Correct Answer: A. [1,2,4,6,7,8]

Explanation

R = {2, 4, 6, 7}; S = {1, 2, 4, 8}

R \(\cup\) S = {1, 2, 4, 6, 7, 8}

Correct Answer: B. 2

Explanation

\(a * b = a^{2} + b + ab\)

\(5 * x = 5^{2} + x + 5x = 37\)

\(25 + 6x = 37 \implies 6x = 12\)

\(x = 2\)

Explanation

(a)

(b)(i) 80th percentile = \(\frac{80 \times 60th}{100} = \text{48th mark}\).

= 31.5.

(ii) Median = \(\frac{60th}{2} = \text{30th mark} = 32.3\)

(iii) Upper quartile = \(\frac{3 \times 60th}{4} = \text{45th mark} = 35.1\)

Lower quartile = \(\frac{60th}{4} = \text{15th mark} = 29.7\)

Semi-interquartile range = \(\frac{35.1 - 29.7}{2} = 2.7\)

Correct Answer: A. 5cm

Correct Answer: D. -12 sin 4x

Correct Answer: A. 4√6 cm

Correct Answer: C. \(\frac{x-11}{6}\)

Explanation

\(\frac{2x - 1}{3} - \frac{x + 3}{2}\)

= \(\frac{2(2x - 1) - 3(x + 3)}{6}\)

= \(\frac{4x - 2 - 3x - 9}{6}\)

= \(\frac{x - 11}{6}\)

Explanation

(a) \(6y + 5x = -12 ... (1)\)

\(4y - 3x = 11 ... (2)\)

Multiply (1) by 2 and (2) by 3,

\(12y + 10x = -24 ... (1a)\)

\(12y - 9x = 33 ... (2a)\)

\((1a) - (2a) : 19x = -57 \implies x = -3\)

\(4y - 3(-3) = 11 \implies 4y + 9 = 11\)

\(4y = 2 \implies y = \frac{1}{2}\)

(b)

\(/BC/^{2} = 36^{2} + 48^{2}\)

\(/BC/^{2} = 1296 + 2304 = 3600\)

\(/BC/ = \sqrt{3600} = 60 cm\)

\(/AC/^{2} = /BC/^{2} + /AB/^{2}\)

\((y + 48)^{2} = 60^{2} + 36^{2} + y^{2}\)

\(y^{2} + 96y + 2304 = 3600 + 1296 + y^{2}\)

\(96y = 3600 + 1296 - 2304 = 2592\)

\(96y = 2592 \implies y = \frac{2592}{96} = 27 cm\)

Correct Answer: A. \(\frac{b(a-x+1)}{a+1}\)

Explanation

If \(\frac{bx+y(a+1)}{b(a+1)}=1\\

bx + ya + y = ba + b; y(a+1) = ba+b-bx\\

y(a+1)=b(a+1-x); y = \frac{b(1+a-x}{a+1}\)

Explanation

(a) \(\frac{3p + 4q}{3p - 4q} = 2\)

\(3p + 4q = 2(3p - 4q)\)

\(3p + 4q = 6p - 8q\)

\(3p - 6p = - 8q - 4q\)

\(-3p = - 12q\)

\(p = 4q\)

\(\frac{p}{q} = \frac{4}{1}\)

\(\therefore p : q = 4 : 1\).

(b)(i)

Perimeter of cross- section = 4 + x + 4 + 2 + y + 2

i.e 34 = 12 + x + y

x + y = 22 ..... (1)

From the diagram, |PQ| = |UR| (opp. sides of a rectangle)

i.e. x = 2 + d + 2

x - 4 = d ..... (2)

From (1), y = 22 - x is the circumference of the semi-circle.

\(22 - x = \frac{2\pi r}{2} = \pi r\)

\(r = \frac{d}{2} \)

\(22 - x = \frac{22}{7} \times \frac{(x - 4)}{2}\)

\(154 - 7x = 11x - 44\)

\(154 + 44 = 11x + 7x \)

\(198 = 18x\)

\(x = \frac{198}{18} = 11 m\)

Hence, |PQ| = 11 m.

(ii) Area of cross section = Area of rectangle PQRU - area of semi-circle

Area of rectangle = \(11 \times 4 = 44 m^{2}\)

Area of semi-circle = \(\frac{\pi r^{2}}{2} \)

= \(\frac{22}{7} \times (\frac{(11 - 4)}{2})^{2} \times \frac{1}{2}\)

= \(19.25 m^{2}\)

Area of cross section = \(44 - 19.25 = 24.75 m^{2}\).