Question The curve C has equation . The point P has coordinates (3, 0). a) Show that P lies on C. b) Find the equation of the tangent to C at P, giving your answer in the form y=mx+c, where m and c are constants. Another point Q also lies on C. The tangent to C […]
Question An arithmetic series has first term and common difference . a) Prove that the sum of the first n terms of the series is Sean repays a loan over a period of months. His monthly repayments form an arithmetic sequence. He repays £149 in the first month, £147 in the second month, £145 in the third month, […]
Question The line passes through the point (9, – 4) and has gradient . a) Find an equation for in the form ax+by+c=0, where a, b and c are integers. The line passes through the origin O and has gradient –2. The lines and intersect at the point P. b) Calculate the coordinates of P. Given that […]
Question a) Show that can be written as . Given that , , and that at , b) Find y in terms of x. Solution a) We are given; We have the algebraic formula; b) We are required to find y in terms of x, when; We are also given that , and that at . Clearly it […]
Question Find the set of values of x for which a) b) c) Both and . Solution a) We are required to solve the inequality; We can rearrange the inequality as; b) We are required to solve the inequality; We solve the following equation to find critical values of ; Now we have two options; Hence the critical […]
Question Solve the simultaneous equations Solution We are given simultaneous equations; Rearranging the first equation we get expression for ; Substituting this for in the second equation; We have the algebraic formula; Therefore; For a quadratic equation , the expression for solution is; For the above equation; Now we have two options. By substituting one-by-one these values of in above derived […]
Question Figure 1 shows a sketch of the curve with equation y = f(x). The curve passes through the origin O and through the point (6, 0). The maximum point on the curve is (3, 5). On separate diagrams, sketch the curve with equation a. y = 3f(x), b. y = f(x + 2). On each […]
Question where a and b are constants. a. Find the value of a and the value of b. b. Hence, or otherwise, show that the roots of are , where c and d are integers to be found. Solution a. We are given; We can write L.H.S of the equation by completing square. We have the […]
Question Given that , , a. Find b. Find Solution a. We are given; We are required to find . Rule for differentiation of is: Rule for differentiation of is: b. We are given; We are required to find . Rule for integration of is: Rule for integration of is: Rule for integration of is:
Question a. Write down the value of . b. Find the value of . Solution a. b.