# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | Oct-Nov | (P1-9709/13) | Q#6

Question A line has equation  and a curve has equation , where k is a constant. i.Find the set of values of  for which the line and curve meet at two distinct points. i.For each of two particular values of , the line is a tangent to the curve. Show that these two tangents meet […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | Oct-Nov | (P1-9709/13) | Q#3

Question The equation of a curve is . The curve has no stationary points in the interval  . Find the least possible value of and the greatest possible value of . Solution We are given; We are given that curve has no stationary point. A stationary point on the curve is the point where gradient […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | Oct-Nov | (P1-9709/13) | Q#2

Question The function g is defined by  for . By first completing the square, find an  expression for and state the domain of . Solution We are given that; We use method of “completing square” to obtain the desired form. We complete the square for the  terms which involve . We have the algebraic formula; […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | Oct-Nov | (P1-9709/12) | Q#9

Question Functions f and g are defined by;  for   for Where  is a constant.     i.      Find the value of for which the line is a tangent to the curve .   ii.     In the case where , find the set of values of for which .  iii.     In the case where , […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | Oct-Nov | (P1-9709/11) | Q#2

Question An increasing function, , is defined for x > n, where n is an integer. It is given that . Find the least possible value of n.  Solution We are given derivative of the function as; We are also given that it is an increasing function. To test whether a function is increasing or […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | May-Jun | (P1-9709/13) | Q#7

Question The coordinates of two points A and B are (1, 3) and (9, −1) respectively and D is the mid-point of  AB. A point C has coordinates (x, y), where x and y are variables. i.State the coordinates of D. ii.It is given that CD2 = 20. Write down an equation relating x and […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | May-Jun | (P1-9709/13) | Q#1

Question The function f is defined by for . i. Express in the form of . ii. Hence find the set of values of for which , giving your answer in exact form.   Solution i. We have the expression; We use method of “completing square” to obtain the desired form. We complete the square […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | May-Jun | (P1-9709/12) | Q#9

Question The curve C1 has equation y = x2− 4x + 7. The curve C2 has equation y2 = 4x + k, where k is a constant. The tangent to C1 at the point where x = 3 is also the tangent to C2 at the point P. Find the  value of k and the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | May-Jun | (P1-9709/11) | Q#5

Question The function f is defined by  for .     i.      Express in the form of where and are constants.   ii.     State the greatest value of .    The function g is defined by  for .  iii.     Find the value of for which . Solution i.   We have the expression;   We use […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | May-Jun | (P1-9709/11) | Q#2

Question The line , where  is a constant, is a tangent to the curve  at the point  on the curve. i.Find the value of . ii.Find the coordinates of . Solution i. We are given equation of the line as; We are given equation of the curve as; It is given that line is tangent […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | Feb-Mar | (P1-9709/12) | Q#8

Question     i.      Express in the form of . The function f is defined by  for , where is constant.   ii.     State the largest value of for which is a decreasing function. The value of is now given to be 1.  iii.     Find an expression for and state the domain of .  iv.     The […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | Feb-Mar | (P1-9709/12) | Q#5

Question Two vectors,  and , are such that and Where is a constant.     i.      Find the values of for which is perpendicular to .   ii.     Find the angle between  and when q = 0. Solution i.   We are given that; If  and & , then  and  are perpendicular. Therefore, if  and are […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | Oct-Nov | (P2-9709/22) | Q#2

Question                     i.       Solve the equation                   ii.       Hence, using logarithms, solve the equation giving the answer correct to 3 significant figures. Solution SOLVING EQUALITION: PIECEWISE      i.   Let, . We have to consider both moduli separately and it leads to following cases;  OR We have the equation; We have to consider both moduli […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | May-Jun | (P2-9709/21) | Q#5

Question The polynomial  is defined by where  and  are constants. It is given that  is a factor of and that  remainder is 27 when  is divided by .     i.       Find the values of a and b.    ii.       Hence factorise   completely.   iii.       State the number of roots of the equation p(2y) = 0, […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | May-Jun | (P2-9709/21) | Q#2

Question      i.       Solve the inequality .    ii.       Hence find the greatest integer satisfying the inequality  Solution SOLVING INEQUALITY: PIECEWISE      i.   Let, . We can write it as; We have to consider both moduli separately and it leads to following cases; When If then above four intervals translate to following with […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | Oct-Nov | (P2-9709/23) | Q#4

Question The polynomial  is defined by where  is a constant. It is given that  is a factor of     i.       Find the value of a .    ii.       Using this value of a, factorise completely.    iii.       Hence solve the equation , giving the answer correct to 2 significant figures. Solution      i.   We are […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | Oct-Nov | (P2-9709/23) | Q#1

Question      i.       Solve the inequality .    ii.       Hence find the largest integer n satisfying the inequality . Solution SOLVING INEQUALITY: PIECEWISE      i.   Let, . We can write it as; We have to consider both moduli separately and it leads to following cases; When If then above four intervals translate to following […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | Oct-Nov | (P2-9709/21) | Q#4

Question The polynomial  is defined by where  is a constant. It is given that  is a factor of .     i.       Find the value of  .    ii.       Using this value of a, factorise  completely.   iii.       Hence solve the equation , giving the answer correct to 2 significant  figures. Solution      i.   We […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | Oct-Nov | (P2-9709/21) | Q#1

Question      i.       Solve the inequality .    ii.       Hence find the largest integer n satisfying the inequality . Solution SOLVING INEQUALITY: PIECEWISE      i.   Let, . We can write it as; We have to consider both moduli separately and it leads to following cases; When If then above four intervals translate to following […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | May-Jun | (P2-9709/23) | Q#2

Question      i.       Solve the equation .    ii.       Hence solve the equation , giving the answer correct to 3  significant figures. Solution i.   SOLVING EQUATION: PIECEWISE Let, . We have to consider both moduli separately and it leads to following cases;  OR We have the equation; We have to consider both moduli separately […]