Q4: (6 Marks) Let Z2 = 5 + 7j . a) Find the modulus and the argument of Zz. b) Express 22 in polar form r cose + r sine i Q5. Let f(x) = 6x2 – 2x – 3, part of the graph of funtion shown below. (1+1+2 Marks) (a) Draw the graph of y = 2 on the same axis. (b) Use the graph to find: (i) The values of of x when 6x2 – 2x - 3...
Q5. Let f(x) = 6x2 - 2x - 3, part of the graph of funtion shown below. (1+1+2 Marks) (a) Draw the graph of y = 2 on the same axis. (b) Use the graph to find (0) The values of of x when 6x2 - 2x - 3 = 2 Find the cordinate of the vertex (1) Find the equation of the axis of symetry.
1+1 5. [10 marks) Find all values of k such that f(x) = -r2 + 2kr +1" has domain (-0,0). 6. [10 marks] Sketch the graph of y = -2 -2 log: (2-2) and find its domain, range, -intercept and y-intercept. Show your transformations step by step. 7. (12 marks] (a) [8 marks) Find the equation of the tangent line to the graph of the function f(x) = 3.0" - 6x2 - 7 at = 2 (b) (4 marks) Find...
Question 4 (2+4+4+1+4 = 15 marks) Consider the function y = 4 sin (2x-π) for-r below to sketch the graph of y. x < π. Follow the steps (a) State the amplitude and period in the graph of this function 4 sin (22-9 ) for-r (b) Solve y π to find the horizontal intercepts x (a-intercepts) of the function. (c) Find the values of x for-π π for which the maximum. and the x minimum values of the function occur...
Question 5 (20 marks) a) [10 marks] Evaluate: 1 el-X Vx+y(y – 2x)2 dydx. 0 Jo (Hint: consider a change of variable.) b) [10 marks) Find the volume of the solid bounded by the sphere x2 + y2 + z2 = aand the cylinder x2 + y2 = ax, a > 0.
Q3(a) Let W be the region above the sphere x2 + y2 + z2 = 6 and below the paraboloid z = 4 - x2 - y2 as shown in Figure Q5(a) below: Z=4-x-y? x2 + y + z = 6 Figure Q3(a) (i) Find the equation of the projection of Won the xy-plane. (ii) Compute the volume of W using polar coordinates. [16 marks] (b) Using double integral in polar coordinates, compute the following: $$*** (2x+3y) dedy [7 marks]...
Let Y = (Yİ Y2 Yn)' be a random vector taking on values in Rn with mean μ E Rn and covariance matrix 2. Also let 1 be the ones vector defined by 1-(1 1) 5.i Find the projection matrix Hy where V is the subspace generated by 1 5.ii Show that Hy is symmetric and idempotent. 5.iii Let x = (a a . .. a)', where a E Rn. Show that Hvx = x. 5.iv Find the projection of...
1. Find the matrices of x and yif.... (23 marks) 2x - y = and x+2y=[] { 5,). 2. Find matrix C if: G 1-6. 9 (18 marks) 3. Given that: b = -32+4- k c = -91 + 12 - 3k Q =(5,4,0) and P = (-1,4,-3) Find a. The unit vector along the direction of a vector has Q as initial point and P as a terminal point Prove that vector b is parallel to vector ĉ using...
Problem 2
(1) Find the area enclosed by the curves y 2 and y-4z-z2 (2) Find the volume of the solid whose base is the triangular region with vertices(0, 0), (2, 0), and (0,1). Cross-sections perpendicular to the y-axis semicircles. are (3) Find the volume of the solid by rotating the region bounded by y=1-z2 and y-0 about the r-axis. 2-z2. Find the volume (4) Let R be the region bounded by y--x2 and y of the solid obtained by...
(3) For the function y =-2(x-1)2 find a) vertex b) axis of symmetry c) maximum or minimum value d) graph the function e) Intervals of increasing and decreasing °For the function y = 0.5(x + 3)2 + 2, find a) vertex b) axis of symmetry c) maximum or minimum value d) graph the function e) Intervals of increasing and decreasing