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Additional Mathematics Revision
1. a) The functions f and g are defined by
π π₯ = 2π₯ β 5 π π₯ = 4π₯ Find the value of:
i. ππ π₯ ii. π!!(β3)
b) If 3π₯! + ππ₯! β ππ₯! + 4 is divisible by π₯! + π₯ β 2, evaluate the values of the constants p and q.
c) i) Find the quotient and the remainder of
(6π₯! β 11π₯! β π₯ + 6)Γ· (2π₯ β 3).
ii) Put in form π π₯ β‘ π π₯ . π π₯ + π (π₯)
iii) Hence, factorize completely the polynomial π π₯ = 6π₯! β 11π₯! β π₯ + 6
2. Determine which of the following linear divisors are factors of 3π₯! β 2π₯! β 13π₯! + 8π₯ + 4
a) 3π₯ + 1 b) 2π₯ β 1 c) π₯ β 1
3. Use the quadratic formula to solve 12π₯! β 16π₯ + 5 = 0
Additional Mathematics Revision
4. Express the quadratic function π π₯ = 5+ 8π₯ β 4π₯! in the form π(π₯ + β)! + π. Hence:
a) State the turning point and state whether it is a maximum or minimum b) State the maximum or minimum value of π₯ c) State the axis of symmetry d) State the range of π for the given domain e) State the π¦ intercept f) Solve π(π₯ + β)! + π = 0 g) State the π₯ intercepts h) Sketch a graph π π₯ = π(π₯ + β)! + π
5. Determine the roots of each of the following quadratic equation.
a) 3π₯! + 17 = 12π₯ b) 4π₯! + 36 = 24π₯ c) 2π₯! + 19 = 12π₯
6. Solve the following equation 3!!!! β 80(3!)β 27 = 0.
7. If πΌ,π½ are the roots of the equation 3π₯! β 5π₯ β 1 = 0, find the value of a) (i) πΌ!π½ + π½!πΌ (ii) (πΌ!π½)( π½!πΌ) (iii) (πΌ + 1)(π½ + 1)
8. Solve the following pair of simultaneous equation π¦ + 7π₯ + 9 = 3π₯! π¦ + 6π₯ = 5
9. Solve the following quadratic inequalities, using an algebraic method and a graphical method for β2π₯! β 3π₯ + 14 < 0
Additional Mathematics Revision
10. Given π: π₯ β !!!!!!!
, π₯ β 2 and π π₯ = 2π₯ β 5, find
a) ππ b) ππ
c) π! d) π!
e) π!!π!!
11. Differentiate the following: a) π¦ = π₯!!
b) π¦ = (π₯ β 15)!"
c) π₯!" sin π₯
d) sin π₯ cos π₯
e) π¦ = !!
!!
f) π¦ = !"# !!
!"# !!
g) π¦ = 2π₯! β 2sin 3π₯! + 5cos 5π₯!
12. Differentiate the following:
a) π = 15(12π‘! β !!π‘)
!!
b) π¦ = !!!!!!
!"#!!!
c) π¦ = !
!!!
13. If 3π₯!π¦ + 5π¦!π₯ = 1π¦. Find !"!"
at (0,1).
14. Find the equation of the tangent in a curve π¦ = π₯! β 6π₯ + 25 at (0,5). Given that π¦ = !!!!
!!!! show that !"
!" is always positive.
Additional Mathematics Revision
15. Find the gradient of the tangent to the curve π¦ = π₯πππ π₯ at the point where π₯ = 0.5 giving your answer to (2dp).
16. A curve of equation π¦! + 3π¦ ln π₯ = 10. Find !"
!" at (π, 2). Note: ππ π = π.
17. Integrate the following
a) 5π!!!
b) ln(π₯! β 5π₯)
c) !!!
!!
d) (6+ 2π₯! β 2sin 3π₯! + 5cos 5π₯!)
e) (2π₯! β 5)!!"
!
18. Solve the following equation:
log!(5π₯ β 4)β log! π₯ + 3 = 1
19. Find the equation of
i. The normal at the given point ii. The tangent at the given point of the following circles.
(π₯ β 1)! + (π¦ β 4)! = 5 ππ‘ (2, 6)
20. a) The position vector of A and B with respect to a fixed origin, O, are given
by ππ΄ = β2π + 3π πππ ππ΅ = 4π + 3π.
Find the unit vector in the direction of AB. b) Given that π = 3π β 4π πππ π = 4π β 2π, determine the angle between p and q.
Additional Mathematics Revision
21. An arc ππ subtends and angle of !!π radians at the center π of a circle of radius
6cm.
Find
(a) the length of PQ
(b) the area of the sector POQ.
ππππ π ππ 3.142
22. a) Prove that !!!"# !!!"# !!
β‘ tan π
b) Prove the identity !!!"# !!!!!"# !!
β‘ π‘ππ2π
23. A particle is projected vertically upwards with a velocity of 40ππ !! from a point O. a) Calculate, for the particle,
i. The maximum height reached ii. The time taken to reach the maximum height
iii. The time taken for it to return to O.
b) Evaluate the time taken for the particle to reach 45 m below O. Take g as 10ππ !!.
24. A particle moves in a straight line so that its displacement, s metres, from a fixed point O on the line is given by the equation π = 9π‘! β π‘!, where t is the time in seconds. Find, for the particle,
a) Its displacement during the first 5 seconds b) Its velocity when the tome is 6 seconds c) Its displacement during the first 8 seconds
25. The volume of a spherical balloon of radius r cm is !
!ππ!ππ!. The radius of the
balloon increases at the rate of 0.3 πππ !!. Determine the rate of increase of the volume of the balloon when its radius is 3.5 cm.