Hi, I'm studying for my test and would like some answers/feedback to check against. Can I get help with the uploaded document (Math 130 2017) particularly 4, 8, 9 &10?

1. [12 Marks] 2. [12 Marks} MATH130 * Mathematics 1E SESSION 1 , 2016 01) b) PART A (Algebra) (Use a separate book.) 4(3m1) Zm SolveT 3 = 2. Leave your answer as a fraction. 1 1 _ _ _ a." (:1: + 1) h fr . Slmphfy t e action $4 + (3: + 1)_1 A 5 litre radiator in a car is lled with a solution of 60% antifreeze, and 40% water. How many litres of the mixture should be removed, and replaced with water, so that the radiator contains 50% antifreeze? Suppose a wooden beam has a square cross section, with the length of the side of the square being 7'. The strength S of the beam is proportional to the square of side length, 7\". Write down an equation that relates S and r. If the side length increases by 20%, say from 10 cm to 12 cm, by what percentage does the strength of the beam increase? The level of pollution, P , in a lake is decreasing by 20% every 3 years, and has a value of P : 500 when t : 0. (i) Without doing any calculation, does the pollution decrease by 40%, less than 40%, or more than 40% every 6 years? Explain. (ii) The formula for P in terms of t, in years, can be written P = 500(0.8)t/3. Calculate the halving time of the function. Give the exact answer, as well as the value correct to 2 decimal places. An exponential function is shown below. Using the graph, estimate the doubling time. Find a formula, of the form y = o- bt, or an equivalent, for the function. Page 2 of 6 3. [12 Marks} MATH130 * Mathematics 1E SESSION 1 , 2016 d) b) (:1) Calculate the exact value of sin(117r/8). The formula sin2 6 = 6) On one graph, draw the functions 25\" and loggm. On the graph, have :1: values between 4 and 4, and also have y values between 4 and 4. Ensure the scale on the 2: axis and the y axis is the same. The temperature T, in degrees celsius, of a metal rod is given by T = 25 + 375e_t/30, where t is the time in minutes since the rod was removed from a blast furnace. How long does it take the rod to cool to 60 degrees? 68 On two separate numberplanes, sketch angles of radian measure %, and 2. Sketch each angle in standard position (with the vertex on the origin, and the initial side along the positive :c axis). \\/ ?. By means of appropriate triangles, show that the exact value of cos(7r/ 6) is Using the unit circle below, carefully estimate sin(117r/8). (This question sheet is not marked. The accuracy of your answer will depend on using the gure below, but draw an appropriate diagram in your answer booklet to demonstrate your method.) 3; 1 cos(26) 2 may be helpful. Using the unit circle denition of sine and cosine, nd a simple formula for cos (0 + g Page 3 of 6 ). MATH130 * Mathematics 1E SESSION 1 , 2016 4. a) What is the discriminant of a quadratic function y = (1:132 + bx + c? What does the [10 Marks} Sign of the discriminant tell you about the roots of the quadratic? b) If the roots of the quadratic y : 2:15:22 i bx _ 10 are #1 and 5, nd the values of a and b. c) Solve the equation 2x2 + 7:3 3 = 0, leaving your answer in exact form. d) Make a sketch of the cubic y 2 72(93 + 1)(.7: 7 4)? There is no need to use calcu- his. 5. a) Evaluate the arithmetic sum 4Mk [ \" A:19+w+41+~~+wz b) Evaluate the geometric sum G=4+4x3+4x+~~+4x9 Page 4 of 6 6. [10 Marks} 7. [14 Marks} 8. [14 Marks] MATH130 * Mathematics 1E SESSION 1 , 2016 PART B (Calculus) (Use a separate book.) Consider the following functions from the real line R into R: (i) M) : cc? + 3 (a) ye) :i a) Find the domain of each of the functions f and g. b) Sketch the curves y : f (2:), y : 9(a) Without using calculus. Indicate the maxi- mum and minimum of the each of the functions (if exist). c) Find the composite functions f 0 g(x) = f(g(3:)) and g o f(:r) = g(f(:c)) a) Consider the function g(a:) : 3:3 7 2w2 + :c a 1. (i) Find the rst derivative 9"(35) and second derivative g\"(:c) . (ii) Find all the stationary points of y : 9(a) and determine the nature of these stationary points. b) Consider the function f (m) : 3 7 it: a e": . (i) Find the equations of the tangent and the normal to the curve y : f(2:) at the point P(0, 2). (ii) Use Newton's method in 2 steps to solve the equation 3 a: es\" = O for an approximation of a root near a: : 0. Give your approximation value in 4 decimal places. 2 Consider the function h(x) = :x. - - . . , 112(2 cc) a) Fmd the domam of h(:c) and show that the derlvative h (ct) = E. e b) Hence nd the 2 stationary points of y = h(:c) and determine their nature. c) Find the second derivative h\"(3:) and show that y : h(3:) has 2 inexjon points at $122+\\/and $222\\/. 332 d) Sketch the curve 3; : [1(1), given that h'm : D. xHoo e\": Page 5 of 6 9. [12 Marks} MATH130 * Mathematics 1E SESSION 1 , 2016 3) Find (i) f(m3 + sinus 251) (in: w/2 (ii) / sin 17 cos :6 dx. 0 b) Find the intersection points of the two curves 3; = 3:1: + 6 and y = :02 + 2. Then nd the area bounded by these two curves. c) The motion of a particle on a straight line has its velocity 1) given as a function of time t by v(t) = 6005(2t). Assume that at time t = 0, the particle is at position a: : 1. Find the expression for the displacement cc as a function of time t and Show that the motion is restricted to the interval 2 3 :3 S 4. Page 6 of 6