[SOLVED] MATHEMATICS Paper 2 Pure Mathematics 2

MATHEMATICS

Paper 2 Pure Mathematics 2

1 The variables x and y satisfy the equation a2y = e3x+k
, where a and k are constants. The graph of y against x is a straight line.

(a) Use logarithms to show that the gradient of the straight line is 2 ln a/3.

(b) Given that the straight line passes through the points (0.4, 0.95) and (3.3, 3.80), find the values of a and k.

2 Solve the inequality |x – 7| > 4x + 3.

3 The function f is defined by f(x) = tan2(2/1x) for 0 ≤ x < π.

(a) Find the exact value of f'(3/2π).

(b) Find the exact value of  (f(x)+sin x)dx.

4 The polynomial p(x) is defined by

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

(a) Find the value of a.

(b) Hence factorise p(x) completely.

(c) Solve the equation 

5 It is given that 2x+1/10 dx = 7, where a is a constant greater than 1.

(a) Show that 

(b) Use an iterative formula, based on the equation in part (a), to find the value of a correct to 3 significant figures. Use an initial value of 2 and give the result of each iteration to 5 significant figures.

6 A curve has parametric equations

(a) Find an expression for dx/dy
in terms of t.

(b) Find the exact gradient of the curve at the point where the curve crosses the y-axis.

7 (a) Prove that 

(b) Solve the equation 5 

(c) Show that the exact value of