# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | May-Jun | (P1-9709/12) | Q#11

Question The function  is defined by  for .      i.       Find the set of values of p for which the equation  has no real roots. The function  is defined by   for .    ii.       Express  in the form , where a, b and c are constants.   iii.       Find the range of . The function  is defined by   for , where k is a constant. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | May-Jun | (P1-9709/12) | Q#10

Question The equation of a curve is  .      i.       Find, showing all necessary working, the volume obtained when the region bounded by the               curve, the x-axis and the lines x=1 and x=2 is rotated through 360O about the x-axis.    ii.     Given that the line  is a normal to the curve, find […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | May-Jun | (P1-9709/12) | Q#9

Question Relative to an origin , the position vectors of points A and B are given by and      i.      Use a vector method to find angle AOB. The point C is such that .    ii.       Find the unit vector in the direction of   iii.     Show that triangle OAC is isosceles. Solution      i.   We recognize that […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | May-Jun | (P1-9709/12) | Q#7

Question The point C lies on the perpendicular bisector of the line joining the points A(4,6) and B(10,2). C  also lies on the line parallel to AB through (3,11).     i.       Find the equation of the perpendicular bisector of AB.    ii.       Calculate the coordinates of C. Solution i.   We are required to find the equation of the perpendicular bisector of […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | May-Jun | (P1-9709/12) | Q#6

Question A tourist attraction in a city centre is a big vertical wheel on which passengers can ride. The wheel  turns in such a way that the height, h m, of a passenger above the ground is given by the formula  . In this formula, k is a constant, t is the time in minutes that has elapsed […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | May-Jun | (P1-9709/12) | Q#5

Question i.          Prove the identity .  ii.       Hence solve the equation , for . Solution i.   We have the trigonometric relation; Therefore; ii.   We are required to solve the equation  for . From (i), we know that; Therefore; Let ; Now we have two options. Since ; Using calculator We utilize the periodic/symmetry property of   […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | May-Jun | (P1-9709/12) | Q#4

Question Variables u, x and y are such that  and . Express u in terms of x and  hence find the stationary value of u. Solution We are given that; We can fin expression for y from this equation and substitute in expression of u to write u in terms of  x. Therefore; Substituting this in expression of u; Now we […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | May-Jun | (P1-9709/12) | Q#2

Question In the diagram, AYB is a semicircle with AB as diameter and OAXB is a sector of a circle with centre  O and radius r. Angle AOB = 2 radians. Find an expression, in terms of r and , for the  area of the shaded region. Solution It is evident from the diagram that; First we calculate area […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | May-Jun | (P1-9709/12) | Q#1

Question The function f is such that  and  is a point on the curve . Find . Solution i.   We are given that for curve ; We are also given that  is a point on the curve . We are required to find the equation of the curve. We can find equation of the curve from its derivative through […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | May-Jun | (P1-9709/12) | Q#3

Question      i.       Find the coefficients of  in the expansion of .    ii.       Find the coefficient of  in the expansion of . Solution i.   Expression for the general term in the Binomial expansion of  is: First we rewrite the expression in the standard form; In the given case: Hence; Since we are looking for the terms with : we can  equate […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | May-Jun | (P1-9709/12) | Q#8

Question a)   The first, second and last terms in an arithmetic progression are 56, 53 and −22 respectively.  Find the sum of all the terms in the progression. b)   The first, second and third terms of a geometric progression are 2k + 6, 2k and k + 2  respectively, where k is a positive constant. i.       Find the […]