KAYONZA DISTRICT
RURAMIRA SECTOR
E.S RURAMIRA
DATE: 28/06/2021
MATHEMATICS EXAMINATION THIRD TERM FOR S6MCB
SECTION A: ATTEMPT ALLL QUESTIONS
1. Given the matrix 𝐴 = (
11 − 𝑥
2
8
2
2−𝑥
−10
8
−10 ) , find the possible values such that
5−𝑥
matrix A is singular if 9 is one those values.
log(𝑥 + 𝑦) = 1
2. Solve the following system {
log 2 𝑥 + log 4 𝑦 = 4
3. Find eccentricity , Centre, foci for the ellipse:𝑥 2 + 4𝑦 2 + 4𝑥 + 8𝑦 + 4 = 0
4. Find the equation of tangent and normal equation to the curve of
3𝑥 2 − 𝑥𝑦 + 2𝑦 2 + 12 = 0 at the point the (2,3)
5. Given that 𝑥̅ = 6, 𝑠𝑥 = 3.03315, 𝑠𝑦 = 0.461519, 𝑦̅ = 2.04 𝑎𝑛𝑑 𝑟𝑥𝑦 = 0.957241 , find
the regression y on 𝑥 where 𝑥̅ , 𝑠𝑥 , 𝑦̅ 𝑠𝑦 𝑎𝑛𝑑 𝑟𝑥𝑦 stand for the mean for 𝑥, the standard
deviation of 𝑥 , the mean of 𝑦 , the standard deviation of 𝑦 and the correlation coefficient
respectively.
6. Solve the following system by Gaussian elimination method
𝑥 + 𝑦 − 𝑧 = −1
{ 3𝑥 − 2𝑦 + 𝑧 = 0
2𝑥 + 3𝑦 − 3𝑧 = −3
7. A point P is 90m away from a vertical flag pole which is 11m high. What is angle of
elevation to the top of the flagpole from p?
sin 𝑥+ sin 2𝑥
8. Prove that 1+cos 𝑥+ cos 2𝑥
𝑥
9. The continuous random variable has probability density function {
a) 𝐸(𝑋)
C) 𝑣𝑎𝑟(𝑋)
b) 𝐸(𝑋 2 )
d) 𝑆𝐷(𝑋)
e) 𝑣𝑎𝑟(3𝑋 + 2)
8
,0 ≤ 𝑥 ≤ 4
0, 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒
find:
2
10. The equation of the curve is 𝑦 = ln 𝑥 + 𝑥 where𝑥 > 0. Find the coordinates of stationary
point of curve and determine whether it is maximum or minimum point.
𝑑𝑥
11. Find ∫ 2+cos 𝑥
12. If you deposits 6500frw into an account paying 8% annual interest compounded monthly,
how much money will be on the account after 7 years?
13. Find derivative of 𝑔(𝑥) = 10𝑥 ln(𝑥 2 + 10)
14. The sum of the first six terms of the arithmetic progression is 72 and the second term is
seven times the fifth term. Find the first term and the common difference.
1
15. Consider the numerical function 𝑓 defined by 𝑓(𝑥) = ln 𝑥
a) Calculate the first derivative of 𝑓 on (0, +∞)
𝑒𝑛
b) If 𝑛 ≥ 2, calculate 𝐼𝑛 = ∫𝑒
1
8𝑥(ln 𝑥)2
𝑑𝑥
c) Calculate lim 𝐼𝑛
𝑛→±∞
SECTION B: ATTEMPT ONLY 3 QUESTIONS
16.a) consider a sample space S on which the probability P is defined . consider also two
5
1
2
events A and B such that:𝑃(𝐴 ∪ 𝐵) = 6 , 𝑃(𝐴 ∩ 𝐵) = 4 , 𝑃(𝐴) = 3 . Find 𝑃(𝐵), 𝑃(𝐵̅ )
̅̅̅̅̅̅̅
and𝑃(𝐴
∩ 𝐵 ).
b) a fair coin is tossed 3times. Find the probability for obtaining two heads.
c) a factor has three machines A,B, C producing large numbers of certain items of the total daily
production of the items, 50% 𝑎𝑟𝑒 produced on A, 30% on B and 20% on C. Records show that
2% of items produced on A are defective , 3% of items o produced on B are defective and 4% of
items produced on C are defective. The occurrence of a defective item is independent of all other
items. One item is chosen at random from a day’s total output.
i.
Show that the probability of its being defective is 0.027
ii.
Given that is defective, find the probability that it was produced on
machine A
17. Consider the function defined on ℛ by 𝑦 = (𝑥 + √(𝑥 2 + 1)2
Calculate the expression (1 + 𝑥 2 )𝑦̈ + 𝑥𝑦̇ − 4𝑦
18. On the same graph, sketch the curves of the functions
𝑦 = 𝑥 2 − 5𝑥 + 4 And 𝑦 = −2𝑥 2 + 5𝑥 + 1. Hence find the area of the region enclosed between
two curves.
19. Determine the values of m such that the system in unknowns
𝑥 + (𝑚 − 1)𝑦 + (2𝑚 − 3)𝑧 = 1
𝑚𝑥 + 2(𝑚 − 1)𝑦 + 2𝑧 = 2
𝑥, 𝑦 𝑎𝑛𝑑 𝑧 {
has unique solution, no solution and more
2
(𝑚 + 1)𝑥 + 3(𝑚 − 1) + (𝑚 − 1)𝑧 = 3
than one solution.
4𝑥
20.a) let 𝑓(𝑥) = (3𝑥+1)(𝑥+1)2 )
i.
Express 𝑓(𝑥) as partial fraction
ii.
Hence show that ∫0 𝑓(𝑥)𝑑𝑥 = 1 − ln 2
1
b) The equation of the curve is 𝑦 = 8𝑥 − 𝑥 2
i.
Express 8𝑥 − 𝑥 2 in the form𝑎 − (𝑥 + 𝑏)2 stating the numerical values of a and b
ii.
Hence or otherwise , find the coordinates of stationary point of the curve
iii.
Find the values of 𝑥 which ≥ −20
GOOD LUCK!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!