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Paper 6Mathematical Formulae
1. ALGEBRA
Quadratic EquationFor the equation ,
Binomial expansion
,
where n is a positive integer and
2. TRIGONOMETRY
Identities
cosec 2 A = 1 + cot 2 A
Formulae for
1 A cup of hot tea was left to cool on the table so that, t minutes later, its temperature, degree
Celsius, is given by = 26 + 64 e .
(a) Find
(i) the temperature of the tea 4 minutes later, [1]
(ii) the time taken for the tea to reach half its original temperature. [4]
(b) State the value which approaches as t becomes very large. [1]
2 (a) The curve y = 3 sin 2x + 1 is defined for 0 ≤ x ≤ 2.
(i) State the amplitude and period of y. [2]
(ii) Sketch the curve of y. [2]
(iii) By adding a suitable straight line on the same graph, find the number of solutions of the equation 6 sin 2x + x = 4 for 0 ≤ x ≤ 2. [2]
(b) Sketch the graph of y = for 0o ≤ x ≤ 360o on a separate diagram. [2]
3 The equation of a curve is y = 2 ex4 + 1.
(i) Find an expression for . [3]
(ii) Find the coordinates of the stationary point and determine the nature of the stationary point. [4]
4 (i) Express in partial fractions. [5]
(ii) Hence evaluate . [3]
5 The equation 3x2 – 5x + 1 = 0 has roots and .
(a) State the value of + and of . [2]
(b) Hence find
(i) the value of 2 + 2, [2]
(ii) the quadratic equation in x whose roots are + 1 and + 1. [4]
6 (a) Express in the form a + b , where a and b are integers. [4]
(b) Given that = a 2n3, find the value of a. [4]
7
In the circle, B, E, C, F and D are points on the circle such that CD is a diameter. The straight lines ABC and FE intersect at the point H. ADG is a tangent to the circle at D.
(a) Show that CD2 = AC.BC [4]
(b) If BDE = FDG, prove that BD is parallel to EF. [2]
(c) State, with reasons, why angle BHF = 90o. [2]
8 A particle moves in a straight line so that, t seconds after leaving a fixed point O, its velocity, v m/s2, is given by v = 5t – 6 – t2.
Find
(i) its initial velocity, [1]
(ii) the accelerations of the particle when it is instantaneously at rest, [4]
(iii) the total distance travelled by the particle in the interval t = 0 to t = 3, [4]
(iv) the maximum velocity. [2]
9
B
E
C
F G
D
A
H
y
xC
D
B0
A(1, 3)
y =
The diagram shows part of the curve y = .
The tangent to the curve at the point A(1, 3) crosses the x-axis at B.
The line CD, with equation x = 3, is parallel to y-axis.
(i) Find the equation of the line AB. [5]
(ii) Calculate the area of the shaded region. [6]
10
The diagram shows a parallelogram ABCD in which A is (3, 7), B is (1, 3) and D is (p, 4), where p is a constant. The perpendicular bisector of AB cuts the x-axis at point C.
(i) Find the coordinates of C. [5]
(ii) Find the value of p. [2]
(iii) Calculate the area of the parallelogram ABCD. [2]
(iv) Hence, or otherwise, find the shortest distance from C to AD. [3]
11
S T
R
Q
P
7 cm
4 cm
A(3, 7)
B(1, 3)
C
D(p, 4)
x
y
In the figure, PQRS is a rectangle of length 7 cm and breadth 4 cm and RST is radians, where is acute.
Let h cm be the perpendicular distance from Q to the line ST.
(i) Show that h = 4cos + 7sin . [3]
(ii) Express h in the form R cos( ), where R is a positive constant and is an acute angle in radians. [6]
(iii) Find the value of when h = 8. [2]
(iv) Find the maximum value of (4cos + 7sin )2 and the corresponding value of .
[2]
Paper 6 (Answers)1 (a) (i) 54.8oC (ii) 6.07 minutes (b) 26
2 (a) (i) Amplitude = 3, period = (ii)
(iii) 4 solutions
(b)
3 (i) = 8x2ex4 (4x4 + 3) (ii) (0, 3) is a minimum point
4 (i) (ii) x – 2ln(x + 2) + 2ln(x – 2) + C
5 (a) + = ; = (b)(i) (ii) 3x2 – 11x + 9 = 0
y
-2
1234
x2
23 2
23 xy
y = 3sin 2x + 1
180o 360o x
y
0o
y =
6 (a) 14 – 6 (b) a = 63
8 (i) –6 m/s (ii) –1 m/s2 & 1 m/s2 (iii) m (iv) 0.25 m/s
9 (i) y = –12x + 15 (ii) units2
10 (i) C = (12, 0) (ii) p = 14 (iii) 50 units2 (iv) 4.39 units
11 (ii) cos( 1.05) (iii) = 1.18 rad (iv) 65; = 1.05 rad