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

Question a.   The functions  and  are defined for by  , where  and  are positive constants  , Given that  and ,       (i)          calculate the values of  and ,      (ii)         obtain an expression for  and state the domain of .   b.   A point P travels along the curve  in such a way that the x-coordinate of P at time t  minutes is […]

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

Question The diagram shows parts of the graphs of  and  intersecting at points A and B.      i.       Write down an equation satisfied by the x-coordinates of A and B. Solve this equation and  hence find the coordinates of A and B.    ii.             Find by integration the area of the shaded region. Solution i. […]

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

Question A curve  has a stationary point at  and is such that .       i.       State, with a reason, whether this stationary point is a maximum or a minimum.    ii.       Find  and . Solution i.   Once we have the coordinates of the stationary point  of a curve, we can determine its  nature, whether minimum or maximum, by finding […]

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

Question Three points, O, A and B, are such that  and  , where is a constant.      i. Find the values of  for which  is perpendicular to .    ii. The magnitudes of   and   are  and  respectively. Find the value of  for which .   iii. Find the unit vector in the direction of  when . Solution i.   We are given that; If  and  & , […]

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

Question  is the point and is the point , where  is a constant.     i.       Find, in terms of a, the gradient of a line perpendicular to .    ii.       Given that the distance  is , find the possible values of . Solution      i.   If two lines are perpendicular (normal) to each other, then product of their slopes  and  is; Since we are […]

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

Question i.       Show that .    ii.  Hence solve the equation  for . Solution      i.   We are given; We choose the left hand side; It can be rewritten as; Utilizing the algebraic formula; Utilizing the trigonometric identity; We can write now; We can rewrite the trigonometric identity as; Therefore; Hence;    ii.   We are required […]

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

Question      i.       Express  in the form  .    ii.       Determine whether  is an increasing function, a decreasing function or  neither. Solution i.   We have the expression; We use method of “completing square” to obtain the desired form. We complete the square for the terms which involve . We have the algebraic formula; For the given case we can compare the […]

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

Question In the diagram, OADC is a sector of a circle with centre O and radius 3 cm. AB and CB are tangents  to the circle and angle  radians. Find, giving your answer in terms of  and ,       i.       the perimeter of the shaded region,    ii.       the area of the shaded region. Solution     i.   It is evident […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | Oct-Nov | (P1-9709/13) | 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 ; Substituting ; […]

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

Question Three geometric progressions, P, Q and R, are such that their sums to infinity are the first three  terms respectively of an arithmetic progression. Progression P is; Progression Q is; i.       Find the sum to infinity of progression R.    ii.       Given that the first term of R is 4, find the sum of the first three terms of R. […]