# 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 […]

# 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 | (P1-9709/13) | Q#1

Question (i)          Expand  in ascending powers of y as far as the term in y2. (ii)       In the expansion of  the coefficient of x2 is 48. Find the value of the positive  constant . Solution i.   We are required to expand . Expression for the Binomial expansion of  is: In the given […]