TMUA Algebraic Techniques & Manipulation 5

Question 1

Rationalise the denominator of: (2 + √3)/(2 - √3)

Accepted Answer

7 + 4√3

Answer

Multiply by (2+√3)/(2+√3): numerator = (2+√3)² = 7+4√3; denominator = 4-3 = 1.

Question 2

Solve: 2^(2x) - 5·2^x + 4 = 0

Accepted Answer

x = 0 or x = 2

Answer

Let u = 2^x: u²-5u+4=0 gives (u-1)(u-4)=0, so 2^x=1→x=0 or 2^x=4→x=2.

Question 3

If x + y = 5 and xy = 4, find x³ + y³.

Accepted Answer

65

Answer

x³+y³ = (x+y)(x²-xy+y²) = (x+y)[(x+y)²-3xy] = 5[25-12] = 65.

Question 4

Simplify: log₂(32) - log₂(4) + log₂(2)

Accepted Answer

4

Answer

log₂(32)=5, log₂(4)=2, log₂(2)=1; so 5-2+1=4.

Question 5

Find the value of the expression: (n+1)! / [(n-1)! · n]

Accepted Answer

n+1

Answer

(n+1)! = (n+1)·n·(n-1)!, so the expression becomes (n+1)·n·(n-1)! / [(n-1)!·n] = n+1.

(TMUA) Test of Mathematics for University Admission Practice Test