ASSIGNMENT COVER SHEET Alpha

Name / ______

*your teacher will explain how to use this cover sheet

Question / Done / Ready for test / Topic / Comment
Drill A / indices
Drill B / factorising quadratics
Drill C / solving equations
Drill D / finding intercepts
Drill E / completing the square
Section A
1 / discriminant
2 / quadratic formula
3 / factorising quadratics
4 / solve completing the sq
5 / surds
6 / laws of indices
7 / solve the quadratic
8 / line & circle intersection
9 / rationalise denominator
10 / quadratics
11 / find the errors
Section B
12 / curve crosses the x axis
13 / sine rule
14 / equation of a circle
15 / cosine rule
16 / equation of a circle
17 / sine or cosine rule
18 / sine or cosine rule
19 / remainder theorem
20 / remainder theorem
21 / remainder theorem
22 / remainder theorem

“Mathematics seems to endow one with something like a new sense” Charles Darwin

Y1 Double Maths Assignment (alpha)

The next tracking test in w/b 10th Oct and you will be examined on everything covered including SWC

Drill

Drill are the very basic techniques you need to solve maths problems.

Section A: Convert these to the form :

(1) (2) (3)

Section B: Factorise and solve the following:

(1) (2) (3)

Section C: Solve each of the following equations for x:

(1) (2) (3)

Section D: Where do these graphs cross the coordinate axes? (see bullet point 1 for a hint)

(1) (2) (3)

Section E: Complete the square of these quadratics, showing all your working:

(1) (2) (3)

Things that might be useful

1.  On the y axis, x = 0. On the x axis, y = 0.

2.  The equation of a circle with centre (a, b) and radius r is

3.  In the discriminant, a represents the amount of x2 in the quadratic, b represents the amount of x in the quadratic and c represents everything else.

4.  If the discriminant of a quadratic is zero then the quadratic has one repeated root. If the discriminant is negative there are no real roots. If the discriminant is positive there are 2 distinct roots.

5.  You can get things into the form by ‘sliding’ for example or

6.  How to complete the square

7.  Quadratic formula

8.  Laws of indices:

The sine rule. If you know an opposite pair in a triangle (A and a, B and b or C and c) you can use the sine rule to find the missing part of another opposite pair (either angle or length)

c A b The sine rule

B C

a

The cosine rule. If you have a problem involving all 3 sides of a triangle and one angle you can use the cosine rule to find the missing value (either the angle or one of the sides)

c A b The cosine rule

B C
a

Section A – Core 1

1 (a) Evaluate the discriminant, , of the quadratic . Hence state which of the following sketches could show the graph of and explain why. Make up a possible equation for each of the other two quadratics (see bullet point 4 below for help).

Quadratic A Quadratic B Quadratic C

(b) Given that the discriminant, , of the quadraticis – 7, find the possible values of k. Make sure you are correctly identifying a, b and c! (See bullet points 3 and 4 below for help).

2 Leaving your answers in the form , solve these quadratic equations using the quadratic formula (see bullet points 7 and 5 below for help):

(a) (b)

3 Solve the following equations by factorising:

(a) (b)

4 Use the method of completing the square to solve the following quadratic equations, giving your answers in the form (see bullet point 6 below for help):

(a) (b)

5 (a) Given that and, calculate the exact values of and

(b) Simplify and

6 Write in the form by using the laws of indices (see bullet point 8 below for help). Show all your steps clearly.

7 Find the values of x for which . Hint: it will be much easier if you replacewith ‘t’, in which case ‘x’ would be written…

8 Carefully substitute y = 3x into the equation x2 + y2 = 120 to find the coordinates of the points where the line y = 3x crosses the circle x2 + y2 = 120.

9 Rationalise the denominator of and simplify:

10 Prove that is a perfect square if, and only if,

11 There are expensive errors in this student’s work. Find as many as you can and correct their work.

(a) / / (b) / / (c) /

Section B – Core 2

CORE 2

12. Find where the circle crosses the x axis

13 Calculate the length of x using the sine rule, giving your answer to 2 significant figures (the sine rule is written at the bottom of this assignment)

C

91° x cm

A 51° B

7.3 cm

14 A circle is given by the equation . Write down the coordinates of the centre of the circle and its radius (see bullet point 2 below for help).

15 Use the cosine rule to calculate the unknown length & the unknown angle, giving your answers to 3sf (the cosine rule is written at the bottom of this assignment)

4 cm x 4 cm 3.5 cm

47° q°

6 cm 5 cm

16 Form the equation of the circle with centre (– 1, 2) and diameter

17 From a lighthouse A at a given time, two ships P and Q lie on bearings 300° and 020° respectively. Given that P is 6 km from the lighthouse and Q is 3.5 km from the lighthouse, calculate the distance between the ships at the given time, giving your answer to 3s.f.

[hint: you will need to use either the sine or the cosine rule]

18

(a)  Find the length of BC to 3sf

(b)  Find the area of the triangle to 3sf

(c)  Find the angle at C to 3sf

Something new…

If a function f (x) is divided by ax + b then there will be a quotient and a remainder just like when a number is divided by another number.

When 13 is divided by 2 there is a quotient of 6 and a remainder of 1.

This is written

When is divided by there is a quotient of (x + 4) and a remainder of 2.

This is written .

Later in the course you will learn how to find the quotient, but for now we will tell you an easy way to find the remainder:

1) Find the value of x which makes the denominator (the thing you are dividing by) equal to 0.

2) Put this value of x into the numerator (the thing you are dividing, on the top)

In the example above, , we would make the denominator, , equal to 0 and solve to get x = 1, then put this value of x into the numerator, f(x) = , giving

f(1) = . The remainder is 2.

i.e. when f(x) is divided by (x – 1), the remainder is f(1)

i.e. when f(x) is divided by (2x + 3), the remainder is f(-3/2)

Now do the questions below

19 Find the remainder when is divided by

20 Find the remainder when is divided by

21 Find the remainder when is divided by

22 The equation has a repeated root, where . When is divided by () the remainder is 1. Find the values of p and q.

Answers Any wrong answers? Check with another student, then email with the correct one, also checked by another student. Thank you!

Answers to Drill

A: (1) 21 (2) 3 (3)

B: (1) (2) (3)

C: (1) (2) (3)

D: (1) (2)

(3)

E: (1) (2) (3)

Answers to Section A

(1a) check using your calculator (1b) k = –1 or k = –3 (2a) (2b)

(3a) 0, 4, 5 (3b) (4a) (4b)

(5a) 10, 72 (5b)51, 14 (6) n= (7)

(8) and (9) (10) discuss in class

(11) discuss in class

Answers to Section B

(12) (7,0) and (-5,0) (13) x = 5.7cm (14) (3, 2) , r=3

(15) x = 4.39 cm, q = 44.0°

(16) (17) 6.40 km (18a) 4.86cm

(18b) 6.02 (18c) 20.7o (19) 17 (20) –11

(21) 2 (22) q = 9 and p = 12 or q = 1 and p = 4

Updated: 13/09/2016