Explanation

(a) \(p \propto r^{2}\)

\(\implies p = kr^{2}\)

\(3.2 = k(4^{2}) \implies 3.2 = 16k\)

\(k = \frac{3.2}{16} = 0.2\)

\(\therefore p = 0.2r^{2}\)

When r = 6.5,

\(p = 0.2 (6.5^{2}) \)

= \(0.2(42.25)\)

\(p = 8.45\)

(b) \(\frac{x}{2} + \frac{y}{4} = 1 ... (1)\)

\(\frac{x}{3} - \frac{y}{4} = -\frac{1}{6} .... (2)\)

Adding equation (1) and (2), we have

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

\(\frac{5x}{6} = \frac{5}{6} \implies x = 1\)

Put x = 1 in (1),

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

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

\((x, y) = (1, 2)\).

Correct Answer: A. 35 mins

Explanation

Since 36km in 21 minutes

1km in \(\frac{21}{36}\)minutes

60km in \(\left(\frac{21}{36}\times 60\right)\)minutes = \(\frac{70}{2}\)min

= 35 mins

Correct Answer: A. -1

Explanation

\(x + y = 7 ------ I\\

3x

Correct Answer: A. 24°C

Correct Answer: B. 7cm

Explanation

The length of the arc subtended by the sector of angle 120° = circumference of the base of the cone.

\(\frac{120}{360} \times 2 \times \frac{22}{7} \times 21 = 2\pi r\)

\(44 = 2\pi r\)

\(r = 22 \div \pi\)

\(r = 22 \times \frac{7}{22}\)

r = 7 cm

Explanation

(a) \(F = xi + 15j\)

\(|F| = \sqrt{x^{2} + 15^{2}} = 17\)

\(x^{2} = 17^{2} - 15^{2} = 64\)

\(x = \pm \sqrt{64} = \pm 8\)

\(x = 8 ; x = -8\)

(b)

\(\tan \alpha = \frac{15}{8} = 1.875\)

\(\alpha = \tan^{-1} (1.875) = 61.9° \approxeq 62°\)

\(\tan \beta = \frac{15}{-8} = -1.875\)

\(\beta = \tan^{-1} (-1.875) = -61.9°\)

= \(180° - 61.9° = 118.1° \approxeq 118°\)

Directions of forces are 62° and 118° each with the positive x- axis.

Correct Answer: c. 16900sqcm

Correct Answer: C. 4

Explanation

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

= \((2^4)^{\frac{5}{4}} \times 2^{-3} \times 1\)

= \(2^5 \times 2^{-3} \times 1\)

= \(2^2\)

= 4.

Correct Answer: C. 148.5cm3

Explanation

Volume of a cone = \(\frac{1}{3} \pi r^2 h\)

r = 4\(\frac{1}{2}\) cm; h = 14 cm

Volume of cone = \(\frac{1}{3} \times \frac{22}{7} \times \frac{9}{2} \times \frac{9}{2} \times 14\)

= 297 cm\(^3\)

When half- filled, the volume of the water = \(\frac{297}{2} = 148.5 cm^3\)

Correct Answer: B. 16\(\frac{7}{8}\)%