CA Foundation. Maths first.

Q. If x = (√3+1)/2 and y = (√3−1)/2 then x² − y² is

Apply the difference-of-squares factorisation and substitute the given expressions for x and y.

1. 01 Factorise x²−y²=(x+y)(x−y).
2. 02 Compute x+y Add the two given expressions for x and y — the irrational parts may cancel, leaving a rational sum.
3. 03 Compute x−y Subtract the expressions — the rational parts may cancel, leaving a surd.
4. 04 Multiply (x+y)(x−y)=4.

Q. Given log 2 = 0.3010 and log 3 = 0.4771 the value of log 6 is

Factor 6 = 2 × 3, then apply log(mn) = log m + log n.

1. 01 Factor the argument 6 = 2 × 3
2. 02 Apply product rule log 6 = log 2 + log 3
3. 03 Substitute = 0.3010 + 0.4771 = 0.7781

Q. If log₂x + log₄x + log₁₆x = 21/4, then x is equal to

Use the change-of-base identity log_(b^n)(x)=(log_b x)/n to unify all three logs to base 2.

1. 01 Convert to base 2 log₄x=(log₂x)/2; log₁₆x=(log₂x)/4
2. 02 Factor out log₂x log₂x(1+1/2+1/4)=log₂x×7/4
3. 03 Solve for log₂x log₂x×7/4=21/4 → log₂x=3
4. 04 Find x x=2³=8

Q. Given a proportion, the value of a certain expression obtained by applying Componendo and Dividendo twice is given by

Apply Componendo and Dividendo to the given proportion, then a second time, and add the results.

1. 01 First C&D application Given a/b=c/d, C&D gives (a+b)/(a−b)=(c+d)/(c−d). Record the first intermediate ratio.
2. 02 Second C&D application Apply C&D to the result of step 1 to obtain a second expression.
3. 03 Sum The sum of results from steps 1 and 2 = 2.

Q. If a = 3 + 2√2 then the value of √a − 1/√a is

Rewrite a=3+2√2 as (√2+1)², take the square root, rationalise, then subtract.

1. 01 Identify perfect-square form 3+2√2 = 2+1+2√2 = (√2)²+2·√2·1+1² = (√2+1)²
2. 02 Find √a √a=√2+1
3. 03 Find 1/√a 1/(√2+1) × (√2−1)/(√2−1) = (√2−1)/(2−1) = √2−1
4. 04 Subtract √a−1/√a = (√2+1)−(√2−1) = 2