CONFIDENTIAL*
950/1, 954/1
1.
Using the laws of the algebra of sets, show that, for any two sets A and B such that A ∪B = ξ , [4 marks] A − B = B′
2.
Find the set of values of k such that k ( x 2 + 2 x) − 4 x + 3k > 2 for all real values of x. [5 marks]
3.
The function f is defined by x <0 xe x f ( x ) = x +1 x ≥0 3 2
2
Evaluate ∫ 2 f ( x ) dx , giving your answer correct to three significant figures.[6
marks] 4.
5.
Given a complex number z = −1 +i 3 . (a) Write z 2 in the form of x + yi .
[2 marks]
5π . 6
(b)
Find the real number p such that arg ( z 2 + pz ) =
(c)
Hence, find the exact value of the modulus of z 2 + pz .
[3 marks] [2 marks]
The functions f and g are defined as f : x →ln x, x > 0 g : x → 1 −x , 0 < x < 1 (a) Find the composite function fg , and state its domain and range. (b) Find the inverse function ( fg ) −1 .
[3 marks] [2 marks]
(c) Sketch the graphs of fg and ( fg ) −1 on the same diagram, showing clearly the relationship between them. [2 marks] n
6.
(a)
Show that ∑(n + r −1)(n + r ) = r =1
n
(b)
Find ∑ r =1
7.
1 n(7n 2 −1) 3
[4 marks]
1 in terms of n. (n + r −1)(n + r ) −
[4 marks] −
−
(a)
If A is a square matrix and A 1 exists, show that (AT) 1 = (A 1)T.
(b)
1 Given the matrix A = 2 2
(i) (ii)
0 a 2
[2 marks]
2 2 , where a and b are constants. b
If A is a singular matrix, express a in terms of b. By taking a = 1 and b = 2, find the matrix B such that AB = 2I.
[3 marks] [4 marks]
8. Find the values of p, q and r such that the circle x 2 + y 2 + px + qy + r = 0 touches the xaxis at the point (1, 0) and passes through the point (4, 3). [6 marks] Express the equation obtained in the standard form and hence find its centre and its radius. [3 marks] [Turn over
CONFIDENTIAL*
2
9. The coordinates of the vertex A of a square ABCD are (4, 0). The diagonal BD lies on the straight line y = 3 x − 2 . (a) Find the equation of the other diagonal. [2 marks] (b) Find the coordinates of the point of intersection of the two diagonals. [2 marks] (c) Find the coordinates of the vertices B, C and D. [6 marks] The function f is defined by f : x →x +2 − x , x ∈ R . (a) Express f as a piecewisedefined function without modulus. [4 marks] (b) Sketch the graph of y = f ( x) . [2 marks] (c) By sketching a suitable graph on the same diagram, find the solution set for x +2 − x > 4 x 2 . [4 marks]
10.
11.
Find the coordinates of the stationary point on the curve y =
nature.
ex and determine its x
[5 marks]
2
d y [2 marks] <0. dx 2 Sketch the curve, showing clearly the stationary point. [2 marks] Using the trapezium rule with four equal intervals, find an approximate value for x 2 e [4 marks] ∫ 1 x dx , giving your answer correct to three significant figures. Determine the interval of x such that
12.
A curve has parametric equations x = t 2 and y = t 3 − t , where t is a parameter. (a) Find the cartesian equation of the curve. Hence, show that the curve is symmetric about the xaxis. [4 marks] (b) Find the coordinates of the points where the curve intersects the xaxis. [2 marks] (c) Sketch the curve. [2 marks] (d) Find the volume of the solid generated when the area bounded by the loop of the curve is rotated through π radians about the xaxis. [4 marks]