An arithmetic sequence has first term \(a_1 = 3\) and common difference \(d = 4\). Find the sum of the first 20 terms, \(S_{20}\).
A. 800
B. 810
C. 820
D. 840
Arithmetic Sequences & Series
Key Formulas to Memorise:
\(u_n = a + (n-1)d\)
\(S_n = \dfrac{n}{2}(2a + (n-1)d) = \dfrac{n}{2}(a + u_n)\)
Quick Example
For \(a=1, d=2, n=5\): \(S_5 = \frac{5}{2}(2+8) = 25\)
1
Use the sum formula: \(S_n = \dfrac{n}{2}(2a + (n-1)d)\)
2
Substitute \(n=20,\ a=3,\ d=4\): \(S_{20} = \dfrac{20}{2}(2(3)+(19)(4))\)
3
\(= 10 \times (6 + 76) = 10 \times 82 = \boxed{820}\)
A geometric sequence has first term \(a = 2\) and common ratio \(r = 3\). Find the sum of the first 6 terms, \(S_6\).
A. 364
B. 486
C. 728
D. 1456
Geometric Sequences & Series
Key Formulas:
\(u_n = ar^{n-1}\)
\(S_n = \dfrac{a(r^n - 1)}{r-1},\quad r \neq 1\)
\(S_\infty = \dfrac{a}{1-r},\quad |r| < 1\)
1
Formula: \(S_6 = \dfrac{2(3^6-1)}{3-1}\)
2
\(3^6 = 729\), so \(S_6 = \dfrac{2 \times 728}{2} = \boxed{728}\)
Find the coefficient of \(x^2\) in the expansion of \((2x+3)^4\).
A. 36
B. 96
C. 144
D. 216
Binomial Theorem
Binomial Expansion:
\((a+b)^n = \displaystyle\sum_{k=0}^n \binom{n}{k} a^{n-k}b^k\)
For the term containing \(x^r\): set the power of \(x\) equal to \(r\) and solve for \(k\).
\(\binom{n}{k} = \dfrac{n!}{k!(n-k)!}\)
1
General term: \(\binom{4}{k}(2x)^{4-k}(3)^k\)
2
For \(x^2\): need \(4-k=2 \Rightarrow k=2\)
3
\(\binom{4}{2}(2)^2(3)^2 = 6 \times 4 \times 9 = \boxed{216}\)
Let \(f(x) = x^3 - 3x^2 + 2x\). Find \(f'(2)\).
A. 0
B. 2
C. 4
D. 6
Differentiation
Power Rule:
\(\dfrac{d}{dx}[x^n] = nx^{n-1}\)
Chain Rule:
\(\dfrac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)\)
Product Rule:
\(\dfrac{d}{dx}[uv] = u'v + uv'\)
1
Differentiate: \(f'(x) = 3x^2 - 6x + 2\)
2
Substitute \(x=2\): \(f'(2) = 3(4) - 6(2) + 2 = 12 - 12 + 2 = \boxed{2}\)
Evaluate \(\displaystyle\int_0^2 (3x^2 - 2x + 1)\,dx\).
A. 4
B. 6
C. 8
D. 10
Integration
Power Rule:
\(\displaystyle\int x^n\,dx = \dfrac{x^{n+1}}{n+1} + C,\quad n\neq -1\)
Fundamental Theorem:
\(\displaystyle\int_a^b f(x)\,dx = F(b) - F(a)\)
1
Antiderivative: \(F(x) = x^3 - x^2 + x\)
2
\(F(2) = 8 - 4 + 2 = 6\)
3
\(F(0) = 0\), so the integral \(= 6 - 0 = \boxed{6}\)
Evaluate \(\log_2 8 + \log_2 4\).
A. 3
B. 4
C. 5
D. 7
Logarithms & Exponentials
Log Laws (memorise all):
\(\log_a(xy) = \log_a x + \log_a y\)
\(\log_a(x/y) = \log_a x - \log_a y\)
\(\log_a(x^n) = n\log_a x\)
\(\log_a a = 1,\quad \log_a 1 = 0\)
1
\(\log_2 8 = \log_2 2^3 = 3\)
2
\(\log_2 4 = \log_2 2^2 = 2\)
Solve \(2\sin^2 x - 1 = 0\) for \(x \in [0,\,\pi]\).
A. \(x = \dfrac{\pi}{4}\) only
B. \(x = \dfrac{\pi}{4}\) and \(x = \dfrac{3\pi}{4}\)
C. \(x = \dfrac{\pi}{6}\) and \(x = \dfrac{5\pi}{6}\)
D. \(x = \dfrac{\pi}{3}\) and \(x = \dfrac{2\pi}{3}\)
Trigonometric Equations
Exact values to memorise:
\(\sin\tfrac{\pi}{6} = \tfrac{1}{2},\ \sin\tfrac{\pi}{4} = \tfrac{\sqrt{2}}{2},\ \sin\tfrac{\pi}{3} = \tfrac{\sqrt{3}}{2}\)
CAST / Unit Circle: In \([0,\pi]\), \(\sin x > 0\) for all \(x\). Solutions come in pairs: if \(\sin\alpha = k\) then also \(\sin(\pi - \alpha) = k\).
1
\(2\sin^2 x = 1 \Rightarrow \sin^2 x = \dfrac{1}{2} \Rightarrow \sin x = \dfrac{1}{\sqrt{2}}\) (positive in \([0,\pi]\))
2
First solution: \(x = \dfrac{\pi}{4}\)
3
Second solution: \(x = \pi - \dfrac{\pi}{4} = \dfrac{3\pi}{4}\)
4
Answer: \(\boxed{x = \dfrac{\pi}{4} \text{ and } \dfrac{3\pi}{4}}\)
Evaluate \((3 + 4i)(2 - i)\), where \(i = \sqrt{-1}\).
A. \(2 + 5i\)
B. \(6 - 4i\)
C. \(10 + 5i\)
D. \(10 - 5i\)
Complex Numbers (HL)
Key Properties:
\(i^2 = -1,\quad i^3 = -i,\quad i^4 = 1\)
Multiply complex numbers using FOIL and replace \(i^2\) with \(-1\).
\((a+bi)(c+di) = (ac-bd)+(ad+bc)i\)
1
Expand: \((3+4i)(2-i) = 6 - 3i + 8i - 4i^2\)
2
Since \(i^2 = -1\): \(= 6 + 5i - 4(-1) = 6 + 5i + 4\)
Find the determinant of the matrix \(A = \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix}\).
A. 2
B. 5
C. 8
D. 11
Matrices (HL)
2×2 Determinant:
\(\det\begin{pmatrix}a&b\\c&d\end{pmatrix} = ad - bc\)
Inverse (if det ≠ 0):
\(A^{-1} = \dfrac{1}{\det A}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}\)
If \(\det A = 0\), the matrix is singular (no inverse).
1
\(\det A = (2)(4) - (3)(1) = 8 - 3 = \boxed{5}\)
Given vectors \(\mathbf{u} = \begin{pmatrix}2\\3\\-1\end{pmatrix}\) and \(\mathbf{v} = \begin{pmatrix}1\\-2\\4\end{pmatrix}\), find \(\mathbf{u} \cdot \mathbf{v}\).
A. \(-8\)
B. \(0\)
C. \(4\)
D. \(8\)
Vectors (HL)
Dot Product:
\(\mathbf{u}\cdot\mathbf{v} = u_1v_1 + u_2v_2 + u_3v_3\)
\(\mathbf{u}\cdot\mathbf{v} = |\mathbf{u}||\mathbf{v}|\cos\theta\)
If \(\mathbf{u}\cdot\mathbf{v} = 0\), vectors are
perpendicular.
1
\(\mathbf{u}\cdot\mathbf{v} = (2)(1) + (3)(-2) + (-1)(4)\)
2
\(= 2 - 6 - 4 = \boxed{-8}\)
Events \(A\) and \(B\) satisfy \(P(A)=0.4\), \(P(B)=0.5\), and \(P(A\cap B)=0.2\). Find \(P(A\cup B)\).
A. 0.6
B. 0.7
C. 0.8
D. 0.9
Probability
Addition Rule:
\(P(A\cup B) = P(A) + P(B) - P(A\cap B)\)
Conditional Probability:
\(P(A|B) = \dfrac{P(A\cap B)}{P(B)}\)
Independence: \(A \perp B \Leftrightarrow P(A\cap B) = P(A)\cdot P(B)\)
1
Apply the addition rule: \(P(A\cup B) = 0.4 + 0.5 - 0.2 = \boxed{0.7}\)
A random variable \(X \sim N(50,\, 16)\). Find \(P(X < 54)\) to 4 decimal places.
A. 0.5000
B. 0.7257
C. 0.8413
D. 0.9772
Normal Distribution
Standardisation:
\(Z = \dfrac{X - \mu}{\sigma},\quad Z \sim N(0,1)\)
Note: \(N(\mu, \sigma^2)\) means variance \(= \sigma^2\), so \(\sigma = \sqrt{\text{variance}}\).
Use GDC or z-tables for \(\Phi(z) = P(Z < z)\).
1
\(X \sim N(50, 16)\) means \(\mu=50,\ \sigma^2=16,\ \sigma=4\)
2
\(Z = \dfrac{54-50}{4} = 1\)
3
\(P(X<54) = P(Z<1) = \Phi(1) \approx \boxed{0.8413}\)
Let \(f(x) = 2x + 3\) and \(g(x) = x^2\). Find \((f \circ g)(x)\).
A. \(4x^2 + 12x + 9\)
B. \((2x+3)^2\)
C. \(2x^2 + 3\)
D. \(2x^3 + 3x\)
Functions & Composition
Composite Function:
\((f \circ g)(x) = f(g(x))\)
Always apply \(g\) first, then \(f\).
Inverse Function:
\(f^{-1}(x)\): swap \(x\) and \(y\), then solve for \(y\).
Domain of \(f^{-1}\) = Range of \(f\).
1
\((f\circ g)(x) = f(g(x)) = f(x^2)\)
2
\(= 2(x^2) + 3 = \boxed{2x^2 + 3}\)
Prove by induction that \(\displaystyle\sum_{k=1}^n k = \dfrac{n(n+1)}{2}\). Which statement correctly describes the inductive step?
A. Assume true for \(n=k\); show true for \(n=k+2\)
B. Assume true for \(n=k\); add \((k+1)\) to both sides to show true for \(n=k+1\)
C. Verify the formula is true only for \(n=1\)
D. Assume true for all \(n\); then verify for \(n=1\)
Proof by Mathematical Induction (HL)
Three Steps:
1. Base case: verify for \(n=1\) (or \(n=0\))
2. Inductive hypothesis: assume true for \(n=k\)
3. Inductive step: prove true for \(n=k+1\)
Must conclude: "By PMI, true for all \(n \in \mathbb{Z}^+\)"
1
Base case \((n=1)\): LHS \(=1\), RHS \(=\frac{1(2)}{2}=1\). True. ✓
2
Inductive hypothesis: Assume \(\sum_{k=1}^n k = \frac{n(n+1)}{2}\)
3
Inductive step: Add \((n+1)\) to both sides: \(\frac{n(n+1)}{2} + (n+1) = \frac{(n+1)(n+2)}{2}\). This is the formula for \(n+1\). ✓
4
The key is option B: assume for \(n=k\), then add \((k+1)\) to prove for \(n=k+1\).
Solve for \(x\): \(2^{x+1} = 8^{x-1}\).
A. \(x = 1\)
B. \(x = 2\)
C. \(x = 3\)
D. \(x = 4\)
Exponential Equations
Strategy: Write all terms as powers of the same base, then equate exponents.
\(a^m = a^n \Rightarrow m = n\)
\(8 = 2^3,\quad 4 = 2^2,\quad 27 = 3^3\)
1
Write \(8 = 2^3\): \(2^{x+1} = (2^3)^{x-1} = 2^{3(x-1)}\)
2
Equate exponents: \(x+1 = 3x - 3\)
3
\(4 = 2x \Rightarrow x = \boxed{2}\)
The function \(f(x) = x^3 - 6x^2 + 9x + 1\) has a local maximum at \(x = p\). Find the value of \(f(p)\).
A. 1
B. 3
C. 5
D. 7
Calculus — Optimisation
Finding Extrema:
1. Solve \(f'(x)=0\) for critical points
2. \(f''(x) < 0 \Rightarrow\) local maximum
3. \(f''(x) > 0 \Rightarrow\) local minimum
1
\(f'(x) = 3x^2 - 12x + 9 = 3(x-1)(x-3)\)
2
Critical points: \(x=1\) and \(x=3\)
3
\(f''(x) = 6x-12\). At \(x=1\): \(f''(1)=-6 < 0\) → local maximum
4
\(f(1) = 1 - 6 + 9 + 1 = \boxed{5}\)
An infinite geometric series has first term 12 and common ratio \(\tfrac{1}{2}\). Find the sum to infinity.
A. 12
B. 18
C. 24
D. 36
Infinite Geometric Series
Converges when \(|r| < 1\):
\(S_\infty = \dfrac{a}{1-r}\)
Diverges (no sum) if \(|r| \geq 1\).
1
Check: \(|r| = \tfrac{1}{2} < 1\), so converges. ✓
2
\(S_\infty = \dfrac{12}{1-\frac{1}{2}} = \dfrac{12}{\frac{1}{2}} = \boxed{24}\)
Find the exact value of \(\sin 75°\).
A. \(\dfrac{\sqrt{6}-\sqrt{2}}{4}\)
B. \(\dfrac{\sqrt{3}+1}{4}\)
C. \(\dfrac{\sqrt{6}+\sqrt{2}}{4}\)
D. \(\dfrac{\sqrt{3}}{2}\)
Trigonometric Identities
Addition Formulae (must memorise):
\(\sin(A+B) = \sin A\cos B + \cos A\sin B\)
\(\cos(A+B) = \cos A\cos B - \sin A\sin B\)
\(\sin 30°=\tfrac{1}{2},\ \cos 30°=\tfrac{\sqrt{3}}{2},\ \sin 45°=\cos 45°=\tfrac{\sqrt{2}}{2}\)
1
Write \(75° = 45° + 30°\)
2
\(\sin 75° = \sin 45°\cos 30° + \cos 45°\sin 30°\)
3
\(= \dfrac{\sqrt{2}}{2}\cdot\dfrac{\sqrt{3}}{2} + \dfrac{\sqrt{2}}{2}\cdot\dfrac{1}{2} = \dfrac{\sqrt{6}}{4} + \dfrac{\sqrt{2}}{4} = \boxed{\dfrac{\sqrt{6}+\sqrt{2}}{4}}\)
Find \(\displaystyle\int x\,e^x\,dx\).
A. \(xe^x + C\)
B. \((x+1)e^x + C\)
C. \((x-1)e^x + C\)
D. \(x^2 e^x + C\)
Integration by Parts
Integration by Parts:
\(\displaystyle\int u\,dv = uv - \int v\,du\)
LIATE rule for choosing \(u\): Logarithm, Inverse trig, Algebraic, Trig, Exponential.
Here: \(u=x\) (algebraic), \(dv=e^x dx\)
1
Let \(u = x,\ dv = e^x dx\). Then \(du = dx,\ v = e^x\)
2
\(\int x e^x dx = xe^x - \int e^x dx = xe^x - e^x + C\)
3
\(= \boxed{(x-1)e^x + C}\)
4
Verify: \(\dfrac{d}{dx}[(x-1)e^x] = e^x + (x-1)e^x = xe^x\) ✓
The volume of a sphere is given by \(V = \dfrac{4}{3}\pi r^3\). At the instant when \(r = 3\) cm, the radius is increasing at \(0.5\) cm s\(^{-1}\). Find \(\dfrac{dV}{dt}\) at that instant.
A. \(6\pi\) cm³ s⁻¹
B. \(12\pi\) cm³ s⁻¹
C. \(18\pi\) cm³ s⁻¹
D. \(36\pi\) cm³ s⁻¹
Related Rates of Change
Chain Rule for rates:
\(\dfrac{dV}{dt} = \dfrac{dV}{dr}\cdot\dfrac{dr}{dt}\)
Steps: (1) differentiate the volume formula with respect to \(r\), (2) multiply by \(\dfrac{dr}{dt}\), (3) substitute the given values.
1
\(\dfrac{dV}{dr} = 4\pi r^2\)
2
\(\dfrac{dV}{dt} = 4\pi r^2 \cdot \dfrac{dr}{dt} = 4\pi(3)^2 \times 0.5\)
3
\(= 4\pi \times 9 \times 0.5 = \boxed{18\pi} \approx 56.55\) cm³ s⁻¹