Algebra, Coordinate Geometry, Calculus & Statistics — Practice

Question 1

The sum of an infinite geometric progression is 64 and the first term is 48. Find the common ratio.

Accepted Answer

S∞ = a/(1−r). Given: S∞ = 64, a = 48. 64 = 48/(1−r). 1 − r = 48/64 = 3/4. r = 1 − 3/4 = 1/4. Verification: |r| = 1/4 < 1 ✓ (series converges). Check: S∞ = 48/(1 − 1/4) = 48/(3/4) = 48 × 4/3 = 64 ✓. The common ratio r = 1/4.

Question 2

Find the term independent of x (the constant term) in the expansion of (x − 1/x)¹².

Accepted Answer

General term: T(r+1) = ¹²Cᵣ (x)^(12−r) (−1/x)^r = ¹²Cᵣ (−1)^r x^(12−r) · x^(−r) = ¹²Cᵣ (−1)^r x^(12−2r). For the term independent of x: power of x = 0 → 12 − 2r = 0 → r = 6. T₇ = ¹²C₆ (−1)⁶ = ¹²C₆ × 1 = 12!/(6!6!) = 924. The constant term (term independent of x) = 924.

Question 3

Find the derivative of f(x) = x² + 3x using first principles (definition of derivative).

Accepted Answer

By first principles: f′(x) = lim(h→0) [f(x+h) − f(x)]/h. f(x+h) = (x+h)² + 3(x+h) = x² + 2xh + h² + 3x + 3h. f(x+h) − f(x) = (x² + 2xh + h² + 3x + 3h) − (x² + 3x) = 2xh + h² + 3h = h(2x + h + 3). [f(x+h) − f(x)]/h = (2x + h + 3). lim(h→0) (2x + h + 3) = 2x + 0 + 3 = 2x + 3. Therefore f′(x) = 2x + 3. (Verified by power rule: d/dx(x²) = 2x, d/dx(3x) = 3, so f′(x) = 2x + 3 ✓)

Practice — Algebra, Coordinate Geometry, Calculus & Statistics

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