4E5N AMP1 Prelim 2021 with Answer Key MSS

3 1 Factorise and hence express in the form , where p and q are integers. [3]

4 2 The equation of a curve is . (a) State the minimum and maximum values of y. [2] (b) Sketch the graph of for . [3]

5 3 The line , where a and b are positive constants, intersects the x-axis at P and the y-axis at Q. Given that the gradient of PQ is and that the distance PQ = , find the value of a and of b. [5]

6 4 In the diagram, A, B, C and D lie on a circle with centre O. E lie on the line BC such that BE = EC. The tangent to the circle at A meets DB produced at T. (a) Show that the length of OE : AB is 1 : 2. [2] (b) Show that angle ATB = . [3] A T B E C D O

7 5 (a) Find the first 3 terms in the expansion, in ascending powers of x, of . Give the terms in their simplest forms. [3] (b) Hence find the term independent of x in the expansion . [3]

8 6 In an experiment, 200 grams of a particular radioactive substance is stored in a container. The amount of substance present after t days is given by , where k is a constant. It is found that the substance is halved after 20 days. (a) Find the amount of substance remaining after 1 week. [4] The radioactive substance is considered to be safe to be discarded using the proper procedures when the amount is reduced to 15 grams. (b) John claimed that it is safe to discard the radioactive substance after 74 days. Is his claim justified? Show the calculations and the reasons on which you based your answer. [2]

9 7 For a particular curve . The curve passes through the point and the gradient of the curve at P is 8. Find the equation of the curve. [6]

10 8 The function f is defined by , . (a) Explain, with working, whether f is an increasing or a decreasing function. [4]

11 (b) A point T moves along the curve in such a way that its y-coordinate is decreasing at a rate of 0.03 units per second. Find the rate of increase of its x- coordinate when x = 3. [2]

12 9 The expression , where a and b are constants, has a factor of and leaves a remainder of when divided by . (a) Find the value of a and of b. [4]