# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | Oct-Nov | (P1-9709/11) | Q#10

Question The diagram shows the curve  and the tangent to the curve at the point . i.         Find the equation of this tangent, giving your answer in the form . ii.       Find the area of the shaded region. Solution i.   We are required to find equation of the tangent to the curve at point […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | Oct-Nov | (P1-9709/11) | Q#8

Question The inside lane of a school running track consists of two straight sections each of  length x metres, and two semicircular sections each of radius r metres, as shown in  the diagram. The straight sections are perpendicular to the diameters of the  semicircular sections. The perimeter of the inside lane is 400 metres. i.       Show that the area, Am2, […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | Oct-Nov | (P1-9709/11) | Q#7

Question The point A has coordinates  and the point B has coordinates . i.       Find the equation of the perpendicular bisector of AB, giving your answer in the form .    ii.       A point C on the perpendicular bisector has coordinates . The distance OC is 2 units, where O is the origin. Write down two equations involving  and […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | Oct-Nov | (P1-9709/11) | Q#6

Question The diagram shows a metal plate made by fixing together two pieces, OABCD  (shaded) and OAED (unshaded). The piece OABCD is a minor sector of a circle with  centre O and radius . The piece OAED is a major sector of a circle with centre O and radius . Angle AOD is  radians. Simplifying your answers […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | Oct-Nov | (P1-9709/11) | Q#2

Question A curve has equation . It is given that  and that . Find . Solution We are given that equation of the curve is . We are also given that  which means there is a point on the curve where  and  i.e. the curve passes through the point . We are also given the derivative of the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | Oct-Nov | (P1-9709/11) | Q#4

Question      i.       Solve the equation  for .  ii.      Hence find the solution of the equation  for . Solution      i.   We have the equation; Since we are required to write it as a quadratic equation in , we need to eliminate . We have the trigonometric identity; We can rewrite the identity as; Hence; Let; Therefore […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | Oct-Nov | (P1-9709/11) | Q#3

Question The diagram shows a pyramid OABCD in which the vertical edge OD is 3 units in length. The point E is the centre of the horizontal rectangular base OABC. The sides OA and AB have lengths of 6 units and 4 units respectively. Unit vectors ,  and  are parallel to ,  and  respectively.     i.       Express each of the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | Oct-Nov | (P1-9709/11) | Q#5

Question The function  is defined by , for ,      i.       Define in a similar way the inverse function .    ii.       Solve the equation . Solution i.   We have the function; We can write it as; We write it as; To find the inverse of a given function  we need to write it in terms of  rather than in terms of . Since given […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | Oct-Nov | (P1-9709/11) | Q#1

Question i.       Find the first three terms in the expansion of  in ascending powers of . ii.    In the expansion of  , the coefficient of  is zero, find the value of . Solution i.   Expression for the Binomial expansion of  is: In the given case: Hence; ii.   To find the value of  in the expansion of  First we know from (i) […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | Oct-Nov | (P1-9709/11) | Q#9

Question a)   In an arithmetic progression the sum of the first ten terms is 400 and the sum of  the next ten terms is 1000. Find the common difference and the first term. b)  A geometric progression has first term , common ratio and sum to infinity 6. A  second geometric progression has first term […]