Q03 Test 30 English

Test No. 30 – Complete Solutions

119

Find the arc length of a sector if radius \(r = 10\) cm and central angle \(\theta = 60°\).

🔑 Key Concept – Arc Length of a Sector:
\[ \ell = r\theta \] where \(\theta\) must be in radians. Convert degrees first: \[ \theta_{\text{rad}} = \theta_{\text{deg}} \times \frac{\pi}{180} \]

Given: \(r = 10\) cm,  \(\theta = 60°\)

Step 1 – Convert \(\theta\) to radians: \[ \theta = 60 \times \frac{\pi}{180} = \frac{\pi}{3}\ \text{rad} \]

Step 2 – Apply the arc length formula: \[ \ell = r\theta = 10 \times \frac{\pi}{3} = \frac{10\pi}{3} \]

Step 3 – Evaluate: \[ \ell = \frac{10 \times 3.14159}{3} \approx 10.47\ \text{cm} \]

\( \boxed{\ell = \dfrac{10\pi}{3} \approx 10.47\ \text{cm}} \)

120

Find the area of a sector with radius \(r = 8\) cm and central angle \(45°\).

🔑 Key Concept – Area of a Sector:
\[ A = \frac{1}{2}r^2\theta \] where \(\theta\) must be in radians. Convert degrees first.

Given: \(r = 8\) cm,  \(\theta = 45°\)

Step 1 – Convert \(\theta\) to radians: \[ \theta = 45 \times \frac{\pi}{180} = \frac{\pi}{4}\ \text{rad} \]

Step 2 – Apply the area formula: \[ A = \frac{1}{2} \times (8)^2 \times \frac{\pi}{4} = \frac{1}{2} \times 64 \times \frac{\pi}{4} \]

Step 3 – Simplify and evaluate: \[ A = \frac{64\pi}{8} = 8\pi \approx 25.13\ \text{cm}^2 \]

\( \boxed{A = 8\pi \approx 25.13\ \text{cm}^2} \)

121

Find the distance between \(A(6,\,7)\) and \(B(0,\,-2)\).

🔑 Key Concept – Distance Formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Given: \(A(x_1, y_1) = (6, 7)\),  \(B(x_2, y_2) = (0, -2)\)

Step 1 – Apply the formula: \[ d = \sqrt{(0-6)^2 + (-2-7)^2} \]

Step 2 – Compute and square the differences: \[ d = \sqrt{(-6)^2 + (-9)^2} = \sqrt{36 + 81} = \sqrt{117} \]

Step 3 – Simplify: \[ d = \sqrt{9 \times 13} = 3\sqrt{13} \approx 10.82\ \text{units} \]

\( \boxed{d = 3\sqrt{13} \approx 10.82\ \text{units}} \)

122

Find the midpoint of the line joining \(A(3,\,1)\) and \(B(-2,\,-4)\).

🔑 Key Concept – Midpoint Formula: \[ M = \left(\frac{x_1 + x_2}{2},\ \frac{y_1 + y_2}{2}\right) \]

Given: \(A(x_1, y_1) = (3, 1)\),  \(B(x_2, y_2) = (-2, -4)\)

Step 1 – Apply the formula: \[ M = \left(\frac{3 + (-2)}{2},\ \frac{1 + (-4)}{2}\right) \]

Step 2 – Simplify: \[ M = \left(\frac{1}{2},\ \frac{-3}{2}\right) = (0.5,\ -1.5) \]

\( \boxed{M = \left(\frac{1}{2},\ -\frac{3}{2}\right) = (0.5,\ -1.5)} \)