Reference no: EM132614362

Question 1. (a) Find the distance between the point P(3,15) and Q(7,12).

(b) Find the gradient ( m ) of the line joining the ponts P(-3, -15) and Q(7,12).

(c) Verify that the line joining the points M (0, 4) and N (5,0) and that joining the points P(0,8) and Q(10,0) are parallel.

(d) Verify that the line joining the points A(3,3) and B(9,12) and that joining the points C(5,11) and D(8,9) are perpendicular.

(e) Find the midpoint of the line segment joining O(0,0) and A(12,12) .

(f) Find the coordinates of the point M that is three-quarters of the distance from A(12,3) to B (8,11).

Question 2. (a) Determine the centre of the circle and the radius, given that equation of the circle is x² + 6x + y2 - 8y -11 = 0.

(b) Find the point of intersection between x² + y² +18x + 20y + 81 = 0 and y = x +1.

(c) Find the equation of the ellipse with foci (0, ±2) and vertices (0, ±3).

(d) Determine the Cartesian version of the parametric equations (hyperbola)

x = 3 + 2 tan t
y = -4 + 3sec t

Question 3. (a) Differentiate y = (3x³ + 5x²)5 .

(b) Given that x² + y² = 16 , find dy/dx.

(c) Find the derivative of |x -1|.

(d) Find the average rate of change of the function f (x) = 3x - 2 over the interval [x, x + h] .

(e) Find f''' (x) if f (x) = (3 - 5x)⁵.

Question 4. (a) A parabola y² = 12x passes through the point P(3,6). Find the

(i) equation of the tangent.
(ii) equation of the normal to the parabola at the point P.

(b) Find the coordinates of the maximum and minimum points of the function y = x3 - 2x2 + x - 7.

Question 5. (a) Find from the first principles, the derivative of f (x) = (x + 3)².

(b) Evaluate :

(i) lim_x→1 x³ -1/(x -1).

(ii) lim_x→0 x(1- x)/( 1-√1- x).

Question 6. (a) The vertices of a triangle are P(1, 2),Q(3,1) and R(-1, -2) . Find

(i) the equation of the line QR .

(ii) the perpendicular distance from P to QR .

(iii) the area of the triangle PQR .

(b) If 1/2 (x² + y² ) = bxy, where b is a constant, find dy/dx.