Question The diagram shows a rectangle ABCD in which AB = 5 units and BC = 3 units. Point P lies on DC and AP is an arc of a circle with centre B. Point Q lies on DC and AQ is an arc of a circle with centre D. i. Show that angle ABP = 0.6435 radians, correct […]
Question i. Show that the equation may be expressed as ii. Hence solve the equation for . Solution i. We are given the equation; We have the trigonometric identity; From this we can substitute in above equation; ii. We are required to solve the equation for . From (i) we know that given equation can be written as; […]
Question a. The function f, defined by for , is such that and . i. Find the values of the constants a and b. ii. Evaluate . b. The function g is defined by for for . The range of g is given by . Find the values of the constants c and d. Solution a. i. We are given the […]
Question i. Show that the equation can be expressed as ii. Hence solve the equation for . Solution i. We are given equation as; Utilising , we can write the given equation as; We have the trigonometric identity; It can be arranged to get; Substituting in the above equation obtained. ii. We […]
Question The diagram shows a semicircle with centre O and radius 6 cm. The radius OC is perpendicular to the diameter AB. The point D lies on AB, and DC is an arc of a circle with centre B. i. Calculate the length of the arc DC. ii. Find the value of giving your answer correct to 3 […]
Question a. The diagram shows part of the graph of y=a+b sin x. Find the values of the constants a and b. b. i. Show that the equation may be expressed as . ii. Hence solve the equation for . Solution a. We are given graph of the equation; y=a+b sin x It is […]
Question The diagram shows an isosceles triangle ABC in which AC = 16 cm and AB = BC = 10 cm. The circular arcs BE and BD have centres at A and C respectively, where D and E lie on AC. i. Show that angle BAC = 0.6435 radians, correct to 4 decimal places. ii. Find the […]
Question The diagram shows two circles with centres A and B having radii 8 cm and 10 cm respectively. The two circles intersect at C and D where CAD is a straight line and AB is perpendicular to CD. (i) Find angle ABC in radians. (ii) Find the area of the shaded region. Solution (i) It is evident from the diagram […]
Question (i) Show that the equation may be expressed as . (ii) Hence solve the equation for 0o < < 180o. Solution (i) We are given that; Utilizing ; (ii) We are required to solve the equation for 0o < < 180o. From (i) we know that given equation can be written as; Now we have two options. Using calculator; […]
Question The function f is defined by , for . i. Solve the equation , giving your answer correct to 1 decimal place. ii. Find an expression for and find the domain of . iii. Sketch, on the same diagram, the graphs of y=f(x) and . Solution i. We are given the function as; We are required to […]
Question The diagram shows a circle with radius r cm and centre O. Points A and B lie on the circle and ABCD is a rectangle. Angle AOB = 2 radians and cm. i. Express the perimeter of the shaded region in terms of r and . ii. In the case where and , find the area of the shaded region. […]
Question i. Prove the identity . ii. Hence solve the equation ,for . Solution i. Utilizing ; We have the trigonometric identity; We can write from this identity; Therefore; We have algebraic identity; Therefore; ii. We are required to solve the equation ,for . From (i) we have found that given equation can be written […]
Question In the diagram, AB=AC=8 cm and angle radians. The circular arc BC has centre A, the circular arc CD has centre B and ABD is a straight line. i. Show that angle radians. ii. Find the perimeter of the shaded region. Solution i. It is evident from the diagram that; Therefore; We are given that ABD is a straight line. […]
Question In the diagram, OAXB is a sector of a circle with centre O and radius 10 cm. The length of the chord AB is 12 cm. The line OX passes through M, the mid-point of AB, and OX is perpendicular to AB. The shaded region is bounded by the chord AB and by the arc of […]
Question The equation of a curve is . i. Sketch the graph of for , stating the coordinates of the point of intersection with the y-axis. Points P and Q lie on the curve and have x-coordinates of and respectively. ii. Find the length of PQ correct to 1 decimal place. The line through P and […]
Question i. Prove the identity ii. Hence solve the equation for . Solution i. We have the trigonometric identity; We can write from this identity; Therefore; ii. We have the equation; From (i) we know that left hand side of given equation can be written as; Therefore; Using calculator we can […]
Question The diagram shows the graphs of and for . The graphs intersect at points A and B. i. Find by calculation the x-coordinate of A. ii. Find by calculation the coordinates of B. Solution i. We are required to find the x-coordinate of point A which is point of intersection of the two given curves. If two lines (or […]