Hits: 1604
Hits: 1604 Question a. Two convergent geometric progressions, P and Q, have the same sum to infinity. The first and second terms of P are 6 and 6r respectively. The first and second terms of Q are 12 and −12r respectively. Find the value of the common sum to infinity. b. The first term of an […]
Hits: 527 Question Showing all necessary working, solve the equation for . Solution We are given equation; We know that ; therefore, Now we have two options. Using calculator we can find that; We have following properties of . Properties of Domain Range Periodicity Odd/Even Translation/ Symmetry We utilize the periodicity property of to find other solutions (roots) of . […]
Hits: 1553 Question A function f is defined by for . i. Find the range of f. ii. Sketch the graph of iii. Solve the equation , giving answers in terms of . The function is defined by for , where k is a constant. iv. State the largest value of k for which g has an inverse. v. For this value of […]
Hits: 2584 Question The diagram shows a metal plate ABCD made from two parts. The part BCD is a semicircle. The part DAB is a segment of a circle with centre O and radius 10 cm. Angle BOD is 1.2 radians. i. Show that the radius of the semicircle is 5.646 cm, correct to 3 decimal places. ii. Find the […]
Hits: 873 Question i. Express the equation in the form , where k is a constant. ii. Hence solve the equation for . Solution i. We are given that; We know that , therefore; Comparison with given yields; ii. We are required to solve for . From (i) we know that can be written in the form; Therefore, we […]
Hits: 1014 Question i. Show that . ii. Hence, or otherwise, solve the equation for . Solution i. We have the trigonometric identity; From this we can substitute ; Hence, L.H.S=R.H.S. ii. We are required to solve the equation for . From (i) we know that; Substituting this in the given equation to […]
Hits: 739 Question i. Show that can be written as a quadratic equation in and hence solve the equation for . ii. Find the solutions to the equation for . Solution i. We have the expression; We know that ; therefore, We have the trigonometric identity; It can be rearranged to; Therefore; To solve this equation for , we can substitute […]
Hits: 2652 Question The diagram shows triangle ABC where AB=5cm, AC=4cm and BC=3cm. Three circles with centres at A, B and C have radii 3 cm, 2 cm and 1 cm respectively. The circles touch each other at points E, F and G, lying on AB, AC and BC respectively. Find the area of the shaded region EFG. Solution […]
Hits: 1388 Question i. Prove the identity . ii. Hence solve, for , the equation Solution i. We are given the identity; We have the trigonometric identity; It can be rearranged as; Therefore; We have ; ii. We are required to solve the equation; We have found in (i) that; Therefore; Using calculator we can find the […]
Hits: 2827 Question The diagram shows a circle with radius r cm and centre O. The line PT is the tangent to the circle at P and angle radians. The line OT meets the circle at Q. i. Express the perimeter of the shaded region PQT in terms of r and . ii. In the case where and , find the […]
Hits: 1324 Question In the diagram, triangle ABC is right-angled at C and M is the mid-point of BC. It is given that angle radians and angle radians. Denoting the lengths of BM and MC by x, i. find AM in terms of x, ii. show that .Solution i. We are required to find AM. Consider right angled […]
Hits: 937 Question The function is defined by for . i. State the range of . ii. Find the coordinates of the points at which the curve intersects the coordinate axes. iii. Sketch the graph of . iv. Obtain an expression for , stating both the domain and range of . Solution […]
Hits: 1523 Question a. The first term of a geometric progression in which all the terms are positive is 50. The third term is 32. Find the sum to infinity of the progression. b. The first three terms of an arithmetic progression are , and respectively, where x is an acute angle. i. Show that . ii. Find the sum […]
Hits: 2938 Question In the diagram, AOB is a quarter circle with centre O and radius r. The point C lies on the arc AB and the point D lies on OB. The line CD is parallel to AO and angle radians. i. Express the perimeter of the shaded region in terms of r, and . ii. For […]
Hits: 703 Question Solve the equation for . Solution We are given; We have the trigonometric identity; From this identity we can have; Substituting in given equation; Let ; Now we have two options. Since; Using calculator we can find the values of . NOT POSSIBLE We utilize the periodic property of to find other solutions (roots) of : Symmetry Property […]
Hits: 3446 Question a. Figure 1 In Fig. 1, OAB is a sector of a circle with centre O and radius r. AX is the tangent at A to the arc AB and angle . i. Show that angle . ii. Find the area of the shaded segment in terms of r and . b. […]
Hits: 1617 Question The diagram shows a pyramid OABC with a horizontal triangular base OAB and vertical height OC. Angles AOB, BOC and AOC are each right angles. Unit vectors , and are parallel to OA, OB and OC respectively, with OA = 4 units, OB = 2.4 units and OC = 3 units. The […]
Hits: 762 Question a. Solve the equation , giving the solution in an exact form. b. Solve, by factorising, the equation for . Solution a. We are given; b. We are required to solve, by factorising, the following equation for . Now we have two options. We utilize the periodic property of to find other solutions (roots) […]