9709/May June/31/2017/Q6

The plane with equation 2x + 2y − z = 5 is denoted by m. Relative to the origin O, the points A and B have coordinates (3, 4, 0) and (−1, 0, 2) respectively.

(i) Show that the plane m bisects AB at right angles.

A second plane p is parallel to m and nearer to O. The perpendicular distance between the planes is 1.

(ii) Find the equation of p, giving your answer in the form ax + by + cz = d.

Solution:

Reference: PYQ - May/Jun 2017 Paper 31 Q6

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/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/Oct Nov/2009/32/Q6

6i) Use the substitution x = 2 tan q to show that

6ii) Hence find the exact value of

Solution:

Reference: PYQ - Oct/Nov 2009 Paper 32 Q6

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9709/Oct Nov/2003/4/Q6

One end of a light inextensible string is attached to a fixed point A of a fixed vertical wire. The other end of the string is attached to a small ring B, of mass 0.2 kg, through which the wire passes.

A horizontal force of magnitude 5N is applied to the mid-point M of the string. The system is in equilibrium with the string taut, with B below A, and with angles ABM and BAM equal to 30◦ (see diagram).

(i) Show that the tension in BM is 5N.

(ii) The ring is on the point of sliding up the wire. Find the coefficient of friction between the ring and the wire.

(iii) A particle of mass m kg is attached to the ring. The ring is now on the point of sliding down the wire. Given that the coefficient of friction between the ring and the wire is unchanged, find the value of m.

Solution:

Reference: PYQ - Oct/Nov 2003 Paper 4 Q6

9709/Oct Nov/2002/3/Q6

Let f(x) = (6+7x) / [(2-x)(1+x^2)]
(i) Express f(x) in partial fractions.
(ii) Show that, when x is sufficiently small for x^4 and higher powers to be neglected, f(x) = 3 + 5x - 1/2 x^2 - 15/4x^3

Solution:
i)

ii)

Reference: PYQ - Oct/Nov 2002 Paper 3 Q6

9709/MayJune/2002/3/Q6

Let f(x) = (4x) / [(3x+1)(x+1)^2]
(i) Express f(x) in partial fractions.
(ii) Hence show that

Solution:

Reference: PYQ - May/June 2002 Paper 3 Q6

You are given the co-ordinates of the four points A(6, 2), B(2, 4), C(−6, −2) and D(−2, −4).

You are given the co-ordinates of the four points A(6, 2), B(2, 4), C(−6, −2) and D(−2, −4).

(i) Calculate the gradients of the lines AB, CB, DC and DA.
Hence describe the shape of the figure ABCD.

(ii) Show that the equation of the line DA is 4y − 3x = −10 and find the length DA.

(iii) Calculate the gradient of a line which is perpendicular to DA and hence find the equation of the line l through B which is perpendicular to DA.

(iv) Calculate the co-ordinates of the point P where l meets DA.

(v) Calculate the area of the figure ABCD.

Solution:

A median of a triangle is a line joining one of the vertices to the mid-point of the opposite side.

A median of a triangle is a line joining one of the vertices to the mid-point of
the opposite side.

In a triangle OAB, O is at the origin, A is the point (0, 6) and B is the point (6, 0).

(i) Sketch the triangle.
(ii) Find the equations of the three medians of the triangle.
(iii) Show that the point (2, 2) lies on all three medians. (This shows that the medians of this triangle are concurrent.)

Solution:

9709/MayJune/13/2015/Q6

The diagram shows the graph of y = f^−1 (x), where f^ −1 is defined by f^−1 (x) = (1 − 5x)/2x for 0<x≤ 2.
(i) Find an expression for f(x) and state the domain of f. [5]
(ii) The function g is defined by g(x) =1/x for x ≥ 1. Find an expression for f^ −1 g(x), giving your
answer in the form ax + b, where a and b are constants to be found. [2]

Solutions:
i)

Explanation:

If x = 1

f^-1 (1) = 1-5(1)  /  2(1)

= -4/2

= -2

If x = 2

f^-1(2) = 1-5(2)   /  2(2)

= 1-10    /   4

= -9/4

= -2  1/4  (smaller than -2)

If x = 0.1

f^-1 (0.1) = 1-5(0.1)  /  2(0.1)

= 2.5

If x = 0.01

f^-1 (0.01) = 1-5(0.01)  /  2(0.01)

= 47.5

If x = 0.00001

f^-1 (0.00001) = 1-5(0.00001)  /  2(0.00001)

= 49997.5

Actually the range of f^-1(x) is the domain of f(x). When I subs x = 2 to the f^-1(x), the range is -9/4 (this one no issue). When subs x = 1, the range become -2.
Why the domain not equal to -9/4 <= x <= -2, but x >= -9/4 only?
You will understand when you see my solutions for x = 0.1, x=0.01, x = 0.00001 ...
x can be 0.000000000000000000000000............1 (it also can be 0.05, 0.12345678 and any number which is more than 0 but less than 2) as long as it is bigger than 0. So the range will be any number which is more than or equal to -9/4

ii)

Reference: PYQ - May/June 2015 Paper 13 Q6