Pure Mathematics — First: Algebra · Mathematics. These are 10 questions from the Magmo3 bank on this lesson, from easy to hard. Solve each one, then open "Show answer and explanation". Magmo3 has 37 questions on this lesson, each with an explanation.
In the expansion of (x + y)⁹ according to the descending powers of x, the ratio T₆ : T₅ equals:
Correct answer: (A) y/x
Explanation: Step 1: Tᵣ₊₁/Tᵣ = (n − r + 1)/r × y/x with n = 9 and r = 5. Step 2: T₆/T₅ = (9 − 5 + 1)/5 × y/x = 5/5 × y/x. Step 3: T₆ : T₅ = y/x.
In the expansion of (1 + 2x)¹⁰ according to the ascending powers of x, the greatest term when x = 1/3 is ..........
Correct answer: (B) T₅
Explanation: Step 1: Tᵣ₊₁/Tᵣ = (n − r + 1)/r × (2x/1) = (11 − r)/r × 2/3. Step 2: Tᵣ₊₁/Tᵣ > 1 ⇒ 22 − 2r > 3r ⇒ r < 4.4, so T₅ > T₄ > … > T₁. Step 3: For r ≥ 5 the ratio is < 1, so T₅ > T₆ > … > T₁₁. ∴ T₅ is the greatest term (T₆ is only the middle term; T₇ comes from inverting 2x).
In the expansion of (1 + x)¹⁰, what is the ratio of the coefficient of the 5th term to that of the 4th term?
Correct answer: (C) 7/4
Explanation: Step 1: In (1 + x)¹⁰ the coefficient of the (r+1)th term is ¹⁰Cᵣ. Step 2: Coefficient of T₅ is ¹⁰C₄ = 210; of T₄ is ¹⁰C₃ = 120. Step 3: Ratio = 210 / 120 = 7/4. ∴ The ratio of the 5th to the 4th coefficient is 7/4.
The value of the greatest term in the expansion of (x + 2)⁹ when x = 1 equals ..........
Correct answer: (D) 5376
Explanation: Step 1: Tᵣ₊₁/Tᵣ = (10 − r)/r × 2/1 = (20 − 2r)/r. Step 2: (20 − 2r)/r > 1 ⇒ 3r < 20 ⇒ r < 6.67, so T₇ > T₆ > … and T₇ > T₈ > … > T₁₀. Step 3: T₇ = ⁹C₆ × 1³ × 2⁶ = 84 × 64 = 5376. ∴ the greatest term = 5376 (4608 is T₈; 672 comes from inverting the ratio 2/x).
In the expansion of (2x − 3/x)⁹, find the ratio between the two middle terms T₅ : T₆.
Correct answer: (B) −2x²/3
Explanation: Step 1: n = 9, the middle terms are T₅ and T₆. Step 2: T₆/T₅ = (9 − 5 + 1)/5 × (−3/x)/(2x) = 1 × (−3/(2x²)) = −3/(2x²). Step 3: T₅ : T₆ = −2x²/3.
In the expansion of (2x + 3)⁸ in descending powers of x, the fifth and sixth terms are equal. Find x (x ≠ 0).
Correct answer: (C) 6/5
Explanation: Step 1: T₆/T₅ = ((8 − 5 + 1)/5)·(3/(2x)) = (4/5)(3/(2x)) = 6/(5x). Step 2: T₆ = T₅ ⇒ 6/(5x) = 1 ⇒ x = 6/5. ∴ x = 6/5
In the expansion of (1 + mx)ⁿ according to the ascending powers of x, the coefficient of T₄ is 16/3 times the coefficient of T₃ and the coefficient of T₆ is 12/5 times the coefficient of T₅. Find n and m.
Correct answer: (B) n = 10, m = 2
Explanation: Step 1: Coefficient T₄/T₃ = (n − 2)/3 × m = 16/3, so (n − 2)m = 16. Step 2: Coefficient T₆/T₅ = (n − 4)/5 × m = 12/5, so (n − 4)m = 12. Step 3: Subtracting: 2m = 4, so m = 2. Step 4: (n − 2) × 2 = 16, so n = 10.
In the expansion of (1 + x)¹² where x > 0, which value of x makes T₅ the greatest term?
Correct answer: (C) x = 1/2
Explanation: Step 1: Tᵣ₊₁/Tᵣ = (13 − r)/r × x. Step 2: T₅ > T₄ ⇒ T₅/T₄ = (9/4)x > 1 ⇒ x > 4/9. Step 3: T₅ > T₆ ⇒ T₆/T₅ = (8/5)x < 1 ⇒ x < 5/8. Step 4: Only 1/2 lies between 4/9 ≈ 0.44 and 5/8 = 0.625 (2/5 is too small, 2/3 and 3/4 make T₆ larger than T₅). ∴ x = 1/2
If T₂ : T₃ in the expansion of (a + b)ⁿ equals T₃ : T₄ in the expansion of (a + b)ⁿ⁺⁵, find n.
Correct answer: (C) 9
Explanation: Step 1: In (a + b)ⁿ: T₃/T₂ = (n − 1)/2 × b/a, so T₂/T₃ = 2a/((n − 1)b). Step 2: In (a + b)ⁿ⁺⁵: T₄/T₃ = (n + 5 − 2)/3 × b/a = (n + 3)/3 × b/a, so T₃/T₄ = 3a/((n + 3)b). Step 3: 2/(n − 1) = 3/(n + 3) gives 2n + 6 = 3n − 3. Step 4: n = 9.
In the expansion of (x + y)ⁿ in descending powers of x, T₂ = 240, T₃ = 720 and T₄ = 1080. Find n.
Correct answer: (C) 5
Explanation: Step 1: T₃/T₂ = ((n − 1)/2)(y/x) = 3 and T₄/T₃ = ((n − 2)/3)(y/x) = 3/2. Step 2: Dividing: 3(n − 1)/(2(n − 2)) = 2 ⇒ 3n − 3 = 4n − 8 ⇒ n = 5 (then y/x = 3/2, x = 2, y = 3 and indeed 5·2⁴·3 = 240). ∴ n = 5
The first 10 questions in every subject are free, each with an explanation.