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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

Hits: 69 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 […]

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

Hits: 81 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 […]

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

Hits: 69 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 […]

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

Hits: 60 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 […]

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

Hits: 82 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 […]

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

Hits: 70 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 […]

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

Hits: 85 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 […]

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

Hits: 86 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 […]

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

Hits: 192 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 […]

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

Hits: 387 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 […]

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

Hits: 142 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 […]