ABSTRACT

This Revision Test covers the material contained in Chapters 28 to 32. The marks for each question are shown in brackets at the end of each question.

Determine the gradient and intercept on the y-axis for the following equations: (a) y = - 5 x + $ y\,{=}\,{-}5x\,{+}\, $ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315561851/4d4f2a98-6163-4fb4-b7c9-27be293e7fad/content/inline-mathrev8_1.tif"/> 2 (b) 3 x + 2 y + 1 = $ x\,{+}\,2y\,{+}\,1\,{=}\, $ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315561851/4d4f2a98-6163-4fb4-b7c9-27be293e7fad/content/inline-mathrev8_2.tif"/> 0(5)

The equation of a line is 2 y = 4 x + $ y\,{=}\,4x\,{+}\, $ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315561851/4d4f2a98-6163-4fb4-b7c9-27be293e7fad/content/inline-mathrev8_3.tif"/> 7. A table of corresponding values is produced and is as shown below. Complete the table and plot a graph of y against x. Determine the gradient of the graph. x - 3 - 2 - 1 0 1 2 3 y - 2.5 7.5 ( 6 ) $$ \begin{aligned}\begin{array}{llllllll} x&-3&-2&-1&0&1&2&3\\ y&-2.5&\,&\,&\,&\,&7.5&\quad (6) \end{array}\end{aligned} $$ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315561851/4d4f2a98-6163-4fb4-b7c9-27be293e7fad/content/mathrev8um_1.tif"/>

Plot the graphs y = 3 x + $ y\,{=}\,3x\,{+}\, $ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315561851/4d4f2a98-6163-4fb4-b7c9-27be293e7fad/content/inline-mathrev8_4.tif"/> 2 and y 2 + x = $ {\dfrac{{y}}{{2}}}\,{+}\,x\,{=}\, $ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315561851/4d4f2a98-6163-4fb4-b7c9-27be293e7fad/content/inline-mathrev8_5.tif"/> 6 on the same axes and determine the co-ordinates of their point of intersection. (7)

The velocity v of a body over varying time intervals t was measured as follows: t seconds 2 5 7 v m/s 15.5 17.3 18.5 $$ \begin{aligned}\begin{array}{llll} t \ \text{ seconds}&2&5&7\\ v \ \text{ m/s}&15.5&17.3&18.5 \end{array}\end{aligned} $$ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315561851/4d4f2a98-6163-4fb4-b7c9-27be293e7fad/content/mathrev8um_2.tif"/> t seconds 10 14 17 v m/s 20.3 22.7 24.5 $$ \begin{aligned}\begin{array}{llll} t \ \text{ seconds}&10&14&17\\ v \ \text{ m/s}&20.3&22.7&24.5 \end{array}\end{aligned} $$ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315561851/4d4f2a98-6163-4fb4-b7c9-27be293e7fad/content/mathrev8um_3.tif"/> Plot a graph with velocity vertical and time horizontal. Determine from the graph (a) the gradient, (b) the vertical axis intercept, (c) the equation of the graph, (d) the velocity after 12.5 s, and (e) the time when the velocity is 18 m/s. (9)

The following experimental values of x and y are believed to be related by the law y = a x 2 + b $ y\,{=}\,ax^{2}\,{+}\,b $ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315561851/4d4f2a98-6163-4fb4-b7c9-27be293e7fad/content/inline-mathrev8_6.tif"/> , where a and b are constants. By plotting a suitable graph verify this law and find the approximate values of a and b. x 2.5 4.2 6.0 8.4 9.8 11.4 y 15.4 32.5 60.2 111.8 150.1 200.9 $$ \begin{aligned}\begin{array}{lllllll} x&2.5&4.2&6.0&8.4&9.8&11.4\\ y&15.4&32.5&60.2&111.8&150.1&200.9 \end{array}\end{aligned} $$ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315561851/4d4f2a98-6163-4fb4-b7c9-27be293e7fad/content/mathrev8um_4.tif"/>

Determine the law of the form y = a e k x $ y\,{=}\,ae^{kx} $ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315561851/4d4f2a98-6163-4fb4-b7c9-27be293e7fad/content/inline-mathrev8_7.tif"/> which relates the following values: y 0.0306 0.285 0.841 x - 4.0 5.3 9.8 $$ \begin{aligned}\begin{array}{llll} y&0.0306&0.285&0.841\\ x&-4.0&5.3&9.8 \end{array}\end{aligned} $$ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315561851/4d4f2a98-6163-4fb4-b7c9-27be293e7fad/content/mathrev8um_5.tif"/> y 5.21 173.2 1181 x 17.4 32.0 40.0 $$ \begin{aligned}\begin{array}{llll} y&5.21&173.2&1181\\ x&17.4&32.0&40.0 \end{array}\end{aligned} $$ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315561851/4d4f2a98-6163-4fb4-b7c9-27be293e7fad/content/mathrev8um_6.tif"/> (9)

State the minimum number of cycles on logarithmic graph paper needed to plot a set of values ranging from 0.073 to 490. (2)

Plot a graph of y = 2 x 2 $ y\,{=}\,2x^{2} $ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315561851/4d4f2a98-6163-4fb4-b7c9-27be293e7fad/content/inline-mathrev8_8.tif"/> from x = - $ x\,{=}\,{-} $ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315561851/4d4f2a98-6163-4fb4-b7c9-27be293e7fad/content/inline-mathrev8_9.tif"/> 3 to x = + $ x\,{=}\,{+} $ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315561851/4d4f2a98-6163-4fb4-b7c9-27be293e7fad/content/inline-mathrev8_10.tif"/> 3 and hence solve the equations:

(a) 2 x 2 - 8 = $ x^{2}\,{-}\,8\,{=}\, $ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315561851/4d4f2a98-6163-4fb4-b7c9-27be293e7fad/content/inline-mathrev8_11.tif"/> 0 (b) 2 x 2 - 4 x - 6 = 0 $ x^{2}\,{-}\,4x\,{-}\,6\,{=}\,0 $ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315561851/4d4f2a98-6163-4fb4-b7c9-27be293e7fad/content/inline-mathrev8_12.tif"/> (9)

Plot the graph of y = x 3 + 4 x 2 + x - $ y\,{=}\,x^{3}\,{+}\,4x^{2}\,{+}\,x\,{-}\, $ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315561851/4d4f2a98-6163-4fb4-b7c9-27be293e7fad/content/inline-mathrev8_13.tif"/> 6 for values of x between x = - $ x\,{=}\,{-} $ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315561851/4d4f2a98-6163-4fb4-b7c9-27be293e7fad/content/inline-mathrev8_14.tif"/> 4 and x = $ x\,{=}\, $ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315561851/4d4f2a98-6163-4fb4-b7c9-27be293e7fad/content/inline-mathrev8_15.tif"/> 2. Hence determine the roots of the equation x 3 + 4 x 2 + x - 6 = 0 $ x^{3}\,{+}\,4x^{2}\,{+}\,x\,{-}\,6\,{=}\,0 $ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315561851/4d4f2a98-6163-4fb4-b7c9-27be293e7fad/content/inline-mathrev8_16.tif"/> . (7)

Sketch the following graphs, showing the relevant points: (a) y = ( x - 2 ) 2 $ y\,{=}\,(x\,{-}\,2)^{2} $ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315561851/4d4f2a98-6163-4fb4-b7c9-27be293e7fad/content/inline-mathrev8_17.tif"/> (b) y = 3 - cos 2 x $ y\,{=}\,3\,{-}\,\cos 2x $ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315561851/4d4f2a98-6163-4fb4-b7c9-27be293e7fad/content/inline-mathrev8_18.tif"/> (c) f ( x ) = - 1 - π ≤ x ≤ - π 2 - x - π 2 ≤ x ≤ π 2 - 1 - π 2 ≤ x ≤ π $ f(x)\,{=}\,{\left\{ \begin{array}{lll} {-}1&{-}\pi \,{\le }\,x\,{\le }\,{-}\,\dfrac{\pi }{2}\\ {{-}}x&{-}\dfrac{\pi }{2}\,{\le }\,x\,{\le }\,\dfrac{\pi }{2}\\ {{-}}1&{{-}}\dfrac{\pi }{2}\,{\le }\,x\,{\le }\,\pi \end{array}\right.} $ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315561851/4d4f2a98-6163-4fb4-b7c9-27be293e7fad/content/inline-mathrev8_19.tif"/> (10)

Determine the inverse of f ( x ) = 3 x + $ f(x)\,{=}\,3x\,{+}\, $ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315561851/4d4f2a98-6163-4fb4-b7c9-27be293e7fad/content/inline-mathrev8_20.tif"/> 1. (3)

Evaluate, correct to 3 decimal places: 2 tan - 1 1.64 + sec - 1 2.43 - 3 cosec - 1 3.85 ( 4 ) $$ 2\tan ^{-1}\!1.64 + \sec ^{-1}\!2.43-3\,\text{ cosec}^{-1}3.85\quad (4) $$ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315561851/4d4f2a98-6163-4fb4-b7c9-27be293e7fad/content/mathrev8um_7.tif"/>