Montfort Sec 3 Express/G3 Add Math EOY 2022 (with answers and solutions)

3 1 Without using a calculator, express 4 7 3 31 − + in the form 3 2 ab + , where a and b are integers. [3] 2 A programmed robot attempts to throw a basketball into a hoop. The height, h feet, of the basketball from the ground at time t seconds, is given by the equation 22 4 7h t t=− + + . The basketball hoop is at a height of 10 feet above the ground. Determine, with calculations, if the robot would be able to throw the basketball into the hoop. [3]

4 3 (a) Sketch the graph of 10xy= and the graph of lgyx= on the same axes. Label the graphs and clearly indicate the intercepts. [3] (b) Solve the equation ( )2 5 3yyee −= . [3]

5 4 Find the coordinates of the points where the line 12 =+ yx meets the curve 134 22 =+ yx . [4]

6 5 (a) Find the range of values of k for which the equation 249x kx+= has no real roots. [3] (b) Hence, or otherwise, find the smallest prime number k so that the equation 249x kx+= has two real and distinct roots. [2]

7 6 (a) A microorganism reproduces where each cell divides into four cells in every hour. If N is the number of cells to begin with and assuming that the cells divide at a constant rate, explain why the number of cells present after h hours is given by ( )4 h N . [2] (b) A scientist discovered that the microorganism has a x% increase in cells after 5 hours. Calculate the value of x. [2]

8 7 Solve, for 0 360x   , the equation 22cot 1 5cosec xx=+ . [4]

9 8 (a) The curve cos 2 xy a c=+ , where a and c are negative integers, is defined for 06 πx . It has an amplitude of 3 and a minimum value of −7. (i) State the value of a and c. [2] (ii) Sketch the graph of y. [2] (b) Given that sin18 p= , express each of the following in terms of p. (i) sec162 [2] (ii) tan( 198 )− [2]

10 9 (a) The remainder when 32 3 2 ,x x x k− − + where k is a constant, is divided by 2x+ is twice of the remainder when it is divided by 3.x− Find the value of k. [3] (b) Solve the equation 32 4 8 8 0,x x x+ − − = expressing non-integer roots in surd form. [4]

11 10 (a) Prove that 2 1 cos(cosec cot ) 1 cos   −−= + . [3] (b) Hence find, for 0 < x < 6, the values of x for which 2 2 5(cosec2 cot 2 )xx =− . [5]

12 11 (a) Show that 1x− is a factor of 32 3 4.xx+− [1] (b) Express 2 32 2 3 8 34 xx xx − + + +− as a sum of three partial fractions. [7]