# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/12) | Q#11

Question The diagram shows a sector of a circle with centre  and radius 20 cm. A circle with centre  and radius  cm lies within the sector and touches it at P, Q and R. Angle POR = 1.2 radians. i. Show that , correct to 3 decimal places. ii. Find the total area of the three parts of the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/12) | Q#10

Question A curve is defined for x > 0 and is such that  . The point P(4, 8) lies on the curve.     i.       Find the equation of the curve.    ii.     Show that the gradient of the curve has a minimum value when x = 2 and state this minimum value. Solution i.   We are given that; Therefore, for the given case; […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/12) | Q#9

Question The diagram shows part of the curve  , crossing the y-axis at the point B(0, 3). The point A on the curve has coordinates (3, 1) and the tangent to the curve at A crosses the y-axis at C.     i.       Find the equation of the tangent to the curve at A.    ii.    Determine, showing all necessary working, whether C […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/12) | Q#7

Question The position vectors of points A and B relative to an origin O are  and  are given by and where  is a constant.      i.       In the case where , calculate angle AOB.    ii.       Find the values of  for which  is a unit vector. Solution      i.   We are given that; In the case where , We recognize that […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/12) | Q#6

Question i.       Show that the equation  can be written as a quadratic equation in . ii.       Solve the equation , for . Solution i.   We have the equation; To write the given equation in terms of , we utilize the trigonometric relation; Hence; Multiplying both sides with ; We have the trigonometric identity; We can rewrite the identity as; Therefore; This is […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/12) | Q#5

Question The diagram shows a triangle ABC in which A has coordinates (1, 3), B has coordinates (5, 11) and angle ABC is  . The point X (4, 4) lies on AC. Find     i.       the equation of BC,    ii.     the coordinates of C. Solution i.   We need to find the equation of BC. To find the equation of the line […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/12) | Q#4

Question The line  , where  is a constant, is a tangent to the curve  at the point P. Find i.          the value of k,  ii.       the coordinates of P. Solution i.   We are given that line is tangent to the curve, therefore, line and the curve intersect at one point only i.e. P. If two […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/12) | Q#3

Question The diagram shows a plan for a rectangular park ABCD, in which AB = 40m and AD = 60m. Points X and Y lie on BC and CD respectively and AX, XY and YA are paths that surround a triangular playground. The length of DY is  m and the length of XC is  m.     i.       Show that […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/12) | Q#2

Question The function  is such that  for . Find      i.     in the form  where ,  and  are constants,    ii.       the domain of . Solution i.   We have the functions;  for 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 . Interchanging ‘x’ with ‘y’; ii. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/12) | Q#1

Question In the expansion of , the coefficient of  is -280. Find the value of the constant . Solution First rewrite the given expression in standard form. Expression for the general term in the Binomial expansion of  is: In the given case: Hence; Since we are looking for the term of   : we can  equate Subsequently substituting  in: Since the coefficient […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/12) | Q#8

Question a)   In a geometric progression, all the terms are positive, the second term is 24 and the fourth term  is . Find i.       the first term, ii.      the sum to infinity of the progression. b)  A circle is divided into n sectors in such a way that the angles of the sectors are in arithmetic  progression. The […]