2014-2-PAHANG-SMK Ketari,Bentong_MATHS QA BY Wong Yew Nung

Section A

Answer all questions.

1 The function h is defined by h(x) = {(𝑥2 + 1)2, 𝑥 > −1

|𝑥 + 1| − 1, 𝑥 ≤ −1.

(a) Determine whether h is continuous at x = – 1.

(b) Sketch the graph of h.

[8]

2 If x = 2t – 1

𝑡 and y = 4t +

1

𝑡 , show that

𝑑𝑦

𝑑𝑥= 2 −

3

2𝑡2+1.

Hence, deduce that −1 ≤𝑑𝑦

𝑑𝑥< 2.

Find the equation of the tangent to the curve when t = 1.

[8]

3 By using the substitution u = ex, evaluate ∫𝑒𝑥

𝑒𝑥+𝑒−𝑥 𝑑𝑥.1

0

[6]

4 The variables x and y, where x > 0 and y > 0, are connected in the differential equation 𝑑𝑦

𝑑𝑥=

𝑦2−𝑥2

2𝑥𝑦.

Using the substitution y = ux, find the particular solution when y = 0 and x = 2.

[8]

5 The variables x and y are related by 𝑑𝑦

𝑑𝑥= 2𝑥𝑦 − 1, and y = 1 when x = 0.

(a) Show that, when x = 0, 𝑑3𝑦

𝑑𝑥3 = – 4. Find the value of 𝑑4𝑦

𝑑𝑥4 .

(b) Find the Maclaurin's series for y up to and including the term in x4, and hence find an

approximation to the value of y when x = 0.1, giving your answer to an appropriate accuracy.

[7]

6 Without sketching the graphs of y = x3 + 1 and y = 2 – x, show that they intersect at one point between

0 and 1.

Use the Newton-Raphson method to find the real root, correct to three decimal places. [8]

Section B

Answer one question.

7 The equation of a curve is y = 𝑥2

𝑥2−5𝑥+6.

(i) State the asymptotes of the curve.

(ii) Find the stationary points and determine their nature.

(iii) Determine the interval where y increases with x.

(iv) Sketch the curve.

(v) Determine the number of real roots of the equation p(x – 2)2 (x – 3) = x2, where p > 0.

[2, 6, 2, 3, 2]

8 Sketch, on the same coordinate axes, the curves of y = ex and y = 2 + 3e – x.

(a) Calculate the area of the region bounded by the x-axis, the line x = 3 and the curve y = ex.

(b) Calculate the area of the region bounded by the y-axis, the curves y = ex and y = 2 + 3e – x.

(c) Calculate the volume of the solid of revolution formed if the area of the region bounded by the

y-axis, the line y = 5 and the curve y = ex is rotated through four right angles about the y-axis.

[2, 3, 6, 4]

THE END OF THE QUESTION PAPER

MARKING SCHEME :

1 a) h is not continuous at x = – 1.

b)

y

y = h(x)

O x

2 Show 𝑑𝑦

𝑑𝑥= 2 −

3

2𝑡2+1.

t = 0, 𝑑𝑦

𝑑𝑥 = – 1

t → ±∞, that 3

2𝑡2+1 → 0,

𝑑𝑦

𝑑𝑥→ 2.

Hence, −1 ≤𝑑𝑦

𝑑𝑥< 2.

Equation of tangent : y = x + 4.

3 1

2𝑙𝑛

1

2(𝑒2 + 1)

4 y2 = 2x – x2.

5 a) 𝑑4𝑦

𝑑𝑥4 = – 4

b) 𝑦 = 1 − 𝑥 + 𝑥2 −2

3𝑥3 +

1

2𝑥4 + ⋯

y = 0.90938

6 f(0) = – 1, f(1) = 1, sign of f(x) changes from negative to positive and f is continuous for x ∈ (0,1), so

there is a real root between 0 and 1.

Estimate = 0.682

7 Asymptotes : x = 2, x = 3, y = 1.

(0, 0) minimum point, (2.4, – 24 ) maximum point.

Interval is (0, 2) ᴗ (2, 2.4).

1 –

o I I

–2 –1

–1

y

O x

Number of real roots is 1.

8 y = 2 + 3e – x y x = 3

5

y = ex

y = 2

O x

a) Area = e3 unit2

b) Area = 2ln 3 unit2

c) Volume = 𝜋 [ 9 – 10ln5 + 5(ln5)2] unit3