# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | May-Jun | (P1-9709/13) | Q#11

Question The function  is defined for . It is given that  has a minimum value when  and that . (i)          Find . It is now given that ,  and  are the first three terms respectively of an arithmetic progression. (ii)        Find the value of . (iii)       Find , and hence find the minimum value of . Solution […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | May-Jun | (P1-9709/13) | Q#10

Question a.     Fig. 1 shows part of the curve  and the line y = h, where h is a constant.        (i)          The shaded region is rotated through 360o about the y-axis. Show that the volume of                         revolution, V, is given by     […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | May-Jun | (P1-9709/13) | Q#8

Question A(-1,1) and P(a,b) are two points, where a and b are constants. The gradient of AP is 2. (i)        Find an expression for b in terms of a. (ii)      B(10,-1) is a third point such that AP = AB. Calculate the coordinates of the possible positions           of P. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | May-Jun | (P1-9709/13) | Q#7

Question The diagram shows two circles with centres A and B having radii 8 cm and 10 cm respectively. The  two circles intersect at C and D where CAD is a straight line and AB is perpendicular to CD. (i)          Find angle ABC in radians. (ii)        Find the area of the shaded region. Solution (i)   It is evident from the diagram […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | May-Jun | (P1-9709/13) | Q#6

Question The line 3y+x=25 is a normal to the curve y= x2 – 5x + k. Find the value of the constant k. Solution If a line  is normal to the curve , then product of their slopes  and  at that point (where line is normal to the curve) is; Therefore, by finding slopes of both the line and […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | May-Jun | (P1-9709/13) | Q#5

Question (i)          Show that the equation  may be expressed as . (ii)        Hence solve the equation  for 0o < < 180o. Solution (i)   We are given that; Utilizing ; (ii)   We are required to solve the equation  for 0o < < 180o. From (i) we know that given equation can be written as; Now we have two options. Using calculator;   […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | May-Jun | (P1-9709/13) | Q#4

Question Relative to an origin O, the position vectors of points A, B and C are given by and The point P lies on AB and is such that . i.       Find the position vector of P. ii.       Find the distance OP. iii.       Determine whether OP is perpendicular to AB. Justify your answer. Solution      i.   We are given that; […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | May-Jun | (P1-9709/13) | Q#3

Question Find the coordinates of the points of intersection of the curve  with the curve . 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 2017 | May-Jun | (P1-9709/13) | Q#2

Question The common ratio of a geometric progression is r. The first term of the progression is  and the sum to infinity is S.      i.       Show that S = 2 − r.    ii.       Find the set of possible values that S can take. Solution      i.   From the given information, we can compile following data about Geometric Progression […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | May-Jun | (P1-9709/13) | Q#1

Question The coefficients of  and  in the expansion of  are equal. Find the value of the non-zero  constant a. Solution We need to equate the coefficients of  and  in the expansion of given expression.  We are given expression as; Expression for the general term in the Binomial expansion of  is: In the given case: Hence; Since we are looking for the coefficients […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | May-Jun | (P1-9709/13) | Q#9

Question      i.       Express  in the form  , where a, b and c are constants. The function  is defined by   for , where p is a constant.    ii.       State the smallest value of  for which  is a one-one function.   iii.       For this value of , obtain an expression for , and state the domain of .   iv.       State the set of values of  for […]