Algebra — Practice

Question 1

If α and β are the roots of x² − 7x + 12 = 0, find α² + β².

Accepted Answer

A

Question 2

Find the domain of f(x) = 1/√(x−3).

Accepted Answer

B

Question 3

The 3rd term of an AP is 10 and the common difference is 4. Find the sum of the first 10 terms.

Accepted Answer

A

Question 4

How many digits does 3^40 have, given log₁₀3 ≈ 0.4771?

Accepted Answer

C

Question 5

Solve x² − x − 6 > 0.

Accepted Answer

B

Question 6

Find the sum to infinity of the GP with first term 12 and common ratio 1/3.

Accepted Answer

C

Question 7

For what value of k does x² − 6x + k = 0 have real and equal roots?

Accepted Answer

B

Question 8

The equations x² − 4x + 3 = 0 and x² + px − 3 = 0 have a common root. Find all possible values of p.

Accepted Answer

Roots of the first equation: x²−4x+3=(x−1)(x−3)=0, giving roots 1 and 3.

If the common root is 1: 1 + p − 3 = 0 ⟹ p = 2.
If the common root is 3: 9 + 3p − 3 = 0 ⟹ p = −2.

Both are valid: p = 2 (shared root 1) or p = −2 (shared root 3).

Question 9

If f(x) = (x−1)/(x+1), find f(f(f(x))) in terms of x.

Accepted Answer

From the chapter's key result, f(f(x)) = −1/x.

So f(f(f(x))) = f(−1/x) = (−1/x − 1)/(−1/x + 1) = [(−1−x)/x] / [(−1+x)/x] = (−1−x)/(x−1) = (1+x)/(1−x).

So f(f(f(x))) = (1+x)/(1−x), which is not simply x or −1/x — the self-inverse property needs exactly two applications together (an even number), not three.

Question 10

Find the minimum value of 2x + 8/x for x > 0.

Accepted Answer

B

Question 11

Solve |3x − 2| = x + 6.

Accepted Answer

Case 1: 3x−2 = x+6 ⟹ 2x = 8 ⟹ x = 4. Check: x+6=10 ≥ 0 and |3(4)−2|=10. Valid.

Case 2: −(3x−2) = x+6 ⟹ −3x+2 = x+6 ⟹ −4x = 4 ⟹ x = −1. Check: x+6=5 ≥ 0 and |3(−1)−2| = |−5| = 5. Valid.

Both x = 4 and x = −1 satisfy the original equation.

Question 12

If a, b, c are in GP, which relationship must hold?

Accepted Answer

A

Practice — Algebra

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