PURE MATH 1 PAST YEARS QUESTIONS – DIFFERENTIATION

1.
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The diagram shows a glass window consisting of a rectangle of height h m and width 2r m
and a semicircle of radius r m. The perimeter of the window is 8 m.
(i) Express h in terms of r. [2]
1
(ii) Show that the area of the window, A m2, is given by A  8r  2r 2  r 2 . [2]
2
Given that r can vary,
(iii) find the value of r for which A has a stationary value, [4]
(iv) determine whether this stationary value is a maximum or a minimum. [2]
1
[ANS: h  4  r  r ,1.12,max]
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Differentiation 1 TanWC/SunwayCollege/2024
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2. A curve has equation y  x 2  .
x
dy d2 y
(i) Write down expressions for and . [3]
dx dx 2
(ii) Find the coordinates of the stationary point on the curve and determine its nature. [4]
(iii) Find the volume of the solid formed when the region enclosed by the curve, the x-axis and
the lines x = 1 and x = 2 is rotated completely about the x-axis. [6]
dy 2 d2 y 4 71
[ANS:  2 x  2 , 2  2  3 ; (1, 3);  ]
dx x dx x 5
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3. The equation of a curve is y = x2 − 3x + 4.
(i) Show that the whole of the curve lies above the x-axis. [3]
(ii) Find the set of values of x for which x2 − 3x + 4 is a decreasing function of x. [1]
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The equation of a line is y + 2x = k, where k is a constant.
(iii) In the case where k = 6, find the coordinates of the points of intersection of the line and
the curve. [3]
(iv) Find the value of k for which the line is a tangent to the curve. [3]
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3
[ANS: x < 1.5; (1, 8), (2, 2); 3 ]
4
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Differentiation 2 TanWC/SunwayCollege/2024
4.
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The diagram shows the cross-section of a hollow cone and a circular cylinder. The cone has
radius 6 cm and height 12 cm, and the cylinder has radius r cm and height h cm. The cylinder
just fits inside the cone with all of its upper edge touching the surface of the cone.
(i) Express h in terms of r and hence show that the volume, V cm3, of the cylinder is given by
V = 12π r2 − 2π r3. [3]
(ii) Given that r varies, find the stationary value of V. [4]
[ANS: V = 64π]
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k
5. A curve has equation y  . Given that the gradient of the curve is −3 when x = 2, find the value
x
of the constant k. [3]
[ANS: k = 12]
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Differentiation 3 TanWC/SunwayCollege/2024
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6. The equation of a curve is y 
5  2x
(i) Calculate the gradient of the curve at the point where x = 1. [3]
(ii) A point with coordinates (x, y) moves along the curve in such a way that the rate
of increase of y has a constant value of 0.02 units per second.
Find the rate of increase of x when x = 1. [2]
(iii) The region between the curve, the x-axis and the lines x = 0 and x = 1 is rotated
12
through 360 about the x-axis. Show that the volume obtained is . [5]
5
4
[ANS: ,0.015]
3
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7. The equation of a curve is y  2x  .
x2
dy d2 y
(i) Obtain expressions for and . [3]
dx dx 2
(ii) Find the coordinates of the stationary point on the curve and determine the nature
of the stationary point. [3]
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(iii) Show that the normal to the curve at the point (−2, −2) intersects the x-axis at the
point (−10, 0). [3]
(iv) Find the area of the region enclosed by the curve, the x-axis and the lines x = 1
and x = 2. [3]
16 48
[ANS: 2  3 , 4 , (2,6) min.,7]
x x
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Differentiation 4 TanWC/SunwayCollege/2024
8. The equation of a curve is y = (2x − 3)3 − 6x.
dy d2 y
(i) Express and in terms of x. [3]
dx dx 2
(ii) Find the x-coordinates of the two stationary points and determine the nature of each stationary
point. [5]
2
dy 2 d y
[ANS:  6  2 x  3  6 , 2  24  2 x  3 ; x = 1 (min), x = 2 (max)]
dx dx
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9. The equation of a curve C is y = 2x2 − 8x +9 and the equation of a line L is x + y =3.
(i) Find the x-coordinates of the points of intersection of L and C. [4]
(ii) Show that one of these points is also the stationary point of C. [3]
3
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[ANS:2, ; 2]
2
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10. The function f is such that f(x) = (3x + 2)3  5 for x ≥ 0.
(i) Obtain an expression for f ’(x) and hence explain why f is an increasing function. [3]
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(ii) Obtain an expression for f 1(x) and state the domain of f 1. [4]
3
x 5 2
[ANS: f 1  x   ; x  3]
3
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Differentiation 5 TanWC/SunwayCollege/2024
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11. The equation of a curve is y  5  .
x
(i) Show that the equation of the normal to the curve at the point P(2, 1) is 2y + x = 4. [4]
This normal meets the curve again at the point Q.
(ii) Find the coordinates of Q. [3]
(iii) Find the length of PQ. [2]
[ANS: (−8,6), 11.2]
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12.
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A wire, 80 cm long, is cut into two pieces. One piece is bent to form a square of side x cm and
the other piece is bent to form a circle of radius r cm (see diagram). The total area of the square
and the circle is A cm2.

(i) Show that A 

  4 x 2  160 x  1600 . [4]

(ii) Given that x and r can vary, find the value of x for which A has a stationary value. [4]
[ANS:11.2]
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Differentiation 6 TanWC/SunwayCollege/2024
13. A curve has equation y = (2x – 1)1 + 2x.
dy d2 y
(i) Find and . [3]
dx dx 2
(ii) Find the x-coordinates of the stationary points and, showing all necessary working,
determine the nature of each stationary point. [4]
2
dy 2 d y 3
[ANS:  2  2 x  1  2 , 2
 8  2 x  1 ; x = 0 (max), x = 1 (min) ]
dx dx
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14. The curve C1 has equation y = x2 − 4x + 7. The curve C2 has equation y2 = 4x + k, where k is a
constant. The tangent to C1 at the point where x = 3 is also the tangent to C2 at the point P.
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Find the value of k and the coordinates of P. [8]
3  1 
[ANS:  ,1 , k  4;  ,  1 , k  1 ]
2  2 
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Differentiation 7 TanWC/SunwayCollege/2024
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15. A curve is such that  3x 2  ax  b . The curve has stationary points at (−1, 2) and (3, k).
dx
Find the values of the constants a, b and k. [8]
[ANS: a = 6, b = 9, k = 30]
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16. The equation of a curve is y  2
.
x 3
dy
(i) Obtain an expression for . [2]
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dx
(ii) Find the equation of the normal to the curve at the point P(1, 3). [3]
(iii) A point is moving along the curve in such a way that the x-coordinate is increasing
at a constant rate of 0.012 units per second. Find the rate of change of the y-coordinate
as the point passes through P. [2]
dy 24 x 2 7
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[ANS:  , y  x  ,–0.018]
dx 
x2  3
2
 3 3
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Differentiation 8 TanWC/SunwayCollege/2024
17.*
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(i) The diagram shows the line 2y = x + 5 and the curve y = x2 − 4x + 7, which intersect at the
points A and B. Find
(a) the x-coordinates of A and B, [3]
(b) the equation of the tangent to the curve at B, [3]
(c) the acute angle, in degrees correct to 1 decimal place, between this tangent and the line
2y = x + 5. [3]
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(ii) Determine the set of values of k for which the line 2y = x + k does not intersect the curve
y = x2 − 4x + 7. [4]
3 31
[ANS:3, ; y = 2x  2; 36.9; k < ]
2 8
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m2  m1
[Note: tan   , where  is the angle between the two lines with gradients m1 and m2 ]
1  m2  m1
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Differentiation 9 TanWC/SunwayCollege/2024
18. The equation of a curve is y = x3 + x2 − 8x + 7. The curve has no stationary points in the interval
a < x < b. Find the least possible value of a and the greatest possible value of b. [4]
4
[ANS: a = 2, b ]
3
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19.
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The dimensions of a cuboid are x cm, 2x cm and 4x cm, as shown in the diagram.
(i) Show that the surface area S cm2 and the volume V cm3 are connected by the relation
2
S  7V 3 . [3]
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(ii) When the volume of the cuboid is 1000 cm3 the surface area is increasing at 2 cm2 s1.
Find the rate of increase of the volume at this instant. [4]
dV 30
[ANS: a = 2,  ]
dt 7
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Differentiation 10 TanWC/SunwayCollege/2024
20. A solid rectangular block has a square base of side x cm. The height of the block is h cm and
the total surface area of the block is 96 cm2.
(i) Express h in terms of x and show that the volume, V cm3, of the block is given by
1
V  24x  x 3 . [3]
2
Given that x can vary,
(ii) find the stationary value of V, [3]
(iii) determine whether this stationary value is a maximum or a minimum. [2]
[ANS:64, max.]
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1 3
21. The equation of a curve is y   2x  3  4x
6
dy
(i) Find . [3]
dx
(ii) Find the equation of the tangent to the curve at the point where the curve intersects the
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y-axis. [3]
1 3
(iii) Find the set of values of x for which  2x  3  4x is an increasing function of x. [3]
6
2 1 1
[ANS:  2x  3  4 ; 2y+9=10x; x  2 , x  ]
2 2
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Differentiation 11 TanWC/SunwayCollege/2024
22.

The diagram shows a metal plate consisting of a rectangle with sides x cm and y cm and a
quarter-circle of radius x cm. The perimeter of the plate is 60 cm.
(i) Express y in terms of x. [2]
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(ii) Show that the area of the plate, Acm2, is given by A = 30x − x2. [2]
Given that x can vary,
(iii) find the value of x at which A is stationary, [2]
(iv) find this stationary value of A, and determine whether it is a maximum or a minimum
value. [2]
x
[ANS: y  30  x  , A = 30x  x2, x=15, Max.]
4
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Differentiation 12 TanWC/SunwayCollege/2024
23. The equation of a curve is y = 3 + 4x − x2.
(i) Show that the equation of the normal to the curve at the point (3, 6) is 2y = x + 9. [4]
(ii) Given that the normal meets the coordinate axes at points A and B, find the coordinates
of the mid-point of AB. [2]
(iii) Find the coordinates of the point at which the normal meets the curve again. [4]
 9 9  1 3
[ANS:   ,  ,  , 4  ]
 2 4 2 4
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24. The length, x metres, of a Green Anaconda snake which is t years old is given approximately by
the formula x = 0.7 ( 2t  1) where 1 ≤ t ≤ 10.
Using this formula, find
dx
(i) , [2]
dt
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(ii) the rate of growth of a Green Anaconda snake which is 5 years old. [2]
1

[ANS: 0.7  2t 1 ,0.23]
2

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Differentiation 13 TanWC/SunwayCollege/2024
25.

The diagram shows an open rectangular tank of height h metres covered with a lid. The base
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1
of the tank has sides of length x metres and x metres and the lid is a rectangle with sides of
2
5 4
length x metres and x metres. When full the tank holds 4 m3 of water. The material from
4 5
which the tank is made is of negligible thickness. The external surface area of the tank together
with the area of the top of the lid is A m2.
3 24
(i) Express h in terms of x and hence show that A  x 2  . [5]
2 x
(ii) Given that x can vary, find the value of x for which A is a minimum, showing clearly
that A is a minimum and not a maximum. [5]
x
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[ANS: y  30  x  , 2, Min.]
4
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Differentiation 14 TanWC/SunwayCollege/2024
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26. A curve has equation y  x.
x 3
dy d2 y
(i) Find and . [2]
dx dx 2
(ii) Find the coordinates of the maximum point A and the minimum point B on the curve. [5]
1 2
[ANS:  2
1, 3
, min(4, 5), max(2,1) ]
 x  3  x  3
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27. The volume of a spherical balloon is increasing at a constant rate of 50 cm3 per second.
Find the rate of increase of the radius when the radius is 10 cm. [4]
4
[Volume of a sphere V   r 3 ]
3
1
[ANS: ]
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8
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28. A curve has equation y  and P(2, 2) is a point on the curve.
3x  4
(i) Find the equation of the tangent to the curve at P. [4]
(ii) Find the angle that this tangent makes with the x-axis. [2]
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[ANS: y  2 =3(x  2); 71.6]
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Differentiation 15 TanWC/SunwayCollege/2024
29.
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The diagram shows the dimensions in metres of an L-shaped garden. The perimeter of the
garden is 48 m.
(i) Find an expression for y in terms of x. [1]
(ii) Given that the area of the garden is A m2, show that A = 48x − 8x2. [2]
(iii) Given that x can vary, find the maximum area of the garden, showing that this is a
maximum value rather than a minimum value. [4]
1
[ANS: y  , A = 48x  8x2, 72]
648  8 x 
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2
30. The equation of a curve is y  4 x  .
x
dy
(i) Obtain an expression for . [3]
dx
(ii) A point is moving along the curve in such a way that the x-coordinate is increasing
at a constant rate of 0.12 units per second. Find the rate of change of the y-coordinate
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when x = 4. [2]
dy 2 1
[ANS:   1.5 ,0.105]
dx x x
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Differentiation 16 TanWC/SunwayCollege/2024
31. A solid rectangular block has a square base of side x cm. The height of the block is h cm and
the total surface area of the block is 96 cm2.
(i) Express h in terms of x and show that the volume, V cm3, of the block is given by
1
V  24 x  x 3 [3]
2
Given that x can vary,
(ii) find the stationary value of V, [3]
(iii) determine whether this stationary value is a maximum or a minimum. [2]
[ANS:64, max.]
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32. An oil pipeline under the sea is leaking oil and a circular patch of oil has formed on the surface
of the sea. At midday the radius of the patch of oil is 50 m and is increasing at a rate of 3 metres
per hour. Find the rate at which the area of the oil is increasing at midday. [4]
[ANS:330π]
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Differentiation 17 TanWC/SunwayCollege/2024
33.

The diagram shows a plan for a rectangular park ABCD, in which AB = 40 m and AD = 60 m.
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Points X and Y lie on BC and CD respectively and AX, XY and YA are paths that surround a
triangular playground. The length of DY is xm and the length of XC is 2x m.
(i) Show that the area, A m2, of the playground is given by A = x2 − 30x + 1200. [2]
(ii) Given that x can vary, find the minimum area of the playground. [3]
[ANS:x = 15, A = 975]
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34. It is given that f(x)  3
 x 3 , for x > 0. Show that f is a decreasing function. [3]
x
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35. It is given that f(x) = (2x – 5)3 + x, for xℝ. Show that f is an increasing function. [3]

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Differentiation 18 TanWC/SunwayCollege/2024
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
36. A curve has equation y = f(x) and is such that f ' ( x )  3 x 2  3x 2
 10 .
1
2
(i) By using the substitution u  x , or otherwise, find the values of x for which the curve
y = f(x) has stationary points. [4]
(ii) Find f "(x) and hence, or otherwise, determine the nature of each stationary point. [3]
(iii) It is given that the curve y = f(x) passes through the point (4, −7). Find f (x). [4]
1 3 3 1
1 3  3 
[ANS: , 9, x 2  x 2 , max, min, 2x 2  6x 2 10x  5 ]
9 2 2
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37. A function f is defined by f  x   , for x ≥ 1.
1  3x
(i) Find an expression for f ’(x). [2]
(ii) Determine, with a reason, whether f is an increasing function, a decreasing function or
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neither. [1]
(iii) Find an expression for f −1(x), and state the domain and range of f −1. [5]
15 x  5 1
[ANS: 2
,increasing, , f ≥ 1, 2.5≤ x < 0]
(1  3 x) 3x
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Differentiation 19 TanWC/SunwayCollege/2024
38.

The inside lane of a school running track consists of two straight sections each of length
x metres, and two semicircular sections each of radius r metres, as shown in the diagram.
The straight sections are perpendicular to the diameters of the semicircular sections. The
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perimeter of the inside lane is 400 metres.
(i) Show that the area, A m2, of the region enclosed by the inside lane is given by
A = 400r −  r2. [4]
(ii) Given that x and r can vary, show that, when A has a stationary value, there are no
straight sections in the track. Determine whether the stationary value is a maximum
or a minimum. [5]
200
[ANS: r  , max]

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Differentiation 20 TanWC/SunwayCollege/2024
39.*.

In the diagram, S is the point (0, 12) and T is the point (16, 0). The point Q lies on ST,
between S and T, and has coordinates (x, y). The points P and R lie on the x-axis and
y-axis respectively and OPQR is a rectangle.
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3
(i) Show that the area, A, of the rectangle OPQR is given by A  12 x  x 2 . [3]
4
(ii) Given that x can vary, find the stationary value of A and determine its nature. [4]
[ANS: A = 48 max]
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k2
40. A curve has equation y   x , where k is a positive constant. Find, in terms of k, the values
x2
of x for which the curve has stationary points and determine the nature of each stationary point.
[8]
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[ANS: x = 2 + k (min); x = 2  k (max)]
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Differentiation 21 TanWC/SunwayCollege/2024
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41. A curve has equation y  2
. Find the equation of the tangent to the curve at the point
 3 x  1
where the line x = −1 intersects the curve. [5]
[ANS: y = 3x + 4]
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42. The base of a cuboid has sides of length x cm and 3x cm. The volume of the cuboid is 288 cm3.
768
(i) Show that the total surface area of the cuboid, A cm2, is given by A  6 x 2  .
x
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(ii) Given that x can vary, find the stationary value of A and determine its nature. [5]
[ANS: A = 288; min]
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Differentiation 22 TanWC/SunwayCollege/2024
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43. A curve has equation y  .
3  2x
dy
(i) Find . [2]
dx
A point moves along this curve. As the point passes through A, the x-coordinate is increasing at
a rate of 0.15 units per second and the y-coordinate is increasing at a rate of 0.4 units per second.
(ii) Find the possible x-coordinates of A. [4]
dy 2
[ANS:  24  3  2 x  ; x = 0 or 3]
dx
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44. (i) Express 9x2 − 12x + 5 in the form_(ax + b)2 + c. [3]
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(ii) Determine whether 3x3 − 6x2 + 5x − 12 is an increasing function, a decreasing function
or neither. [3]
2
[ANS: (3x 2) + 1; x ℝ increasing function]
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Differentiation 23 TanWC/SunwayCollege/2024
45*.

The diagram shows the curve y = 2x2 and the points X(−2, 0) and P(p, 0). The point Q lies
on the curve and PQ is parallel to the y-axis.
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(i) Express the area, A, of triangle XPQ in terms of p. [2]
The point P moves along the x-axis at a constant rate of 0.02 units per second and Q moves
along the curve so that PQ remains parallel to the y-axis.
(ii) Find the rate at which A is increasing when p = 2. [3]
1
[ANS: A  2  p   2 p 2 , 0.4]
2
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Differentiation 24 TanWC/SunwayCollege/2024
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46.* The function f is defined by f(x) =  for x > 1.
x  1  x  12
(i) Find f ’(x). [3]
(ii) State, with a reason, whether f is an increasing function, a decreasing function or neither. [1]
1 1
The function g is defined by g(x) =  for x < 1
x  1  x  12
(iii) Find the coordinates of the stationary point on the curve y = g(x). [4]
2 3  1
[ANS:   x  1  2  x  1 ; decreasing;  3,   ]
 4
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Differentiation 25 TanWC/SunwayCollege/2024
47.

Fig. 1 shows an open tank in the shape of a triangular prism. The vertical ends ABE and DCF
are identical isosceles triangles. Angle ABE = angle BAE = 30. The length of AD is 40 cm.
The tank is fixed in position with the open top ABCD horizontal. Water is poured into the tank at
a constant rate of 200 cm3 s−1. The depth of water, t seconds after filling starts, is h cm (see Fig. 2).
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(i) Show that, when the depth of water in the tank is h cm, the volume, V cm3, of water in the
tank is given by V = (403)h2. [3]
(ii) Find the rate at which h is increasing when h = 5. [3]
[ANS: 0.289]
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48. (i) Express 3x2 − 6x + 2 in the form a(x + b)2 + c, where a, b and c are constants. [3]

(ii) The function f, where f(x) = x3 − 3x2 + 7x − 8, is defined for xℝ. Find f ’(x) and state,
with a reason, whether f is an increasing function, a decreasing function or neither. [3]
[ANS: 3( x  1)2  1 ; 3( x  1)2  4 , increasing ]
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Differentiation 26 TanWC/SunwayCollege/2024
 4
49*. A curve has equation y  3x  and passes throught the points A(1, 1) and B(4, 11). At
x
each of the points C and D on the curve, the tangent is parallel to AB. Find the equation of the
perpendicular bisector of CD. [7]
1
[ANS: y   x ]
2
[9709 S2016_11]
TA
50.
N
A farmer divides a rectangular piece of land into 8 equal-sized rectangular sheep pens as shown
in the diagram. Each sheep pen measures x m by y m and is fully enclosed by metal fencing.
The farmer uses 480 m of fencing.
(i) Show that the total area of land used for the sheep pens, A m2, is given by
A = 384x − 9.6x2. [3]
W
(ii) Given that x and y can vary, find the dimensions of each sheep pen for which the value
of A is a maximum. (There is no need to verify that the value of A is a maximum.) [3]
[ANS:x = 20, y = 24]
[9709 S2016_11]
C

Differentiation 27 TanWC/SunwayCollege/2024
51. A curve has equation y = 8x + (2x  1)1. Find the values of x at which the curve has a
stationary point and determine the nature of each stationary point, justifying your answers.
[7]
[9709 S2016_13]

3
10
TA
2
52*. The point P(x, y) is moving along the curve y  x  x 2  5 x in such a way that the rate of
3
change of y is constant. Find the values of x at the points at which the rate of change of x is equal
to half the rate of change of y. [7]
9
[ANS: , 1]
4
[9709 S2016_13]
N
53. The point A(2, 2) lies on the curve y = x2 − 2x + 2.
(i) Find the equation of the tangent to the curve at A. [3]
The normal to the curve at A intersects the curve again at B.
W
(ii) Find the coordinates of B. [4]
The tangents at A and B intersect each other at C.
(iii) Find the coordinates of C. [4]
 1 13 
[ANS:y=2x2;   ,  ]
 2 4
[9709 M2017_12]
C

Differentiation 28 TanWC/SunwayCollege/2024
54. The horizontal base of a solid prism is an equilateral triangle of side x cm. The sides of the
prism are vertical. The height of the prism is h cm and the volume of the prism is 2000 cm3.
(i) Express h in terms of x and show that the total surface area of the prism, Acm2, is given by
3 2 24000 1
A x  x . [3]
2 3
(ii) Given that x can vary, find the value of x for which A has a stationary value. [3]
(iii) Determine, showing all necessary working, the nature of this stationary value. [2]
[ANS: x=20; Min.]
[9709 S2017_11]
TA

55. The equation of a curve is y  8 x  2 x.

N
(i) Find the coordinates of the stationary point of the curve. [3]
d2 y
(ii) Find an expression for and hence, or otherwise, determine the nature of the stationary
dx 2
point. [2]
W
(iii) Find the values of x at which the line y = 6 meets the curve. [3]
d2 y 3
[ANS: (4, 8); 2
 2 x 2 (max); x = 1 or 9]
dx
[9709 S2017_12]
C

Differentiation 29 TanWC/SunwayCollege/2024
56. Machines in a factorymake cardboard cones of base radius r cmand vertical height h cm. The
1
volume, V cm3, of such a cone is given by V   r 2 h .The machines produce cones for which
3
h + r = 18.
1
(i) Show that V  6 r 2   r 3 . [1]
3
(ii) Given that r can vary, find the non-zero value of r for which V has a stationary value and
show that the stationary value is a maximum. [4]

(iii) Find the maximum volume of a cone that can be made by these machines. [1]
[ANS: r =12 (max); 288]
[9709 W2017_11]
TA
N
dy
57. A curve is such that   x2  5x  4 .
dx
(i) Find the x-coordinate of each of the stationary points of the curve. [2]
d2 y
(ii) Obtain an expression for 2 and hence or otherwise find the nature of each of the
dx
stationary points. [3]
W
(iii) Given that the curve passes through the point (6, 2), find the equation of the curve. [4]
d2 y x3 5x 2
[ANS: x = 1 (min), x = 4 (max); r =12 (max);  2 x  5 ; y     4x  8 ]
dx 2 3 2
[9709 W2017_12]
C

Differentiation 30 TanWC/SunwayCollege/2024
 3
58. The function f is such that f  x    2 x  1 2  6 x for 1
2  x  k , where k is a constant.
Find the largest value of k for which f is a decreasing function. [5]
5
[ANS: k  ;]
2
[9709 W2017_13]

3
1 2
TA
59. A curve has equation y  x  4 x 2  8x .
2
(a) Find the x-coordinates of the stationary points. [5]
2
d y
(b) Find . [1]
dx 2
(b) Find, showing all necessary working, the nature of the stationary point. [2]
2 1
d y 
[ANS: max. 4, min.16; 2  1  3 x 2 ; min.(0, 2)]
dx
[9709 M2018_12]
N
W
5
60. A point is moving along the curve y  2 x  in such a way that the x-coordinate is increasing
x
at a constant rate of 0.02 units per second. Find the rate of change of the y-coordinate when x = 1.
[4]
C
[ANS:0.06]
[9709 S2018_11]

Differentiation 31 TanWC/SunwayCollege/2024
61. (i) The tangent to the curve y = x3 − 9x2 + 24x − 12 at a point A is parallel to the line y = 2 − 3x.
Find the equation of the tangent at A. [6]

(ii) The function f is defined by f(x) = x3 − 9x2 + 24x − 12 for x > k, where k is a constant.
Find the smallest value of k for f to be an increasing function. [2]
[ANS:y6=3(x3); k=4]
[9709 S2018_13]
TA
1 1  1
62. A curve has equation y   4 x  3 . The point A on the curve has coordinates  1,  .
2  2
(i) (a) Find and simplify the equation of the normal through A. [5]
N
(b) Find the x-coordinate of the point where this normal meets the curve again. [3]
(ii) A point is moving along the curve in such a way that as it passes through A its x-coordinate
is decreasing at the rate of 0.3 units per second. Find the rate of change of its y-coordinate
at A. [2]
1 1
W
[ANS: y  x ; x   ; 0.6]
2 4
[9709 W2018_11]
C

Differentiation 32 TanWC/SunwayCollege/2024
63*.

The diagram shows part of the curve y = x(9 − x2) and the line y = 5x, intersecting at the origin
O and the point R. Point P lies on the line y = 5x between O and R and the x-coordinate of P is t.
TA
Point Q lies on the curve and PQ is parallel to the y-axis.
(i) Express the length of PQ in terms of t, simplifying your answer. [2]
(ii) Given that t can vary, find the maximum value of the length of PQ. [3]
16 3
[ANS: PQ=4t  t3 ; ]
9
[9709 W2018_12]
N
W
64*. The function f is defined by f(x) = x3 + 2x2 − 4x + 7 for x ≥ −2. Determine, showing all
necessary working, whether f is an increasing function, a decreasing function or neither. [4]
2 2 2
[ANS:x=2 & x= neither; 2<x< decreasing; x > increasing]
3 3 3
[9709 W2018_13]
C

Differentiation 33 TanWC/SunwayCollege/2024
65. A curve has equation y = (2x − 1)−1 + 2x.
dy d2 y
(i) Find and . [3]
dx dx 2
(ii) Find the x-coordinates of the stationary points and, showing all necessary working,
determine the nature of each stationary point. [4]
2
dy 2 d y 3
[ANS:  2  2 x  1  2 ; 2  8  2 x  1 ; min x  1, max x  0 ]
dx dx
[9709 M2019_12]
TA
N
66*. The curve C1 has equation y = x2 − 4x + 7. The curve C2 has equation y2 = 4x + k, where k is a
constant. The tangent to C1 at the point where x = 3 is also the tangent to C2 at the point P.
Find the value of k and the coordinates of P. [8]
3 
[ANS: k=5,  ,1 ]
W
2 
[9709 S2019_12]
C

Differentiation 34 TanWC/SunwayCollege/2024
67.

The diagram shows a solid cone which has a slant height of 15 cm and a vertical height of h cm.
1
(i) Show that the volume, V cm3, of the cone is given by V    225h  h3  . [2]
TA
3
1
[The volume of a cone of radius r and vertical height h is V   r 2 h .]
3
(ii) Given that h can vary, find the value of h for which V has a stationary value.
Determine, showing all necessary working, the nature of this stationary value. [5]
[ANS: h  5 3 , max.]
[9709 W2019_12]
N
W
68*. The equation of a curve is y = x3 + x2 − 8x + 7. The curve has no stationary points in the interval
a < x < b. Find the least possible value of a and the greatest possible value of b. [4]
C
4
[ANS: a  2, b ]
3
[9709 W2019_13]

Differentiation 35 TanWC/SunwayCollege/2024
69. A curve has equation y = x2 − 2x − 3. A point is moving along the curve in such a way that at P
the y-coordinate is increasing at 4 units per second and the x-coordinate is increasing at 6 units
per second.
Find the x-coordinate of P. [4]
4
[ANS: ]
3
[9709 M2020_12]
TA
1
70. The function f is defined by f  x    x 2 for x < −1.
3x  2
Determine whether f is an increasing function, a decreasing function or neither. [3]
[ANS:decreasing]
[9709 M2020_12]
N
71. The equation of a curve is y = (3 − 2x)3 + 24x.
dy d2 y
(a) Find expressions for and . [4]
W
dx dx 2
(b) Find the coordinates of each of the stationary points on the curve. [3]
(c) Determine the nature of each stationary point. [2]
dy 2 d2 y 1  5 
[ANS:  6  3  2 x   24 ; 2  24  3  2 x  ; min  , 20  , max  ,52  ]
dx dx 2  2 
[9709 S2020_11]
C

Differentiation 36 TanWC/SunwayCollege/2024
72. A weather balloon in the shape of a sphere is being inflated by a pump. The volume of the balloon
is increasing at a constant rate of 600 cm3 per second. The balloon was empty at the start of
pumping.
(a) Find the radius of the balloon after 30 seconds. [2]
(b) Find the rate of increase of the radius after 30 seconds. [3]
[ANS:16.3; 0.181]
[9709 S2020_12]
TA

73. The equation of a curve is y = 54x  (2x  7)3.

N
dy d2 y
(a) Find and . [4]
dx dx 2
(b) Find the coordinates of each of the stationary points on the curve. [3]
(c) Determine the nature of each stationary point. [2]
dy 2 d2 y
[ANS:  54  6  2 x  7  ; 2  24  2 x  7  ; min  2,135  , max  5, 243 ]
W
dx dx
[9709 S2020_12]
C

Differentiation 37 TanWC/SunwayCollege/2024
74. A point P is moving along a curve in such a way that the x-coordinate of P is increasing at a
1
constant rate of 2 units per minute. The equation of the curve is y   5 x  1 2 .
(a) Find the rate at which the y-coordinate is increasing when x = 1. [4]
5
(b) Find the value of x when the y-coordinate is increasing at units per minute. [3]
8
5
[ANS: ; 13]
2
[9709 S2020_13]
TA
75. Air is being pumped into a balloon in the shape of a sphere so that its volume is increasing at a
constant rate of 50 cm3 s−1.
Find the rate at which the radius of the balloon is increasing when the radius is 10 cm. [3]
1
[ANS:  ]
8
[9709 W2020_11]
N
W
76. The equation of a curve is y  2  25  x 2 .
4
Find the coordinates of the point on the curve at which the gradient is . [5]
3
[ANS: (4, 5)]
[9709 W2020_12]
C

Differentiation 38 TanWC/SunwayCollege/2024
 1 1
77. The equation of a curve is y  2 x  1  for x   .
2x 1 2
dy d2 y
(a) Find and . [3]
dx dx 2
(b) Find the coordinates of the stationary point and determine the nature of the stationary
point. [5]
2
dy 2 d y 3
[ANS:  4  2 x  1 ; 2  8  2 x  1 ; min.(0, 2)]
dx dx
[9709 W2020_13]
TA
78. The equation of a curve is y  2 3x  4  x .
(a) Find the equation of the normal to the curve at the point (4, 4), giving your answer in the
form y = mx + c. [5]
N
(b) Find the coordinates of the stationary point. [3]
(c) Determine the nature of the stationary point. [2]
(d) Find the exact area of the region bounded by the curve, the x-axis and the lines x = 0 and
x = 4. [4]
W
 5 13  8
[ANS:y=4x12; max.  ,  ; 16 ]
3 3  9
[9709 S2021_11]
C

Differentiation 39 TanWC/SunwayCollege/2024
79. The equation of a curve is y   x  3 x  1  3 . The following points lie on the curve.
Non-exact values are rounded to 4 decimal places.
A(2, k) B(2.9, 2.8025) C (2.99, 2.9800) D(2.999, 2.9980) E (3, 3)
(a) Find k, giving your answer correct to 4 decimal places. [1]
(b) Find the gradient of AE, giving your answer correct to 4 decimal places. [1]
The gradients of BE, CE and DE, rounded to 4 decimal places, are 1.9748, 1.9975 and 1.9997
respectively.
(c) State, giving a reason for your answer, what the values of the four gradients suggest about
the gradient of the curve at the point E. [2]
[ANS: 1.2679; 1.7321; 2 because the gradient at E is the limit of the gradients of the
chords as the x-value tends to 3 or ꝺx tends to 0.]
TA
[9709 S2021_12]
N
W
3
1 1
80. The function f is defined by f  x    2 x  1 2  2 x for  x  a . It is given that f is a decreasing
3 2
function.
Find the maximum possible value of the constant a. [4]
C
1
[ANS: a  2 ]
2
[9709 S2021_13]

Differentiation 40 TanWC/SunwayCollege/2024
 4
81. A curve has equation y = f(x), and it is given that f '  x   2 x 2  7  .
x2
1
(a) Given that f 1   find f(x). [4]
3
(b) Find the coordinates of the stationary points on the curve. [5]
(c) Find f ′′(x). [1]
(d) Hence, or otherwise, determine the nature of each of the stationary points. [2]
2 3 4  14   26 
[ANS: f  x   x  7 x   2 ; min.  2,   , max .  2,  ]
3 x  3  3 
[9709 W2021_11]
TA
82. A curve has equation y = f(x) and it is given that
2
1  2
f '  x    x  k   1  k  ,
2 
N
where k is a constant. The curve has a minimum point at x = 2.
(a) Find f ′′(x) in terms of k and x, and hence find the set of possible values of k. [3]
 1
It is now given that k = −3 and the minimum point is at  2, 3  .
 2
(b) Find f(x). [4]
W
(c) Find the coordinates of the other stationary point and determine its nature. [4]
3
1  2 x  1
[ANS: f "  x     x  k  , k<1; f  x     3 ;  10,   ]
2  1  4  2
 x  3
2 
[9709 W2021_13]
C

Differentiation 41 TanWC/SunwayCollege/2024
83. (a) Express 5y2 − 30y + 50 in the form 5(y + a)2 + b, where a and b are constants. [2]
(b) The function f is defined by f(x) = x5 − 10x3 + 50x for xℝ.
Determine whether f is an increasing function, a decreasing function or neither. [3]
2
[ANS: 5  y  3  5 ; increasing function]
[9709 W2021_13]
TA
84. It is given that a curve has equation y = k(3x − k)1 + 3x, where k is a constant.
(a) Find, in terms of k, the values of x at which there is a stationary point. [4]
The function f has a stationary value at x = a and is defined by
3
f(x) = 4(3x – 4)1 + 3x for x  .
2
(b) Find the value of a and determine the nature of the stationary value. [3]
(c) The function g is defined by g(x) = −(3x + 1)1 + 3x for x ≥ 0.
N
Determine, making your reasoning clear, whether g is an increasing function, a decreasing
function or neither. [2]
k k
[ANS: x  ; a = 2 (min); increasing function]
3
[9709 M2022_12]
W
C

Differentiation 42 TanWC/SunwayCollege/2024
85.

1 4 2 5
[ANS: min. point ; y  x  2; ]
TA
2 x 4
[9709 S2022_11]
N
1 1
86. The equation of a curve is y  3 x  1  4  3x  1 2 for x   .
3
2
dy d y
(a) Find and . [3]
W
dx dx 2
(b) Find the coordinates of the stationary point of the curve and determine its nature. [4]
dy 1 d2 y 3
[ANS:  3  6  3x  1 2 , 2  9  3x  1 2 ; (1, 4) ~min]
dx dx
[9709 S2022_12]
C

Differentiation 43 TanWC/SunwayCollege/2024
87.

1
[ANS: ;  0, 14  ]
24
[9709 S2022_13]
TA
N
1
88. A curve has equation y  ax 2  2 x , where x > 0 and a is a constant. The curve has a stationary
point at the point P, which has x-coordinate 9.
W
Find the y-coordinate of P. [5]
[ANS:y = 18]
[9709 W2022_11]
C

Differentiation 44 TanWC/SunwayCollege/2024
89.

[ANS:0.24]
[9709 W2022_13]
TA
N
90.

[ANS: x = 3]
W
[9709 M2023_12]
C

Differentiation 45 TanWC/SunwayCollege/2024
91.

[ANS: y  2 x  2 x  3 ;  14 ,3 12  ; max. pt.]

[9709 M2023_12]
TA

92.
N
W
[ANS: 0.102; V = 0.69900]
[9709 S2023_11]
C

Differentiation 46 TanWC/SunwayCollege/2024
93.

[ANS: a = 4; min. pt.; y = 2x3  15x2 + 24x + 1; (1, 12)]

[9709 S2023_11]
TA

94.
N
W
dy 1 4k 2  1 1
[ANS:  2k  4 x  1 2 ; x  ; y  66.5    x  12  ]
dx 4 2
[9709 S2023_12]
C

Differentiation 47 TanWC/SunwayCollege/2024
95.
TA
dy 3 dx 5
[ANS: y = 2x + 11;  ;  ]
dt 2 dt 2
[9709 S2023_13]
N
W
96.

[ANS: 3x  8y + 2 = 0]
C
[M2024_12]

Differentiation 48 TanWC/SunwayCollege/2024
97.
TA
N
[ANS: 2 < x < 0, 0< x < 2; 6.51]
[9709 S2024_11]
W
C

Differentiation 49 TanWC/SunwayCollege/2024
98.

[ANS: ]
[9709 S2024_12]
TA
99.
N
[ANS: ]
[9709 S2024_12]

5
[ANS: 0.556; ; 2x  13y + 247 = 0 ]
9
W
[9709 S2024_12]
C

Differentiation 50 TanWC/SunwayCollege/2024
 1
99. The equation of a curve is y  2 x 2  3.
2x
(a) Find the coordinates of the stationary point. [3]
(b) Determine the nature of the stationary point. [2]
(c) For positive values of x, determine whether the curve shows a function that is increasing,
decreasing or neither. Give a reason for your answer. [2]
1 9
[ANS: x   , y  ; min. pt.; increasing]
2 2
[9709 S2024_13]
TA
N
100.
W
1 5 28
[ANS: x   , x  ; k  ]
3 3 9
[9709 W2024_11]
C

Differentiation 51 TanWC/SunwayCollege/2024
101.
TA
1 4 5
[ANS: y   x  1 ; f ( x)  3  2 x  3 3  6 x 3  3 ; Decreasing]
18
[9709 W2024_12]
N
W
C

Differentiation 52 TanWC/SunwayCollege/2024
102.

dy 1  12 d2 y 1 3  1 11 
TA
[ANS:  kx  8 x, 2   kx 2  8 ; max.  ,  ; k = 3]
dx 2 dx 4 4 4 
[9709 W2024_13]
N
W
C

Differentiation 53 TanWC/SunwayCollege/2024