Number & Algebra - Practice Set 1 | Math AA

Q1

Solve the equation 3^(2x) − 12·3^x + 27 = 0, giving your answers in exact form.

(a) Let y = 3^x. Write the equation in terms of y.

(b) Solve for y and hence find the values of x.

Answer

(a) y² − 12y + 27 = 0 (b) x = 1 or x = 2. Explanation: y² - 12y + 27 = (y-3)(y-9)=0 → y=3 or y=9 → 3^x=3 → x=1; 3^x=9 → x=2.

Q2

Prove by mathematical induction that Σ(r=1 to n) r · 2^r = 2 + (n − 1) · 2^(n+1) for all n ∈ ℤ⁺.

(a) Verify the result for n = 1.

(b) State the inductive hypothesis for n = k.

(c) Complete the inductive step to show the result is true for n = k + 1.

Answer

(a) LHS = 1·2 = 2; RHS = 2 + 0·4 = 2 ✓ (b) Assume Σᵣ₌₁ᵏ r·2ʳ = 2 + (k−1)·2ᵏ⁺¹ for some k ≥ 1 (c) Add (k+1)·2ᵏ⁺¹: RHS = 2 + (k−1)·2ᵏ⁺¹ + (k+1)·2ᵏ⁺¹ = 2 + 2ᵏ⁺¹·2k = 2 + k·2ᵏ⁺² = 2 + ((k+1)−1)·2^((k+1)+1) ✓

Q3

Express (5x + 1) / [(x + 3)(x − 1)] in partial fractions.

(a) Write the expression in the form A/(x + 3) + B/(x − 1).

(b) Find the values of A and B.

(c) Hence find ∫ (5x + 1) / [(x + 3)(x − 1)] dx.

Answer

(a) A/(x+3) + B/(x−1) (b) A = -7/2, B = 3/2 (c) -(7/2) ln|x+3| + (3/2) ln|x-1| + C

Q4

An infinite geometric series has first term a and common ratio r, where |r| < 1. The sum to infinity is 20 and the sum of the first 3 terms is 19.

(a) Show that a = 20(1 − r).

(b) Hence form an equation in r and solve to find r.

(c) Find the first term a.

Answer

(a) S∞ = a/(1−r) = 20 ⇒ a = 20(1−r) ✓ (b) r = (−20 ± √(400−80))/40 → r ≈ -0.0513 or r ≈ -0.9487 (c) For r ≈ -0.0513, a ≈ 21.03; For r ≈ -0.9487, a ≈ 38.97

Q5

A bouncing ball is dropped from a height of 4 m. Each bounce reaches 75% of the previous height.

(a) Find the height of the 6th bounce.

(b) Find the total distance travelled by the ball before it comes to rest.

(c) After how many bounces does the total distance first exceed 24 m?

Answer

(a) h₆ ≈ 0.949 m (b) Total distance = 28 m (c) After 7 bounces

Q6

Let z = 2(cos(π/6) + i·sin(π/6)).

(a) Write z in Cartesian form a + bi.

(b) Find z⁴ using De Moivre's theorem. Give your answer in Cartesian form.

(c) Find all cube roots of z³. Write answers in polar form.

Answer

(a) z = √3 + i (b) z⁴ = −8 + 8√3·i (c) w₀ = 2(cos(π/6)+i·sin(π/6)), w₁ = 2(cos(5π/6)+i·sin(5π/6)), w₂ = 2(cos(3π/2)+i·sin(3π/2))

Q7

Prove by deduction that for any integer n, the expression n³ − n is always divisible by 6.

(a) Factorise n³ − n completely.

(b) Using part (a), show that n³ − n is divisible by 6 for all integers n.

Answer

(a) n³ − n = (n−1)n(n+1) (b) Product of 3 consecutive integers: divisible by 2 (one even) and 3 (one multiple of 3) ⇒ divisible by 6 ✓

Q8

Let m and n be odd integers.

(a) Prove that mn is odd.

(b) Prove that m² + n² is even but not divisible by 4.

Answer

(a) m = 2a+1, n = 2b+1; mn = 2(2ab+a+b)+1 ⇒ odd ✓ (b) m²+n² = 4(a²+a+b²+b)+2; divisible by 2, not by 4 ✓

Q9

Prove by mathematical induction that Σᵣ₌₁ⁿ (2r − 1) = n² for all n ∈ ℤ⁺.

(a) Verify the base case n = 1.

(b) State the inductive hypothesis for n = k.

(c) Complete the inductive step to prove true for n = k + 1.

Answer

(a) LHS = 1; RHS = 1² = 1 ✓ (b) Assume Σᵣ₌₁ᵏ (2r−1) = k² for some k ≥ 1 (c) Add (2k+1): k² + 2k + 1 = (k+1)² ✓

Q10

Prove by contradiction that √3 is irrational.

(a) State the assumption for proof by contradiction.

(b) By writing √3 = p/q in lowest terms (gcd(p,q) = 1), derive a contradiction.

Answer

(a) Assume √3 = p/q where p,q∈ℤ, q≠0, gcd(p,q)=1 (b) 3q² = p² ⇒ p divisible by 3. Let p=3k: q²=3k² ⇒ q divisible by 3. Contradicts gcd=1 ✓

Q11

(a) Use the binomial theorem with negative/fractional indices to expand (1 + x)^(−2) up to and including the term in x³, stating the range of x for which the expansion is valid.

(b) Use your expansion to find an approximation for 1/(1.02)² correct to 4 decimal places.

(c) Compare with the exact value and comment.

Answer

(a) 1 − 2x + 3x² − 4x³ + … valid for |x| < 1 (b) x = 0.02 → 1 − 0.04 + 0.0012 − 0.000032 = 0.961168 ≈ 0.9612 (4 d.p.) (c) Exact: 1/1.0404 = 0.961168... Very close — small x gives excellent approximation.

Q12

Eiffel Tower replicas exist at different scales. The largest is 324 m tall. A collection of 8 replicas has heights in geometric sequence. The smallest replica is 0.5 m and the largest is 324 m.

(a) Show that the common ratio is r = (648)^(1/7).

(b) Find the heights of all 8 replicas. Give answers to 3 significant figures.

(c) Find the total height of all 8 replicas.

(d) Which replica has height closest to 10 m?

Answer

(a) 0.5·r⁷ = 324 ⇒ r⁷ = 648 ⇒ r = 648^(1/7) ✓ (b) u₁=0.500, u₂=1.45, u₃=4.22, u₄=12.3, u₅=35.6, u₆=103, u₇=301, u₈=324 (c) S₈ ≈ 169.8 m (d) u₄ ≈ 12.3 m is closest to 10 m.

Q13

(a) Express z = −1 + i√3 in modulus-argument form.

(b) Find z⁶ and show it is a real number.

(c) The 4th roots of z are w₀, w₁, w₂, w₃. Find all four roots and plot them on an Argand diagram. Describe which geometric shape they form.

Answer

(a) |z|=2, arg(z)=2π/3 → z = 2(cos(2π/3)+i·sin(2π/3)) (b) z⁶ = 64 (real) ✓ (c) wₖ = 2^(1/4)[cos((2π/3+2kπ)/4)+i·sin((2π/3+2kπ)/4)] for k=0,1,2,3. They form a square on the Argand diagram.

Q14

A bank offers compound interest at rate r% per annum. Alice invests $P at the start of each year for n years.

(a) Show that the total value after n years is V = P·[(1+r/100)^n − 1]/(r/100).

(b) Alice wants $50 000 after 10 years. If r = 5%, find the annual payment P.

(c) If instead Alice invests a lump sum of $20000 at 5%, find the number of years for her investment to exceed $50000.

Answer

Q15

The arithmetic sequence has sixth term u₆ = ln 2 and eighth term u₈ = (3/2) ln 4.

(ii) Find u₁.

(iii) Hence show that the sum of n terms is given by Sₙ = (n² − 9n) ln √2.

(ii) Sketch f(x) on the grid below.

$Number and algebra study illustration$

(ii) Solve f(x) < g(x).

(iii) Find the range of values of n that satisfy Sₙ < g(n).

(a) (i) Show that the common difference is ln 2.

(b) (i) Write down the equation of the axis of symmetry of f(x).

(c) (i) Find the minimum value of Sₙ, giving your answer in the form a ln b where a,b∈ℤ.

(d) Consider g(x) = (6 − 3x) ln √2.

(i) Solve f(x)=g(x).

Answer

(a)(i) d = ln 2; (ii) u₁ = -4 ln 2; (iii) Sₙ = (n²-9n) ln √2 ✓ (b)(i) Axis of symmetry: x = 9/2 = 4.5 (c)(i) Minimum value = -20 ln √2 = -10 ln 2 = ln(2⁻¹⁰) (d)(i) x = 0 or x = 12; (ii) 0 < x < 12; (iii) 1 ≤ n ≤ 11, n ∈ ℤ⁺

Q16

$Number and algebra2 study illustration$

$Number and algebra3 study illustration$

(a) The diagram shows [PQ], with length 4 cm. The line is divided into an infinite number of line segments. The lengths of the line segments are m cm, m² cm, m³ cm, …, where 0 < m < 1. Show that m = 4/5.

(b) The following diagram shows [RS], with length l cm, where l > 1. Squares with side lengths n cm, n² cm, n³ cm, …, where 0 < n < 1, are drawn along [RS]. The total sum of the areas of all the squares is 25/11. Find the value of l.

Answer

(a) m + m² + m³ + … = 4 ⇒ m/(1-m) = 4 ⇒ m = 4/5 ✓ (b) n²/(1-n²) = 25/11 ⇒ 11n² = 25 - 25n² ⇒ 36n² = 25 ⇒ n = 5/6; l = n/(1-n) = (5/6)/(1/6) = 5

Q17

Bill takes out a bank loan of $100 000 at an annual interest rate of 5.49%. The interest is calculated at the end of each year and added to the amount outstanding.

To pay off the loan, Bill makes quarterly deposits of $P at the end of every quarter in a savings account, paying a nominal annual interest rate of 3.2%. He makes his first deposit at the end of the first quarter after taking out the loan.

$Number and algebra4 study illustration$

(a) Find the amount of money Bill would owe after 10 years.

(b) Show that the total value after 10 years is P[(1.008⁴⁰ − 1)/(1.008 − 1)].

(c) Find the value of P to the nearest dollar.

(d) Melinda makes a single deposit of $Q at 3.5% annual interest. She wishes to withdraw $8000 at the end of each year for n years. Find the minimum value of Q that permits indefinite withdrawals.

Answer

(a) 100000 × (1.0549)¹⁰ ≈ $170,763 (b) Geometric series with first term P(1.008) and ratio 1.008, 40 terms → P[(1.008⁴⁰−1)/(0.008)] ✓ (c) P ≈ $3,552 (d) Q_min = 8000/0.035 = $228,571

Q18

(a) Use the binomial theorem to expand (2 + 3x)⁵, writing each coefficient as an integer.

(b) The coefficient of x³ in the expansion of (2 + kx)⁵ is 2160. Find the value of k.

(c) Consider the expansion of (1 + x)⁻³. Write down the first four terms and state the range of x for which the expansion is valid.

(d) Use your expansion to find an approximation of (0.97)⁻³ correct to 4 significant figures.

Answer

(a) 32 + 240x + 720x² + 1080x³ + 810x⁴ + 243x⁵ (b) C(5,3)·2²·k³ = 10·4·k³ = 2160 ⇒ k³ = 54 ⇒ k = 3·³√2 (c) 1 − 3x + 6x² − 10x³ + …, valid for |x| < 1 (d) x = −0.03 → 1 + 0.09 + 0.0054 + 0.00027 ≈ 1.0957 (Exact: 1.09274…)

Q19

Consider the system: x + 2y + z = 4; 2x − y + 3z = 9; x + 5y + kz = 1, where k ∈ ℝ.

(a) Find the value of k for which the system does not have a unique solution.

(b) When k = 2, use row reduction to find the unique solution.

(c) For the value of k found in part (a), determine whether the system has no solution or infinitely many solutions.

Answer

(a) k = −1 (b) x = 2, y = −1, z = 3 (c) Augmented matrix is inconsistent → no solution

Q20

Prove by mathematical induction that 4ⁿ − 1 is divisible by 3 for all n ∈ ℤ⁺.

Answer

(a) n=1: 4¹−1=3 ✓ (b) Assume 4ᵏ−1=3m for some integer m (c) 4^(k+1)−1 = 4(3m+1)−1 = 3(4m+1), divisible by 3 ✓

Q21

Prove by induction that Σᵣ₌₁ⁿ r·r! = (n+1)! − 1 for all n ∈ ℤ⁺.

Answer

(a) n=1: LHS=1·1!=1, RHS=2!−1=1 ✓ (b) Assume Σᵣ₌₁ᵏ r·r! = (k+1)! − 1 (c) Add (k+1)·(k+1)!: (k+1)![1+(k+1)]−1 = (k+2)!−1 ✓

Q22

Let z = 2 + 3i and w = 1 − i.

(a) Find z + w and z × w.

(b) Find |z| and arg(z), giving arg(z) in radians to 3 significant figures.

(c) Write z in modulus-argument (polar) form.

Answer

(a) z+w = 3 + 2i; z×w = 5 + i (b) |z| = √13; arg(z) ≈ 0.983 rad (c) z = √13 (cos 0.983 + i sin 0.983)

Q23

Let z = √3 + i.

(a) Write z in the form r(cosθ + i sinθ), where r > 0 and −π < θ ≤ π.

(b) Use De Moivre's theorem to find z⁶, giving your answer in Cartesian form.

(c) Find the three cube roots of z, giving your answers in polar form.

Answer

(a) r = 2, θ = π/6; z = 2(cos(π/6)+i sin(π/6)) (b) z⁶ = −64 (c) √2 (cos(π/18 + 2kπ/3) + i sin(π/18 + 2kπ/3)) for k = 0,1,2

Q24

(a) Show using De Moivre's theorem that cos(4θ) = 8cos⁴θ − 8cos²θ + 1.

(b) Hence solve cos(4θ) = 1 for 0 ≤ θ < 2π.

(c) Find the exact roots of 8x⁴ − 8x² + 1 = 0 in trigonometric form.

Answer

(a) Expand (cosθ + i sinθ)⁴, take real part; use sin²θ = 1 − cos²θ ✓ (b) θ = 0, π/2, π, 3π/2 (c) x = cos(π/8), cos(3π/8), cos(5π/8), cos(7π/8)

Q25

Consider the system: x + 2y + z = 4; 2x − y + 3z = 9; x + 5y + kz = 1, where k ∈ ℝ.

(a) Find the value of k for which the system has no unique solution.

(b) When k = 2, use row reduction to find the unique solution.

(c) When k takes the value from (a), determine whether the system has no solution or infinitely many solutions.

Answer

(a) k = −1 (b) x = 2, y = −1, z = 3 (c) Augmented matrix is inconsistent ⇒ no solution

Q26

(a) Write down the first four terms of the binomial expansion of (1 + x)^(−2) for |x| < 1.

(b) By substituting a suitable value of x, show that Σₙ₌₀^∞ (n+1)(1/2)ⁿ = 4.

Answer

(a) 1 − 2x + 3x² − 4x³ + … (b) Substitute x = 1/2: (3/2)^(−2) = 4/9; series = Σ(n+1)(1/2)ⁿ = 4 ✓

Q27

A committee of 4 is to be chosen from 6 men and 5 women.

(a) Find the number of ways the committee can be selected with no restrictions.

(b) Find the number of ways the committee can be selected if it must contain at least 2 women.

Answer

(a) C(11,4) = 330 (b) C(5,2)·C(6,2) + C(5,3)·C(6,1) + C(5,4)·C(6,0) = 150 + 60 + 5 = 215

Q28

Six students are to be seated in a row of 6 chairs.

(a) How many different arrangements are possible?

(b) Two particular students, Alex and Beth, must not sit next to each other. Find the number of valid arrangements.

(c) The 6 students now sit at a circular table. Find the number of distinct arrangements.

Answer

(a) 6! = 720 (b) 720 − 2 × 5! = 720 − 240 = 480 (c) (6−1)! = 120

Q29

Express (5x + 1)/[(x + 3)(x − 1)] in partial fractions.

(a) Write the expression in the form A/(x + 3) + B/(x − 1).

(b) Find the values of A and B.

(c) Hence find ∫ (5x + 1)/[(x + 3)(x − 1)] dx.

Answer

(a) A/(x+3) + B/(x−1) (b) A = −7/2, B = 3/2 (c) −(7/2) ln|x+3| + (3/2) ln|x−1| + C

Q30

(a) Express (3x² + 4x − 2)/[(x − 1)(x² + 2)] in partial fractions. Write it in the form A/(x − 1) + (Bx + C)/(x² + 2).

(b) Hence evaluate ∫₂³ (3x² + 4x − 2)/[(x − 1)(x² + 2)] dx, giving your answer to 3 significant figures.

Answer

(a) A = 5/3, B = 4/3, C = 1/3 (b) ≈ 2.01 (to 3 s.f.)

Number and Algebra