Page 54 - Focus SPM 2022 - Additional Mathematics

Additional Mathematics SPM Chapter 2 Differentiation

7. Given a formula H = 3 2 which is obtained from 10. (a) The diagram below shows a pattern in the
an experiment. 5 + g shape of a square PQRS with side of 8 cm.
(a) Express dH in terms of g. P Q
dg
(b) Ali read the measurement g from a faulty M
instrument with error and recorded the reading
as 2.98 instead of the real reading of 3. Find the
Penerbitan Pelangi Sdn Bhd. All Rights Reserved.
estimated error in the measurement of H. S N R
x y
8. (a) The straight line – = 1 is the normal to
2 6 M and N are two points lie on the side PS
2
3
the curve y = x – 4x + 5x – 3 25 at point X. and SR respectively such that PM = x cm and
27
Find SN = 3x cm.
(i) the coordinates of point X. (i) Express the area of triangle QMN in terms
(ii) the equation of the tangent to the curve at of x.
point X. (ii) Hence, find the minimum area of triangle
(b) In an experiment concerning the focal length of QMN. State the value of x when this
a lens, an object is placed u cm from a lens minimum area occurs.
and the image distance v cm is measured. It (b) Given y = 5 and s = 8 – 3x. Find
is given that u and v are related by the formula 4s 2
1 + 1 = 1 . (i) the rate of change of the value s when x
u v 6 changes with a rate of 1 unit s ,
–1
(i) Express du in terms of v. 3
dv (ii) the approximate change in the value of y
(ii) Hence, find the rate of change of v when when x increase from 2.50 units to 2.56
u increases with a rate of 1.8 cm s when units.
–1
u = 5 cm.
11. The diagram below shows the front elevation
9. (a) The diagram below shows a hemispherical SPM of part of the rail of a roller coaster in an indoor
container with a radius of 12 cm. 2018 themepark inside a shoping mall.
12 cm
Ceiling

10 cm
h cm
Floor
x
Izzatul pours a type of liquid into the container
such that the height, h cm, of the liquid in the
container increases with a rate of 0.8 cm s . The curve part of the roller coaster is
–1
(i) Express the area, in cm , of the liquid represented by the equation y = 1 x – 1 x – 6x,
2
2
3
surface in the container in terms of h. 3 2
(ii) Hence, find the rate of change of the such that the point x is the origin. Find the shortest
surface area of the liquid when the height vertical distance, in m, from the rail to the ceiling of
of the liquid is 8 cm. the building. Form 5
(b) In the laboratory, it is discovered that the
4
duration of the oscillation, T seconds of an 12. The curve y = hx – 5x has a turning point (2, k).
SPM
oscillating object which is hung by using a 2019 Find the values of h and k.
string, l cm in length is given by
2
3
l 13. The curve y = 2x + 3x – 12x – 15 passes through
T = 2p point P(–3, –6) and has turning points A(1, –22)
10
(i) Find dT . and B. Find
dl (a) the gradient of the curve at point P.
(ii) Estimate the small change in the length (b) the equation of the normal to the curve at point
of the string when the duration of the P.
oscillation changes from 8p seconds to (c) the coordinates of point B and determine the
8.02p seconds. nature of this turning point B.
(iii) Find the small increase in the duration of
the oscillation when the length of the string
increases from 40 cm to 43 cm.
263