# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | May-Jun | (P1-9709/13) | Q#11

Question The diagram shows a parallelogram ABCD, in which the equation of AB is  and the equation  of AD is . The diagonals AC and BD meet at the point E . Find, by calculation, the coordinates of A, B, C  and D. Solution It is evident from the diagram that point A is the intersection point of sides AD […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | May-Jun | (P1-9709/13) | Q#10

Question The diagram shows the curve  and the line . Find, showing all necessary working, the area of the shaded region. Solution It is evident from the diagram that; First we find area under the curve. We are given equation of the curve as; We are also given equation of the line as; To find the area of region under the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | May-Jun | (P1-9709/13) | Q#9

Question The base of a cuboid has sides of length  cm and  cm. The volume of the cuboid is 288 cm3.      i.       Show that the total surface area of the cuboid, A cm2, is given by     ii.       Given that  can vary, find the stationary value of A and determine its nature. Solution From the given information we can compile following data; […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | May-Jun | (P1-9709/13) | Q#8

Question      i.       Express  in the form , where a, b and c are constants, and use your  answer to state the minimum value of .    ii.       Find the set of values of  for which the equation  has no real roots. Solution      i.   We have the expression; We use method of “completing square” to obtain the desired form. We take out […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | May-Jun | (P1-9709/13) | Q#7

Question The position vectors of points A, B and C relative to an origin O are given by and i.       Show that angle .    ii.  Use the result in part (i) to find the exact value of the area of triangle ABC. Solution      i.   We recognize that angle BAC is between  and . Therefore, we need the scalar/dot […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | May-Jun | (P1-9709/13) | Q#6

Question A curve is such that  , where a is  constant. The point  lies on the curve and the normal to the curve at  is .      i. Show that .    ii. Find the equation of the curve. Solution i.   If two lines (or one line and a curve) are perpendicular (normal) to each other, then product of […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | May-Jun | (P1-9709/13) | Q#5

Question A functions  is such that   for   .      i.       Find an expression for  and use your result to explain why  has an inverse.    ii.       Find an expression for , and state the domain and range of . Solution i.   We have; Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of curve  with […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | May-Jun | (P1-9709/13) | Q#4

Question      i.       Prove the identity    ii.       Hence solve the equation  for . Solution i.   We utilize the relation; We have the trigonometric identity; 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 other solutions (roots) of :  Periodic/Symmetry […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | May-Jun | (P1-9709/13) | Q#3

Question The diagram shows part of a circle with centre O and radius 6 cm. The chord AB is such that angle  AOB = 2.2 radians. Calculate     i.       the perimeter of the shaded region,    ii.       the ratio of the area of the shaded region to the area of the triangle AOB, giving your answer in  the form k : 1. Solution […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | May-Jun | (P1-9709/13) | Q#1

Question Find the coefficient of  in the expansion of . 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 terms with  i.e. : we can  equate Now we can find the term with; Substituting ; Hence the coefficient of the term containing […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | May-Jun | (P1-9709/13) | Q#2

Question The first term in a progression is 36 and the second term is 32.      i.       Given that the progression is geometric, find the sum to infinity.     ii.       Given instead that the progression is arithmetic, find the number of terms in the progression if  the sum of all the terms is 0. Solution i.   From the given information, […]