Factorise fully:
(a) $3p(a-8b)-7q(8b-a)$

(a) Express $\frac{x-1}{3}-\frac{2x-1}{2}$ as a single fraction.
Write your answer as simply as possible.

(b) $\frac{1}{x-1}-\frac{3}{x}+1$ can be written as $\frac{x^2+bx+c}{x(x-1)}$.
Find the value of $b$ and the value of $c$.

Simplify $\frac{12x^2+ax-2a^2}{3x-2a}$.

The letter $x$ represents a digit in the number " $x.6$ ".
In the expression $\frac{2.45+1(x.6)}{1.71}$, when each digit is estimated to one significant figure, the answer becomes 6.
Find the digit $x$.

Leaving your answer in terms of $x$, state the reading shown on the scale.

(a) Expand $(a-5)(a+3)$.

(a)(i) Factorise $a^2-b^2$.

(ii) Use your answer to part (a)(i) to evaluate ${97}^2-9$.

(a) Expand and collect like terms:
(i) $4(k-1)-(3k+2)+14$

(ii) $(7x-2y)(3x+y)$

(iii) ${\left(\frac{5}{4^{-1}-9^{-1}}\right)}^{\frac{1}{2}}$

(a) (i) Simplify
$\frac{25x^2-49}{5x^2-12x+7}$

(ii) Hence or otherwise express as a single fraction in its simplest form.
$\frac{25x^2-49}{5x^2-12x+7}\times (\frac{2}{x-3}-\frac{3}{5x+7})$

(b) Find the value of x for which (152x) has no inverse.

(c) The two inequalities $2x+3y\le 4$ and $2x+3y\le 1$ are constraints to a linear program.
Which of the constraints is redundant? Explain your answer.

(a) A whole number formed by using the digit in the hundreds place and digit in the tens place is 100h + 10t.
(i) Reversing the digits gives a new number that is 540 less in size.
Show that h − t = 6.

(ii) The two numbers in the digit h and t have a sum of 440.
Show that h + t = 4.

(iii) Hence or otherwise, find the value of h and value of t.

(b) The diagram shows a shape of area 70 cm² formed by cutting off a rectangle of length x cm and width 3 cm from a bigger rectangle of length (3x + 2) cm and width x cm. (i) Show that $3x^2-x-70=0$.

(ii) Find the length of the rectangle.

The diagram shows a rectangle of length (x + 9) cm and width of x cm. (i) Write an expression in terms of x for the perimeter of the rectangle.
Give your answer in its simplest form.

(ii) The length and the width of the rectangle are in the ratio 3:1.
Calculate the width of the rectangle.

The diagram shows triangle PQR.
The size of angle QPR = 30°, PQ = (x + 1) m and PR = (4 − x)² m. (a) Show that the area of the triangle QPR is given by $\frac{1}{4}(x^3-7x^2+8x+16)$.

(b) It is given that x = 10.
Find:
(i) The area of PQR,

(ii) The length of QR.

(c) A vertical pole is placed at Q.
The height of the pole is 8 m.
A bird is at the top of the pole and sees a prey along PR.
Find PT, the maximum angle of depression of the prey from the bird.

On the 1st January 2000, Tau was x years old, Pitso was 5 years older than Tau while Neo was twice as old as Tau.
(a) Write expressions, in terms of x, for ages of Pitso and Neo on the 1st January 2000.

(b) The product of Neo's age and Tau's age on the 1st January 2002 is the same as the square of Pitso's age on the 1st January 2000.
Write down an equation in x and show that it simplifies to $x^2-4x-21=0$.

(c) (i) Solve $x^2-4x-21=0$.

(ii) How old is Pitso on the 1st January 2002?

(d) Neo's height, h metres, is one of the solutions of $h^2+8h-17=0$.
(i) Solve $h^2+8h-17=0$.
Show all your working and give your answers correct to 2 decimal places.

(ii) Write down Neo's height in centimetres.

(a) y is directly proportional to $\sqrt{x-3}$.
When y = 10, x = 28.
Find y when x = 147.

(b) Solve $2x^2+7x-1=0$.
Give your answer correct to 2 decimal places.

(c) Simplify.
$\frac{x^2+5x+6}{x^2-9}$

Thabiso runs 34 km at an average speed of x km/h.
(a) Write down, in terms of x, an expression for the time taken.

(b) Lipuo runs 34 km at an average speed of 2 km/h greater than Thabiso's speed.
Write down, in terms of x, an expression for the time taken by Lipuo.

(c) It is given that Lipuo took 15 minutes less than Thabiso to complete the 34 km.
Write down an equation in terms of x and show that it simplifies to $x^2+2x-272=0$.

(d) Solve the equation $x^2+2x-272=0$.
Give your answer correct to one decimal place.

(e) Calculate, in hours and minutes, the time taken by Thabiso to complete the 34 km.

Anna types x words in one minute.
(a) Show that Anna takes $\frac{12000}{x}$ seconds to type 200 words.

(b) Puleng types 10 fewer words than Anna in one minute.
Write an expression, in terms of x for:
(i) The number of words Puleng types in one minute,

(ii) The number of seconds Puleng takes to type 200 words.

(c) It takes Puleng 100 seconds longer to type 200 words than it takes Anna.
(i) Write an equation in x to represent this information and show that it simplifies to $x^2-10x-1200=0$.

(ii) Solve the equation $x^2-10x-1200=0$.

(iii) Find the time taken, in minutes, for Puleng to type 3510 words.

Thabo bought x eggs at y maloti per dozen.
He sold each egg for P maloti.
Write an expression, in terms of x, y and P, for the profit he made.

The diagram shows an arrangement of three circles which are touching. The radii of the circles, centre A and centre B, are 3 cm and 5 cm respectively.
The radii of the circle centre C is x cm.
(a) Given that angle ACB = 90°, write down an equation in x and show that it reduces to $x^2+8x-15=0$.

(b) Solve the equation $x^2+8x-15=0$.
Show all your working and give each answer correct to 2 decimal places.

(c) Write down the lengths of AC and BC.