27 - 27 + 3 + c = 5 \implies 3 + c = 5\)

\(c = 2\)

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

Correct Answer: B. \(\beta\)

Explanation

From the diagram, \(g(3) = s; h(s) = \beta\).

Correct Answer: C. y - 4x + 9 = 0

Explanation

\(y = 4x^{2} - 12x + 7\)

\(\frac{\mathrm d y}{\mathrm d x} = 8x - 12\)

At x = 2, y = 8(2) - 12 = 4

Equation of the tangent to the curve: \(y - (-1) = 4(x - 2)\)

\(y + 1 = 4x - 8 \implies y - 4x + 1 + 8 = y - 4x + 9 = 0\)

Correct Answer: D. 8

Explanation

The gradient of the equation = \(\frac{y_{1} - y_{0}}{x_{1} - x_{0}} = \frac{\frac{5}{2} - 0}{\frac{5}{4} - 0}\)

= \(\frac{\frac{5}{2}}{\frac{5}{4}} = 2\)

The line passes through the origin, therefore the equation of the line is y = mx

\(y = 2x\)

When x = 4, y = 2 x 4 = 8.

Correct Answer: B. \(q \vee \sim p\)

Correct Answer: A. 23.4

Explanation

\(Mean = \frac{\sum fx}{\sum f}\)

= \(\frac{1170}{50} = 23.4\)

Explanation

|\(\over{OP}\)| = \(\sqrt{(-3)^2 + (-4)^2}\) = 5

|\(\over{OQ}\)| = \(\sqrt{(8)^2 +(-15)^2}\) = 17

and |\(\over{OP}\)|.|\(\over{OQ}\)| = (\(^{-3}_{-4}\)).(\(^8_{15}\)) = -24 + 60 = 36

Then 36 = (5)(17) cos \(\theta\) and finding cos \(\theta\) = \(\frac{36}{85}\) = 0.4235 and taking the inverse cosine to yield

\(\theta\) = cos\(^{-1}\)(0.4235)

= 64.94\(^o\)

Correct Answer: D. \(\frac{56}{33}\)

Explanation

\(\tan (x + y) = \frac{\tan x + \tan y}{1 - \tan x\tan y}\)

\(\tan x = \frac{5}{12} ; \tan y = \frac{3}{4}\)

\(\tan (x + y) = \frac{\frac{5}{12} + \frac{3}{4}}{1 - (\frac{5}{12} \times \frac{3}{4}})\)

= \(\frac{\frac{14}{12}}{\frac{33}{48}}\)

= \(\frac{56}{33}\)

Correct Answer: C. 18

Explanation

\(\int^0_0 \sqrt{x} dx\)

\(\frac{x^{\frac{1}{2}} + 1}{\frac{1}{2} + 1}\) |\(^9_0\)

\(\frac{2x^{\frac{3}{2}}}{3}\)|\(^9_0\)

= \(\frac{2(9)^{\frac{3}{2}}}{3}\)

= \(\frac{2(3)^3}{3}\)

= \(\frac{2 \times 27}{3}\)

= 2 x 9

= 18

Correct Answer: C. \(\frac{5}{4}\)

Explanation

(1 + \(\frac{1}{2} x )^5\)

1(1)\(^3\)(\(\frac{1}{2} x)^0\) - 5(1)\(^4\)(\(\frac{1}{2} x)^1\)

+ 10 (10)\(^3\)(\(\frac{1}{2} x )^2\) + 10(1)\(^2\) (\(\frac{1}{2} x\))\(^3\)

Coefficient of x\(^3\)

10(\(\frac{1}{2} x)^3\) = \(\frac{10}{8} x^3\) =

= \(\frac{5}{4}\)