Sample Aptitude Test Questions 1. (a) Prove, by completing the square, that the roots of the equation x 2 + 2kx + c = 0, where k and c are constants, are k ± (k 2 c). The equation x 2 + 2kx ± 81 = 0 has equal roots. (b) Find the possible values of k. 2. h x 2x Fig. 4 A manufacturer produces cartons for fruit juice. Each carton is in the shape of a closed cuboid with base dimensions 2x cm by x cm and height h cm, as shown in Fig. 4. Given that the capacity of a carton has to be 1030 cm 3, (a) express h in terms of x, (b) show that the surface area, A cm 2, of a carton is given by 3090 A 4x 2. x 1
3. The equation x 2 + 5kx + 2k = 0, where k is a constant, has real roots. (a) Prove that k(25k 8) 0. (b) Hence find the set of possible values of k. (c) Write down the values of k for which the equation x 2 + 5kx + 2k = 0 has equal roots. (1) 4. The points A( 1, 2), B (7, 2) and C(k, 4), where k is a constant, are the vertices of ABC. Angle ABC is a right angle. (a) Find the gradient of AB. (b) Calculate the value of k. (c) Show that the length of AB may be written in the form p 5, where p is an integer to be found. (d) Find the exact value of the area of ABC. (e) Find an equation for the straight line l passing through B and C. Give your answer in the form ax + by + c = 0, where a, b and c are integers. [P1 June 2001 Question 8*] 5. The straight line l 1 has equation 4y + x = 0. The straight line l 2 has equation y = 2x 3. (a) On the same axes, sketch the graphs of l 1 and l 2. Show clearly the coordinates of all points at which the graphs meet the coordinate axes. The lines l 1 and l 2 intersect at the point A. (b) Calculate, as exact fractions, the coordinates of A. (c) Find an equation of the line through A which is perpendicular to l 1. Give your answer in the form ax + by + c = 0, where a, b and c are integers. 2
6. Initially the number of fish in a lake is 500 000. The population is then modelled by the recurrence relation u n + 1 = 1.05u n d, u 0 = 500 000. In this relation u n is the number of fish in the lake after n years and d is the number of fish which are caught each year. Given that d = 15 000, (a) calculate u 1, u 2 and u 3 and comment briefly on your results. Given that d = 100 000, (b) show that the population of fish dies out during the sixth year. (c) Find the value of d which would leave the population each year unchanged. 7. (a) By completing the square, find in terms of k the roots of the equation x 2 + 2kx 7 = 0. (b) Prove that, for all values of k, the roots of x 2 + 2kx 7 = 0 are real and different. (c) Given that k = 2, find the exact roots of the equation. [P1 June 2002 Question 4] 3
8. Figure 3 B A The points A( 3, 2) and B(8, 4) are at the ends of a diameter of the circle shown in Fig. 3. (a) Find the coordinates of the centre of the circle. (b) Find an equation of the diameter AB, giving your answer in the form ax + by + c = 0, where a, b and c are integers. (c) Find an equation of tangent to the circle at B. The line l passes through A and the origin. (d) Find the coordinates of the point at which l intersects the tangent to the circle at B, giving your answer as exact fractions. 9. (a) Solve the inequality (b) Solve the inequality 3x 8 > x + 13. x 2 5x 14 > 0. 4
10. (a) An arithmetic series has first term a and common difference d. Prove that the sum of the first n terms of the series is 1 n[2a + (n 1)d]. 2 A company made a profit of 54 000 in the year 2001. A model for future performance assumes that yearly profits will increase in an arithmetic sequence with common difference d. This model predicts total profits of 619 200 for the 9 years 2001 to 2009 inclusive. (b) Find the value of d. Using your value of d, (c) find the predicted profit for the year 2011. 11. f(x) = 9 (x 2) 2 (a) Write down the maximum value of f(x). (b) Sketch the graph of y = f(x), showing the coordinates of the points at which the graph meets the coordinate axes. (5) The points A and B on the graph of y = f(x) have coordinates ( 2, 7) and (3, 8) respectively. (1) (c) Find, in the form y = mx + c, an equation of the straight line through A and B. (d) Find the coordinates of the point at which the line AB crosses the x-axis. The mid-point of AB lies on the line with equation y = kx, where k is a constant. (e) Find the value of k. 5
12. The straight line l 1 with equation y = 23 x 2 crosses the y-axis at the point P. The point Q has coordinates (5, 3). The straight line l 2 is perpendicular to l 1 and passes through Q. (a) Calculate the coordinates of the mid-point of PQ. (b) Find an equation for l 2 in the form ax + by = c, where a, b and c are integer constants. The lines l 1 and l 2 intersect at the point R. (c) Calculate the exact coordinates of R. 13. Find the set of values for x for which (a) 6x 7 < 2x + 3, (b) 2x 2 11x + 5 < 0, (c) both 6x 7 < 2x + 3 and 2x 2 11x + 5 < 0. (1) 6
14. In the first month after opening, a mobile phone shop sold 280 phones. A model for future trading assumes that sales will increase by x phones per month for the next 35 months, so that (280 + x) phones will be sold in the second month, (280 + 2x) in the third month, and so on. Using this model with x = 5, calculate (a) (i) the number of phones sold in the 36th month, (ii) the total number of phones sold over the 36 months. The shop sets a sales target of 17 000 phones to be sold over the 36 months. Using the same model, (b) find the least value of x required to achieve this target. 15. The points A and B have coordinates (4, 6) and (12, 2) respectively. The straight line l 1 passes through A and B. (a) Find an equation for l 1 in the form ax + by = c, where a, b and c are integers. The straight line l 2 passes through the origin and has gradient 4. (b) Write down an equation for l 2. (1) The lines l 1 and l 2 intercept at the point C. (c) Find the exact coordinates of the mid-point of AC. (5) 7
16. (a) Solve the equation 4x 2 + 12x = 0. f(x) = 4x 2 + 12x + c, where c is a constant. (b) Given that f(x) = 0 has equal roots, find the value of c and hence solve f(x) = 0. 17. Solve the simultaneous equations x 3y + 1 = 0, x 2 3xy + y 2 = 11. (7) 8
18. Figure 1 C M B (8, 5) A (2, 3) The points A and B have coordinates (2, 3) and (8, 5) respectively, and AB is a chord of a circle with centre C, as shown in Fig. 1. (a) Find the gradient of AB. The point M is the mid-point of AB. (b) Find an equation for the line through C and M. (5) Given that the x-coordinate of C is 4, (c) find the y-coordinate of C, (d) show that the radius of the circle is 5 17. 4 9
19. The first three terms of an arithmetic series are p, 5p 8, and 3p + 8 respectively. (a) Show that p = 4. (b) Find the value of the 40th term of this series. (c) Prove that the sum of the first n terms of the series is a perfect square. 20. f(x) = x 2 kx + 9, where k is a constant. (a) Find the set of values of k for which the equation f(x) = 0 has no real solutions. Given that k = 4, (b) express f(x) in the form (x p) 2 + q, where p and q are constants to be found, 10
21. Figure 2 y Q y 2 = 4(x 2) 2x 3y = 12 O A x P Figure 2 shows the curve with equation y 2 = 4(x 2) and the line with equation 2x 3y = 12. The curve crosses the x-axis at the point A, and the line intersects the curve at the points P and Q. (a) Write down the coordinates of A. (b) Find, using algebra, the coordinates of P and Q. (c) Show that PAQ is a right angle. (1) (6) 22. f(x) = ( x 2 3) x 3 2, x 0. (a) Show that f(x) x 6x 1 + 9x 3. (b) Hence, or otherwise, differentiate f(x) with respect to x. 11
23. The points A and B have coordinates (1, 2) and (5, 8) respectively. (a) Find the coordinates of the mid-point of AB. (b) Find, in the form y = mx +c, an equation for the straight line through A and B. 24. Giving your answers in the form a + b 2, where a and b are rational numbers, find (a) (3 8) 2, 1 (b) 4. 8 25. The width of a rectangular sports pitch is x metres, x > 0. The length of the pitch is 20 m more than its width. Given that the perimeter of the pitch must be less than 300 m, (a) form a linear inequality in x. Given that the area of the pitch must be greater than 4800 m 2, (b) form a quadratic inequality in x. (c) by solving your inequalities, find the set of possible values of x. 26. The curve C has equation y = x 2 4 and the straight line l has equation y + 3x = 0. (a) In the space below, sketch C and l on the same axes. (b) Write down the coordinates of the points at which C meets the coordinate axes. (c) Using algebra, find the coordinates of the points at which l intersects C. 12