# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | May-Jun | (P1-9709/13) | Q#11

Question In the diagram, OAB is a sector of a circle with centre O and radius r. The point C on OB is such  that angle ACO is a right angle. Angle AOB is  radians and is such that AC divides the sector into  two regions of equal area.     i.       Show that   It is given […]

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

Question Points A(2,9) and B(3,0) lie on the curve y=9+6x−3×2, as shown in the diagram. The tangent at A  intersects the x-axis at C. Showing all necessary working,     i.       find the equation of the tangent AC and hence find the x-coordinate of C,    ii.       find the area of the shaded region ABC. Solution      i.   To find the equation of […]

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

Question The function  is defined by   for x > −1.      i.       Find .    ii.       State, with a reason, whether f is an increasing function, a decreasing function or neither. The function  is defined by  for x < −1   iii.       Find the coordinates of the stationary point on the curve . Solution      i.   We are given that; We can […]

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

Question The point A has coordinates  and the point B has coordinates , where p is a  constant.      i.       For the case where the distance AB is 13 units, find the possible values of p.    ii.       For the case in which the line with equation 2x+3y=9 is perpendicular to AB, find the value of p. Solution      i. […]

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

Question The diagram shows the graph of , where  is defined by  for 0 < x ≤ 2.      i.       Find an expression for  and state the domain of .    ii.       The function  is defined by  for x ≥ 1. Find an expression for , giving your  answer in the form ax + b, where a and b are constants to be […]

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

Question Relative to an origin O, the position vectors of the points A, B and C are given by and      i.       Show that angle ABC is  .    ii.       Find the area of triangle ABC, giving your answer correct to 1 decimal place. Solution      i.   We recognize that angle ABC is between  and . Therefore, we need […]

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

Question        i.      Express the equation  in the form  and solve the equation for  .             ii.               Solve the equation  for .   Solution i.   We are given; We know that; Therefore; Now we solve this equation for . 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 | (P1-9709/13) | Q#2

Question A curve is such that  and the point (4, 7) lies on the curve. Find the equation of the curve. Solution We are required to find the equation of the curve whose derivative is given as below. We can find equation of the curve from its derivative through integration; Rule for integration of  is: If […]

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

Question Express  in the form , where a, b and c are constants. Solution  We have the expression; We use method of “completing square” to obtain the desired form. We take out factor ‘2’ from the terms which involve ; Next we complete the square for the terms which involve . We have the algebraic formula; For the given case we […]

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

Question      i.      Write down the first 4 terms, in ascending powers of , of the expansion of .    ii.       The coefficient of  in the expansion of  is −200. Find the possible values of the constant . Solution i.   Expression for the Binomial expansion of  is: First we rewrite the expression in the standard form; In the given case: […]

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

Question a)   The first term of an arithmetic progression is −2222 and the common difference is 17. Find the  value of the first positive term. b)   The first term of a geometric progression is  and the second term is  , where  . Find the set of values of  for which the progression is convergent. Solution a)     From […]