Mathematics Sample of Question
Grade 12 Advanced
Choose the correct answer:
1-The complete factorization of x4- 81 is :
a)(x – 9)2( x+ 9 )2 b)(x-9)(x+9) c)(x-3)(x+3)(x2+9) d)(x-3)(x+3)(x2-9)
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2-Which of the following is the simplest form of +
a) b) c) d)
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3- The partial fraction decomposition of
a) - b) + c) - d) +
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4-When 8x3 + 2x -1 is divided by 2x + 1 the remainder is :
a) -13 b) 9 c) 3 d) - 3
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5-If (x – 2) is a factor of x 4 + x 2 + k then k =
a ) 4 b) - 4 c) - 20 d) 20
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6- The simplest form of is:
a) b) c) d)
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7- Simplify
a) b) c) d)
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8- Find :
a) b) c) d)
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9- If : then
a) A=2 ; B=2 b) A=2 ; B=-2 c) A=-2 ; B=-2 d) A=-2 ; B=2
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10- When we divide f(x) = 3x3 + 5x2 – 4x – 7 by x + 2 the remainder is:
a) 3 b) 29 c) – 3 d) - 45
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11- If P(x) = x3 – 2x2 – x + 2 , one of the following is a factor of P(x), which one?
a) x + 3 b) x + 2 c) x – 2 d) x
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12- Evaluate:
a) b) c) d)
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13- Simplify: =
a) b) c) d)
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14- If 7= a and 5= b , then =
a) a + b b) a + b + 2 c) a + b -2 d) a – b - 2
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15- Find 25 to 2 decimal places
a) 1.18 b) 1.65 c) 0.55 d) 2.24
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16- If ln(x) = -2 ,then x =
a) b) e2 c) – 2e d)
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17- The graph of the function y = is :
a) always increasing b) always decreasing
c) constant d) cuts y-axis at (0,0)
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18- =
a) b) c) d)
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19– Given then =
a) 0 . 4 b) 0.36 c ) 0.216 d ) 1.8
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20- If then x =
a) 17 b) 9 c) 15 d) 7
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21- The graph of y = e2x has:
a)Horizontal b)Horizontal c) Vertical d)Vertical
asymptote x =2 asymptote y = 0 asymptote x = 0 asymptote x = 2
22- One of the following is the graph of y = e-x:
a- b- c- d-
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23- The domain of the function f(x) = ln (x-2 ) is :
a) ]2,[ b) 2, [ c) ]-, 2[ d) ], -2
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24- The inverse of the function f (x) = ln x is :
a) (x) = b) (x) = c) (x) = d) (x) =
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25- If f(x) = 3x-1 and g (x) = , then f (g(3)) =
a) 2 b) 5 c) -5 d)
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26- If f (x) = 4x - 3 , then (x) =
a) b) c) d) 4(x+3)
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27- If f(x) and g(x) are inverse for each other, then the graph of f (x) is the reflection
ofthe graph of g (x) about :
a) y = -x b) y = x c) x-axis d) y-axis
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28- The domain of the function y = ln x is :
a) R b) R-{0} c) x ≥ 0 d) x 0
29- If then f(x) is continuous
a) everywhere except at x = 1 b) everywhere except at x = 0
c) everywhere except at x = -2d) everywhere.
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30- If f(x) = ln x then f-1(2)
a) 0.69 b) 0.3 c) 7.39 d) 2.7
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31- If fog(x) = , x ≥1 then one of the following must be true :
a) g(x) = b) g(x) = x – 1 c) f(x) = (x – 1)2 d ) f(x) = + 1
f(x) = x - 1 f(x) = g(x) = g(x) = x -2
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32 - Which of the following functions have inverse function:
A B C D
33- For the given graph of f(x)
the graph of f-1(x) is :
a- b- c- d-
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34– The given graph of f(x) :
a) Continuous on R. b) Discontinuous on R.
c)Discontinuous at x= -1 d)Discontinuous at x=1.
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35- solve the equation 2= 10 is
a) ln 5 b) ln 10 c) ln 5 d) 2ln 5
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36- The solution of the equation 3 e2x-1 = 3 is :
a)- 0.5 b) 1 c) 0 d) 0.5
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37- The solution of ln2x = 1 is :
a)e b) c)2e d) - e
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38- For , then
a ) 4 b) -2 c) 2 d) – 4
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39- A function whose derivative is given by :
has a local minimum
at x =
a) 1 b) c) 0 d)
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40- If the graph of has a local minimum at x = -1, then the value of k is:
a) 1 b) 4 c) -1 d) -4
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41- If , then the function f has:
a)a local maximum at x = a b) a local maximum at x = 7
c) a local minimum atx = a d) a local minimum at x = 7
42- The slope of y = ex is equal to 1 at the point :
a) (e,0) b)(0,e) c)(1,0) d) (0,1)
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43- One of the following functions its second derivative gives an expression the same
as the original function :
a) f(x) = cos x b) f(x) = sin x c) f(x) = e-x d) f(x) =xe
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44- The derivative function of is:
a) b) c) d)
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45- If y = 2sin π – 5cos θ , then
a) 5sin θ b) 2cosπ - 5sin θ c) -5sin θ d) 2-5sin θ
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46- The third derivative of y = cos
a) b) c) d)
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47- The derivative of y= tan (4θ) is:
a) b) c) d)
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48- If f() = cos – sin, then the slope of the tangent to the curve of f() at = 0 is :
a)1 b) -1 c) 0 d) 2
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49- If h(x) = f(x) g(x) , and it is known that f(3) = -4, f`(3) = 5, g(3) = 1 , g`(3) = 2
Then h`(3) =
a) -1 b)1 c) 3 d) -3
50-
a) 0 b) c) d)
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51- The graph alongside represents a function f(x):
One of the following is not true
a) f (x) 0 for all x - 2
b) f (x) 0 for all - 2 x 1
c) f (x) = 0 for all - 2 x 1
d) f (3) 0
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52- The function represented by the graph alongside has :
a) Only one local maximum, one local minimum and
one inflection point.
b) Two local maximum and one inflection point.
c) Three local maximum
d) No local maximum, no local minimum no inflection point
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53- If c is a stationary value and f(c) 0 ,then :
a) f(x) has a local maximum at x = c. b) f(x) has a local minimum at x = c .
c) The second derivative test fails. d) f(x) has an inflection point at x = c.
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54- The slope of the tangent line to the curve of at x = 0 is
a) 1 b) 5 c) e d) 6
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55- One of the following functions satisfies f ( x) = f (x ) ,which one ?
a) f (x) = x2 b) f (x) = ln (x) c) f (x) = e x d) f (x) = ex
56- If g'(2) = 5 and f(x) = -3 g(x) +5 , then
f'(2) =
a)10 b) -10 c) 15 d) -15
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57- If
a) b) c) d)
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58- If y = (x3- 5)4 , then =
a) 108x6 b) 4(x3-5)3 c) 12x2(x3-5)3 d) 81x3
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59- One of the following functions has the derivative function
a) b) c) d)
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60- One of the following is equal to the slope of the tangent to y2-x2 = 1 at (1,)
a) b) - c) d) -
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61- If y is a function of x such that > 0 for all x and < 0 for all x , which of the
following could be part of the graph of y = f(x)
a b c d
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62- If y = ln (x ) then
a) b) c) d) ex
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63- If y = 2sin x + cos 3x , then =
a) 2 cos x b) 2sin x + 3cos x c) 2cos x + sin3 x d) 2cos x 3sin3 x
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64- If y=tan 5x , then =
a) 5 cot 5x b) 5sec2 5x c) sec25x d)
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65- If f ( 1) = 3 and g ( 1) = 5 then (f + g) ( 1) =
a) 8 b) 15 c) 3 d) 5
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66- If y = x2.lnx then =
a) 2 b) 2x.lnx x c) 2x.lnx + x d) x
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67- If y = f(x). g (x) then y =
a) f (x). g (x) b) f (x). g (x) - f(x). g (x)
c) d) f (x). g (x) + f(x). g (x)
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68- If a flu is spreading at the rate of , where P(t) is the total number of
students infected t days after the flu first started to spread, then one of the
following is the initial number of students infected.
a) 1 b) 8 c) 7 d) 3
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69- If particle’s position is given by
s(t) = 3t2 + 2t – 4 , then the acceleration at time t=2
a) 0 b) 6 c) 12 d) 14
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70-
a) b) c) d)
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71-
a) b) c) d)
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72-
a) f ( g(x)) .g (x) b) f ( g(x)) c) f ( g (x)) .g (x) d) f ( g (x)) .g (x)
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73-
a) b) c) d)
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74- If x2 + y2 = 5 then =
a) b) c ) –x d) 0
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75- If the slope of the tangent line to the graph of y= eax at x = 0 is 3 ,
then the constant a =
a) 1 b) 3 c) d) – 1
76- A particle moves along a straight line. Its position function, in cm, is S(t) = t3 – 3t 1
Where the time is in seconds. The initial velocity for the particle is :
a) 1 cm/s to the left from the origin
b) 1 cm/s to the right from the origin
c) 3 cm/s to the left from the origin
d) 3 cm/s to the right from the origin
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77- The antiderivative of g(x) = 3x2 + 5 is :
a)6x + 5 b) 6x c) x3 + c d) x3 + 5x + c
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78- =
a) ln│x│+c b) + c c) 2+ c d) + c
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79- If = + c then f(x) =
a)+c b) c) + c d) 2( )
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80- =
a) cos 2x +c b) cos 2x +c c) 2cos 2x+c d) - 2 cos 2x+c
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81- If then =
a ) 0 b) 10 c) 15 d) -10
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82- If , then k =
a) - b) c) d)
83- The area of the region bounded by y = , the x-axis from x = 1 to x = 2 is :
a)- ln 2 b) ln 2 c) 0.5 d) 2
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84- If the area of the given region is 8 unit2 then one of the following must be true:
a) b) c) d)
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85- A particle moves in a straight line with acceleration a(t) = 2 - 6t cm/s2.
If v(1) = 2 cm/s , Then its velocity when t= 3 seconds is :
a) 20 cm/s b) -20 cm/s c ) -18 cm/s d ) -24 cm/s
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86- particle moves with velocity function v(t) = cos( cm/s . The total distance
travelled from t= 0 to t = π second is :
a) 0 cm b) 2 cm c) 1 cm d) -1cm
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87-=
a) b) c) d)
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88-
a) b) c) d)
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89- =
a) b) c) d)
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90- If then
a) ln│x2 -3x│ b) ln│x2-3x│+c c) ln│x│+ ln│x-3│+c d) ln│x│+ 2 ln│x-3│+c
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91-
a) 3 x2 b) x3 + c c) 6 x d) 3x2 + c
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92-
a) b)x4 + x2 + c c) x5 + c d) x4 + 2 ln∣x∣+ c
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93- ∫e2xdx=
a) e2x + c b) e2x+1 + c c) e2x + c d) 2 e2x + c
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94- ∫ cos(5x – 1) dx =
a) -sin(5x – 1) + c b) sin(5x – 1) + c
b) sin(5x – 1) + c d) - sin(5x – 1) + c
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95-
a) 2 b) 1 c) 0 d)-1
96-
a) - 9 b) 5 c) 9 d ) - 5
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97- One of the following represents the shaded area:
a) b)
c) d)
98- If f(x) 0 for all a ≤ x ≤ b , then
a) b)
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c) d)
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99- An object is moving along a line with acceleration a(t) = 9.8 m /s2 .
its initial velocity is 3 m /s2 .Its velocity function is v(t) =
a) 0 b) 9.8 t + 3 c) 4.9t2 +3t +c d) 9.8 t
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100- The velocity of an object is v(t) = sin(t) m/s. Its initial position is 4 m .
Its position afterseconds is :
a) 5 m b) 1 m c) 4 m d) 6 m
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101-
a) b)
c) d)
102-
a) b) c) d)
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103-
a) b) ln( 2x2+3)+c
c) 2x2 ln( 2x2+3) + c d)
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104- If and ,then
a) b) c) d)
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105- If P (−2 , 3) and Q (4 , −5) , then =
a) b) c) d)
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106- If the vectors and are parallel, then k =
a) 4 b) – 4 c) – 10 d) 10
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107- One of the following vectors is a unit vector, which one ?
a) b) c) d)
108- Find m given that is a unit vector:
a) b) c) d)
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109- The length of the vector
a) – 4 b) 4 c) – 2 d) 2
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110- If the vectors and are perpendicular, then t =
a) – 8 b) 8 c) 1 d) – 2
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111- If is parallel to then
a) a = 2 , b = -3 b) a = -2 , b =3 c) a = 3 , b =-2 d) a = -1 , b = 6
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112-
a) b) c) d)
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113- The magnitude for the vector: is :
a) b) 14 c) 0 d)
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114- Express the vector: in component form
a) b) c) d)
115- If:
a) 2 b) 1 c) 3 d) 0
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116- then
a) b) c) 10 d)
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117- If A(3 , 2) and B(-1 , 2), then
a) b) c) d)
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118- If then
a) 625 b) - 5 c) 5 d) 5
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119- Construct vector equation for
a) + + = 0
b) = +
c) + − = 0
d) = +
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120- The measure of angle between the vectors and is
a) 30° b) 150° c) 60° d) 120°
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121- If is a unit vector ,then the value of k is:
a) b) - c) d)
122- On of the following pairs of vectors are parallel:
a) b) c) d)
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123- The unit vector form of
a) 3i + j +2 k b) 3i-j+2k c) 2i-k+j d) -1i+2j+3k
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124- The points A and B have the coordinates (5, -4), (-2, 3) respectively .
a) b) c) d)
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125- Which equation represent the diagram:
a)t = r-s
b)t = r +s
c)t = -r-s
d)t = s-r
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126- One of the following is a vector in the direction of and has a length of 2 units:
a) b) c) d)
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127- Find the magnitude of the vector
a)25 b)5 c) -5 d) -25
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128- If are perpendicular vectors what is the value of t =
a) 0.4 b)2 c) 5 d) -0.4
129- If the vector a = , b = then
a)a = b b) a⊥b c) a // b d) a∦ b
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130- The exact value of tan 15 is :
a) 2 + b) 2- c) 2 + d) 2 -
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131- The exact value of sin is :
a) 0.5 b) -0.5 c) d)
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132- cos2x =
a) 2cos22x -1 b) 2sin22x -1 c) 2cos2x-1 d) 2sin2x-1
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133- sin 17° cos 27° + cos 17° sin 27°=
a)sin 10° b)cos 44° c) cos 10° d) sin 44°
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134- cos 2x =
a) sin2x –cos2x b) cos2x-sin2x c) 1-2cos2x d) 2sin2x-1
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135- cos 50° cos 31° – sin 50° sin 31° =
a) cos 19° b) cos 81° c) sin 19° d) sin 81°
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136- =
a) -1 b) 1 c) tan 5° d)
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137- If sin x = , then cos x =
(a) (b) - (c) (d)
138- 2cos2 4x -1 =
a) cos 4x b) sin 8x c) cos 8x d) cos 16 x
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139- If 2sin x = , 0 ≤x≤ 2π then x =
a) π , 2π b) π , 5π (c) π , 5π (d) π , 11π
3 3 3 3 6 6 6 6
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140- The solution of sin2x = 0 , -π ≤ x ≤ π is :
a) - π b) π and -π c) π and- π d) π , - π, π , -π , 0
6 4 4 3 3 2 2
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141- The general solution of the equation tan x = 1 is :
a) b) c) d)
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142- The solution of cos( x +π) = -1 -2π ≤ x ≤ 0 is :
a) 0 b) 0 and π c) 0 and - 2π d) 0 , -π , -2π
Answer the following questions:
(1) Factor completely f(x) = 5x3 – 10x2 + 6x – 12
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(2) Divide:
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(3) Decompose to partial fractions: ( find A,B )
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(4) When we divide f(x) = x3 – 3x2 – kx + 5 by (x – 2) the remainder is – 7 ,
find the value of k.
(5) If (x – 3) is a factor of f(x) = x3 – 2x2 – 9x + 18, factorize completely.
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(6) Simplify :
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(7) Solve the equation:
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(8)The weight Wt grams, of bacteria in a culture t hours after establishment is given
by:
.
Find the time for the weight of the culture to reach 30 grams.
(9) A man jumps from an aeroplane and his speed of descent is given by
V = 50 ( 1 – e- 0 . 2t ) m/s where t is the time in seconds. Find his speed after 10
seconds.
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(10) If f(x) = and g(x) = ln x, find
1) g(f(x))
2) f(g())
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(11) If f(x) = ,Find (x)
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(12) Simplify:
(13)Express the following fraction in partial fraction:
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( 14) Solve for x :
(x – 1) 2 + 1 = ( x + 2)2
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(15) When px2 + q x + 1 is divided by (x – 1) the remainder is 2 , when divided by
(x + 1) the remainder is 4 . Find p and q .
(16) Factories completely x3 + 6 x2 + 11x + 6 .
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(17) Solve for x :
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(18) Solve for x : ln(x-3) + ln(x+3) = ln 8x
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(19) Given f(x) = x + 2 , g(x) = x2 + 1 solve the equation f o g (x) = 4
(20) Plot the graph of the function y = 2+1
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(21) Plot the graph of the function: y = – ln x
(22) Draw the graph of (x)
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(23) From the graph,
a)Write an interval, where f (x) is
continuous
b)Find a point where f (x) is not
continuous
( 24 ) a- Sketch the graph of y = -│x - 2│.
b- Determine the domain and the
range.
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(25) The temperature T of a liquid which has been placed in a refrigerator is given by
T = 4 + 96 e -0.03t 0c , where t is the time in minutes.
Find :
a-The initial temperature .
b-The time required for the temperature to reach 5 0c.
26)The graph of f(x) = ex shown below:)
a) State the following :
Domain :------
Range:------
Intercepts : ------
Asymptotes: ------
b) On the same set of axis sketch the graph
of g(x)= ln x
c) What is the relation between f(x) and g(x) .
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(27) If
a-Find the inverse function g-1(x).
b-Find the domain and the range of g-1(x).
)28) The following figure represents graph of the function of f(x).
Find all points of discontinuity.
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(29)A particle P moves in a straight line where position given by ,
where t is the time in seconds,
a) Find the velocity and acceleration functions.
b) Find the initial position, velocity and acceleration of P.
(30) For the function: f(x) = 2x3 – 3x2 -36x + 7
a)Find and classify all stationary points and points of inflection.
b)Find intervals where the function is increasing and decreasing.
c)Find intervals where the function is concave up or down.
d)Sketch the graph of the function showing all important features.
( Note: x and y-intercepts are plotted)
(31) Differentiate with respect to x:
a)
b)
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(32) Find the slope of tangent function of f(x) :
a) at x = 1
b)
c)
(33) A book is being prepared for publication. Each page has height y cm, width x cm
and a fixed total area of 294 cm2. Each paper has 3 cm margin at both top and
bottom and a 2 cm margin on each side. Let A be the area of that part of the page
which is available for printing.
a)Express A in terms of x.
b)Find the dimensions of a page which will provide maximum space for printing.
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(34) Find, from the first principle, the slope function of y = ex
(35) Let f(x) = x3 – 3x2 +2
a)State the intervals on which the function is increasing or decreasing and classify thestationary points.
b)State the inflection points and the intervals on which the curve of f(x) is concave up or down.
c)Use the information from a) and b) to sketch the graph of f(x) .
(36) Using the second derivative test ,find the local maximum and local minimum
for f(x) = x3 – 6x2 + 5
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(37) Show that if y = then ( hint : )
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(38) Find the slope of the tangent line to the graph of y = ln( x2 + 3 ) + x at x = 1.
(39) Find when y = 5x2tanx .
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(40) Find for each of the following functions:
a)
b)
c) y = sin(x2 – 4 ) + (1-3x2)6
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(41)
(42) Find when y2 + 3x2 y + x = 7.
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(43) Find two positive numbers their sum is 34 and their product is maximum.
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(44) A psychologist claims that the ability A ( t ) to memorize simple facts during
infancy years can be calculated using the formula where 0 <t ≤5 ,
t being the age of the child in years. At what age is the child's memorizing
a maximum?
(45) An object moves in a straight line with position function given by
S(t) = t3 – 12t +3 cm.
Find an expression for its velocity and acceleration at any instant t.
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(46) If g'(x) = sec2x + ex + 1 and g(0) = 2 , find g(x)
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(47) Find :
a)
b)
(48) If
find .
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(49) Find the area of the region enclosed by the function f(x) = x – x3 and
the x-axis .
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(50)
(51) Find the area of the region bounded by the graph of and the x-axis
from x = 1 to x = e.
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(52) A function has slope function and passes through
thepoints(0,2) and (1,4). Find the value of a and the function f(x) .
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(53) Use the integration by parts to find
sec2x dx .
(54) Find
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(55) Find
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(56) Find
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(57) Evaluate :
=
(58) Use integration by partial fraction to find :
dx
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(59) Find the general solution of :
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(60) A small object is moving along a line with a velocity V(t) = 6t +5 m/s .
Find its position function S(t) given that S(1) = 2.
(61)Find k 1 if the area of the region enclosed by and the
x-axis from x = 1 to x = k is units2. ( Note : f(x) 0 )
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(62)Find the volume of the solid of revolution when the region between
and is revolved about the x-axis .
(63) Find
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(64) By using suitable substitution, find
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(65) Find the general solution of the equation
(66) Using partial fractions, find
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(67) The tube cross-section shown has inner radius of 3 cm and outer radius 6 cm and
within the tube water is maintained at a temperature of 100C.Within the metal
the temperature falls off at a rate according to where x is
the distance from the central axis O and 3 ≤ x ≤ 6. Find the
temperature of the outer surface of the tube.
(68) Find the volume of revolution ofaround x-axes from x = 1 to x = 3
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(69)A particle moves in a straight line with acceleration function a(t) = m/s2
and has initial velocity 2m/s . Find :
a- An expression of velocity function . .
b- The displacement of the particle at the end of one second .
(70) The number of bacteria present in a culture increases at a rate proportional to the
number present according to ,where N is the number of bacteria
present at time t hours. If the initial population of bacteria is 100 ,find the
number of bacteria at any time t.
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(71) Find the area of the region enclosed by the graphs of y = x2 + 4x and y = 5
(72) 100 meters of fences available for making the sides of a rectangular enclosure
against an existing wall . Find the greatest area that can be enclosed
Wall
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(73) A particle moving such that the displacement from the origin ( O ) is given by the
formula P(t) = a t3 + 2t + 1 . if the acceleration of the particle after 2 seconds
is 24 m/s2 , then find the value of ( a )
(74) Find scalars r and s such that:
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(75)
Find x and y .
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(76) If A(6,-1,0) , B(5,1,2) , C(3,2, d ) ,and ∆ABC is right-angled triangle at B.
Find the value of d.
(77) For , ,
Find:
a)
b)The angle between p and r.
c)
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(78) If :
Represent on the grid alongside:
1)
2)
(79) If : and Find algebraically:
1)
2) =
3)
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(80) If :
find the following:
a)
b)
c)
d) The angle between the two vectors.
(81) If ABCD is a parallelogram, A (3, 0), B (2,-1) and C (8,-2).Using vectors,
find the coordinates of D
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(82) Given that A ( -2 ,3,0) , B ( 3 , 4,-5 ) and C ( 5 , -4,6) Find :
a) in component form
b) The distance from A to B .
c) The position vector from C to A.
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(83) Use vectors to show that the points A(0,1) , B( -1 , -2) and C( 2,7) are
collinear.
(84) If P ( 3,4) , Q(-3,-1) ,R (-2,5) and S (4,10).
a) Use vectors to show that PQRS is a parallelogram.
b) State if PQRS is a rhombus or not.
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(85) solve the following equations :
1) Sin2x = Cosx 0 ≤ x ≤ π
2) tan 2x = 0 -2π ≤ x ≤ π
(86) Show that Cos 2x = 1 – 2 Sin2x
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(87) Prove the following identities
a)
b)
(88) If Sin = , Find the value of
a) Sin 2
b) Cos 2
c) Sin ( + )
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(89) Without using calculator find:
a)Cos(75◦ )
b) tan (15 ◦)
(90) Solve the following equations
a)
b)
c)
1