1. [Maximum mark: 3]
Solve the equation \(8 \sin^2 \theta + 6 \cos \theta + 1 = 0\) for \(0^\circ < \theta < 180^\circ\).
▶️Answer/Explanation
Steps:
1. Substitute \(\sin^2 \theta = 1 – \cos^2 \theta\): \[8(1 – \cos^2 \theta) + 6 \cos \theta + 1 = 0\]
2. Simplify to quadratic: \[8 \cos^2 \theta – 6 \cos \theta – 9 = 0\]
3. Solve using quadratic formula: \[\cos \theta = -0.75\]
4. Find \(\theta\): \[\theta \approx 138.6^\circ\]
2. [Maximum mark: 6]
The function \(f(x) = -2x^2 – 8x – 13\) for \(x < -3\).
(a) Express \(f(x)\) in the form \(-2(x + a)^2 + b\).
(b) Find the range of \(f\).
(c) Find \(f^{-1}(x)\).
▶️Answer/Explanation
(a)
\[f(x) = -2(x + 2)^2 – 5\] (b) f ( x ) < − 5 f(x)<−5 (c) f − 1 ( x ) = − 2 − − x − 5 2 f −1 (x)=−2− 2 −x−5
3. [Maximum mark: 6]
(a) Expand \((1 + 2x)^5\) up to \(x^2\).
(b) Expand \((1 – 3x)^4\) up to \(x^2\).
(c) Find the coefficient of \(x^2\) in \((1 + 2x)^5(1 – 3x)^4\).
▶️Answer/Explanation
(a)
\[1 + 10x + 40x^2\] (b) 1 − 12 x + 54 x 2 1−12x+54x 2 (c) Coefficient: − 26 −26
4. [Maximum mark: 5]
Find the rate of increase of water depth when depth is 4m, in cm/min.
▶️Answer/Explanation
1. Differentiate volume: \[\frac{dV}{dx} = (9 – x)^2 = 25\]
2. Apply chain rule: \[\frac{dx}{dt} = 0.144 \, \text{m/hr}\]
3. Convert: \[0.24 \, \text{cm/min}\]
5. [Maximum mark: 6]
(a) Scale factor of stretch: 3
(b) Original radius: 2
(c) Centre after translation: (1, 15)
(d) Original centre: (1, 6)
▶️Answer/Explanation
(a) 3
(b) 2
(c) (1, 15)
(d) (1, 6)
6. [Maximum mark: 5]
Find exact value of \(\frac{1}{\sin \alpha} + \frac{1}{\tan \alpha}\) where \(\alpha = \arccos\left(\frac{8}{17}\right)\).
▶️Answer/Explanation
1. Calculate \(\sin \alpha = \frac{15}{17}\), \(\tan \alpha = \frac{15}{8}\)
2. Sum: \[\frac{17}{15} + \frac{8}{15} = \frac{5}{3}\]
7. [Maximum mark: 7]
(a) Find \(a\) where tangent gradient is \(-\frac{16}{27}\).
(b) Find \(f(x)\) passing through \((-1, 5)\).
▶️Answer/Explanation
(a)
\[a = -\frac{1}{2} \text{ or } -\frac{7}{2}\] (b) f ( x ) = 1 ( x + 2 ) 3 + 4 f(x)= (x+2) 3 1 +4
8. [Maximum mark: 7]
(a) Perimeter of shaded region: \(2r \cdot 2 \arccos\left(\frac{5}{6}\right)\)
(b) Area of shaded region: \(2 \left[ \frac{1}{2} r^2 \theta – \frac{1}{2} r^2 \sin \theta \right]\)
▶️Answer/Explanation
(a)
\[4r \arccos\left(\frac{5}{6}\right)\] (b) r 2 ( θ − sin θ ) r 2 (θ−sinθ)
9. [Maximum mark: 9]
(a) Sum to infinity: 648
(b) Sum of first 21 terms of AP: 0
▶️Answer/Explanation
(a)
\[S_\infty = 648\] (b) S 21 = 0 S 21 =0
10. [Maximum mark: 10]
(a) Coordinates of A: (1, 1)
(b) Volume: \(\frac{\pi}{3}(4\sqrt{2} – 5)\)
(c) Perimeter: \(\frac{\pi \sqrt{2}}{4} + \sqrt{2}\)
▶️Answer/Explanation
(a)
\[(1, 1)\] (b) π 3 ( 4 2 − 5 ) 3 π (4 2 −5) (c) π 2 4 + 2 4 π 2 + 2
11. [Maximum mark: 11]
(a) \(p = \frac{8}{3}\)
(b)(i) \(p = 4\)
(b)(ii) Circle equation: \(x^2 + y^2 – 15x – y + 44 = 0\)
▶️Answer/Explanation
(a)
\[p = \frac{8}{3}\] (b)(i) p = 4 p=4 (b)(ii) x 2 + y 2 − 15 x − y + 44 = 0 x 2 +y 2 −15x−y+44=0