Cambridge International AS and A Level Mathematics Pure Mathematics 1

Co-ordinate geometry P1^ 2 Different techniques to solve problems The following examples illustrate the different techniques and ...

Finding the equation of a line P1^ 2 SOLUTION The line 2y = x + 12 has gradient^12 and passes through the point (0, 6) (since di ...

Co-ordinate geometry P1^ 2 The gradient of the line PQ is 35 24 2 6 1 3 – –(–) = =– – and so the gradient of the perpendicular b ...

Exercise 2C 55 P1^ 2 2 Find the equations of the following lines. (i) parallel to y = 2 x and passing through (1, 5) (ii) parall ...

Co-ordinate geometry P1^ 2 The intersection of two lines The intersection of any two curves (or lines) can be found by solving t ...

The intersection of two lines P1^ 2 EXAMPLE 2.12 Find the co-ordinates of the vertices of the triangle whose sides have the equa ...

Co-ordinate geometry 58 P1^ 2 ●?^ The line l has equation 2x^ −^ y^ = 4 and the line m has equationy^ =^2 x^ − 3. What can you ...

Exercise 2D P1^ 2 5  A median of a triangle is a line joining a vertex to the mid-point of the opposite side. In any triangle ...

Co-ordinate geometry P1^ 2 The lines AL and BM meet at H. (iv) Find the co-ordinates of H. (v) Show that CH is perpendicular to ...

Exercise 2D P1^ 2 10 Three points have co-ordinates A(2, 6), B(8, 10) and C(6, 0). The perpendicular bisector of AB meets the li ...

Co-ordinate geometry P1^ 2 13 In the diagram, the points A and C lie on the x and y axes respectively and the equation of AC is ...

Drawing curves 63 P1^ 2 Drawing curves You can always plot a curve, point by point, if you know its equation. Often, however, al ...

Co-ordinate geometry P1^ 2 There are some polynomials for which not all the stationary points materialise, as in the case of y = ...

Drawing curves P1^ 2 Behaviour for large x (positive and negative) What can you say about the value of a polynomial for large po ...

Co-ordinate geometry 66 P1^ 2 EXAMPLE 2.13 Sketch the curve y = x^3 − 3 x^2 − x + 3 = (x + 1) (x − 1) (x − 3). SOLUTION Since th ...

Drawing curves P1^ 2 In this example the polynomial x^3 − 3 x^2 − x + 3 has three factors, (x + 1), (x − 1) and (x − 3). Each of ...

Co-ordinate geometry P1^ 2 EXERCISE 2E Sketch the following curves, marking clearly the values of x and y where they cross ...

Drawing curves 69 P1^ 2 An important feature of these curves is that they approach both the x and the y axes ever more closely b ...

Co-ordinate geometry P1^ 2 The intersection of a line and a curve When a line and a curve are in the same plane, there are three ...

The intersection of a line and a curve P1^ 2 (iii) Thelineandthecurvedonotmeet(see figure 2.36). The co-ordinates of the ...