9709/May June/33/2017/Q5

A curve has equation y = 2/3 ln(1 + 3 cos² x) for 0 ≤ x ≤ ½

(i) Express dy/dx in terms of tan x.

(ii) Hence find the x-coordinate of the point on the curve where the gradient is −1. Give your answer correct to 3 significant figures.

Solution:

Reference: PYQ - May/Jun 2017 Paper 33 Q5

Exercise 1 from note (Question 3)

Find the area of the shaded region for each of the following graphs.

Solution:

Vector: The intersection of two lines (Exercise 2, Question 3)

In this question the origin is taken to be at a harbour and the unit vectors i and j to have lengths of 1 km in the directions E and N.

A cargo vessel leaves the harbour and its position vector t hours later is given by r1 = 12ti + 16tj.

A fishing boat is trawling nearby and its position at time t is given by r2 = (10 - 3t)i + (8 + 4t)j.

Solution:

 

9709/May June/3/2002/Q8

The straight line l passes through the points A and B whose position vectors are i + k and 4i - j + 3k respectively. The plane p has equation x + 3y - 2z = 3

i) Given that l intersects p, find the position vector of the point of intersection.

ii) Find the equation of the plane which contains l and is perpendicular to p, giving your answer in the form ax + by + cz = 1.

Solution:

Reference: PYQ - May/Jun 2002 Paper 3 Q8

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9709/Oct Nov/33/2010/Q10

The polynomial p(z) is defined by

p(z) = z^3 + mz^2 + 24z + 32,

where m is a constant. It is given that (z + 2) is a factor of p(z).

(i) Find the value of m.

(ii) Hence, showing all your working, find

(a) the three roots of the equation p(z) = 0,

(b) the six roots of the equation p(z^2) = 0.

Solution:

Reference: PYQ - Oct/Nov 2010 Paper 33 Q10

Fixed point iteration on Scientific Calculator

Please refer to the video below. 

9709/Oct Nov/43/2011/Q6

The diagram shows a ring of mass 2 kg threaded on a fixed rough vertical rod. A light string is attached to the ring and is pulled upwards at an angle of 30◦ to the horizontal. The tension in the string is T N. The coefficient of friction between the ring and the rod is 0.24. Find the two values of T for which the ring is in limiting equilibrium. 

Solution:

Reference: PYQ - Oct/Nov 2011 Paper 43 Q6

9709/Oct Nov/43/2011/Q4

ABC is a vertical cross-section of a surface. The part of the surface containing AB is smooth and A is 4m higher than B. The part of the surface containing BC is horizontal and the distance BC is 5m (see diagram). A particle of mass 0.8 kg is released from rest at A and slides along ABC. Find the speed of the particle at C in each of the following cases.

(i) The horizontal part of the surface is smooth.

(ii) The coefficient of friction between the particle and the horizontal part of the surface is 0.3.

Solution:

(i) 

mgh = 0.8 ×  10 × 4   = 32

For using ½ mv^2 = PE

[½ 0.8v^2 = 32]

Speed at C, v = 8.94 ms^–1

(ii) 

R = mg

= 0.8 x 10

8 N

Friction force, 

F = μ mg

   = 0.3 (8)

   = 2.4 N

Force from B to C = 2.4N (opposite direction of friction force)

Work done from B to C 

= 2.4 x 5

= 12 J

Conservation of Energy

½ mv^2 + 12 = PE

½ mv^2 = 32 - 12

Speed at C, v = 7.07 ms^-1

Reference: PYQ - Oct/Nov 2011 Paper 43 Q4

9709/Oct Nov/13/2016/Q8

(i) Express 4x² + 12x + 10 in the form (ax + b)² + c, where a, b and c are constants.

(ii) Functions f and g are both defined for x > 0. It is given that f(x) = x² + 1 and fg(x) = 4x² + 12x + 10. Find g(x).

(iii) Find (fg)⁻¹ (x) and give the domain of (fg)⁻¹.

Solution:

Reference: PYQ - Oct/Nov 2016 Paper 13 Q8

9709/May June/2011/42/Q6

A small smooth ring R, of mass 0.6 kg, is threaded on a light inextensible string of length 100 cm. One end of the string is attached to a fixed point A. A small bead B of mass 0.4 kg is attached to the other end of the string, and is threaded on a fixed rough horizontal rod which passes through A. The system is in equilibrium with B at a distance of 80 cm from A (see diagram).

(i) Find the tension in the string. 

(ii) Find the frictional and normal components of the contact force acting on B. 

(iii) Given that the equilibrium is limiting, find the coefficient of friction between the bead and the rod.

Solution:

Reference: PYQ - May/Jun 2011 Paper 42 Q6

9709/May June/2011/42/Q3

The velocity-time graph shown models the motion of a parachutist falling vertically. There are four
stages in the motion:
• falling freely with the parachute closed,
• decelerating at a constant rate with the parachute open,
• falling with constant speed with the parachute open,
• coming to rest instantaneously on hitting the ground.

(i) Show that the total distance fallen is 1048m.
The weight of the parachutist is 850N.
(ii) Find the upward force on the parachutist due to the parachute, during the second stage.

Solution:

Reference: PYQ - May/Jun 2011 Paper 42 Q3

9709/May June/2011/42/Q2

An object of mass 8 kg slides down a line of greatest slope of an inclined plane. Its initial speed at the top of the plane is 3ms−1 and its speed at the bottom of the plane is 8ms−1. The work done against the resistance to motion of the object is 120 J. Find the height of the top of the plane above the level of the bottom.

Solution:

Reference: PYQ - May/Jun 2011 Paper 42 Q2

9709/Oct Nov/2003/3/Q9

Compressed air is escaping from a container. The pressure of the air in the container at time t is P, and the constant atmospheric pressure of the air outside the container is A. The rate of decrease of P is proportional to the square root of the pressure difference (P − A). Thus the differential equation connecting P and t is

where k is a positive constant.
(i) Find, in any form, the general solution of this differential equation.
(ii) Given that P = 5A when t = 0, and that P = 2A when t = 2, show that k =√A.
(iii) Find the value of t when P = A.
(iv) Obtain an expression for P in terms of A and t.

Solution:

Reference: PYQ - Oct/Nov 2003 Paper 3 Q9

9709/May Jun/2008/3/Q10

The points A and B have position vectors, relative to the origin O, given by

OA = i + 2j + 3k  and  OB = 2i + j + 3k.

The line l has vector equation

r = (1 − 2t)i + (5 + t)j + (2 − t)k.

(i) Show that l does not intersect the line passing through A and B.

(ii) The point P lies on l and is such that angle PAB is equal to 60^o. Given that the position vector

of P is (1 − 2t)i + (5 + t)j + (2 − t)k, show that 3t² + 7t + 2 = 0. Hence find the only possible

position vector of P.

Solution:

Reference: PYQ - May/Jun 2008 Paper 3 Q10