University:
University of Cambridge
Course:
0580 IGCSE | Mathematics
Academic year:

2024
Views:

477

Pages:

5
Author:

Shahbaz A.

6663/01 Edexcel GCE Core Mathematics C1 Advanced Subsidiary shahbaz ahmed July 2024 Synthetic Division Consider the polynomial P (x) = x3 − x2 − 5x − 3 = 0 Putting x=-1 P (−1) = (−1)3 − (−1)2 − 5(−1) − 3 P (−1) = (−1) − 1 + 5 − 3 = 0 Hence x=-1 is one of the roots −1 1 1 −1 −5 −3 −1 2 3 −2 −3 0 Factorising x2 − 2x − 3 = 0 x2 − 3x + x − 3 = 0 x(x − 3) + 1(x − 3) = 0 (x + 1)(x − 3) = 0 1 x + 1 = 0 =⇒ x = −1 x − 3 = 0 =⇒ x = 3 Hence roots are -1 and 3 Q 6. The figure above shows part of the curve C with equation y = (x − 1)(x2 − 4) (1) The curve cuts the x-axis at the points P, (1,0) and Q, as shown in the figure above. (a) Write down the x-coordinate of P, and the x-coordinate of Q. (b) Show that dy dx = 3x2 − 2x − 4 (c) Show that y = x + 7 is an equation of the tangent to 2 C at the point (–1, 6). The tangent to C at the point (–1, 6) cuts C again at the point R. (d) Find the exact coordinates of R. Solution (a) y = (x − 1)(x2 − 4) Putting y=0 (x − 1)(x2 − 4) = 0 x − 1 = 0 =⇒ x = 1 x2 − 4 = 0 =⇒ x2 = 4 =⇒ x = ±2 x-coordinate of P= -2 the x-coordinate of Q=2 (b) y = (x − 1)(x2 − 4) d[(x−1)(x2 −4)] dx dy dx = dy dx = (x − 1) d[(xdx−4)] + (x2 − 4) d[(x−1)] dx dy dx ) = (x − 1)[ d(x dx − dy dx = (x − 1)[2x − 0] + (x2 − 4)[1 − 0] 2 2 d(4) dx ] + (x2 − 4)[ d[x] dx − 3 d[(1)] dx ] dy dx = 2x(x − 1) + (x2 − 4) dy dx = 2x2 − 2x + x2 − 4 dy dx = 3x2 − 2x − 4 (c) Since dy dx = 3x2 − 2x − 4 Putting x=-1 dy dx = 3(−1)2 − 2(−1) − 4= 3+2-4=1=m Where m is the slope of tangent at the point (−1, 6) Equation of the tangent to C at the point (−1, 6) with slope m = 1 y − 6 = 1[x − (−1)] y−6=x+1 y =x+7 (d) Since y = (x − 1)(x2 − 4) Putting y = x + 7 x + 7 = (x − 1)(x2 − 4) (x − 1)(x2 − 4) = x + 7 4 x3 − 4x − x2 + 4 − x − 7 = 0 x3 − x2 − 5x − 3 = 0 Finding roots using synthetic division Roots are -1 and 3 Putting x=3 in y = (x − 1)(x2 − 4) y = (3 − 1)[(3)2 − 4] = (2)(5) = 10 Exact coordinates of R. x=3. and y=10 Or R(3,10) is another point on the curve C where tangent intersects the curve 5