# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2020 | Feb-Mar | (P1-9709/12) | Q#12

Question A diameter of a circle C1 has end-points at (−3, −5) and (7, 3). a)   Find an equation of the circle C1. The circle C1 is translated by  to give circle C2, as shown in the diagram. b)  Find an equation of the circle C2.  The two circles intersect at points R and S. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2020 | Feb-Mar | (P1-9709/12) | Q#11

Question     i.      Solve the equation  for .   ii.      Find the set of values of  for which the equation has no solution.  iii.      For the equation , state the value of for which there are three solutions in the interval , and find these solutions. Solution i.   We have the equation; Let ; […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2020 | Feb-Mar | (P1-9709/12) | Q#10

Question The gradient of a curve at the point (x, y) is given by . The curve  has a stationary point at (a, 14), where a is a positive constant. a)Find the value of . b)Determine the nature of the stationary point. c)Find the equation of the curve. Solution a) We are given that gradient […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2020 | Feb-Mar | (P1-9709/12) | Q#9

Question a)Express in the form , where  and  are constants. The function f is defined by  for . b)Find an expression for and state the domain of . The function is defined by  for . c)For the case where k = −1, solve the equation . d)State the largest value of possible for the composition […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2020 | Feb-Mar | (P1-9709/12) | Q#8

Question A woman’s basic salary for her first year with a particular company is $30000 and at the end of the year she also gets a bonus of$600. a)For her first year, express her bonus as a percentage of her basic salary. At the end of each complete year, the woman’s basic salary will […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2020 | Feb-Mar | (P1-9709/12) | Q#7

Question The diagram shows a sector AOB which is part of a circle with centre O and radius 6  cm and with angle AOB = 0.8 radians. The point C on OB is such that AC is  perpendicular to OB. The arc CD is part of a circle with centre O, where D lies on  […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2020 | Feb-Mar | (P1-9709/12) | Q#6

Question The coefficient of  in the expansion of  is 720. a)Find the possible values of the constant . b) Hence find the coefficient of  in the expansion. Solution a) We are given expression as; Expression for the general term in the Binomial expansion of is: In the given case: Hence; Since we are looking for […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2020 | Feb-Mar | (P1-9709/12) | Q#5

Question Solve the equation for . Solution We are given the equation; Since , therefore; Now we have two options. If we utilize the odd/even property of and   to find other solutions (roots),  then these solutions will lie outside desired range .

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2020 | Feb-Mar | (P1-9709/12) | Q#4

Question A curve has equation y = x2 − 2x − 3. A point is moving along the curve in such a  way that at P the y-coordinate is increasing at 4 units per second and the x- coordinate is increasing at 6 units per second.  Find the x-coordinate of P. Solution We are required […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2020 | Feb-Mar | (P1-9709/12) | Q#3

Question The diagram shows part of the curve with equation y=x2+1. The shaded region  enclosed by the curve, the y-axis and the line y = 5 is rotated through 360o about the y-axis. Find the volume obtained. Solution Expression for the volume of the solid formed when the shaded region under the  curve is rotated […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2020 | Feb-Mar | (P1-9709/12) | Q#2

Question The graph of  is transformed to the graph of . Describe fully the two single transformations which have been combined to give the  resulting transformation. Solution We are given function; The following function represents the vertical translation (along y-axis) of graph by units upwards; Therefore, for the given case, following will represent vertical translation […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2020 | Feb-Mar | (P1-9709/12) | Q#1

Question The function f is defined by  for . Determine 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 a particular point  , we […]

# 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#11

Question The diagram shows part of the curve , and the lines x = 1 and x = 3. The point A  on the curve has coordinates (2, 3). The normal to the curve at A crosses the line x = 1 at B. (i)       Show that the normal AB has equation . (ii)    […]

# 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#10

Question Relative to an origin O, the position vectors of points A, B and X are given by and i.Find and show that AXB is a straight line. The position vector of point C is given by . ii.Show that CX is perpendicular to AX. iii.Find the area of triangle ABC. Solution i. First, we […]

# 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#9

Question The first, second and third terms of a geometric progression are ,  and  respectively. (i)       Show that  satisfies the equation 7k2 − 48k + 36 = 0. (i)       Find, showing all necessary working, the exact values of the common ratio corresponding to  each of the possible values of k. (ii)        One of […]

# 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#8

Question A function  is defined for  and is such that .     i.      Find the set of values of  for which f is decreasing.   ii.      It is now given that . Find . Solution      i.   We are given derivative of the function as; We are also given that it is a decreasing […]

# 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#7

Question     i.      Show that the equation  can be expressed as Where   ii.      Hence solve the equation   for . Solution i.   We are given the equation; We have the trigonometric identity; From this we can substitute in above equation; Let ;      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 | 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#5

Question The dimensions of a cuboid are x cm, 2x cm and 4x cm, as shown in the diagram. i.Show that the surface area S cm2 and the volume V cm3 are connected by the relation  ii.When the volume of the cuboid is 1000 cm3 the surface area is increasing at 2 cm2 s−1. Find  […]

# 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#4

Question The diagram shows a semicircle ACB with centre O and radius . Arc OC is part of a circle with  centre A. (i)Express angle CAO in radians in terms of . (ii)Find the area of the shaded region in terms of ,  and , simplifying your answer. Solution (i)   We are required to […]