Question The equation of a curve is y = x + 2cos x. Find the x-coordinates of the stationary points of the curve for 0 ≤ x ≤ 2π, and determine the nature of each of these stationary points. Solution We are required to find the x-coordinates of stationary points of the curve. A […]
Question The diagram shows the roof of a house. The base of the roof, OABC, is rectangular and horizontal with and . The top of the roof is 5m above the base and . The sloping edges , , and are all equal in length. Unit vectors and are parallel to , respectively and is vertically upwards. i. […]
Question Solve the equation for . Solution First we need to manipulate the given equation to write it in a single trigonometric ratio i.e. ; Dividing the entire equation with ; Using the relation ; To solve this equation for , we can substitute . Hence, Since given interval is , for interval can be found as follows; Multiplying the entire inequality with […]
Question Functions and are defined by for where is constant for , i. Find the values of for which the equation has two equal roots and solve the equation in these cases. ii. Solve the equation when . iii. Express in terms of . Solution i. We have the functions; We can write these functions as; We can equate both functions; Standard form […]
Question The diagram shows the curve , where is a constant. The curve has a minimum point on the x-axis. i. Find the value of . ii. Find the coordinates of the maximum point of the curve. iii. State the set of values of for which is a decreasing function of . iv. Find the area of the shaded region. […]
Question The diagram shows a circle with centre O and radius 8 cm. Points A and B lie on the circle. The tangents at A and B meet at the point T, and cm. i. Show that angle AOB is 2.16 radians, correct to 3 significant figures. ii. Find the perimeter of the shaded region. iii. Find the area of […]
Question In the diagram, ABC is a triangle in which cm, cm and angle . The line CX is perpendicular to the line ABX. i. Find the exact length of BX and show that . ii. Show that the exact length of AC is Solution From the given information we can compile following data; i. To find we consider the […]
Question A curve is such that and P (1, 8) is a point on the curve. i. The normal to the curve at the point P meets the coordinate axes at Q and at R. Find the coordinates of the mid-point of QR. ii. Find the equation of the curve. Solution i. To find the mid-point […]
Question The curve intersects the line at two points. Find the distance between the two points. Solution Expression to find distance between two given points and is: So first we need to find the coordinates of points of intersection of the curve and the line. If two lines (or a line and a curve) intersect each other […]
Question A curve has equation . . Given that the gradient of the curve is −3 when x = 2, find the value of the constant k. Solution Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of curve with respect to is: In this case; Therefore; We can rewrite the expression; Rule for […]
Question The first three terms in the expansion of , in ascending powers of , are . Find the values of the constants . Solution Expression for the Binomial expansion of is: In the given case: Hence; We are given the first three terms of the expansion: Comparing 1st terms of both above equations: Comparing 2nd terms of both above […]
Question Each year a company gives a grant to a charity. The amount given each year increases by 5% of its value in the preceding year. The grant in 2001 was $5000. Find i. the grant given in 2011, ii. the total amount of money given to the charity during the years 2001 to 2011 inclusive. […]