# 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 | (P1-9709/11) | Q#2

Question The equation of a curve is . The equation of a line is . On the same diagram, sketch the curve and the line for . Solution First we sketch  for . We can find the points of the graph as follows. Now we sketch the line  for . We can write it as; Slope-Intercept form of the […]

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

Question The diagram shows the graph of  for .      i.       Find the values of ,  and .    ii.      Find the smallest value of  in the interval  for which . Solution i.   We are given that; When we observe given graph of this function, it is evident that y-intercept of the graph, i.e. when , is 3. Hence for […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2009 | Oct-Nov | (P1-9709/11) | Q#5

Question The diagram shows a semicircle ABC with centre O and radius 6 cm. The point B is such that angle BOA is 90o and BD is an arc of a circle with centre A. Find i.       the length of the arc BD,    ii.       the area of the shaded region. Solution      i.   Expression for length of a […]

# 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 | (P1-9709/11) | Q#1

Question Solve the equation  for . Solution To solve this equation for , we can substitute . Hence, Since given interval is  , for  interval can be found as follows; Multiplying the entire inequality with 2; Adding  to entire inequality; Since ; Hence the given interval for  is . To solve  equation for interval , Using calculator we can find the value of . To find all […]

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

Question Prove the identity Solution We are given the equation; We can rewrite it as; We have the trigonometric identity; We can rewrite it as; Therefore; We have the trigonometric relation; Therefore;