Correct Answer: A. \(5\sqrt{3} - 3\)

Explanation

\(F = F \cos \theta i + F \sin \theta j\)

\(10N = 10 \cos 60 i + 10 \sin 60 j\)

\(6N = -6 \cos 330 i - 6 \sin 330 j\)

\(R_{x} = 10 \cos 60 - 6 \cos 330 \)

= \(10 \times \frac{1}{2} - 6 \times \frac{\sqrt{3}}{2}\)

= \(5 - 3\sqrt{3}\)

Correct Answer: C. 225 litres

Correct Answer: D. 56.25 m

Explanation

\(h = 45t - 9t^{2}\)

\(\frac{\mathrm d h}{\mathrm d t} = 45 - 18t = 0\)

\(45 = 18t \implies t = 2.5 s\)

\(h(2.5) = 45(2.5) - 9(2.5)^{2} = 112.5 - 56.25\)

= \(56.25 m.\)

Correct Answer: A. 2.4

Explanation

\(Average \hspace{1mm}speed = \frac{Distance}{Time\hspace{1mm} taken}\\

Distance = 800m = \frac{800}{1000}km = 0.8km\\

Time\hspace{1mm}taken = 20\hspace{1mm}minutes = \frac{20}{60}hr=\frac{1}{3}hr\\Average \hspace{1mm} speed = \left(0.8\div \frac{1}{3}\right)km/hr = (0.8\times3)km/hr\\=2.4km/hr\)

Correct Answer: D. 2 units

Explanation

\(\sqrt{(x_2 - x_1)^2 + (y_2 - Y_1)^2}\)

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

= \(\sqrt{0^2 + 2^2}\)

= \(\sqrt{4}\)

= 2 units

Explanation

(a) \((PT)^{2} + (TQ)^{2} = (PQ)^{2}\)

\((PT)^{2} + 6^{2} = 10^{2}\)

\((PT)^{2} = 100 - 36 = 64\)

\(\therefore PT = 8 cm\)

\(\therefore \frac{8}{6} = \frac{14.4}{SR}\)

\(SR = \frac{6 \times 14.4}{8}\)

\(SR = 10.8 cm\)

Area of the quadrilateral QRST = \(\frac{1}{2} (6 + 10.8) \times 6.4\)

= \(53.76 cm^{2}\)

(b) Opp sides of a square = \((\frac{(100 - 10)x}{100} = 0.9x\)

other two sides of the square = \(\frac{(100 + 15)x}{100} = 1.15x\)

Area of the square : \(x \times x = x^{2}\)

Area of the rectangle : \(0.9x \times 1.15x = 1.035x^{2}\)

\(\therefore \text{The ratio of the area of the rectangle to the square} = 1.035x^{2} : x^{2}\)

= \(1.035 : 1\)

Correct Answer: (a) 577.5km

Explanation

\(\frac{3x^2 + 3x - 2}{(x - 1)(x + 1)}\) = P + \(\frac{Q}{x - 1} + \frac{R}{x - 1}\)

Long division

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

= \(\frac{Q}{x - 1} + \frac{R}{x + 1}\)

6x - 2 = Q (x + 1) + R(x - 1)

Let x = -1

6(-1) - 2 = Q(-1 + 1) + R (-1 - 1)

-6 - 2 = -2R

\(\frac{-8}{-2} = \frac{-2R}{-2}\)

R = 4

From 6x - 2 = Q(x - 1) + R(x - 1)

Let x = 1

6(1) - 2 = Q(1 + 1) + R(1 - 1)

6 - 2 = 2Q

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

= Q = 2

Correct Answer: B. √6

Correct Answer: C. 44°

Explanation

2x + m = 180

x + m = 112

x = 122 - m

2(112 - m) + m = 180

224 - 2m + m = 180

224 - m = 180

224 - 180 = m

m = 44\(^o\)