P.R.O. Academy Inspire Logical Thinking Thinking
EDEXCEL London Examinations International Advanced Level (IAL)
Mathematics WMA01 Core Math C12
Class Handouts
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Declaration: Most of the materials in these class handouts are reproduced from the course textbook. Any mistakes appeared are not the responsibility of the authors of the textbook. Students are encouraged to read further from the course text book and reference book to gain more information.
Chapter 1: Algebra and Functions (Core 1)
1. Simplify a) b)
2. Factorize a) b)
3. Evaluate a) b)
4. Simplify a) b) 5. 6.
√ √ √ Express √ in the form √ , where a is an integer. Express ( √ ) in the form √ , where b and c are integers.
7. Rationalize the denominator for 8. Given that 9. Given that
√ √ . √ √
write down the value of t . , , find the value of x.
Past year
1. Solve a) Write down the value of b) Find the value of
2. Solve a) Expand and simplify b)
Jan05 Q1 (3marks)
√ √ .
Express √ in the form √ , where a and b are integers.
Jun06 Q6 (4marks)
Algebra and Function (Core 2 chapter 1)
1. 2.
Simplify . Divide by
3. Show that is a factor of in the form a) by long division method b) expand & compare coefficient.
. Hence express , where A, B and C are to be found.
completely. When is divided by the remainder is 2. Find the value
4. Factorize 5.
of a. 6.
. When f(x) is divided by (x+1) the remainder is 6. When
f(x) is divided by (x-1) the remainder is 2. Find the value of p and the value of q.
Chapter 2 Quadratic Functions (Core 1)
1. Solve the following equation by factorization a) b) c)
2. Completing the square for a) b) 3.
Solve , leaving your answers in surd form.
4. Solve a) By completing the square, find in term of k , the roots of the equation . b) Show that, for all values of k , the roots of are real and different.
The equation , where p is a constant, has real roots. a) Show that b) Write down the values of p for which the equation has
5. Sketch the graph of 6.
equal roots.
Past year
1. The equation , where p is a positive constant, has equal roots. a) Find the value of p. b) For this value of p, solve the equation Jun06 Q8 (6marks)
2. Given that a) Express f (x) in the form The curve C with equation y = f(x) , minimum point at Q.
, where a and b are integers. , meets the y axis at P and has a
b) Sketch the graph of C , showing the coordinates of P and Q. The line y = 41 meets C at the point R.
√
c) Find the x-coordinate of R, giving your answers in the form , where p and q are integers. Jan05 Q10 (12marks) Chapter 3: Equation and Inequalities (Core 1)
1. Solve the simultaneous equations.
a) by elimination method b) by substitution method
2. Solve the simultaneous equations.
3. 4. 5. 6.
Find the set of values of x for which Find the set of the values of x for which Find the set of values of k for which has real roots. Find the range of values of x for which a) b) c) both and
Past year 1. By eliminating y from the equations
Show that . Hence, or otherwise, solve the simultaneous equations
Giving your answer in the form √ , where a and b are integers.
Jun07 Q6 (7marks)
2. Find the set of values of x for which a) b) c) both and
June 05 Q06 (8marks)
Chapter 4: Sketching Curves (Core 1) 1. Sketch the curve with equation 2. Sketch the following curves and indicate the points where the curves cross the coordinate axes. a) b) 3. Sketch the following curves and indicate coordinate of points where the y cross the axes. In (b) state the equations of any asymptotes. a)
b)
4. Solve a) On the same axes, sketch the curves with equations
.
and
Make and label the points of intersection b) Use algebra to find the coordinates of the points of intersection. c) State the solutions to the equation . 5.
The figure shows a sketch of the curve with equati on . The point is a maximum point and the line is a horizontal asymptote to the
curve. On separate diagrams, sketch the following graphs and in each case indicate the coordinates of the maximum point and the equation of the horizontal asymptote. a) b) c) d)
6. Solve a) Factorize completely b) Sketch the curve with equation , showing the coordinates of the points where the curve crosses the x-axis. c) On the separate diagram, sketch the curve with equation showing the coordinates of the points where the c urve crosses the x-axis.
Past year 1. Figure shows a sketch of the curve with equation . The curve crosses the x-axis at the point (2,0) and (4,0). The minimum point on the curve is P (3, -2).
In separate diagrams sketch the curve with equation a) b)
On each diagram, give the coordinates of the points at which the curve crosses the x-axis, and the coordinates of the image of P under the given transformation. Jan 05 Q6 (6marks)
2. Given that , a) Express in the form , where a, b and c are constants. b) Hence factorize completely. c) Sketch the graph of , showing the coordinates of each point at which the graph meets the axes. Jun 06 Q9 (8marks)
Chapter 5: Coordinate geometry in the ( x,y ) plane (Core 1)
1. The line with gradient passes through the point (2,-5) and meets the x-axis at A. a) Find an equation of the line b) Work out the coordinate of A. 2. Show the lines
are perpendicular.
3. Find an equation of the line that passes through the point ( -4, -5) and is perpendicular to the line
. Write your answer in the form
, where a, b and c are integers.
4. The line p passes through the points (1,-3) and (4,6), and the line q has equation
. The lines p and q intersect at T .
a) Find an equation for p. b) Calculate the coordinate of T .
5. The points P(3,-2), Q(1,2) and R(9,r) are the vertices of a triangle, where . a) Find the gradient of the line PQ b) Calculate the value of r .
Past Year 1. The point A (1,7), B (20,7) and C(p,q) form the vertices of a triangle ABC , as shown in figure below. The point D (8,2) is the mid point of AC .
a) Find the value of p and the value of q. The line l , which passes through D and is perpendicular to AC , intersects AB at E . b) Find an equation for l , in the form , where a, b and c are integers. c) Find the exact x-coordinate of E . Jan05 Q8 (9marks)
2. The line l 1 passes through the points P (-1,2) and Q (11,8). a) Find an equation for l 1 in the form of y = mx + c, where m and c are constants The line l 2 passes through the point R (10, 0) and is perpendicular to l 1. The line l 1 and l 2 intersect at point S . b) Calculate the coordinates of S . c) Show that the length of RS is . d) Hence, or otherwise, find the exact area of triangle PQR. June 06 Q11 (15marks)
√
Coordinate geometry in the ( x,y ) plane (Core 2 chapter 4) 1. Work out the coordinates of the centre and the redius of the circle
2.
Work out the coordinates of the points where the circle meets the y-axis.
3. The point P (-1, 6) lies on the circle centre (3,4). a. Work out the radius of the circle. b. Hence, write down an equation of the circle c. Find and equation of the tangent to the circle at P . 4. Work out the coordinates of the points where the line y = x +10 meets the circle . 5. A circle has centre C (-2, 0) and radius 5. A tangent is drawn from the point P (7,4) to touch the circle at T . find the length of PT .
Chapter 6: Sequences and Series (Core 1)
. Find
1. The nth term in a sequence is given by a. the first three terms of the sequence b. the 50th term of the sequence c. the value of n for which U n = 87
* +
2. A sequence of term is defined by the recurrence relation where p is a constant. Given also that U 1 = 1 and U 2 = 6. a. Find an expression in terms of p for U 3 b. Find an expression in terms of p for U 4 Given that U 4 = 29 c. Find possible values of p.
3. For the following arithmetic series. 4 + 7 + 10 + 13 + 16 + …. Find a. the 30th term b. the sum to 30 terms. 4. Given that the 2rd term of an arithmetic series is 8 and the 6 th term is -12, a. find the first term b. find the sum to 10 terms.
5. Each year, Peter pays into a savings scheme, in t he first year he pays in £500. His payments then increase by £200 each year, so that he pays £700 in the second year, £900 in the third year and so on. How many years will it be before he has saved £32000? 6. Show that
∑
Past Year
1. The rth term of arithmetic series is (2r, 5). a. Write down the first three terms of this series. b. State the value of the common difference. c. Show that
∑
Jan05 Q5 (6marks)
2. A sequence a1 , a2 , a3,…. is defined by a. b.
Find the value of a and the value of a . Calculate the value of ∑ 2
3
June06 Q4 (5marks) 3. An athlete prepares for a race by completing a practice run on each 11 consecutive days. On each day after the first day, he runs further than he ran on the previous day. The lengths of his 11 practice runs form an arit hmetic sequence with first term a km and common difference d km He runs 9 km on the 11 th day, and he runs a total of 77 km over the 11days period. Find the value of a and the value of d . June 06 Q7 (7marks) 4. A sequence a1 , a2 , a3,…. is defined by
Where k is a positive integer. a. Write down an expression for a2 in terms of k . b. Show that a3 = 9k + 20 c. Find in terms of k . Hence, show that
∑
∑ is divisible by 10. June 07 Q8 (7marks)
