Question 1. (a) For the function y = f(x) = |2x + 3|:

(i) sketch the function;

(ii) find the domain and range;

(iii) show, using algebra, that f(x) is not one-to-one;

(iv) find a restriction of the domain such that the function is one-to-one.

(b) Evaluate the following limits

(i)  lim   (3x² - 7x + 1)/(5x² - 2)
     x→∞

(ii) lim   (x² - 7x + 6)/(x² - 4x -12)
     x→6

(iii) lim   (3e^7x + 5x - 3cosx)/(6x - sin2x)
     x→0

Question 2.

(a) Find dy/dx in the following cases:

(i) y = (4 + x³ln x)⁸

(ii) y = ( sin x - 4 cos x)e^-4x

(iii) 3x⁵-2xy + y³ = 12

(iv) y = (1 + x²)tan x^-1.

(b) (i) State the definitions of cosh x and sinh x.

(ii) Using the definitions in (i), prove that

ln( cosh 3x - sinh 3x) = -3x

Question 3.

(a) Find

(i) I = ∫x √(2x² + 3) dx .

(ii) I = ∫(4 - 3x)e^-3x dx.

(iii) I = ∫10sin x/(6 + 5 cos x)³. dx

(b)

Use a standard integral to evaluate I = ₃∫⁶ 1/√(36-x²) dx, and express the answer in terms of Π.

(c) Use partial fractions to find I = ∫ 5/(x² + x - 6) dx.

Question 4. (a) (i) Convert I = ₁∫³ 2x/√(x⁴ + 63) dx to a standard integral by making the substitution u = x².

(ii) Use the standard integral obtained in (i) to evaluate I, and express the answer in terms of the natural logarithm.

(b) Find the volume formed when the area above the curve y = x(x-2)√x, and below the x axis, is rotated about the x axis.

(c) Solve

(i) dy/dx =  8x(3x - 1)y3/4; y(2) = 1

(ii) dy/dx + 5y/x = (4x² - 3x + 6)/x⁴; y(1) = 7.

Question 5. (a) Find the open interval of convergence for the power series

∞
Σ  (x + 3)^{n + 1}/(n + 1)4ⁿ
n = 0

(b) Derive the first three terms of the MacLaurin series for

f(x) = (1 + x)^-1/2 .

(c) Given that sin x = x - x³/3! + x⁵/5! - x⁷/7! +....., for -∞ < x < ∞ :

(i) write down the first three non-zero terms of the power series for (x - sin x)/x³;

(ii) use the first three non-zero terms of the series for (x - sin x)/x³ to approximate

     1
I = ∫ (x - sin x)/x³ .dx
    0

Question 6.

(a) Given A = 4i + 3j - 5k , and B = 9i + 4j + k ,find:
                ˜       ˜    ˜      ˜            ˜      ˜     ˜     ˜

(i) A and B ;
     ˜       ˜
(ii) A . B ;
     ˜    ˜
(iii) the cosine of the angle between A and B .
                                                        ˜         ˜

(b) (i) Find a vector perpendicular to the plane containing the points P(1, 1, 1), Q(5, -3, 3) and R(6, -3, 4).

(ii) Hence find the area of the triangle PQR.

(iii) Find the equation of the plane containing the points P, Q, and R.

(c) Find parametric equations of the line through the points A(2, 7, 9) and B(3, 4, 6).