Question The diagram shows the curve and the tangent to the curve at the point . i. Find the equation of this tangent, giving your answer in the form . ii. Find the area of the shaded region. Solution i. We are required to find equation of the tangent to the curve at point […]
Question The inside lane of a school running track consists of two straight sections each of length x metres, and two semicircular sections each of radius r metres, as shown in the diagram. The straight sections are perpendicular to the diameters of the semicircular sections. The perimeter of the inside lane is 400 metres. i. Show that the area, Am2, […]
Question The point A has coordinates and the point B has coordinates . i. Find the equation of the perpendicular bisector of AB, giving your answer in the form . ii. A point C on the perpendicular bisector has coordinates . The distance OC is 2 units, where O is the origin. Write down two equations involving and […]
Question The diagram shows a metal plate made by fixing together two pieces, OABCD (shaded) and OAED (unshaded). The piece OABCD is a minor sector of a circle with centre O and radius . The piece OAED is a major sector of a circle with centre O and radius . Angle AOD is radians. Simplifying your answers […]
Question A curve has equation . It is given that and that . Find . Solution We are given that equation of the curve is . We are also given that which means there is a point on the curve where and i.e. the curve passes through the point . We are also given the derivative of the […]
Question i. Solve the equation for . ii. Hence find the solution of the equation for . Solution i. We have the equation; Since we are required to write it as a quadratic equation in , we need to eliminate . We have the trigonometric identity; We can rewrite the identity as; Hence; Let; Therefore […]
Question The diagram shows a pyramid OABCD in which the vertical edge OD is 3 units in length. The point E is the centre of the horizontal rectangular base OABC. The sides OA and AB have lengths of 6 units and 4 units respectively. Unit vectors , and are parallel to , and respectively. i. Express each of the […]
Question The function is defined by , for , i. Define in a similar way the inverse function . ii. Solve the equation . Solution i. We have the function; We can write it as; We write it as; To find the inverse of a given function we need to write it in terms of rather than in terms of . Since given […]
Question i. Find the first three terms in the expansion of in ascending powers of . ii. In the expansion of , the coefficient of is zero, find the value of . Solution i. Expression for the Binomial expansion of is: In the given case: Hence; ii. To find the value of in the expansion of First we know from (i) […]
Question a) In an arithmetic progression the sum of the first ten terms is 400 and the sum of the next ten terms is 1000. Find the common difference and the first term. b) A geometric progression has first term , common ratio and sum to infinity 6. A second geometric progression has first term […]