Page 52 - Focus SPM 2022 - Additional Mathematics

Additional Mathematics SPM Chapter 2 Differentiation

18. Given y = 1 . 20. The relation between the focal length, f m, of a
√x convex lens with its object length, u m and image
dy length, v m, is given by the formula:
(a) Express in terms of x.
dx
(b) Hence, calculate the approximate value of 1 = 1 + 1
1 . f v u
√80.98
If the focal length of the lens is 4 m, find the small
19. A filler funnel in a cone shape has a height of 8 cm. change in its object length when the image length
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If the radius of the base of cone changes from 6 cm changes from 28 to 27.8 m.
to 6.04 cm, find the approximate change in
(a) the volume of the cone,
(b) the curve surface area of the cone.

SPM Practice 2
SPM Practice

PAPER 1 (b) f"(–2) if f(x) = x 2x – 3x + 4 2 .
1
2
lim
1. (a) Find the value of x → 0 (8 – √x ). x
SPM 9. The gradient function of a curve is (3x – 5). Given
2018 (b) Differentiate the following function by using the the point Q(–2, 6) lies on the curve, find
first principle. (a) the gradient of the tangent at point Q,
y = 5 (b) the equation of the normal to the curve at
x 2
2 – 3x 2 point Q.
2. (a) Given f(x) = , find f '(x).
5x – 4 10. Given that the gradient of the normal to the curve
2
3
(b) Given f(x) = 18 , evaluate f'(2). y = 9x – mx at x = –3 is 4 .
x 4 3
(a) Find the value of m.
3. (a) Find d 1 1 2 . (b) Find the equation of the tangent to the curve at
dx 8x – 5 x = –3.
(b) Given r = 3x – 2 and y = – 5 2 . Find dy in terms 11. Given that hx – x is the gradient function of a
2
of x. r dx
curve such that h is a constant. If y – 7x + 5 = 0 is
4. (a) Differentiate x (1 + 5x) with respect to x. the equation of the tangent at point (–1, –12), find
6
3
(b) Given f(x) = (1 – 3x) , evaluate f'(–1). (a) the value of h.
5
5. Given the function h(x) = px – 8x + 6x, find (b) the equation of the normal to the curve at point
2
3
(a) h'(x), (–1, –12).
(b) the value of p if h'(–1) = 5. 12. Find the equations of two tangent lines from the point Form 5
d y dy 5 11
2
2
6. Given y = x(5 – x), express y + x + 24 in 1 , 2 to the curve y = 4 + 3x – x . HOTS
dx 2 dx 2 2 Applying
terms of x in the simplest form. Hence, solve the
equation 13. A car moves along a straight road. The
2
y d y + x dy + 24 = y. displacement, J metre, of the car from a fixed point
dx 2 dx X along the straight road is given by
7. (a) Given f(x) = (7x – 4) , find f"(x). J = 3t + 23 2
t – 3t – 3

6
3
(b) Given h(x) = kx – 5x + 6x, find the value k if 4
3
2
h"(–1) = 5. such that t is the time, in second, after passing
through point X. Find HOTS
8. Find the value of dJ Applying
SPM 4 (a) dt .
lim
x 2x – 3x +
2018 (a) x → 0 1 2 x 2 . (b) the maximum displacement of the car.
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