JAMB Mathematics Past Questions & Answers - Page 99

\( 5^{(x − 2)} = (125^{−1} )^{(x + 3)}\\

5^{(x − 2)} = (5− 3)^{x + 3}\\

5^{(x − 2)} = 5^{− 3x − 9}\\

x + 3x = − 9 + 2\\

4x = − 7\\

x = \frac{−7}{4} \)

Let x represent the first number;

Then, the other is (5 − x) , since their sum is 5 and their product is 14

x(5 − x) = 14

5x − x2 = 14

x2 − 5x − 14 = 0

(x2 − 7x) + (2x − 14) = 0

x(x − 7) + 2(x − 7) = 0

(x − 7)(x + 2) = 0

Either (x − 7) = 0 or (x + 2) = 0

x = 7 or x = − 2

The two numbers are 7 and − 2

√(2x + 2 ) − √x = 1

√(2x + 2) = 1 + √x

Square both sides

√(2x + 2)² = (1 + √x)²

((2x + 2)²)^½ = 1 + √x + √x+ x

2x + 2 = 1 + 2√x +x

Collect the like term together

2x − x + 2 − 1 = 2 √x
x + 1 = 2√x

Square both sides

(x + 1)² = (2√x)²

x² + 2x + 1 = 4x

x² + 2x − 4x + 1 = 0

(x²− x)(x + 1)= 0

x(x − 1) −1(x − 1)

(x − 1)(x −1)

Either (x − 1) = 0 or (x − 1) = 0

x = 1 or 1

x = 1 twice

√30 × √40

√(30 × 40)

√(3 × 10 × 4 × 10)

√(400 × 3)

√400 × √3

20 × √3

20√3

\(27^x ÷ 81^{(x + 2y)} = 9 \\

(27)x = 9 × 81^{(x+2y)} \\

(3^3 )^x =32 \times 3^{4(x + 2y)} \\

=3^{(2 + 4x + 8y)}\\

3^{3x} = 3^{ (2 + 4x + 8y)}\\

3x = 2 + 4x + 8y\\

3x − 4x − 8y = 2 … … … (1)\\

x + 4y = 0 … … … (2)\\

− 4y = 2\\

y = (− 2) ÷ 4 = − ½\\

y = − ½\\ \)

Substitute the value of y into equation (2)

i.e x + 4y = 0

x + 4( − 1/2) = 0

x − 2 = 0

x = 2

∴ x = 2,y = − ½)

Method II

\( 27^x ÷ 31^{(x + 2y) }= 9\\

3^{3x} × 3^{( − 4x − 8y)} = 32\\

3^{(3x − 8y)} = 32\\

− x − 8y=2 ……… (1)\\

x + 4y = 0 ……… (2)\\

− 4 = 2\\

y= 2/4 = ½\\

y = ½ \)

Substitute the value of y into equation 2

x + 4y=0

x + 4 (− 1) ÷ 2) = 0

x − 2 = 0

x = 2

x = 2, y = ½