2022: TEST 2 L6
PURE MATHEMATICS
Mr. SHARE
1. Solve the following equations
(a)
𝑥 2 +4𝑥
3
84
= 11 Hint: Use 𝑦 = 𝑥 2 + 4𝑥
+ 2
𝑥 +4𝑥
(b) 2𝑙𝑜𝑔3 (𝑥 − 5) − 𝑙𝑜𝑔3 (2𝑥 − 13) = 1
2. Express
6
3+√5+√14
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in the form 𝑎 + 𝑏√𝑐 + 𝑑 √𝑒 where 𝑎, 𝑏, 𝑐, 𝑑 and 𝑒 are real
numbers.
3. Express
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3𝑥 3 −2𝑥 2 −16𝑥+20
(𝑥 2 +1)(𝑥+2)
in partial fractions.
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4. If $ 6000 is invested in a bank which pays 3% per year compounded continuously.
How much will be in the account after 8 years?
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5. Solve the following inequality,
(a) |𝑥 + 6| < |3𝑥 + 2|
(b) |𝑥 + 6| < 3𝑥 + 2
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6.
a. If (𝑥 2 − 1) is a factor of 𝑓(𝑥), where 𝑓(𝑥) = 3𝑥 4 + 𝑎𝑥 3 + 𝑏𝑥 2 − 7𝑥 − 4, find
the value of 𝑎 and 𝑏.
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b. Factorise 𝑓(𝑥) completely, using the factor theorem.
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c. Hence, sketch the graph of 𝑦 = 𝑓(𝑥).
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7. Find the set of values of 𝑘 for which the line 4𝑦 + 3𝑥 = 𝑘 is a tangent to the curve
𝑥 2 + 𝑦 2 − 4𝑥 − 21 = 0.
8. Find the domain and range of the following functions.
(a) 𝑦 = 𝐼𝑛 (4𝑥 − 5)
1
(b) 𝑦 = 9𝑥+3
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[6]
2022: TEST 2 L6
PURE MATHEMATICS
Mr. SHARE
(c) 𝑦 = √3𝑥 + 71
1
(d) 𝑦 = √𝑥 2
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−9
9. Given that 𝑓(𝑥) = 2𝑥 2 − 7𝑥 − 4. Sketch the following graphs after stating
transformaton(s).
(a) 𝑓(𝑥)
(b) 𝑓(𝑥 + 2)
(c) 𝑓(2𝑥)
(d) 𝑓(𝑥) − 3
(e) 2𝑓(𝑥) + 3
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10. Prove the following identities.
(i)
(ii)
sin(𝐴+𝐵)
𝑐𝑜𝑠𝐴𝑐𝑜𝑠𝐵
≡ 𝑡𝑎𝑛𝐴 + 𝑡𝑎𝑛𝐵
cot(𝐴 + 𝐵) ≡
𝑐𝑜𝑡𝐴𝑐𝑜𝑡𝐵−1
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𝑐𝑜𝑡𝐴+𝑐𝑜𝑡𝐵
11. Show that the equation tan(30° + 𝜃) = 2 tan(60° − 𝜃) can be written in the form
tan2 𝜃 + (6√3) tan 𝜃 − 5 = 0, hence or otherwise, solve the equation
tan(30° + 𝜃) = 2 tan(60° − 𝜃) for 0° ≤ 𝜃 ≤ 180°.
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1
1
12. Find the solutions, in the range 0 to 𝜋 of the equation tan (2𝑥 − 𝜋) = √3
2
3
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