Q03 Test 41 English

Test No. 41 – Complete Solutions

164

Find the value of \(\cos 60°\cos 30° + \sin 60°\sin 30°\).

🔑 Key Concept – Cosine Subtraction Formula: \[ \cos(A - B) = \cos A\cos B + \sin A\sin B \] So \(\cos 60°\cos 30° + \sin 60°\sin 30° = \cos(60° - 30°) = \cos 30°\).
Standard values: \[ \cos 60° = \frac{1}{2},\quad \cos 30° = \frac{\sqrt{3}}{2},\quad \sin 60° = \frac{\sqrt{3}}{2},\quad \sin 30° = \frac{1}{2} \]

Method 1 – Using cosine subtraction formula: \[ \cos 60°\cos 30° + \sin 60°\sin 30° = \cos(60° - 30°) = \cos 30° = \frac{\sqrt{3}}{2} \]

Method 2 – Direct substitution: \[ = \frac{1}{2} \times \frac{\sqrt{3}}{2} + \frac{\sqrt{3}}{2} \times \frac{1}{2} \] \[ = \frac{\sqrt{3}}{4} + \frac{\sqrt{3}}{4} = \frac{2\sqrt{3}}{4} = \frac{\sqrt{3}}{2} \]

\( \boxed{\cos 60°\cos 30° + \sin 60°\sin 30° = \dfrac{\sqrt{3}}{2} \approx 0.866} \)

165

Find equation of line with y-intercept \(-7\) and slope \(5\).

🔑 Key Concept – Slope-Intercept Form: When slope \(m\) and y-intercept \(c\) are known, use: \[ y = mx + c \] The y-intercept is the point where the line crosses the y-axis, i.e. where \(x = 0\).

Given: Slope \(m = 5\),  y-intercept \(c = -7\)

Step 1 – Apply slope-intercept form directly: \[ y = 5x + (-7) \] \[ y = 5x - 7 \]

Step 2 – Write in standard form: \[ 5x - y - 7 = 0 \]

\( \boxed{y = 5x - 7 \quad \text{or} \quad 5x - y - 7 = 0} \)

166

The ratio of radii of two spheres is \(3:4\). Find the ratio of their volumes.

🔑 Key Concept – Volumes of Similar Solids: For two similar solids with corresponding lengths \(r_1\) and \(r_2\): \[ \frac{V_1}{V_2} = \left(\frac{r_1}{r_2}\right)^3 \] Volumes scale as the cube of the ratio of corresponding lengths.

Given: \(\dfrac{r_1}{r_2} = \dfrac{3}{4}\)

Step 1 – Apply the volume ratio formula: \[ \frac{V_1}{V_2} = \left(\frac{r_1}{r_2}\right)^3 = \left(\frac{3}{4}\right)^3 \]

Step 2 – Evaluate the cube: \[ \frac{V_1}{V_2} = \frac{3^3}{4^3} = \frac{27}{64} \]

\( \boxed{\dfrac{V_1}{V_2} = \dfrac{27}{64} \quad \text{i.e. } 27:64} \)

167

Convert to degrees, minutes and seconds:  (i) \(\dfrac{17\pi}{4}\)    (ii) \(\dfrac{7\pi}{12}\)

🔑 Key Concept – Radians to DMS: Step 1: Multiply by \(\dfrac{180°}{\pi}\) to get decimal degrees.
Step 2: Multiply decimal part by 60 → minutes.
Step 3: Multiply decimal part of minutes by 60 → seconds.

(i) Convert \(\dfrac{17\pi}{4}\) to DMS

Step 1 – Convert to decimal degrees: \[ \frac{17\pi}{4} \times \frac{180°}{\pi} = \frac{17 \times 180°}{4} = \frac{3060°}{4} = 765° \]

Step 2 – Reduce to within 360° (find equivalent angle): \[ 765° = 2 \times 360° + 45° \] Equivalent angle \(= 45°\) (exact whole number — no minutes or seconds needed)

\( \boxed{\dfrac{17\pi}{4} = 765° = 45°\ \text{(equivalent angle)}} \)

(ii) Convert \(\dfrac{7\pi}{12}\) to DMS

Step 1 – Convert to decimal degrees: \[ \frac{7\pi}{12} \times \frac{180°}{\pi} = \frac{7 \times 180°}{12} = \frac{1260°}{12} = 105° \] This is an exact whole number — no decimal, so no minutes or seconds.

\( \boxed{\dfrac{7\pi}{12} = 105°\ 0'\ 0''} \)