# 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#6

Question      i.       Prove the identity    ii.       Hence solve the equation  for Solution i.   First, we are required to show that; Since ; We have the trigonometric identity; From this we can write; Therefore; We have the algebraic formula;      ii.   We are required to solve the equation;   For . […]

# 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#4

Question The diagram shows a trapezium ABCD in which the coordinates of A, B and C are (4, 0), (0, 2) and  (h, 3h) respectively. The lines BC and AD are parallel, angle ABC = 90o and CD is parallel to the x- axis.      i.       Find, by calculation, the value of h.    ii. […]

# 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#3

Question A sector of a circle of radius cm has an area of  cm2. Express the perimeter of the sector in  terms of and . Solution We are given that for the circular sector; We are required to find expression for perimeter of the circular sector. Expression for perimeter of a circular sector with radius […]

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

Question The term independent of x in the expansion of , where k is a constant, is 540. (i)               Find the value of k. (ii)             For this value of k, find the coefficient of x2 in the expansion. Solution (i)   Expression for the general term in the Binomial expansion of  is: In the given […]

# 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#10

Question The diagram shows the curve with equation . i.The straight line with equation y = x + 3 intersects the curve at points A and B. Find the length of  AB. ii.The tangent to the curve at a point T is parallel to AB. Find the coordinates of T. iii.Find the coordinates of 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#9

Question The diagram shows part of the curve with equation . The shaded region is bounded by  the curve, the x-axis and the line x = 3.      i.       Find, showing all necessary working, the volume obtained when the shaded region is rotated  through 360O about the x-axis.    ii.       P is the point on […]

# 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#7

Question a)  Solve the equation   for . b)    The diagram shows part of the graph of , where is measured in radians and and are constants. The curve intersects the x-axis at  and the y-axis at . Find the values of and . Solution a)   We are required to solve the equation for […]

# 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#6

Question      i.       The first and second terms of a geometric progression are p and 2p respectively, where p is a  positive constant. The sum of the first n terms is greater than 1000p. Show that 2n > 1001.    ii.       In another case, p and 2p are the first and second terms respectively of […]

# 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 1 (P1-9709/01) | Year 2019 | Feb-Mar | (P1-9709/12) | Q#4

Question A curve has equation .     i.       Find  and  .     ii.       Find the x-coordinates of the stationary points and, showing all necessary working, determine  the nature of each stationary point. Solution i.   We are required to find and  . Therefore, we find the derivative of equation of the curve. Gradient (slope) […]

# 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#3

Question In the diagram, CXD is a semicircle of radius 7 cm with centre A and diameter CD. The straight line  YABX is perpendicular to CD, and the arc CYD is part of a circle with centre B and radius 8 cm. Find the total area of the region enclosed by the two arcs. Solution […]

# 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#2

Question A curve with equation y = f(x) passes through the points (0, 2) and (3, −1). It is given that  , where  is a constant. Find the value of . Solution i.   We can find equation of the curve from its derivative through integration; For the given case; Therefore; Rule for integration of […]

# 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#1

Question The coefficient of  in the expansion of  is -2160. Find the value of the constant . Solution We are given expression as; Expression for the general term in the Binomial expansion of  is: First rewrite the given expression in standard form. In the given case: Hence; Since we are looking for the coefficient of […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | Feb-Mar | (P2-9709/22) | Q#4

Question     i.       Find the quotient when   is divided by  ,  and show  that the remainder is 5.    ii.       Show that the equation  has exactly one real root. Solution      i.   Hence quotient is and remainder is .    ii.   We are required to show that following equation has exactly one real […]