(a) Simplify : \(\frac{x^{2} - y^{2}}{3x + 3y}\)
(b)
In the diagram, PQRS is a rectangle. /PK/ = 15 cm, /SK/ = /KR/ and <PKS = 30°. Calculate, correct to three significant figures : (i) /PS/ ; (ii) /SK/ and (iii) the area of the shaded portion.
(b)
In the diagram, PQRS is a rectangle. /PK/ = 15 cm, /SK/ = /KR/ and <PKS = 30°. Calculate, correct to three significant figures : (i) /PS/ ; (ii) /SK/ and (iii) the area of the shaded portion.
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Correct Answer: Option
Explanation:
(a) \(\frac{x^{2} - y^{2}}{3x + 3y}\)
\(\frac{(x + y)(x - y)}{3(x + y)}\) (Using difference of two squares)
= \(\frac{x - y}{3}\)
(b)(i)
\(\sin 37 = \frac{/PS/}{15}\)
\(/PS/ = 15 \times 0.6018\)
= \(9.03 cm\)
(ii) \(\cos 37 = \frac{/SK/}{15}\)
\(/SK/ = 15 \times 0.7986\)
= \(11.98 cm \)
\(\approxeq 12.0 cm\)
(iii) Area of the shaded portion = Area of rectangle PQRS - Area of triangle PSK.
/SR/ = 2(/SK/) = 2(11.98)
= 23.96 cm
Area of rectangle PQRS = \(23.96 \times 9.03 \)
= \(216.3588 cm^{2}\)
Area of triangle PKS = \(\frac{1}{2} \times 11.98 \times 9.03\)
= \(54.088 cm^{2}\)
Area of shaded portion : \((216.3588 - 54.088)cm^{2}\)
\(162.2708 cm^{2}\)
\(\approxeq 162 cm^{2}\)
(a) \(\frac{x^{2} - y^{2}}{3x + 3y}\)
\(\frac{(x + y)(x - y)}{3(x + y)}\) (Using difference of two squares)
= \(\frac{x - y}{3}\)
(b)(i)
\(\sin 37 = \frac{/PS/}{15}\)
\(/PS/ = 15 \times 0.6018\)
= \(9.03 cm\)
(ii) \(\cos 37 = \frac{/SK/}{15}\)
\(/SK/ = 15 \times 0.7986\)
= \(11.98 cm \)
\(\approxeq 12.0 cm\)
(iii) Area of the shaded portion = Area of rectangle PQRS - Area of triangle PSK.
/SR/ = 2(/SK/) = 2(11.98)
= 23.96 cm
Area of rectangle PQRS = \(23.96 \times 9.03 \)
= \(216.3588 cm^{2}\)
Area of triangle PKS = \(\frac{1}{2} \times 11.98 \times 9.03\)
= \(54.088 cm^{2}\)
Area of shaded portion : \((216.3588 - 54.088)cm^{2}\)
\(162.2708 cm^{2}\)
\(\approxeq 162 cm^{2}\)