# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | Oct-Nov | (P1-9709/13) | Q#11

Question     i.      Express in the form of where a, b and c are constants. The function f is defined by  for .    ii.       State the largest value of the constant k for which f is a one-one function.  iii.     For this value of k find an expression for and state the domain of […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | Oct-Nov | (P1-9709/13) | Q#9

Question A curve has equation  and a line has equation , where  is a constant. i.Show that, for all values of k, the curve and the line meet. ii.State the value of k for which the line is a tangent to the curve and find the coordinates of the  point where the line touches the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | Oct-Nov | (P1-9709/13) | Q#2

Question The function f is defined by  for . Determine, showing all necessary working, whether f is an increasing function, a decreasing  function or neither. Solution We are given function;   We are required to find whether is an increasing function, decreasing function or neither. To test whether a function is increasing or decreasing at […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | Oct-Nov | (P1-9709/12) | Q#10

Question The equation of a curve is  and the equation of a line is , where k is a constant. i.       Find the set of values of k for which the line does not meet the curve. In the case where k = 15, the curve intersects the line at points A and B. ii. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | Oct-Nov | (P1-9709/12) | Q#9

Question The function f is defined by  for .     i.      Express in the form of where a and b are constants.   ii.     State the range of .  The function g is defined by  for .  iii.       State the largest value of k for which g has an inverse.   iv.     Given that g has […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | Oct-Nov | (P1-9709/12) | Q#6

Question The diagram shows a triangle ABC in which BC = 20 cm and angle ABC = 90o. The perpendicular  from B to AC meets AC at D and AD = 9 cm. Angle BCA = .     i.      By expressing the length of BD in terms of in each of the triangles ABD and […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | Oct-Nov | (P1-9709/11) | Q#11

Question The one-one function f is defined by  for , where  is a constant. a.                    i.     State the greatest possible value of .              ii.     It is given that takes this greatest possible value. State the range of f and find an expression for .   b.    The function g is defined by […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | Oct-Nov | (P1-9709/11) | Q#2

Question A line has equation y = x + 1 and a curve has equation y = x2 + bx + 5. Find the set of values of the  constant b for which the line meets the curve. Solution If two lines (or a line and a curve) intersect each other at a point then […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | Oct-Nov | (P1-9709/11) | Q#1

Question Showing all necessary working, solve the equation . Solution We are required to solve the equation; Let , then taking square of both sides results in . Hence the given equation becomes; Now we have two options. Since ;

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | Feb-Mar | (P1-9709/12) | Q#10

Question The functions f and g are defined by  for  for . i.   a.   State the range of the function f. b.   State the range of the function g. c.   State the range of the function fg.    ii.       Explain why the function gf cannot be formed.   iii.       Find the set of values […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | Feb-Mar | (P1-9709/12) | Q#9

Question A curve has equation  and a line has equation , where  is a constant. i.Find the set of values of  for which the curve and the line meet. ii.The line is a tangent to the curve for two particular values of . For each of these values find  the x-coordinate of the point at […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | May-Jun | (P1-9709/13) | Q#10

Question The one-one function f is defined by  for , where c is a  constant.   i.       State the smallest possible value of c. In parts (ii) and (iii) the value of c is 4.    ii.       Find an expression for  and state the domain of .   iii.       Solve the equation , giving your […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | May-Jun | (P1-9709/13) | Q#8

Question i.       The tangent to the curve y = x3 − 9×2 + 24x − 12 at a point A is parallel to the line  y = 2 − 3x. Find the equation of the tangent at A.    ii.       The function f is defined by f(x) = x3 − 9×2 + 24x − 12 […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | May-Jun | (P1-9709/13) | Q#1

Question Express 3×2 − 12x + 7 in the form a(x + b)2 + c, where a, b and c are constants.  Solution We have the expression; We use method of “completing square” to obtain the desired form. Next we complete the square for the terms which involve . We have the algebraic formula; For […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | May-Jun | (P1-9709/12) | Q#7

Question The function is defined by  for .      i.       Express   in the form , where a and b are constants.    ii.       State the coordinates of the stationary point on the curve y = f(x). The function is defined by  for .   iii.       State the smallest value of k for which g […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | Feb-Mar | (P1-9709/11) | Q#2

Question The equation of a curve is  , where  is a constant.     i.       Find the set of values of  for which the whole of the curve lies above the x-axis.    ii.       Find the value of  for which the line y + 2x = 7 is a tangent to the curve. Solution i.   […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | May-Jun | (P1-9709/11) | Q#10

Question The curve with equation  passes through the origin.     i.       Show that the curve has no stationary points.    ii.       Denoting the gradient of the curve by m, find the stationary value of m and  determine its nature. Solution i.   We are required to show that curve has no stationary points. A stationary […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | May-Jun | (P1-9709/11) | Q#9

Question Functions f and g are defined for  by;     i.Find the points of intersection of the graphs of y = f(x) and y = g(x). ii.Find the set of values of x for which f(x) > g(x). iii.Find an expression for fg(x) and deduce the range of fg. The function h is defined […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2018 | Oct-Nov | (P2-9709/22) | Q#7

Question     i.       Use the factor theorem to show that (2x+3) is a factor of    ii.       Show that the equation  can be expressed as   iii.       Solve the equation For . Solution      i.   We are given that; We are also given that is a factor of . When a polynomial, , is […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2018 | Oct-Nov | (P2-9709/22) | Q#2

Question Given that , find the value of 3x and hence, using logarithms, find the  value of x correct to 4 significant figures. Solution We are given that; Let ; Now we have two options. Since ; Taking logarithm of both sides; Since logarithm of negative number is nt possible, only possible solution is; Power […]