Question i. Solve the equation for . ii. Find the set of values of for which the equation has no solution. iii. For the equation , state the value of for which there are three solutions in the interval , and find these solutions. Solution i. We have the equation; Let ; […]
Question Solve the equation for . Solution We are given the equation; Since , therefore; Now we have two options. If we utilize the odd/even property of and to find other solutions (roots), then these solutions will lie outside desired range .
Question i. Show that the equation can be expressed as Where 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; Let ; ii. We are required to solve the equation for […]
Question a) Given that x > 0, find the two smallest values of x, in radians, for which . Show all necessary working. b) The function is defined for . i. Express f(x) in the form , where a and b are constants. ii. Find the range of f. Solution a) We […]
Question The diagram shows a sector OAC of a circle with centre O. Tangents AB and CB to the circle meet at B. The arc AC is of length 6 cm and angle radians. i. Find the length of OA correct to 4 significant figures. ii. Find the perimeter of the shaded region. iii. […]
Question i. Given that , show, without using a calculator, that . ii. Hence, showing all necessary working, solve the equation for Solution i. We are given that; Since ; We have the trigonometric identity; From this we can write; Therefore; Let ; Now we have two options. Since ; We […]
Question The functions is defined for , where and are positive constants. The diagram shows the graph of y = f(x). i. In terms of and state the range of . ii. State the number of solutions of the following equations. a) […]
Question The diagram shows triangle ABC which is right-angled at A. Angle radians and AC = 8 cm. The points D and E lie on BC and BA respectively. The sector ADE is part of a circle with centre A and is such that BDC is the tangent to the arc DE at D. i. […]
Question The equation of a curve is and the equation of a line is . i.State the smallest and largest values of y for both the curve and the line for . ii.Sketch, on the same diagram, the graphs of and for . iii.State the number of solutions of the equation for . Solution i. […]
Question Angle x is such that sin x = a + b and cos x = a − b, where a and b are constants. A curve is such that . The point P (2,9) lies on the curve. i. Show that a2 +b2 has a constant value for all values of x. […]
Question The functions f is defined by for , i. State the range of . ii. Sketch the graph of . The functions g is defined by for , where is a constant. iii. State the largest value of for which g has an inverse. iv. For this value of , find an […]
Question i. Prove the identity ii. Hence solve the equation for Solution i. First, we are required to show that; Since ; We have the trigonometric identity; From this we can write; Therefore; We have the algebraic formula; ii. We are required to solve the equation; For . […]
Question a) Solve the equation for . b) The diagram shows part of the graph of , where is measured in radians and and are constants. The curve intersects the x-axis at and the y-axis at . Find the values of and . Solution a) We are required to solve the equation for […]
Question i. Show that ii. Hence, showing all necessary working, solve the equation For . Solution i. We are given the expression; We have the algebraic identity; We have the trigonometric identity; Since ; We can write; Therefore; ii. We are required to solve […]
Question The diagram shows an arc BC of a circle with centre A and radius 5 cm. The length of the arc BC is 4 cm. The point D is such that the line BD is perpendicular to BA and DC is parallel to BA. i. Find angle BAC in radians. ii. Find […]
Question The diagram shows a triangle ABC in which BC = 20 cm and angle ABC = 90o. The perpendicular from B to AC meets AC at D and AD = 9 cm. Angle BCA = . i. By expressing the length of BD in terms of in each of the triangles ABD and […]
Question The functions f and g are defined by for , for i. Solve the equation . ii. Sketch the graph of . Solution i. We are given; for for We are required to solve equation ; To solve this equation for , we can substitute . Hence, Since given interval is […]
Question The diagram shows a triangle OAB in which angle ABO is a right angle, angle AOB = radians and AB = 5 cm. The arc BC is part of a circle with centre A and meets OA at C. The arc CD is part of a circle with centre O and meets OB at […]
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 […]
Question The diagram shows a sector POQ of a circle of radius 10 cm and centre O. Angle POQ is 2.2 radians. QR is an arc of a circle with centre P and POR is a straight line. i. Show that the length of PQ is 17.8 cm, correct to 3 significant figures. […]