Correct Answer: D. 7.0

Explanation

\(\frac{2 + 5+ (x + 2) + 7 + 9}{5} = 6 \implies 25 + x = 30\)

\(x = 5 \therefore x + 2 = 5 + 2 = 7\)

Arranging the numbers in ascending order: 2, 5, 7, 7, 9.

Median = 7.0

Correct Answer: B. x< -2 or x > 4

Explanation

(a)

(b)

(i) Median, \(Q_{2} = \frac{1}{2} \text{Nth score}\)

= \(\frac{1}{2} \text{60th score} = 30th\) score.

= \(48.4\)

(ii) \(Q_{1} = 35.5 ; Q_{3} = 57.5\)

Quartile deviation = \(\frac{1}{2} (Q_{3} - Q_{1})\)

= \(\frac{1}{2} (57.5 - 35.5) = \frac{1}{2} (22)\)

= \(11.0\)

Correct Answer: B. 6

Explanation

Remainder for f(2) = 20.

\(f(2) = 2(2^{3}) + 2^{2} - 3(2) + p = 20\)

\(16 + 4 - 6 + p = 20\)

\(14 + p = 20\)

\(p = 6\)

Explanation

(a) \((f^{-1})^{-1}(x)=f(x)\)

\(f^{-1}(x)=\frac{5x - 6}{4 - x},x ≠ 4\)

Let \(y=\frac{5x - 6}{4 - x}\)

\(=y(4-x)=5x-6\)

\(=4y-xy=5x-6\)

\(=-xy-5x=-6-4y\)

\(=x(-y-5)=-6-4y\)

\(=x=\frac{-6 - 4y}{-y - 5}=\frac{-(6 + 4y)}{-(y + 5)}\)

\(=x=\frac{6 + 4y}{y + 5}\)

\(∴f(x)=\frac{6 + 4x}{x + 5},x≠-5\)

(b) \(f(x)=5\)

\(=\frac{6 + 4x}{x + 5}=5\)

\(=6+4x=5(x+5)\)

\(=6+4x=5x+25\)

\(=4x-5x=25-6\)

\(=-x=19\)

\(∴x=-19\)

Explanation

p = 8cos30i + 8sin30j

q = 9cos150i + 9sin150j

p = 6.9282i + 4j

q = -7.7942i + 4.5j

pq = (-7.7942 - 6.9282)i + (4.5 - 4)j

pq = -14.7224i + 0.5j

|pq| = √((-14.7224)2 + (0.5)2)

|pq| = √(216.7491 + 0.25)

|pq| = √216.9991

|pq| = 14.7309

the unit vector in the direction of p - q = \(\frac{-14.7224i+0.5j}{14.7309}\)

-0.9994i + 0.0339j ? -i + 0.0339j

Explanation

\(2^{(2y+1)} - 5(2^y) + 2\) = 0

Let p = 2\(^y\)

\(2^{2y} (2^1) - 5(2^y)\) + 2 = 0

2p\(^2\) - 5p + 2 = 0

2p\(^2\) - p - 4p + 2 = 0

p (2p - 1) - 2(2p - 1) = 0

(p - 2)(2p - 1) = 0

p = 2 or \(\frac{1}{2}\)

p = 2\(^y\)

when p = 2

2\(^y\) = 2

y = 1

when p = \(\frac{1}{2}\)

2\(^y\) = \(\frac{1}{2}\)

2\(^y\) = 2\(^{-1}\)

y = -1

y = -1 or 1

Correct Answer: D. \(10 ms^{-2}\)

Explanation

\(v(t) = (3t^{2} - 2t) ms^{-1}\)

\(a(t) = \frac{\mathrm d v}{\mathrm d t} = (6t - 2) ms^{-2}\)

\(a(2) = 6(2) - 2 = 12 - 2 = 10 ms^{-2}\)

Correct Answer: C. 2

Explanation

\(log_{10}(3x+1) + log_{10}4 = log_{10}(9x+2)\)

\(log_{10}4(3x+1) = log_{10}(9x+2)\)

4(3x+1) = 9x + 2

12x -4 = 9x + 2

12x - 9x = 2 + 4

3x = 6

x = 2

Explanation

(a) \(y = 2x^{3} - 3x^{2} - 12x + 4\)

At stationary points, \(\frac{\mathrm d y}{\mathrm d x} = 0\).

\(\frac{\mathrm d y}{\mathrm d x} = 6x^{2} - 6x - 12 = 0\)

\(\implies x^{2} - x - 2 = 0\)

\(x^{2} - 2x + x - 2 = 0 \implies x(x - 2) + 1(x - 2) = 0\)

\(x = -1 ; x = 2\)

When \(x = -1\), \(y = 2(-1)^{3} - 3(-1)^{2} - 12(-1) + 4 = 11\)

\(\frac{\mathrm d^{2} y}{\mathrm d x^{2}} = 12x - 6 = 12(-1) - 6 = -18 < 0\)

\(\therefore (-1, 11) = \text{maximum point}\)

When \(x = 2\), \(y = 2(2)^{3} - 3(2)^{2} - 12(2) + 4 = -16\)

\(\frac{\mathrm d^{2} y}{\mathrm d x^{2}} = 12x - 6 = 12(2) - 6 = 18 > 0\)

\(\therefore (2, -16) = \text{minimum point}\)

(b)