9709/May June/33/2016/Q2

The variables x and y satisfy the relation 3^y = 4^(2−x).

(i) By taking logarithms, show that the graph of y against x is a straight line. State the exact value of the gradient of this line. [3]

(ii) Calculate the exact x-coordinate of the point of intersection of this line with the line with equation y = 2x, simplifying your answer. [2]

Solutions:

Reference: PYQ - May/Jun 2016 Paper 33 Q2

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/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