# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | May-Jun | (P1-9709/13) | Q#10

Question Functions  and  are defined by ,        i.       Evaluate .  ii.       Sketch in a single diagram the graphs of  and , making clear the relationship between the graphs.   iii.      Obtain an expression for  and use your answer to explain why  has an inverse.   iv.       Express each of  and , in terms of . Solution i.   We have functions; We […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | May-Jun | (P1-9709/13) | Q#9

Question A curve is such that  and P(9, 5) is a point on the curve.     i.       Find the equation of the curve.    ii.       Find the coordinates of the stationary point on the curve.   iii.       Find an expression for  and determine the nature of the stationary point.   iv.       The normal to the curve at P makes an angle of  with the positive x-axis. Find […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | May-Jun | (P1-9709/13) | Q#8

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 […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | May-Jun | (P1-9709/13) | Q#7

Question In the diagram, AB is an arc of a circle, centre O and radius 6 cm, and angle  radians. The line AX is a tangent to the circle at A, and OBX is a straight line.     i.       Show that the exact length of AX is  cm.  Find, in terms of  and ,    ii.       the area of the shaded region,   […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | May-Jun | (P1-9709/13) | Q#5

Question In the diagram, OABCDEFG is a rectangular block in which  cm and  cm. Unit vectors ,  and  are parallel to ,  and  respectively. The point P is the mid-point of DG, Q is the centre of the square face CBFG and R lies on AB such that AR = 4 cm.     i.       Express each of the vectors  and […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | May-Jun | (P1-9709/13) | Q#4

Question a)   Differentiate  with respect to . b)   Find   and hence find the value of a . Solution a)   We are given that; Rule for differentiation of  is: Rule for differentiation of  is: b)     We are given that; Rule for integration of  is: Now we find the value of;

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | May-Jun | (P1-9709/13) | Q#3

Question The line  , where a and b are positive constants, meets the x-axis at P and the y-axis at Q. Given that   and that the gradient of the line PQ is  , find the values of a and b. Solution We are given that points P and Q are x and y intercepts, respectively. First we find the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | May-Jun | (P1-9709/13) | Q#2

Question Find the set of values of  for which the line  intersects the curve  at two distinct points. 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 […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | May-Jun | (P1-9709/13) | Q#1

Question The coefficient of  in the expansion of  , where is positive, is 90. Find the value of . Solution Expression for the Binomial expansion of  is: We need to expand both terms one-by-one. First we expand In the given case: Hence; Similarly we also need to expand First rewrite the given expression in standard form. In the given case: Hence; […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | May-Jun | (P1-9709/13) | Q#6

Question a)   A geometric progression has a third term of 20 and a sum to infinity which is three times the first  term. Find the first term. b)   An arithmetic progression is such that the eighth term is three times the third term. Show that the  sum of the first eight terms is four times the sum of […]