Hits: 68

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | May-Jun | (P1-9709/13) | Q#9

Hits: 68 Question The diagram shows a pyramid OABCD with a horizontal rectangular base OABC.  The sides OA and AB have lengths of 8 units and 6 units respectively. The point E on  OB is such that OE = 2 units. The point D of the pyramid is 7 units vertically  above E. Unit vectors […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | May-Jun | (P1-9709/12) | Q#5

Hits: 154 Question The diagram shows a three-dimensional shape. The base OAB is a horizontal  triangle in which angle AOB is 90o. The side OBCD is a rectangle and the side OAD  lies in a vertical plane. Unit vectors  and are parallel to OA and OB respectively  and the unit vector  is vertical. The position […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | May-Jun | (P1-9709/11) | Q#7

Hits: 203 Question Relative to an origin O, the position vectors of points A, B and C are given by and                     i.       Find .                ii.       The point M is the mid-point of AC. Find the unit vector in the direction of  .                  iii.       Evaluate and hence […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | Oct-Nov | (P1-9709/13) | Q#9

Hits: 1615 Question Relative to an origin O, the position vectors of points A, B and C are given by and A fourth point, D, is such that the magnitudes ,  and  are the first, second and third  terms respectively of a geometric progression.  i.       Find the magnitudes ,  and . ii.       Given that D is a point lying on […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | Oct-Nov | (P1-9709/12) | Q#9

Hits: 1102   Question The diagram shows a trapezium OABC in which OA is parallel to CB. The position vectors of A and  B relative to the origin O are given by and                           i.       Show that angle OAB is 90o. The magnitude of  is three times the […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | Oct-Nov | (P1-9709/11) | Q#8

Hits: 918   Question a.   Relative to an origin O, the position vectors of two points P and Q are p and q respectively. The  point R is such that PQR is a straight line with Q the mid-point of PR. Find the position vector of R in  terms of p and q, simplifying your answer. b.   The vector […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | May-Jun | (P1-9709/13) | Q#4

Hits: 686   Question Relative to an origin O, the position vectors of points A, B and C are given by and The point P lies on AB and is such that . i.       Find the position vector of P. ii.       Find the distance OP. iii.       Determine whether OP is perpendicular to AB. Justify your answer. Solution      i.   We are […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | May-Jun | (P1-9709/12) | Q#8

Hits: 788   Question Relative to an origin , the position vectors of points A and B are given by and where p and q are constants.      i.       In the case where , use a scalar product to find angle AOB.    ii.       In the case where  is parallel to , find the values of p and q. Solution      i. […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | Feb-Mar | (P1-9709/12) | Q#6

Hits: 669   Question Relative to an origin , the position vectors of points A and B are given by and i.       Use a scalar product to find angle OAB. ii.       Find the area of triangle OAB. Solution      i.   It is evident that angle OAB is between  and . We are given that; and Next, we need scalar/dot product of  and . […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | May-Jun | (P1-9709/11) | Q#2

Hits: 808   Question Relative to an origin O, the position vectors of points A and B are given by and and angle AOB=90o. i.       Find the value of . The point C is such that .    ii.       Find the unit vector in the direction of . Solution      i.   It is evident that angle AOB is between  and . […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/13) | Q#7

Hits: 496   Question The diagram shows a triangular pyramid ABCD. It is given that i.       Verify, showing all necessary working, that each of the angles DAB, DAC and CAB is 90o.    ii.       Find the exact value of the area of the triangle ABC, and hence find the exact value of the volume of the pyramid.  [The volume V of a […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/12) | Q#9

Hits: 795 Question Relative to an origin O, the position vectors of points A, B and C are given by and i.       Use a scalar product to find angle AOB. ii.       Find the vector which is in the same direction as  and of magnitude 15 units.  iii.       Find the value of the constant p for which  perpendicular to . Solution      i.   It […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | May-Jun | (P1-9709/13) | Q#9

Hits: 483 Question The position vectors of points A, B and C relative to an origin O are given by and where p is a constant. i.       Find the value of p for which the lengths of AB and CB are equal. ii.       For the case where p=1, use a scalar product to find angle ABC. Solution      i.   […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | May-Jun | (P1-9709/12) | Q#3

Hits: 600 Question Relative to an origin , the position vectors of points A and B are given by and The point C is such that . Find the unit vector in the direction of . Solution A unit vector in the direction of  is; Therefore for the given case; Therefore, we need to find . We are given that; A […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | May-Jun | (P1-9709/11) | Q#10

Hits: 858   Question Relative to an origin O, the position vectors of points A, B and C are given by and respectively, where k is a constant. i.       Find the value of k in the case where angle AOB=90o. ii.       Find the possible values of k for which the lengths of AB and OC are equal. The point D is such […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Feb-Mar | (P1-9709/12) | Q#7

Hits: 1176 Question The diagram shows a pyramid OABC with a horizontal triangular base OAB and vertical height OC.  Angles AOB, BOC and AOC are each right angles. Unit vectors ,  and  are parallel to OA, OB  and OC respectively, with OA = 4 units, OB = 2.4 units and OC = 3 units. The point P on CA […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | Oct-Nov | (P1-9709/13) | Q#5

Hits: 1606 Question Relative to an origin O, the position vectors of the points A and B are given by and    i.       For the case where OA is perpendicular to OB, find the value of p.    ii.       For the case where OAB is a straight line, find the vectors  and . Find also the length of        […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | Oct-Nov | (P1-9709/12) | Q#7

Hits: 874 Question Relative to an origin O, the position vectors of the points A, B and C are given by and i.         In the case where ABC is a straight line, find the values of p and q. ii.       In the case where angle BAC is 90o, express q in terms of p.  iii.   In […]