# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | May-Jun | (P1-9709/12) | Q#10

Question Functions and  are defined by  for  for        i.       Solve the equation .    ii.       Find the range of .   iii.       Find the set of values of x for which .   iv.       Find the value of the constant  for which the equation  has two equal roots. Function  is defined by  for , and it is given that  has an inverse.    v.       State […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | May-Jun | (P1-9709/12) | Q#9

Question The diagram shows part of the curve   and the tangent to the curve at .      i.       Find expressions for  and .    ii.       Find the equation of the tangent to the curve at P in the form .   iii.       Find, showing all necessary working, the area of the shaded region. Solution i.   First we find the expression […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | May-Jun | (P1-9709/12) | Q#8

Question The equation of a curve is such that . Given that the curve has a  minimum point at , find the coordinates of the maximum point. Solution To find the coordinates of a stationary point (in this case a maximum point) we need derivative of equation of the curve. We are given the second derivative of the equation of […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | May-Jun | (P1-9709/12) | Q#7

Question The diagram shows a trapezium ABCD in which BA is parallel to CD. The position  vectors of A, B and C relative to an origin O are given by and      i.       Use a scalar product to show that AB is perpendicular to BC.    ii.       Given that the length of CD is 12 units, find the position […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | May-Jun | (P1-9709/12) | Q#5

Question      i.       Prove the identity    ii.       Solve the equation  for . Solution i.   We have the trigonometric identity; From this we can substitute  in the above equation. Since ; ii.   We are required to solve the equation  for . From (i), we know that; Therefore; Using calculator; We utilize the periodic/symmetry property of   to find […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | May-Jun | (P1-9709/12) | Q#4

Question The diagram shows a sector of a circle with radius  cm and centre . The chord AB  divides the sector into a triangle AOB and a segment AXB. Angle AOB is  radians.      i.       In the case where the areas of the triangle AOB and the segment AXB are  equal, find the value of the constant  for which .    ii.  […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | May-Jun | (P1-9709/12) | Q#3

Question The reflex angle  is such that , where .      i.       Find an expression, in terms of , for a.   b.      ii.       Explain why  is negative for . Solution i.   A Reflex Angle is one which is more than 180° but less than 360°. a.     We are given that; We have the trigonometric identity; From this we can […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | May-Jun | (P1-9709/12) | Q#1

Question Find the coordinates of the point at which the perpendicular bisector of the line  joining  to  meets the x-axis. Solution We are required to find the coordinates of the x-intercept of the perpendicular  bisector of the line joining  to . The point  at which curve (or line) intercepts x-axis, the value of . So we  can find the value […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | May-Jun | (P1-9709/12) | Q#2

Question Find the coefficient of  in the expansion of . Solution It is evident that to get the terms containing  in the product of we need; This will result; Hence we need first to find the terms with  and  i.e.  in the expansion of . First rewrite the given expression in standard form. Expression for the general term in the Binomial expansion of  is: In the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | May-Jun | (P1-9709/12) | Q#6

Question The 1st, 2nd and 3rd terms of a geometric progression are the 1st , 9th and 21st terms respectively of  an arithmetic progression.  The 1st term of each progression is 8 and the common ratio of the geometric progression is r, where  r ≠ 1. Find     i.       the value of r,     ii.     the 4th term of each […]