2014 2 Pahang SMK Ketari,Bentong_Maths QA
Transcript of 2014 2 Pahang SMK Ketari,Bentong_Maths QA
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) = {(π₯2 + 1)2, π₯ > β1
|π₯ + 1| β 1, π₯ β€ β1.
(a) Determine whether h is continuous at x = β 1.
(b) Sketch the graph of h.
[8]
2 If x = 2t β 1
π‘ and y = 4t +
1
π‘ , show that
ππ¦
ππ₯= 2 β
3
2π‘2+1.
Hence, deduce that β1 β€ππ¦
ππ₯< 2.
Find the equation of the tangent to the curve when t = 1.
[8]
3 By using the substitution u = ex, evaluate β«ππ₯
ππ₯+πβπ₯ ππ₯.1
0
[6]
4 The variables x and y, where x > 0 and y > 0, are connected in the differential equation ππ¦
ππ₯=
π¦2βπ₯2
2π₯π¦.
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 ππ¦
ππ₯= 2π₯π¦ β 1, and y = 1 when x = 0.
(a) Show that, when x = 0, π3π¦
ππ₯3 = β 4. Find the value of π4π¦
ππ₯4 .
(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 β 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 = π₯2
π₯2β5π₯+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 β 2)2 (x β 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 β 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 β 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 = β 1.
b)
y
y = h(x)
O x
2 Show ππ¦
ππ₯= 2 β
3
2π‘2+1.
t = 0, ππ¦
ππ₯ = β 1
t β Β±β, that 3
2π‘2+1 β 0,
ππ¦
ππ₯β 2.
Hence, β1 β€ππ¦
ππ₯< 2.
Equation of tangent : y = x + 4.
3 1
2ππ
1
2(π2 + 1)
4 y2 = 2x β x2.
5 a) π4π¦
ππ₯4 = β 4
b) π¦ = 1 β π₯ + π₯2 β2
3π₯3 +
1
2π₯4 + β―
y = 0.90938
6 f(0) = β 1, f(1) = 1, sign of f(x) changes from negative to positive and f is continuous for x β (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, β 24 ) maximum point.
Interval is (0, 2) α΄ (2, 2.4).
1 β
o I I
β2 β1
β1