# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | Oct-Nov | (P1-9709/13) | Q#10

Question The function  is defined by  for .      i.       Find  and  and hence verify that the function  has a minimum value at . The points  and  lie on the curve , as shown in the diagram.      i.       Find the distance AB.    ii.       Find, showing all necessary working, the area of the shaded region. Solution i.   We are given; Rule for differentiation […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | Oct-Nov | (P1-9709/13) | Q#9

Question A curve passes through the point A(4,6) and is such that  . A point P is moving along  the curve in such a way that the x-coordinate of P is increasing at a constant rate of 3 units per  minute.      i.       Find the rate at which the y-coordinate of P is increasing when P is at A.    […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | Oct-Nov | (P1-9709/13) | Q#8

Question The function  is defined by  for , where a is a constant. The function  is  defined  for .      i.       Find the largest value of a for which the composite function can be formed. For the case where ,    ii.       solve the equation ,   iii.       find the set of values of  which satisfy the inequality . Solution i.   We are […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | Oct-Nov | (P1-9709/13) | Q#7

Question a.   Show that the equation  can be expressed as and hence solve the equation  for . b.     The diagram shows part of the graph of , where a and b are constants. The graph  crosses the x-axis at the point  and the y-axis at the point ,. Find c and d in  terms of a and b. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | Oct-Nov | (P1-9709/13) | Q#6

Question A ball is such that when it is dropped from a height of 1 metre it bounces vertically from the ground  to a height of 0.96 metres. It continues to bounce on the ground and each time the height the ball  reaches is reduced. Two different models, A and B, describe  this. Model A : The height reached […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | Oct-Nov | (P1-9709/13) | Q#5

Question Relative to an origin O, the position vectors of the points A and B are given by and    i.       For the case where OA is perpendicular to OB, find the value of p.    ii.       For the case where OAB is a straight line, find the vectors  and . Find also the length of            […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | Oct-Nov | (P1-9709/13) | Q#4

Question The diagram shows a metal plate OABCDEF consisting of 3 sectors, each with centre O. The radius of sector COD is 2r and angle COD is  radians. The radius of each of the sectors BOA and FOE is  r, and AOED and CBOF are straight lines. i. Show that the area of the metal plate is . ii. Show that […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | Oct-Nov | (P1-9709/13) | Q#3

Question i.       Express  in the form , where a, b and c are constants.  ii.   The function, where , is defined for . Find  and state, with       a reason, 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 […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | Oct-Nov | (P1-9709/13) | Q#2

Question Find the coefficient of x in the expansion of . Solution 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 terms with  i.e. : we can  equate; Now we can find the term with; Substituting ; Hence the coefficient of the term containing  is 7.

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | Oct-Nov | (P1-9709/13) | Q#1

Question A line has equation  and a curve has equation , where c is a constant. Find the set of possible values of c for which the line does not intersect 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.  […]