Pure Mathematics 1

1. Find the value of the term which is independent of x in the expansion of x³ x‾⁴.
 
2. A geometric progression, for which the common ratio is positive, has a second term of 18 and a fourth term of 8.

Find

(i) the ?rst term and the common ratio of the progression,

(ii) the sum to in?nity of the progression.

3.

In the diagram, OPQ is a sector of a circle, centre O and radius r cm. Angle QOP Θ radians. The tangent to the circle at Q meets OP extended at R.

(i) Show that the area, A cm², of the shaded region is given by A    1 r²₂(tan Θ    Θ).

(ii) In the case where Θ  0.8 and r  15, evaluate the length of the perimeter of the shaded region.

4. The gradient at any point (x, y) on a curve is (1  2x). The curve passes through the point (4, 11).

Find

(i) the equation of the curve,

(ii) the point at which the curve intersects the y-axis. 

5.  (i) Show that the equation 3 tan Θ 2 cos Θ can be expressed as 2 sin² Θ  3 sin Θ  2  0.

(ii)  Hence solve the equation 3 tan Θ  2 cos Θ, for 0^Æ  Θ 360^Æ.

6.

In the diagram, triangle ABC is right-angled and D is the mid-point of BC.  Angle DAC  30^Æand angle BAD  x^Æ.  Denoting the length of AD by l,

(i) Express each of AC and BC exactly in terms of l, and show that  AB  ¹ L/2   7,
 
(ii) Show that x   tan ¹    2/3   30.
 
7.  Given that a   2/ 2   , b    2/6  and c  p/p    , ?nd 1    3    p¹ 
 
(i) the angle between the directions of a and b,

(ii) the value of p for which b and c are perpendicular.

8. A curve has equation y x³ 3x² 9x k, where k is a constant.

(i) Write down an expression for  dy/dx.

(ii) Find the x-coordinates of the two stationary points on the curve.

(iii) Hence ?nd the two values of k for which the curve has a stationary point on the x-axis.

9.

The diagram shows a rectangle ABCD, where A is (3, 2) and B is (1, 6).

(i) Find the equation of BC.

Given that the equation of AC is y x 1, ?nd

(ii) the coordinates of C,

(iii) the perimeter of the rectangle ABCD.

10.

The diagram shows the points A (1, 2) and B (4, 4) on the curve y  2x.  The line BC is the normal to the curve at B, and C lies on the x-axis. Lines AD and BE are perpendicular to the x-axis.

(i) Find the equation of the normal BC.

(ii) Find the area of the shaded region.

11 (i)  Express 2x²    8x    10 in the form a(x    b)²    c.

(ii) For the curve y    2x²    8x    10, state the least value of y and the corresponding value of x.

(iii) Find the set of values of x for which y    14. Given that f : x 2x² 8x 10 for the domain x k,

(iv) Find the least value of k for which f is one-one,

(v) Express f ¹(x) in terms of x in this case.