MPOMA SCHOOL SATELLITE CAMPUS MUKONO
SENIOR SIX PURE MATHEMATICS PAPER 1
DURATION 3 HOURS
-Answer all the questions in Section A and only five questions in Section B
-Show all the necessary working clearly
-Silent non- programmable scientific calculators and mathematical tabled with a list of formula may be used.
- Solve the simultaneous equations
8x-y=4x+y, 5x2-y2= 15625 (5 mks)
- The second term of an Arithmetic progression is -4 and its sixth term is -24. Find the fifteenth term and the sum of the first fifteen terms of the progression. (5 mks)
- Find the volume generated when the area bounded by the curve y=5cos 2x, the x-axis and the ordinates x=0 and x= is rotated about x-axis through a complete rotation
- Differentiate with respect to x:
- If Sin 2 Find values of, hence solve the equation
-=0 for 0=0 for (5 mks)
- Evaluate by use of the substitution x=
- Solve the differential equation
=e2+xt, given that R (0) = 3 (5 mks)
- Find the acute angle between the line
== and the plane 6x+2y-z=-4 (5 mks)
SECTION B (60 MARKS)
Answer any five questions from this section
All questions carry equal marks
- (a) Given that the vectors i-pj+K and 3i +2j +4K are perpendicular, determine the value of p (02 marks)
(b) Find the angle between the lines;
r1= (1+λ) i+ (1-j+ (2+λ)K and
(c) Show that the line x+1=y=
Is parallel to the plane
- (i+j-K) =3 and find the distance between them
- (a) if z=x+1y, show that
arg= is a circle . Find its centre and its radius.
(b) Z and are conjugate complex numbers. Find the values of z that satisfies the equation; 3Z+2(Z-39+12I (6 mks)
- If y =, Express y in partial fractions. Hence determine (12 mks)
- (a) If y =x, find in its simplest form (6 mks)
(b) Using small changes, evaluate correct your answer to four decimal places (6 mks)
- (a) solve the equation
tan-1(2x+1) + tan -1(2x-1) = tan -1(2) (05 mks)
(b) Solve the equation
tan2x- sin2 x =1, 0X2 (6 MKS)
- (a) obtain the expression in ascending powers of X of (1+2x) 15 as far the term in X3. Hence evaluate (1.002)15 correct to 5 decimal places.
(b) An amount of shs 2000 is invested at an interest of 5 per month. if Shs 2000 is added at the beginning of each successive month but no withdrawals.
(i) Give an expression for the value accumulated after n months
(ii) After how many months will the amount have accumulated first exceed Shs. 42000?
- Newton’s law of cooling states that the rate at which a body cools is directly proportional to the excess temperature of the body over the temperature of its surroundings. Given that at time t minutes a body has a temperature T and its surroundings a constant temperature from a differential equation in terms of T,,t and the constant of proportionality k,K0. Integrate this equation to show that (N(T-)=Kt+C where C is a constant.
At 2:33 pm, the water in a kettle boils at 100 in a room of constant temperature 21. After 10 minutes, the temperature of the water in the kettle is 84. Use this information to find C and k, hence find the time taken for the water in the kettle to have the temperature of 70. (12 mks)
- (a) find the values of m for which the line y=mx is a tangent to the circle
(b) Find the points where the line 2y-x+5=0 meets the circle x2+y2-4x+3y-5=0.obtain the equation of the tangents and normals to the circle at these points (09 marks)
A cat likes to eat fresh fish but it will not go into the water