Practice Paper C Sec 3 Math

3 Practice Paper C Sec 3 1 (a) Find the product of 1 1 1 1 3 3 3 3( )( )x y x y and 2 1 1 2 2 1 1 2 3 3 3 3 3 3 3 3( )( )x x y y x x y y    and simplify the answer. [2] (b) If x a x ay x a x a       , simplify 1y y . [3] (c) Solve the equation 5log 1 6log 5 xx  . [3] 2 Angles A and B are in different quadrants such that 3tan 4A and 4sin 5B . Find, without using calculator, the possible values of ( 2sin tan )AB in the exact form. [6] 3 (i) Find 1A and 1B , the inverses of the matrices 43 22   A and 54 11   B . [2] (ii) Use 1A and 1B from (i), to find the matrix M 4 3 5 4 4 0 2 2 1 1 6 2                  M . [3] 4 (a) Find the range of values of h , 0h , for which the line y = x – h meets the curve 2 7y hx x . [3] (b) Find the range of values of k for which 2 ( 4) 1y x k x    is always positive for all real values of x. [3]

4 5 Given that 2 6xx is a factor of 4 3 22 1x x ax bx a b      , find the value of a and of b. [4] 6 (a) Simplify + ( )( ) ( )( ) ( )( ) a b b c c a b c a c a c a b a b b c          . [3] (b) Simplify 22 22 2 6 4 20 6 5 3 6 24 x x x x x x x x        . [3] 7 Sketch the graph 1 4cos 2yx for 0 360x   . [3] 8 Solve the equations for 0 180x   . (a) sin 3 cos 2xx , [3] (b) tan(2 40 ) 2.718x    . [3] 9 Prove the identities. (a) 4 2 2 2sin + sin cos cos 1 x x x x  , [2] (b) 2 2 2 cosec sin cos cosec 2cot sin cos x x x x x x x   . [4] 10 It is given that z varies jointly as the square of x and inversely as the square root of y, for all positive values of x, y and z. When x = 2 and y = 4, 6z . (a) Find the value of x when y = 9 and z = 25. [4] (b) Find the percentage change in z when x is increased to 200% and y is reduced to 25%. [2] 11 Solve the equation 2 3 2xx [4]

5 12 In the diagram, PQR represents a horizontal triangular field such that PR = 85 m and PQ = 40 m. The bearing of R from P and the bearing of Q from P is 125 and 195 respectively. 40 m 85 m N T R Q P (a) Calculate the distance from Q to R. [2] (b) Calculate the area of the triangular field PQR. [2] (c) Calculate the shortest distance from P to the road QR. [2] TP represents a building at the corner P. Given that the angle of depression of the top of the building T to the corner R is 20 , (d) calculate the height of the building TP. [2] A man walks along the road from Q to R, (e) calculate the greatest angle of elevation of the top of the building. [2] 125 195

6 13 (i) The equation of a circle is given by 22 10 2 1 0x y x y     . Find the centre and the radius of the circle. [2] (ii) The tangent to the circle at a point A is parallel to the straight line 20xy . Find the equation of the diameter of the circle through A. [2] (iii) The circle is then reflected on the line yx . State the equation of the new circle formed. [1] 14 Circles A, B and C are externally tangent to each other and internally tangent to circle D. Circles B and C are congruent. E, F and G are the centres of circles A, B and C respectively. Circle A has radius 1 and passes through O, the centre of circle D. Both circles B and C have radii y. By considering two right-angled triangles, find the radius y of the circle B . [6] O y 1 D C B A G F E

7 15 (a) The table shows the experimental values of two variables x and y . x 1 2 3 4 5 y 3.8 11.2 20.5 33.2 50.6 y x 3.8 5.6 8.3 It is known that x and y are related by the equation 2y ax bx Plot y x against x and draw a suitable straight line. Use the graph to estimate (i) the value of a and of b, [4] (ii) the value of x when y = 8x. [1] (b) The diagram shows the straight line graph of xy against 2x passing through the points A(0, 6) and B(8, 4). Express y in terms of x. [3] O B (8, 4) A (0, 6) xy

8 16 Solutions to this question by accurate drawing will not be accepted The diagram shows a quadrilateral PQRS in which QS is the perpendicular bisector of PR. The mid-point of PR is M. The coordinates of R and M are (11, 4) and (9, 7) respectively. RQ is parallel to the line 54yx and the equation of SR is 8 21 0xy   . Find (i) the coordinates of P, [2] (ii) the equations of QR and QS, [4] (iii) the coordinates of Q and of S, [3] (iv) the area of PQRS. [2] End Of Paper 54yx 8 21 0xy   (9,7)M (11,4) y x O