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

Question The diagram shows parts of the curves  and  intersecting at points  and . The angle between the tangents to the two curves at  is .      i.       Find , giving your answer in degrees correct to 3 significant figures.    ii.       Find by integration the area of the shaded region. Solution i.   Angle between two curves is the angle […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | Oct-Nov | (P1-9709/11) | Q#10

Question      i.       Express  in the form .  The function  is defined for , where  and  are positive constants, by The range of  is given by , where  and  are constants.    ii.       State the smallest possible value of . For the case where  and ,   iii.       find  and ,   iv.       find an expression for . Solution i.   We have the expression; We […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | Oct-Nov | (P1-9709/11) | Q#9

Question The function  is defined for  and is such that . The curve  passes through the point .      i.       Find the equation of the normal to the curve at P.    ii.       Find the equation of the curve.    iii.     Find the x-coordinate of the stationary point and state with a reason whether this point is a maximum or a […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | Oct-Nov | (P1-9709/11) | Q#8

Question In the diagram, AB is an arc of a circle with centre O and radius 4 cm. Angle AOB is  radians. The  point D on OB is such that AD is perpendicular to OB. The arc DC, with centre O, meets OA at C.      i.       Find an expression in terms of  for the perimeter of the shaded region […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | Oct-Nov | (P1-9709/11) | Q#6

Question Relative to an origin O, the position vector of A is  and the position vector of B is  .      i.       Show that angle OAB is a right angle.    ii.       Find the area of triangle OAB. Solution i.   It is evident that angle OAB is the angle between vectors  and . We are given  but we need to find […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | Oct-Nov | (P1-9709/11) | Q#5

Question Find the set of values of  for which the line  meets the curve                     at two distinct points. Solution If two lines (or a line and a curve) intersect each other at a point then that point lies on both lines i.e.  coordinates of that point have same values on both lines (or on the line and the curve).  Therefore, […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | Oct-Nov | (P1-9709/11) | Q#4

Question The line  passes through the points  and , where  and  are constants.      i.       Find the values of  and .    ii.       Find the coordinates of the mid-point of . Solution i.   Since the line  through the points  and , coordinates of both points must satisfy equation of the line. For point For point Substituting  in equation ; ii.   We are given the coordinates […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | Oct-Nov | (P1-9709/11) | Q#3

Question Solve the equation  for . Solution We are given the equation; We have the trigonometric identity; We can rewrite it as; Substituting this in above equation; Now we have two options. Using calculator we can find that; Since we are required to solve the equation  for , we consider angles only in this range. Therefore; We utilize  the symmetry property […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | Oct-Nov | (P1-9709/11) | Q#2

Question Find the value of  satisfying the equation . Solution We are given the equation; Using calculator we can find; Therefore; Using calculator we can find; Therefore;

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

Question In the expansion of , the coefficient of  is equal to the coefficient of . Find the value of the non-zero constant . Solution Expression for the general term in the Binomial expansion of  is: In the given case: Hence; Since we are looking for the terms with  :  we can  equate Now we can find the term with; Substituting ; […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | Oct-Nov | (P1-9709/11) | Q#7

Question i.       A geometric progression has first term , common ratio  and sum to infinity . A  second geometric progression has first term , common ratio  and sum to infinity . Find the  value of .    ii.       An arithmetic progression has first term 7. The nth term is 84 and the (3n)th term is 245. Find  the value of […]