HOLIDAYS PACKAGE FOR S4 MCB&PCB
𝑥+1
1) for what interval the inequality > 1 hold? 3marks
2−𝑥
2) solve |𝑥 + 3| = 5 3marks
3) find the value of y if 81−𝑦 = 42𝑦+3 3marks
4) a)convert 30 degrees 15 minutes and 50 seconds to decimal degrees . 2marks
1 1
b) show that (𝑡𝑎𝑛𝑥 + 𝑐𝑜𝑡𝑥)𝑠𝑖𝑛𝑥𝑐𝑜𝑠𝑥 = 1 3marks
𝑥 2 −𝑥+1
5) Determine the value(s) of k for which the equation = 𝑘 has repeated roots 4marks
𝑥−1
6) Transform the radical √9 + √80 to simple radical . 3marks
3𝜋
sin(𝜋−𝑎) cot( +𝑎)cos (𝑎−2𝜋)
7) Simplify 2
𝜋 3𝜋 4marks
tan(𝜋+𝑎) tan( +𝑎)cos ( −𝑎)
2 2
𝑥 2 −9
8) 𝐿𝑒𝑡 𝑡ℎ𝑒 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑓(𝑥) 𝑏𝑒 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑏𝑦 𝑓(𝑥) = { 𝑥 ≠ −3 3marks
𝑥+3
𝑘 𝑥 = −3
Determine k if 𝑓(−3) = lim 𝑓(𝑥)
𝑥→−3
9. The following equations show the model of a given market for a commodity x
𝑄𝑑 = −𝑥 + 6
𝑄𝑠 = 4𝑥 − 4
Determine a) the equilibrium cost of the commodity for the model 2marks
b) the equilibrium quantity of commodity 1mark
10 ) Given that 𝛼 𝑎𝑛𝑑 𝛽 are the roots of the quadratic equation 2𝑥 2 + 2𝑥 + 2 = 0
Find a) 𝛼 2 + 𝛽 2 2marks
b) 𝛼 3 + 𝛽 3 2marks
c) 𝛼 2 𝛽 + 𝛼𝛽 2 2marks
11. 𝑠𝑜𝑙𝑣𝑒 log2 + log(x + 2) − log(3x − 5) = log3 3marks
√5𝑥−4−√𝑥
12. 𝑒𝑣𝑎𝑙𝑢𝑎𝑡𝑒 𝑙𝑖𝑚 3marks
𝑥→1 𝑥−1
13. for what values of 𝑥 the equation 100𝑥 − 12(10𝑥 ) + 32 = 0
4marks
14) Find the constants 𝑎 𝑎𝑛𝑑 𝑏 such that the function 𝑓(𝑥) is continuous on entire real line
2 , 𝑥 ≤ −1
{𝑎𝑥 + 𝑏 , −1 < 𝑥 < 3 3marks
−2, 𝑥≥3
𝐵+𝐶 𝐴
15) If 𝐴, 𝐵𝑎𝑛𝑑 𝐶 are interior angles of a triangle ABC ,Show that sin ( ) = 𝑐𝑜𝑠 2 3marks
2
𝑑𝑦
16. a) find 𝑑𝑥 if 𝑦 = (𝑥 2 + 3)4
b) Given that 𝑥 2 + 𝑥𝑦 + 𝑦 2 − 3𝑥 − 𝑦 = 3
𝑑𝑦 3−2𝑥−𝑦
i) show that 𝑑𝑥 = 𝑥+2𝑦−1
ii) find ,and classify ,the maximum and minimum values of y
iii) Determine the coordinates of the points on the curve where the tangents are parallel to y axis.
17. solve in the set of real numbers 6𝑥 4 − 5𝑥 3 − 38𝑥 2 − 5𝑥 + 6 = 0 15marks
18. Let the binary operation ∗ be defined on ℤ ring of integers by
𝑥 ∗ 𝑦 = 𝑥 + 𝑥𝑦 + 𝑦
a) calculate 2 ∗ (−1) 𝑎𝑛𝑑 (−1) ∗ 9
b) determine whether ∗ is commutative or not
c) determine whether ∗ is associative or no
d) determine whether or not there exist an identity element
19. a) Moses is standing on the 1stfloor of a building .Suppose Moses can oversee the bottom of a
tree 60 metres from the building at an angle of depression 0f 750.find the height from the ground
,where Moses is standing
b) From a ship the angle of elevation of a point A at the top of a cliff is 190. After the ship has
sailed 800m directly towards the foot of the cliff .the angle of elevation of A is 440 . 5marks
c) Felix has 19 coins in his pocket some of them are 50frw and the rest are 100frw coins . the
total value of coins is 1550 frw . find the number of 50 frw coins and the 100frw coins he has.
5marks
3
20. a) let f(x)= √𝑥 3 + 2𝑥 + 1 , 𝑔(𝑥) = 𝑥 − 7 and ℎ(𝑥) = √2𝑥 + 8
𝑔(𝑥)
i) find the domain of definition of 𝑓(𝑥) + ℎ(𝑥) 2marks
ii) find 𝑓 ∘ 𝑔(𝑥) 2marks
𝑖𝑖𝑖)𝑓𝑖𝑛𝑑 𝑔−1 (𝑥) 2marks
𝑥 2 −2
b) Find the equations of asymptotes for the function 𝑥 2 −𝑥−2 and study the parity of f(x)
c) state the intermediate value theorem and show that 𝑓(𝑥) = 𝑥 3 + 3𝑥 − 1 has a
root in the interval ]0,1[
d) state the Rolle’s theorem and interpret it by using graph.
21. Describe in two pages atleast 10 strategies that a student who studies
Mathematics in s4 can pay attention in order to obtain his/her marks in interval
[85 100] especially in Mathematics course.
N.B :your research includes also ideas from your neighbor whom you know are
bright in Mathematics. /20marks
