# iGCSE Mathematics (0580) :E2.3 Manipulate algebraic fractions.iGCSE Style Questions Paper 4

### 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 = ………………………………………….

(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)
………………………………………….

(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.
r = …………………………………………

(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)$$\frac{-200+\sqrt{200^{2}-4(1)(323)}}{2\times 1}$$