**MPOMA SCHOOL SATELLITE CAMPUS MUKONO**

**LOCKDOWN EXAMINATION**

** SENIOR SIX PURE MATHEMATICS PAPER 1**

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**DURATION 3 HOURS **

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**INSTRUCTIONS **

**-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

8^{x-y}=4^{x+y}, 5^{x2-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

(5 mks)

- 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

=e^{2+}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;

r_{1}= (1+λ) i+ (1-j+ (2+λ)K and

r_{2}= (1-µ)i+(1-2µ)j+(1+µjK)

(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

tan^{2}x- sin^{2} x =1, 0X2 (6 MKS)

- (a) obtain the expression in ascending powers of X of (1+2x)
^{ 15}as far the term in X^{3}. 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?

(6 mks)

- 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

x^{2}+y^{2}+fy+C=0

(b) Find the points where the line 2y-x+5=0 meets the circle x^{2}+y^{2}-4x+3y-5=0.obtain the equation of the tangents and normals to the circle at these points (09 marks)

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**A cat likes to eat fresh fish but it will not go into the water**

**END**[i]

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