Dcribe the transformation that transforms the curve y

Question

1. Simplify (x - 5)(x² + 3) - (x + 4)(x -1).

2. Express each of the following in the form 7^k :

(i) ⁴√7,
(ii) 1/ 7√7,
(iii) 7⁴ x 49¹⁰.

3 (i) Find the gradient of the line 1 which has equation 3x - 5y - 20 = 0.

(ii) The line l crosses the x-axis at P and the y-axis at Q. Find the coordinates of the mid-point of PQ.

4 (i) Express 2x² - 20x + 49 in the form p(x - q)² + r.

(ii) State the coordinates of the vertex of the curve y = 2x² - 20x + 49.

5. (i) Sketch the curve y = √x.

(ii) Describe the transformation that transforms the curve y = √x to the curve y = √(x-4)

(iii) The curve y = √x is stretched by a scale factor of 5 parallel to the x-axis. State the equation of the transformed curve.

6. Find the equation of the normal to the curve y = (6/x²) - 5 at the point on the curve where x = 2. Give your answer in the form ax + by + c = 0, where a, b and c are integers.

7. Solve the equation x - 6x^1/2 + 2 = 0, giving your answers in the form p ± q√r, where p, q and r are integers.

8. (i) Find the coordinates of the stationary point on the curve y = x⁴ + 32x.

9. (i) A rectangular tile has length 4x cm and width (x + 3)cm. The area of the rectangle is less than 112 cm². By writing down and solving an inequality, determine the set of possible values of x.

(ii) A second rectangular tile of length 4ycm and width (y + 3)cm has a rectangle of length 2y cm and width y cm removed from one corner as shown in the diagram.

Given that the perimeter of this tile is between 20 cm and 54cm, determine the set of possible values of y.

10. A circle has equation (x - 5)² + (y + 2)² = 25.

(i) Find the coordinates of the centre C and the length of the diameter.

(ii) Find the equation of the line which passes through C and the point P (7, 2).

(iii) Calculate the length of CP and hence determine whether P lies inside or outside the circle.

(iv) Determine algebraically whether the line with equation y = 2x meets the circle.