Hits: 1544

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/13) | Q#9

Hits: 1544   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 […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/13) | Q#3

Hits: 510   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 . […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/12) | Q#10

Hits: 1506 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 […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/12) | Q#6

Hits: 2514 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 […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/12) | Q#2

Hits: 842 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 […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/11) | Q#6

Hits: 993 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 […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | May-Jun | (P1-9709/13) | Q#8

Hits: 712 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 […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | May-Jun | (P1-9709/13) | Q#6

Hits: 2579 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 […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | May-Jun | (P1-9709/12) | Q#7

Hits: 1334 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 […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | May-Jun | (P1-9709/12) | Q#6

Hits: 2739 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 […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | May-Jun | (P1-9709/12) | Q#5

Hits: 1245 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 […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | May-Jun | (P1-9709/11) | Q#11

Hits: 895 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 […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | May-Jun | (P1-9709/11) | Q#9

Hits: 1468 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 […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | May-Jun | (P1-9709/11) | Q#7

Hits: 2861 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 […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | May-Jun | (P1-9709/11) | Q#2

Hits: 675 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 […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Feb-Mar | (P1-9709/12) | Q#9

Hits: 3373 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.     […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Feb-Mar | (P1-9709/12) | Q#7

Hits: 1571 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 point P on CA […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Feb-Mar | (P1-9709/12) | Q#4

Hits: 727 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) […]