# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | May-Jun | (P1-9709/12) | Q#9

Question The function f is such that , for , where  is a constant.      i.       In the case where ,         a.   Find the range of ,        b.   Find the exact solutions of the equation .    ii.       In the case where ,        a.   Sketch the graph of  ,        b.   […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | May-Jun | (P1-9709/12) | Q#4

Question A curve has equation  and P(2, 2) is a point on the curve.     i.       Find the equation of the tangent to the curve at P.    ii.       Find the angle that this tangent makes with the x-axis. Solution i.   To find the equation of the line either we need coordinates of the two points on the line (Two-Point form […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | May-Jun | (P1-9709/12) | Q#11

Question The diagram shows part of the curve . The curve has a maximum point at M and meets the x-axis at O and A.     i.       Find the coordinates of A and M.    ii.       Find the volume obtained when the shaded region is rotated through 360o about the x-axis, giving your answer in terms of . Solution     i. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | May-Jun | (P1-9709/12) | Q#8

Question Relative to an origin , the position vectors of the points ,  and  are given by and i.       Find angle ABC. The point D is such that ABCD is a parallelogram.    ii.       Find the position vector of D. Solution      i.   We recognize that angle ABC is between  and . Therefore, we need the scalar/dot product of these two […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | May-Jun | (P1-9709/12) | Q#7

Question The line passes through the points  and . The line  is parallel to  and passes through the origin. The point C lies on  such that AC is perpendicular to . Find     i.       the coordinates of C,    ii.       the distance AC. Solution     i.   Consider the diagram below. It is evident that point  is the intersection of lines  and AC. If […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | May-Jun | (P1-9709/12) | Q#5

Question      i.       Prove the identity    ii.       Hence solve the equation  , for . Solution i.   We have the equation; We have the relation , therefore, We have the trigonometric identity; We can rewrite it as; Therefore; Using formula;      ii.   To solve the equation  , for , as demonstrated in (i), we can write the given […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | May-Jun | (P1-9709/12) | Q#3

Question The equation , where  and  are constants, has roots −3 and 5.     i.       Find the values of  and .    ii.       Using these values of  and , find the value of the constant  for which the equation  has equal roots. Solution     i.   We have the equation;   If the equation has roots −3 and 5 then; We can expand this product; […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | May-Jun | (P1-9709/12) | Q#1

Question Find . Solution Rule for integration of  is: Rule for integration of  is:

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | May-Jun | (P1-9709/12) | Q#6

Question The function f is defined by , ,      i.     Show that .    ii.     Hence, or otherwise, obtain an expression for . Solution i.   We have function; We can write these as; For , we can write; ii.   We have functions;  , We can write these as; To find the inverse of a given function  we […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | May-Jun | (P1-9709/12) | Q#2

Question      i.       Find the terms in  and  in the expansion of .    ii.       Given that there is no term in  in the expansion of , find the value of the constant . Solution i.   First rewrite the given expression in standard form. Expression for the Binomial expansion of  is: In the given case: Hence; Hence the required terms in […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | May-Jun | (P1-9709/12) | Q#10

Question a)   A circle is divided into 6 sectors in such a way that the angles of the sectors are in arithmetic  progression. The angle of the largest sector is 4 times the angle of the smallest sector. Given that  the radius of the circle is 5 cm, find the perimeter of the smallest sector. b)   The first, […]