(a) Write as a single fraction in its simplest form
\(\frac{x+3}{x-3}-\frac{x-2}{x+2}\)
(b) \(2^{12}\div 2^{\frac{k}{2}}=32\)
Find the value of \(k\).
(c) Expand and simplify.
\((y+3)(y-4)(2y-1)\)
(d) Make \(x\) the subject of the formula.
\(x=\frac{3+x}{y}\)
▶️ Answer/Explanation
(a) Ans: \(\frac{10x}{(x-3)(x+2)}\) or \(\frac{10x}{x^2 – x – 6}\)
Combine fractions: \(\frac{(x+3)(x+2)-(x-2)(x-3)}{(x-3)(x+2)}\).
Expand numerators: \(x^2 + 5x + 6 – (x^2 – 5x + 6) = 10x\).
Denominator remains \((x-3)(x+2)\).
(b) Ans: 14
Simplify equation: \(2^{12 – \frac{k}{2}} = 2^5\).
Set exponents equal: \(12 – \frac{k}{2} = 5 \Rightarrow k = 14\).
(c) Ans: \(2y^3 – 3y^2 – 23y + 12\)
First multiply \((y+3)(y-4) = y^2 – y – 12\).
Then multiply by \((2y-1)\): \(2y^3 – y^2 – 24y – y^2 + y + 12\).
Combine like terms for final answer.
(d) Ans: \(x = \frac{3}{y-1}\)
Multiply both sides by \(y\): \(xy = 3 + x\).
Rearrange: \(xy – x = 3 \Rightarrow x(y-1) = 3\).
Divide by \((y-1)\) to isolate \(x\).
(a) \(s = ut + \frac{1}{2}at^{2}\)
Find the value of \(s\) when \(u = 5.2\), \(t = 7\), and \(a = 1.6\).
(b) Simplify.
(i) \(3a – 5b – a + 2b\)
(ii) \(\frac{5}{3x} \times \frac{9x}{20}\)
(c) Solve
(i) \(\frac{15}{x} = -3\)
(ii) \(4(5 – 3x) = 23\)
(d) Simplify.
\((27x^{9})^{\frac{2}{3}}\)
(e) Expand and simplify.
\((3x – 5y)(2x + y)\)
▶️ Answer/Explanation
(a) Ans: 75.6
Substitute \(u = 5.2\), \(t = 7\), \(a = 1.6\) into \(s = ut + \frac{1}{2}at^2\).
Calculate \(s = (5.2)(7) + \frac{1}{2}(1.6)(49) = 36.4 + 39.2 = 75.6\).
(b)(i) Ans: \(2a – 3b\)
Combine like terms: \(3a – a – 5b + 2b = 2a – 3b\).
(b)(ii) Ans: \(\frac{3}{4}\)
Multiply numerators and denominators: \(\frac{45x}{60x} = \frac{3}{4}\).
(c)(i) Ans: \(-5\)
Multiply both sides by \(x\): \(15 = -3x\), then divide by \(-3\) to get \(x = -5\).
(c)(ii) Ans: \(-\frac{1}{4}\)
Expand: \(20 – 12x = 23\), solve \(-12x = 3\), so \(x = -\frac{1}{4}\).
(d) Ans: \(9x^6\)
Simplify using exponent rules: \(27^{\frac{2}{3}} = 9\) and \(x^{9 \times \frac{2}{3}} = x^6\).
(e) Ans: \(6x^2 – 7xy – 5y^2\)
Expand using distributive property: \(6x^2 + 3xy – 10xy – 5y^2 = 6x^2 – 7xy – 5y^2\).