text: You are on page 1of 9
  
  # 1
  
  Q1: [ 3 Marks]
  CLO 1: Perform algebraic manipulations including working with exponents,
  radicals, polynomial operations, factoring and algebraic fractions.
  Simplify the following expression:
  
  Solution:
  ( )
  ( )( )
  ( )
  ( ) ( )
  1
  1
  1
  1
  1 ) 1 (
  1 1
  1
  2 2
  2 3
  =
  +
  
  + +
  + +
  
  +
  +
  x x x
  x x x
  x x
  x
  
  ( )
  ( ) ( ) ( ) 1 2
  1
  1
  ) 1 (
  1
  1
  2 2
  3
  2
  3
  + +
  
  + +
  
  +
  x x x x
  x
  x
  x
  2
  
  Q2: [ 4 Marks]
  Present methods for solving first and second degree equations and inequalities
  and related topics.
  Find the values of (x) that satisfy the following equation:
  0 1
  2
  = + + x x
  Are these values (roots) real, imaginary or complex? What is the relation between
  these roots?
  1 = a , 1 = b , 1 = c
  a
  ac b b
  x
  2
  4
  2
  
  =
  1 2
  1 1 4 1 1
  2
  
  = x
  2
  3 1
  = x
  i x .
  2
  3
  2
  1
  1
  |
  |
  .
  |
  
  \
  |
  + =
  i x .
  2
  3
  2
  1
  2
  |
  |
  .
  |
  
  \
  |
  =
  these values (roots) are complex.
  These two roots are complex conjugates.
  
  3
  
  Q3: [ 6 Marks]
  Recognize and graph linear functions, rational functions, absolute value
  functions and graph inequalities in two variables.
  1. Graph the inequality : [ 3 Marks]
  x y x 2 3 >
  x x y + s 2 3
  x y 3 3 s
  x y s
  x y =
  Plot y against x and the region below the line y=x (including line points) is
  the solution.
  
  4
  
  2. [ 3 Marks]
  The manager of a weekend flea market knows from past experience that if
  he charges (x) dollars for a rental space at the market, the number (y) of
  spaces he can rent is given by the equation:
  x y 2 100 =
  a) Sketch a graph of this linear function.
  b) What do the slope, the y-intercept and the x-intercept of the graph
  represent?
  Slope = -2 and it represents the number of spaces per dollar.
  y-intercept = 100 and it represents the number of spaces if he charges
  (zero dollars.)
  x-intercept = 50 dollars for a rental space at the market when the number
  of spaces is zero.
  
  5
  
  Q4: [ 5 Marks]
  CLO 4: form and solve matrices and operations involving matrices.
  
  Find the value of (x) that satisfies the solution for the following system of linear
  equations using Cramers Rule:
  
  3 2 4 = + z y x
  
  z y x 7 1 3 = +
  z y x = 9 8 2
  
  |
  |
  |
  .
  |
  
  \
  |
  =
  1 9 2
  7 3 1
  2 4 1
  A
  , det (A) =1(3-63)-1(4+18)+2(28+6) = -14
  
  |
  |
  |
  .
  |
  
  \
  |
  =
  8
  1
  3
  b
  
  |
  |
  |
  .
  |
  
  \
  |
  =
  1 9 8
  7 3 1
  2 4 3
  x
  A
  , det (Ax) = 3(3-63)-1(4+18)+8(28+6) = -180 22 + 272 = 70
  
  5
  14
  70
  =
  
  = =
  A
  A
  x
  x
  
  6
  
  Q5: [5 Marks]
  CLO 5: Identify limits of algebraic, trigonometric, and composite functions.
  1. On what intervals is the following function continuous? [3 Marks]
  
  x
  x
  x g
  2 cos 2
  2 sin
  ) (
  +
  =
  
  Then find:
  
  x
  x
  x
  2 cos 2
  2 sin
  lim
  4
  +
  
  t
  
  Solution:
  The function is continuous on the open interval ( + , )
  2
  1
  0 2
  1
  2
  cos 2
  2
  sin
  4
  2 cos 2
  4
  2 sin
  2 cos 2
  2 sin
  lim
  4
  =
  +
  =
  +
  =
  |
  .
  |
  
  \
  |
  +
  |
  .
  |
  
  \
  |
  
  =
  +
  
  t
  t
  t
  t
  t x
  x
  x
  
  2. Find the value of the following limit:
  [2 Marks]
  
  2
  4
  2
  2
  lim
  
  x
  x
  x
  
  Solution:
  ( )
  ) 2 (
  2 ) 2 (
  lim
  2
  
  +
  =
  
  x
  x x
  x
  
  ( ) 4 2 2 2
  lim
  2
  = + = +
  =
  
  x
  x
  
  7
  
  Q6: [6 Marks]
  CLO 6: solve for the derivatives of algebraic, trigonometric and composite
  functions.
  Find the equation of the tangent line to the curve:
  2
  3 6 x x y =
  at the point where x=2.
  Also find the point at which the rate of change of (y) with respect to (x) is equal to
  zero. Is this point a maximum, a minimum or a turning point?
  Solution:
  x
  dx
  dy
  6 6 =
  = = = |
  .
  |
  
  \
  |
  =
  6 2 6 6
  2 x
  dx
  dy
  slope of tangent to curve at x = 2.
  0 12 12 2 3 2 6
  2
  = = = y
  Equation of straight line is:
  c mx y + =
  ( ) 12 2 6 0 = + = c c
  12 6 + = x y
  1 0 6 6 = = = x x
  dx
  dy
  
  3 1 3 1 6
  2
  = = y y so the point is (1,3)
  6
  2
  2
  =
  dx
  y d
  
  Since second derivative is zero, therefore, the point (1,3) is a maximum.
  8
  
  Q7: [6 Marks]
  CLO 7: find indefinite and definite integrals of algebraic, trigonometric, and
  composite functions.
  1. Find the area between the curve [3 Marks]
  1
  2
  + = x y
  And the x-axis from x=0 to x=3.
  ( ) 12 0 ) 3
  3
  3
  (
  3
  1
  3
  3
  0
  3 3
  0
  2
  =
  (
  
  + =
  (
  
  + = + =
  }
  x
  x
  dx x A
  
  2. Find the value of the following integral: [3 Marks]
  
  ( )dx x x
  }
  
  t
  0
  sin 5
  ( ) ( ) ( )
  (
  
  + =
  (
  
  + + =
  (
  
  + =
  }
  ) 1 ( 5 1 5
  2
  0 cos 5 0 ) cos 5
  2
  ( cos 5
  2
  sin 5
  2 2
  0
  2
  0
  t
  t
  t
  t
  t
  x
  x
  dx x x
  
  10
  2
  2
  =
  t
  
  9
  
  Q8: [5 Marks]
  CLO 8: Graph and solve the equations of lines, circles, parabolas, ellipses and
  Hyperbolas.
  Decide which of the following equations represents a straight line, a parabola, a
  circle, an ellipse or a hyperbola. Then sketch the graph of the equation that
  represents an ellipse.
  Circle x y = + 1
  2 2
  
  Line Straight x y . 1 + =
  Hyperbola x y = 1
  2 2
  
  Parabola x y =
  2
  
  Ellipse
  y x
  = + 1
  9 16
  2 2
metadata: {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}