Solvelog2 (2x + 1) – log2 x = 2

Log tipsWhen solving, you can often either:a) Get in the form logab = c. Then rearrange as ac = bb) Get in the form logab = logac. Then b = c.

Actual exam questions:

log 2 (x + 1) – log 2 x = log 2 7

2 log3 x – log3 7x = 1 Solve simultaneous equations:a = 3b,log3 a + log3 b = 2

log5 (4 – x) – 2 log5 x = 1

log2y = –3

xx 2

2

22 loglog

16log32log

log2 (11 – 6x) = 2 log2 (x – 1) + 3

logx 64 = 2

Log tipsOther types of question:

3x = 10 52x – 12(5x) + 35 = 0

What if you had 52x+1 here instead?

The points A and B have coordinates (–2, 11) and (8, 1) respectively. Given that AB is a diameter of the circle C, (a) show that the centre of C has coordinates (3, 6),

(1)(b) find an equation for C.

(4)(c) Verify that the point (10, 7) lies on C.

(1)(d) Find an equation of the tangent to C at the point (10, 7), giving your answer in the form y = mx + c, where m and c are constants.

(4)

Circles

Remember that you need the centre (a,b) and the radius r, which gives the equation:

(x-a)2 + (y-b)2 = r2

The tangent is perpendicular to the radius at the point of contact.

CirclesHow could you tell if a line and a circle intersect:

0 times twice once

Equate the expressions then look at the discriminant:

b2 – 4ac < 0 b2 – 4ac > 0 b2 – 4ac = 0? ? ?

x2 + y2 = 1

y = 4-x

8. The circle C, with centre at the point A, has equation x2 + y2 – 10x + 9 = 0. Find (a) the coordinates of A,

(2)(b) the radius of C,

(2)(c) the coordinates of the points at which C crosses the x-axis.

(2) Given that the line l with gradient is a tangent to C, and that l touches C at the point T, (d) find an equation of the line which passes through A and T.

(3)

Circles

Circles

If PR is the diameter, how would you prove that a = 13?

Circles

Circles

(a) Use the factor theorem to show that (x + 4) is a factor of 2x3 + x2 – 25x + 12.

(2)(b) Factorise 2x3 + x2 – 25x + 12 completely.

(4)

Factor Theorem

f(x) = x3 + ax2 + bx + 3, where a and b are constants.Given that when f (x) is divided by (x + 2) the remainder is 7,(a) show that 2a − b = 6

(2)Given also that when f (x) is divided by (x −1) the remainder is 4,(b) find the value of a and the value of b.

(4)

(a) Find all the values of , to 1 decimal place, in the interval 0 < 360 for which5 sin ( + 30) = 3.

(4)(b) Find all the values of , to 1 decimal place, in the interval 0 < 360 for which tan2 = 4.

(5)

(a) Given that sin = 5 cos , find the value of tan .

(1)(b) Hence, or otherwise, find the values of in the interval 0 < 360 for which

sin = 5 cos , giving your answers to 1 decimal place. (3)

Trigonometric Solutions

Find all the solutions, in the interval 0 ≤ x < 2, of the equation 2 cos2 x + 1 = 5 sin x, giving each solution in terms of .

(a) Sketch, for 0 ≤ x ≤ 2, the graph of y = sin ( x + ( /6)).

(2) (b) Write down the exact coordinates of the points where the graph meets the coordinate axes.

(3)(c) Solve, for 0 ≤ x ≤ 2, the equationsin (x + ( /6)) = 0.65, giving your answers in radians to 2 decimal places.

Solve, for 0 x < 360°, (a) sin(x – 20) = 1/√2,

(4)(b) cos 3x = –1/2 .

(6)

Trigonometric Solutions(a) Show that the equation 3 sin2 – 2 cos2 = 1 can be written as 5 sin2 = 3.

(2)(b) Hence solve, for 0 < 360, the equation 3 sin2 – 2 cos2 = 1, giving your answer to 1 decimal place.

(7)

(a) Show that the equation4 sin2 x + 9 cos x – 6 = 0can be written as4 cos2 x – 9 cos x + 2 = 0.

(2)(b) Hence solve, for 0 x < 720°,4 sin2 x + 9 cos x – 6 = 0, giving your answers to 1 decimal place.

(6)

Show that the equationtan 2x = 5 sin 2x

can be written in the form(1 – 5 cos 2x) sin 2x = 0

(2)(b) Hence solve, for 0 ≤ x ≤180°,

tan 2x = 5 sin 2xgiving your answers to 1 decimal place where appropriate.You must show clearly how you obtained your answers.

(5)

Trigonometric Solutions

(i) Find the solutions of the equation sin(3x - 15 ) = ½ for which 0 ≤ x ≤ 180

Trigonometric Solutions

Summary of tips:• To get all your solutions when you do you inverse sin/cos/tan:

• Remember that sin(180-x) = sin(x) and cos(360-x) = cos(x)• sin and cos repeat every 360 (i.e. you can add 360 to your

solution as many times as you like).• But tan repeats every 180.• If you’re working in radians, then sin(pi – x) = sin(x), etc.

• Look at the range the question gives: if it’s in radians, give your answers in radians.

• Ensure your calculator is correctly set to either radians or degrees mode.

• If you have sin2, then make sure you get both positive and negative solution. Likewise for cos2 and tan2.

• Make sure you write out enough solutions before you start manipulating: if you had sin(3x) = ½ for example and had the range 0 < x < 360, then you’d initially need to write values up to 1080 since you’re going to be dividing by 3.

Areas of sector/Arc lengths/Sine and Cosine Rule

Areas of sector/Arc lengths/Sine and Cosine Rule

AreasOnly 1 in 36 candidates (across the country) got this question fully correct.

Optimisation

Optimisation

Optimisation

Optimisation

Integration

Figure 1 shows part of a curve C with equation , x > 0. The points P and Q lie on C and have x-coordinates 1 and 4 respectively. The region R, shaded in Figure 1, is bounded by C and the straight line joining P and Q.

(a) Find the exact area of R.(8)

(b) Use calculus to show that y is increasing for x > 2.(4)

Integration

Integration

Integration

Examiner’s Report:(a) A pleasing majority of the candidates were able to differentiate these fractional powers correctly, but a sizeable group left the constant term on the end. They then put the derivative equal to zero. Solving the equation which resulted caused more problems as the equation contained various fractional powers. Some tried squaring to clear away the fractional powers, but often did not deal well with the square roots afterwards. There were many who expressed 6x-1/2 = 1/(6x1/2) and tended to get in a muddle after that. Those who took out a factor x1/2 usually ended with x = 0 as well as x = 4 and if it was not discounted, they lost an accuracy mark. Those who obtained the solution x = 4 sometimes neglected to complete their solution by finding the corresponding y value. Some weaker candidates did not differentiate at all in part (a), with some integrating, and others substituting various values into y.

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Geometric Series

Geometric Series

Geometric Series

Binomial Expansion