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CAIE IGCSE Additional Mathematics Calculus Question Bank

CAIE IGCSE Additional Mathematics Calculus Question Bank
Cambridge IGCSE Additional Mathematics 0606 syllabus for exams in 2028, 2029 and 2030Exams 2028-2030

Practise differentiation and integration with standard and composite functions, then apply derivatives, areas and kinematics to optimisation and rate problems.

Covered syllabus points

14. Calculus

Question 6

A curve has equation y=ln⁡(5−3x)y=\ln(5-3x), where x< rac53. The normal to the curve at the point where x=-5 cuts the x-axis at the point P.
Find the equation of the normal and the x-coordinate of P.

Question 8

[Maximum number: 13]

In this question, the units are metres and seconds.
A particle P is travelling in a straight line through a fixed point O.
At time t its acceleration, a, is given by a=(2t−3)2a=(2 t-3)^{2}, where t⩾0t \geqslant 0.
When t=3, P has a velocity of 6.

Question 8(a)(i)

(a)

Find an expression for the velocity, v, of P at time t.

[ 3 ]

Question 8(a)(ii)

(b)

Find the time when P is at rest.

When t=52t=\frac{5}{2}, the displacement of P from O is 4.

[ 3 ]

Question 8(a)(iii)

(c)

Find the displacement of P from O when t=3.

[ 4 ]

Question 8(b)

(d)

Use calculus to find the approximate change in v when t increases from 52\frac{5}{2} by the small amount 0.02.

[ 3 ]

Question 6

[Maximum number: 8]

A curve has equation y=(x2−1x2+1)4y=\left(\frac{x^{2}-1}{x^{2}+1}\right)^{4}.

Question 6(a)

(a)

Show that dy dx\frac{\mathrm{d} y}{\mathrm{~d} x} can be written as Ax(x2−1)3(x2+1)5\frac{A x\left(x^{2}-1\right)^{3}}{\left(x^{2}+1\right)^{5}}, where A is a positive integer to be found.

[ 5 ]

Question 6(b)(i)

(b)

Show that the curve has stationary points where x=-1, x=0 and x=1.

[ 1 ]

Question 6(b)(ii)

(c)

Use the first derivative test to determine which two stationary points have the same nature and state whether they are maximum or minimum points.

[ 2 ]

Question 4(b)

[Maximum number: 3]

Show that ∫π/3π/2sec⁡2(12x) dx=2(1−33)\int_{\pi/3}^{\pi/2}\sec^2\left(\frac12x\right)\,dx=2\left(1-\frac{\sqrt3}{3}\right).

Question 8

[Maximum number: 6]

It is given that y=ln⁡(3x2+16)x+2y=\frac{\ln \left(3 x^{2}+16\right)}{x+2}.

Question 8(a)

(a)

Find dy dx\frac{\mathrm{d} y}{\mathrm{~d} x} when x=0.

Give your answer in the form ln⁡p\ln p, where p is a constant.

[ 5 ]

Question 8(b)

(b)

Given that x increases from 0 to h, where h is small, write down the approximate change in y.

[ 1 ]

Question 7

Question 7(a)

(a)

Given that y=xcos⁡2xy=x \cos 2 x, find dy dx\frac{\mathrm{d} y}{\mathrm{~d} x}.

[ 2 ]

Question 7(b)

(b)

Hence find ∫xsin⁡2x dx\int x \sin 2 x \mathrm{~d} x.

[ 4 ]

Question 11(b)

[Maximum number: 5]

A cylinder has radius r cmr \mathrm{~cm} and height h cmh \mathrm{~cm}.
The total surface area, including the two ends, is A cm2A \mathrm{~cm}^{2}.
The volume of the cylinder is 330 cm3330 \mathrm{~cm}^{3}.

Given that r can vary, find the value of r that gives a stationary value for A and show that this value is a minimum.

Question 12

Question 12(a)

Question 12(b)

Question 12

[Maximum number: 7]

It is given that y=xe3x+2y=x \mathrm{e}^{3 x+2}.

Question 12(a)

(a)

Find dy dx\frac{\mathrm{d} y}{\mathrm{~d} x}.

[ 3 ]

Question 12(b)

(b)

Hence find ∫xe3x+2 dx\int x\mathrm{e}^{3x+2}\,\mathrm{d}x.

[ 4 ]

Question 11

[Maximum number: 8]

In this question, all lengths are in centimetres.
The diagram shows a cone of base radius x, height y, and sloping edge x2+y2\sqrt{x^2+y^2}. The volume of the cone is 10π cm310\pi\,\mathrm{cm}^3.

Figure for Question 11 — CAIE IGCSE Additional Math

Question 11(a)

(a)

Show that the curved surface area, S, of the cone is given by S= rac{\pi\sqrt{x^6+900}}{x}.

[ 3 ]

Question 11(b)

(b)

Given that x can vary and that S has a minimum value, find the value of x for which S is a minimum.

[ 5 ]