4e3 a Maths Prelim Exam Paper 8 With Ans (1)
Transcript of 4e3 a Maths Prelim Exam Paper 8 With Ans (1)
Paper 8Mathematical Formulae
1. ALGEBRAQuadratic Equation
For the equation ax2 + bx + c = 0,
x = .
Binomial expansion
,
where n is a positive integer and
2. TRIGONOMETRYIdentities
sin2 A + cos2 A = 1.sec2 A = 1 + tan2 A.
cosec2 A = 1 + cot2 A.
sin 2A = 2 sin A cos A.
cos 2A = cos2 A – sin2 A = 2 cos2 A – 1 = 1 – 2 sin2 A
.
sin A + sin B = 2 sin ( A + B ) cos ( A – B )
sin A – sin B = 2 cos ( A + B ) sin ( A – B )
cos A + cos B = 2 cos ( A + B ) cos ( A – B )
cos A – cos B = –2 sin ( A + B ) sin ( A – B )
Formulae for ΔABC
.
.
Δ = .
1 (a) Express in partial fractions. [4]
(b) Evaluate , giving your answer correct to 2 decimal places.
[4]
2 (a) Given that and , find in terms of x and y,
(i) ,
(ii) . [5]
(b) Simplify and give your answer in form where
a and b are rational numbers. [3]
3 (a) The equation has roots and while the roots of the
equation are Find the value of p and of k. [4]
(b) The roots of the equation differ by 2, find the possible values of h. [4]
4 The equation of a function is defined by for all values of x.
(a) State the amplitude and period of f. [2]
(b) Find the values of x, for , where the curve cuts the x-axis. Give your answers in terms of . [4]
(c) Sketch the graph of [2]
5 The equation of a curve is Find an expression for Hence find the
equation of the normal to the curve at the point [4]
6 To measure the thyroid condition in a person, a small dose of a radioactive iodine is injected into the blood stream. The amount of iodine A remaining in the blood after x hours is given by the equation where D is a constant. If a dose of iodine is injected into the bloodstream at noon on Monday, find
(a) the percentage of iodine remaining in the bloodstream at 6 pm on the same day, [2]
(b) the number of hours it takes for the percentage level of iodine to drop to 50% of its original dose. [3]
7 In the diagram, PAQ is a tangent to the circle at point A and AC is a diameter of the circle.
(a) Prove that . Hence or otherwise, prove that [4]
(b) Prove that . [4]
A
B
O
C
P Q
8 Express in the form where R is positive and is acute. [4]
Hence find (a) the acute angle x for which , [3]
(b) the maximum value of and the angle x between and which gives this value, [2]
(c) the minimum value of and the angle x between and which gives this minimum value. [3]
9 The equation of a curve is .
(a) Find expressions for [2]
(b) Find the co-ordinates of the stationary points and determine the nature of the stationary points. [4]
10 (a) Evaluate giving your answer correct to 2 decimal places. [4]
(b) The diagram shows part of the graphs of and . Find
(i) the coordinates of A, B and C,[3](ii) the area of the shaded region giving your answer correct to 2 decimal
places. [4]
11 A particle moves in a straight line so that, at time t seconds after leaving a fixed point
O , its velocity , v m s1, is given by .
(a) Find (i) the initial acceleration of the particle, [2](ii) the value of t when the particle is instantaneously at rest, [3]
(iii) the distance of the particle from O when t = 2. [3]
x
y
A
C
B
12
xy
xy 2cosO
(b) Sketch the velocity-time curve for t 0, indicating the coordinates of the points of intersection with the axes. [3]
12 The diagram shows a quadrilateral ABCD where AC is parallel to the line and A is on the x-axis. The coordinates of P and D are and
respectively, PC=2AP, , units and APC is a straight lines. Find
(a) the equation of AC, [2](b) the coordinates of A, [1](c) the equation of AB, [2](d) the coordinates of C and B, [5](e) the area of the quadrilateral ABCD. [1]
END OF PAPER
Answers (Paper 8)
1. (a) (b) 48.49
2. (a)(i) (ii) (b)
A
C
D
P
x
y
O
B
3. (a) (b)
4. (a) amplitude=3, period= radian (b)
5. , .
6. (a) 95.3% (b) 86.6 hrs
7. (a) (b)
8. (a) (b) (c) 0;
9. (a) (b) (3, 36) max; (-3, -36) min
10. (a) 1.14 (b) (i) A(0, 1) , B( , C (ii) 0.45 unit
11. (a)(i) 1 (ii) t=2.77 sec (iii) 1.53 m
12 (a) (b) A(-6, 0) (c)
(d) C(3, 6), B(-2, -6) (e) 67.5 units
END