P425/1

Pure math

Paper 1

3 hours

GREENHILL ACADEMY

MOCK EXAMINATIONS 2016

UGANDA ADVANCED CERTIFICATE OF EDUCATION

INSTRUCTIONS TO CANDIDATES:

-  Attempt all the eight questions in section A and any five questions from section B.

-  Clearly show all the necessary working

-  Begin each answer on a fresh sheet of paper

-  Silent, simple non-programmable scientific calculators may be used.

SECTION A (40 MARKS)

1. Find the fourth root of

2. The 5th term of an arithmetic progression is 12 and the sum of the first five terms is 100. Determine the first term and the common difference

3. Solve the inequality to four significant figures3

4. Find the angle subtended by the plane and a line .

5. Express in the form, hence, solve the equation , for .

6. Evaluate:

7. A is a point and B is the point . P is the variable point which moves such that. Show that the locus of P describes the circle, hence state the centre and the radius of the circle

8. Solve the differential equation given that when .

SECTION B (60MARKS)

9a) Use a substitution to solve the equation , for .

b) Show that for any triangle ABC, .

10a) Given that is a complex number, find the equation of the locus .

b) Solve the equation

11a) Find: using

b) Evaluate: .

12a) Prove by induction that.

b) Given that . Find the values of the constants .

13a) Given vectors and . show that a and b are perpendicular.

b) Two planes have equation and 3x-3y+6z=3. Find

i) The acute angle between the two planes

ii) The line of intersection in vector form of the two planes.

14a) Find the equation of a circle through the points and .

b) Given that and .

i) Show that x and y represents a circle. Hence its state the centre and radius

ii) Find the equation of the common chord of intersection of the circles in (b) (i) above and a circle x2+y2-2x+6y=0

15. Sketch the curve

16a) Find the particular solution of the equation given

b) The rate at which mangoes fall off a tree is directly proportional to the number of mangoes remaining on the tree. a certain tree initially had 100 fruits, 2hours later the fruits had reduced to 80mangoes

(i)  Find the numbers of mangoes which had remained after 312 hours

(ii)  calculate the time it takes for just 1 mango to remain

END

S 6 MARKING GUIDE MOCK TERM 2 2016

1. / Let
2. /
3. / (0.6)-2x<3.6

4. /
5. /
6 /

7. /
Circle of center (2.5,3.5) and radius 3.54
8. /
9a) /
b) /
10(a) /
b) /
11a) /
b) /
12a) / r=1n(2r-1)2= 13n(4n2-1)it holds for n=1, n=2,n=3,Therefore it holds for all positive values of n
b) /
1+ax1+bx=1-4x+12x2+cx3

13a) /
b) / a=3i+2j+k, b= -2i-j+4k .

14.(a) / A(0,1) , B( 2,3) and C(0,5) generally,

Is the equation of the circle
(b) /
(i)  x= 1+ 2 sin θ and y= -3+ 2 cos θ.

this is the eqn of the circle of centre (1,-5) with radius 2
(ii)
Is the eqn of the common chord
15 / 15 / Intercepts: For , thus .
Turning points:
,
and
L / / R / L / / R
Sign of / / / /

As , so, vertical Asymptotes:
As . is the horizontal asymptote.
Region table:
/ / /
/ / / /
/ / / /
/ / / /
/ / / /
/ Very popular but low scores, need to do better.
16(a) /
For
(b) / Let N= number of mangoes present at any time t

20minutes will elapse for 1 mango to remain