Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2002 | Oct-Nov | (P1-9709/01) | Q#7

Question Given that   ,  and  , find i.         the angle between the directions of  and , ii.       the value of  for which  and  are perpendicular. Solution We have the position vectors; i.   To find the direction (i.e. angle) between  and  we first find the scalar/dot product of  and . The scalar or dot product of two vectors […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2002 | Oct-Nov | (P1-9709/01) | Q#5

Question      i.       Show that the equation  can be expressed as    ii.       Hence solve the equation , for . Solution i.   We have; Using the relation ; We have the trigonometric identity; It can be written as; Therefore; Becomes; ii.   As demonstrated in (i) we can write the equation  as,  To solve this equation for , let; Therefore  can be written as, […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2002 | Oct-Nov | (P1-9709/01) | Q#11

Question      i. Express  in the form .    ii. For the curve , state the least value of  and the corresponding value of .   iii. Find the set of values of  for which . Given that  for the domain .   iv. Find the least value of  for which  is one-one,    v. Express  in terms of  in this case. Solution i.   We have the expression; […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2002 | Oct-Nov | (P1-9709/01) | Q#10

Question The diagram shows the points  and  on the curve . The line BC is the normal to the curve at B, and C lies on the x-axis. Lines AD and BE are perpendicular to the x-axis. i.       Find the equation of the normal BC.    ii.       Find the area of the shaded region. Solution i.   To find the equation […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2002 | Oct-Nov | (P1-9709/01) | Q#9

Question The diagram shows a rectangle ABCD, where A is (3, 2) and B is (1, 6). i.       Find the equation of BC. Given that the equation of AC is , find    ii.       the coordinates of C,   iii.       the perimeter of the rectangle ABCD.  Solution i.   To find the equation of the line either we need coordinates of thetwo points […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2002 | Oct-Nov | (P1-9709/01) | Q#8

Question A curve has equation  , where k is a constant. i.       Write down an expression for .    ii.       Find the x-coordinates of the two stationary points on the curve.   iii.       Hence find the two values of  for which the curve has a stationary point on the x-axis. Solution i.   Rule for differentiation of  is: In the given case; […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2002 | Oct-Nov | (P1-9709/01) | Q#6

Question In the diagram, triangle ABC is right-angled and D is the mid-point of BC. Angle  and angle . Denoting the length of AD by ,      i.       express each of AC and BC exactly in terms of , and show that .     ii.       show that . Solution i.   Expression for  trigonometric ratio in right-triangle is; It is […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2002 | Oct-Nov | (P1-9709/01) | Q#4

Question The gradient at any point  on a curve is . The curve passes through the point . Find i.           the equation of the curve,  ii.       the point at which the curve intersects the y-axis. Solution i.   To find the equation of the curve: We can find equation of the curve from its derivative […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2002 | Oct-Nov | (P1-9709/12) | Q#3

Question In the diagram, OPQ is a sector of a circle, centre O and radius r cm. Angle QOP= radians. The tangent to the circle at Q meets OP extended at R. i.     Show that the area, Acm2, of the shaded region is given by .    ii.       In the case where   and , evaluate the length of the […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2002 | Oct-Nov | P1-9709/01 | Q#1

Question Find the value of the term which is independent of  in the expansion of Solution Expression for the general term in the Binomial expansion of  is: In the given case: Hence; Since we are looking for value of the term independent of , i.e. : we can  equate Finally substituting  in: Therefore the value of the term independent of   […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2002 | Oct-Nov | P1-9709/01 | Q#2

Question A geometric progression, for which the common ratio is positive, has a second term of 18 and a  fourth term of 8. Find                     i.         the first term and the common ratio of the progression         ii.        the sum to infinity of the progression. Solution From the given information, we can compile […]