Question
(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.
k = ………………………………………….
(c) Expand and simplify.
(y+3)(y-4)(2y-1)
………………………………………….
(d) Make x the subject of the formula.
\(x=\frac{3+x}{y}\)
x = ………………………………………….
Answer/Explanation
(a)\(\frac{10x}{(x-3)(x+2)}\) or\( \frac{10x}{x^{2}-x-6}\) final answer
(b) 14
(c) \(2y^{3}-3y^{2}-23y+12 \)final answer
(d)\(x=\frac{3}{y-1}\) final answer
Question
(a)\(s=ut+\frac{1}{2}at^{2}\)
Find the value of s when u = 5.2 , t = 7 and a = 1.6
s = …………………………………………
(b) Simplify.
(i) 3a-5b-a+2b
………………………………………….
(ii)\(\frac{5}{3x}\times \frac{9x}{20}\)
………………………………………….
(c) Solve
(i)\( \frac{15}{x}=-3\)
x = …………………………………………
(ii) 4(5-3x)=23
x = …………………………………………
(d) Simplify.
\((27x^{9})^{\frac{2}{3}}\)
………………………………………….
(e) Expand and simplify.
(3x – 5y)(2x + y)
………………………………………….
Answer/Explanation
(a) The formula is:
\(s = ut + (\frac{1}{2})at^2\)
Substituting u = 5.2, t = 7, and a = 1.6, we get:
\(s = (5.2)(7) + (\frac{1}{2})(1.6)(7)^2\)
Simplifying this expression, we get:
s = 36.4 + 39.2
Therefore, the value of s is:
s = 75.6
(b)(i)3a-a- 5b+2b=(3a-a)+(-5b+2b)=2a-3b
Therefore, the simplified expression is:
3a-5b-a+2b=2a-3b
(b)(ii)\(\frac{5}{3x}\times \frac{9x}{20}=\frac{\left ( 5\times 9x \right )}{\left ( 3x\times 20 \right )}=\frac{45x}{60x}\)
Simplifying further by canceling out the common factor of x in the numerator and denominator,
\(\frac{45x}{60x}=\frac{3}{4}\)
(c)(i)\( \frac{15}{x}=-3\)
Multiplying both sides by x, we get
15 = -3x
Dividing both sides by -3, we get
x = -5
(ii) 4(5 – 3x) = 23
20 – 12x = 23
-12x=3
\(x=-\frac{1}{4} \)or -0.25
(d) To simplify the expression \(\left ( 27x^{9} \right )^{\frac{2}{3}}\),we can use the property of exponents that says \((a^b)^c = a^(b\times c)\), and simplify the base and exponent separately
\((27x^{9})^{\frac{2}{3}} = 27^{\frac{2}{3}}\times(x^{9})^{\frac{2}{3}}=(3^{3})^{\frac{2}{3}}\times x^{3\times 2}=9x^{6}\)
(e)To expand the given expression, we can use the distributive property of multiplication,
(3x – 5y)(2x + y) = 3x(2x + y) – 5y(2x + y)
Now, we can use the distributive property again to simplify each of these products,
=(3x.2x) + (3x.y)-(5y.2x)-(5y.y)
\(= 6x^2+3xy-10xy-5y^2\)
\(= 6x^2 -7xy-5y^2\)
Therefore, the expanded and simplified expression is \(6x^{2} – 7xy – 5y^{2}\)
Question
(a) Factorise completely.
\(3a^{2}b-ab^{2}\)
…………………………………………
(b) Solve the inequality.
3x+12< 5x-3
………………………………………….
(c) Simplify.
\((3x^{2}y^{4})^{3}\)
………………………………………….
(d) Solve.
\(\frac{2}{x}=\frac{6}{2-x}\)
x = …………………………………………
(e) Expand and simplify.
(x-2)(x+5)(2x-1)
………………………………………….
(f) Alan invests \($200\) at a rate of r% per year compound interest.
After 2 years the value of his investment is\( $206.46 .\)
(i) Show that \(r^{2}+200r-3232=0.\)
(ii) Solve the equation\( r^{2}+200r-323=0 \)to find the rate of interest.
Show all your working and give your answer correct to 2 decimal places.
r = …………………………………………
Answer/Explanation
(a) ab(3a – b) final answer
(b) x > 7.5 final answer
(c) \(27x^{6}y^{12}\)
(d)0.5 or \(\frac{1}{2}\)
(e) \(2x^{3}+ 5x^{2}– 23x + 10\) final answer
(f)(i) \(200\left ( 1+\frac{r}{100} \right )^{2}=206.46\)
\(1+\frac{2r}{100}+\frac{r^{2}}{100^{2}}\)
\(r^{2}+200r-323=0\)
(ii)\(\frac{-200+\sqrt{200^{2}-4(1)(323)}}{2\times 1}\)
1.60 final answer