# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2009 | Oct-Nov | (P1-9709/12) | Q#4

Question Functions  and  are defined by  for .      i.       Find the range of .    ii.       Sketch the graph of .   iii.       State, with a reason, whether  has an inverse. Solution i.   We have the function; We can write it as; We know that; Hereby; We can find the range of   by substituting extreme possible values of ; Therefore; ii.   […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2009 | Oct-Nov | (P1-9709/12) | Q#10

Question i.       The diagram shows the line  and the curve , which intersect at the points A and B. Find a.   the x-coordinates of A and B, b.   the equation of the tangent to the curve at B, c.   the acute angle, in degrees correct to 1 decimal place, between this tangent and the line .    ii.       Determine […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2009 | Oct-Nov | (P1-9709/12) | Q#9

Question The diagram shows a rectangle . The point  is  and  is . The diagonal  is parallel to the x-axis.     i.       Explain why the y-coordinate of  is 6. The x-coordinate of  is .    ii.       Express the gradients of  and  in terms of .   iii.       Calculate the x-coordinates of  and .   iv.       Calculate the area of the rectangle . Solution i.   Consider the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2009 | Oct-Nov | (P1-9709/12) | Q#7

Question A piece of wire of length 50 cm is bent to form the perimeter of a sector POQ of a circle. The radius of the circle is r cm and the angle POQ is q radians (see diagram). i.       Express  in terms of r and show that the area, Acm2, of the sector is given by    .    […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2009 | Oct-Nov | (P1-9709/12) | Q#6

Question In the diagram, OABCDEFG is a cube in which each side has length 6. Unit vectors ,  and  are parallel to ,  and  respectively. The point  is such that   and the point  is the mid-point of . i.       Express each of the vectors  and  in terms of ,  and .    ii.       Find the angle OQP. Solution i.   […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2009 | Oct-Nov | (P1-9709/12) | Q#1

Question The equation of a curve is such that  . Given that the curve passes through the point (4, 6), find the equation of the curve. Solution We can find equation of the curve from its derivative through integration; For the given case; Therefore; Rule for integration of  is: Rule for integration of  is: Rule for integration of […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2009 | Oct-Nov | (P1-9709/12) | Q#5

Question i.               Prove the identity . ii.               Solve the equation  for . Solution i.   We have the trigonometric identity; We can rewrite it in two ways; ii.   To solve the equation  for , we can express, as demonstrated in (i), the right hand side of given equation as; Therefore the given equation can be written as; We have […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2009 | Oct-Nov | (P1-9709/12) | Q#8

Question The function  is such that  for ,      i.       Obtain an expression for  and explain why  is a decreasing function.    ii.       Obtain an expression for .   iii.       A curve has the equation . Find the volume obtained when the region bounded by the curve, the coordinate axes and the line is rotated through  about the x-axis. Solution i.   We have the function; […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2009 | Oct-Nov | (P1-9709/12) | Q#2

Question i.   Find, in terms of the non-zero constant ,  the first four terms in the expansion of  in ascending powers of .    ii.     Given that the coefficients of  and  in this expansion are equal, find the value of . Solution i.   Expression for the Binomial expansion of  is: In the given case: Hence; ii.   Since the coefficients […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2009 | Oct-Nov | (P1-9709/12) | Q#3

Question A progression has a second term of 96 and a fourth term of 54. Find the first term of the progression  in each of the following cases:     i.       the progression is arithmetic,    ii.       the progression is geometric with a positive common ratio. Solution i.   From the given information, we can compile following data for Arithmetic Progression (A.P); Expression for […]