Additional Mathematics Revision

1. a) The functions f and g are defined by

𝑓 𝑥 = 2𝑥 − 5 𝑔 𝑥 = 4𝑥 Find the value of:

i. 𝑓𝑔 𝑥 ii. 𝑓!!(−3)

b) If 3𝑥! + 𝑝𝑥! − 𝑞𝑥! + 4 is divisible by 𝑥! + 𝑥 − 2, evaluate the values of the constants p and q.

c) i) Find the quotient and the remainder of

(6𝑥! − 11𝑥! − 𝑥 + 6)÷ (2𝑥 − 3).

ii) Put in form 𝑝 𝑥 ≡ 𝑔 𝑥 . 𝑞 𝑥 + 𝑟 (𝑥)

iii) Hence, factorize completely the polynomial 𝑝 𝑥 = 6𝑥! − 11𝑥! − 𝑥 + 6

2. Determine which of the following linear divisors are factors of 3𝑥! − 2𝑥! − 13𝑥! + 8𝑥 + 4

a) 3𝑥 + 1 b) 2𝑥 − 1 c) 𝑥 − 1

3. Use the quadratic formula to solve 12𝑥! − 16𝑥 + 5 = 0

Additional Mathematics Revision

4. Express the quadratic function 𝑓 𝑥 = 5+ 8𝑥 − 4𝑥! in the form 𝑎(𝑥 + ℎ)! + 𝑘. Hence:

a) State the turning point and state whether it is a maximum or minimum b) State the maximum or minimum value of 𝑥 c) State the axis of symmetry d) State the range of 𝑓 for the given domain e) State the 𝑦 intercept f) Solve 𝑎(𝑥 + ℎ)! + 𝑘 = 0 g) State the 𝑥 intercepts h) Sketch a graph 𝑓 𝑥 = 𝑎(𝑥 + ℎ)! + 𝑘

5. Determine the roots of each of the following quadratic equation.

a) 3𝑥! + 17 = 12𝑥 b) 4𝑥! + 36 = 24𝑥 c) 2𝑥! + 19 = 12𝑥

6. Solve the following equation 3!!!! − 80(3!)− 27 = 0.

7. If 𝛼,𝛽 are the roots of the equation 3𝑥! − 5𝑥 − 1 = 0, find the value of a) (i) 𝛼!𝛽 + 𝛽!𝛼 (ii) (𝛼!𝛽)( 𝛽!𝛼) (iii) (𝛼 + 1)(𝛽 + 1)

8. Solve the following pair of simultaneous equation 𝑦 + 7𝑥 + 9 = 3𝑥! 𝑦 + 6𝑥 = 5

9. Solve the following quadratic inequalities, using an algebraic method and a graphical method for −2𝑥! − 3𝑥 + 14 < 0

Additional Mathematics Revision

10. Given 𝑓: 𝑥 → !!!!!!!

, 𝑥 ≠ 2 and 𝑔 𝑥 = 2𝑥 − 5, find

a) 𝑔𝑓 b) 𝑓𝑔

c) 𝑓! d) 𝑔!

e) 𝑔!!𝑓!!

11. Differentiate the following: a) 𝑦 = 𝑥!!

b) 𝑦 = (𝑥 − 15)!"

c) 𝑥!" sin 𝑥

d) sin 𝑥 cos 𝑥

e) 𝑦 = !!

!!

f) 𝑦 = !"# !!

!"# !!

g) 𝑦 = 2𝑥! − 2sin 3𝑥! + 5cos 5𝑥!

12. Differentiate the following:

a) 𝑝 = 15(12𝑡! − !!𝑡)

!!

b) 𝑦 = !!!!!!

!"#!!!

c) 𝑦 = !

!!!

13. If 3𝑥!𝑦 + 5𝑦!𝑥 = 1𝑦. Find !"!"

at (0,1).

14. Find the equation of the tangent in a curve 𝑦 = 𝑥! − 6𝑥 + 25 at (0,5). Given that 𝑦 = !!!!

!!!! show that !"

!" is always positive.

Additional Mathematics Revision

15. Find the gradient of the tangent to the curve 𝑦 = 𝑥𝑐𝑜𝑠𝑥 at the point where 𝑥 = 0.5 giving your answer to (2dp).

16. A curve of equation 𝑦! + 3𝑦 ln 𝑥 = 10. Find !"

!" at (𝑒, 2). Note: 𝒍𝒏 𝒆 = 𝟏.

17. Integrate the following

a) 5𝑒!!!

b) ln(𝑥! − 5𝑥)

c) !!!

!!

d) (6+ 2𝑥! − 2sin 3𝑥! + 5cos 5𝑥!)

e) (2𝑥! − 5)!!"

!

18. Solve the following equation:

log!(5𝑥 − 4)− log! 𝑥 + 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.

(𝑥 − 1)! + (𝑦 − 4)! = 5 𝑎𝑡 (2, 6)

20. a) The position vector of A and B with respect to a fixed origin, O, are given

by 𝑂𝐴 = −2𝑖 + 3𝑗 𝑎𝑛𝑑 𝑂𝐵 = 4𝑖 + 3𝑗.

Find the unit vector in the direction of AB. b) Given that 𝒑 = 3𝑖 − 4𝑗 𝑎𝑛𝑑 𝒒 = 4𝑖 − 2𝑗, determine the angle between p and q.

Additional Mathematics Revision

21. An arc 𝑃𝑄 subtends and angle of !!𝜋 radians at the center 𝑂 of a circle of radius

6cm.

Find

(a) the length of PQ

(b) the area of the sector POQ.

𝑇𝑎𝑘𝑒 𝜋 𝑎𝑠 3.142

22. a) Prove that !!!"# !!!"# !!

≡ tan 𝜃

b) Prove the identity !!!"# !!!!!"# !!

≡ 𝑡𝑎𝑛2𝜃

23. A particle is projected vertically upwards with a velocity of 40𝑚𝑠!! 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 10𝑚𝑠!!.

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 𝑠 = 9𝑡! − 𝑡!, 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 !

!𝜋𝑟!𝑐𝑚!. The radius of the

balloon increases at the rate of 0.3 𝑐𝑚𝑠!!. Determine the rate of increase of the volume of the balloon when its radius is 3.5 cm.