# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | Oct-Nov | (P1-9709/13) | Q#7

Question The diagram shows the function  defined for  by  for  for      i.       State the range of .    ii.       Copy the diagram and on your copy sketch the graph of .   iii.       Obtain expressions to define , giving the set of values of  for which each expression is valid. Solution i.   We have the function;  for  for It is evident from […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | Oct-Nov | (P1-9709/11) | Q#7

Question A function is defined by   for .    i.     State the range of .  ii.     State the exact value of .  iii.      Sketch the graph of .  iv.      Obtain an expression, in terms of , for Solution i.   We have the function;  for This can be written as; We know that range […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | May-Jun | (P1-9709/12) | Q#11

Question The function  is defined for the domain .      i.       Solve the equation .    ii.       Sketch the graph of .   iii.       Find the set of values of  for which the equation  has no solution. The function  is defined for the domain .   iv.       State the largest value of  for which g has an inverse.    v.        For […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | Oct-Nov | (P1-9709/13) | Q#6

Question A curve has equation . It is given that .     i.       Find the set of values of  for which  is an increasing function.    ii.       Given that the curve passes through (1, 3), find . Solution i.   To test whether a function  is increasing or decreasing at a particular point , we take derivative of a function at that point. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | May-Jun | (P1-9709/12) | Q#10

Question The equation of a curve is .     i.        Find .    ii.      Find the equation of the tangent to the curve at the point where the curve intersects the y-axis.   iii.       Find the set of values of  for which  is an increasing function of . Solution     i.   Rule for differentiation of  is: Rule […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | Oct-Nov | (P1-9709/12) | Q#7

Question The function f is defined by  for .      i.       Express  in the form of  and hence state the range of .    ii.       Obtain an expression for  and state the domain of . The function g is defined by  for . The function  is such that  and the domain of  is .   iii.       Obtain an expression for . Solution i. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | Oct-Nov | (P1-9709/11) | Q#3

Question Functions  and  are defined for  by  Express  in the form , where a, b and c are constants. Solution i.   We have the functions; For ; We have the expression; We use method of “completing square” to obtain the desired form. We take out factor ‘4’ from the terms which involve ; Next we complete the square for the terms […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | May-Jun | (P1-9709/13) | Q#10

Question The function  is defined for .      i.       Find the values of the constant  for which the line  is a tangent to the curve .    ii.       Express  in the form of , where ,  and  are constants.   iii.       Find the range of . The function g is defined by  is defined for .   iv.       Find the smallest value of  for which g has […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | May-Jun | (P1-9709/13) | Q#3

Question The function  is defined for . Given that  and  find      i.       the values of  and ,    ii.       the range of .   iii.       the exact value of . Solution i.   We have the function; We are given that; Since Since Subtracting both equation; Substituting  in anyone of the above equations will yield value of . We choose; ii.   We have the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | May-Jun | (P1-9709/12) | Q#3

Question The functions  and  are defined for  by      i.       Find the range of .    ii.       Find the value of the constant  for which the equation  has equal roots. Solution i.   Standard form of quadratic equation is; The graph of quadratic equation is a parabola. If  (‘a’ is positive) then parabola opens upwards and its vertex is the minimum […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | May-Jun | (P1-9709/11) | Q# 9

Question The function  is defined by  for .      i.       Express  in the form .    ii.       State the range of .   iii.        Find the set of values of  for which . The function g is defined by  for .   iv.       Find the value of the constant  for which the equation  has two equal roots. Solution i.   We have the expression; We use method […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | May-Jun | (P1-9709/11) | Q#5

Question The function  is such that    for .      i.       Express  in the form , stating the values of  and .    ii.       State the greatest and least values of .   iii.       Solve the equation . Solution i.   We have the function; We have the trigonometric identity; We can rewrite it as; Therefore the function becomes; Hence; ii.   From (i), we can rewrite […]