Subject

Quantitative Aptitude

Class

SSCCGL Class 12

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 Multiple Choice QuestionsMultiple Choice Questions

108.

ABCD is a cyclic trapezium with AB || DC and AB a diameter of the circle. If angle CAB space equals space 30 degree comma then angle ADC is

  • 60°

  • 120°

  • 150°

  • 30°


B.

120°



angle ACB space equals space 90 degree space left parenthesis Angle space in space straight a space semi minus circle right parenthesis
angle CAB space equals space 30 degree space left parenthesis given right parenthesis
Now, in Δ ACB,
  angle ACB space plus space angle CAB space plus space angle ABC space equals space 180 degree
90 degree space plus space 30 degree space plus space angle ABC space equals space 180 degree
space space therefore space space space angle ABC space equals space 180 degree space minus space 120 degree space equals space 60 degree
As ABCD is a cyclic trapezium,
therefore space space angle ADC space equals space 180 degree space minus space 60 degree space equals space 120 degree space left parenthesis Sum space of space opposite space angles space of space straight a space cyclic space quadrilateral space is space 180 degree right parenthesis

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105.

Two circles touch each other internally, Their radii are 2cm and 3 cm. The biggest chord of the greater circle which is outside the inner circle is of length

  • 2 square root of 2 space cm

  • 3 square root of 2 space cm

  • 2 square root of 3 space cm

  • 4 square root of 2 space cm


D.

4 square root of 2 space cm


From the figure, 
MN is the diameter fo the smaller circle and PQ is the biggest chord of the greater circle. O is the centre of the greater circle and O' is the centre of the inner circle.
Now, 
OM = MN - ON = 4 - 3 = 1 cm
Now, 
   In Δ PMO,
         PM space equals space square root of OP squared space minus space OM squared space end root space equals space square root of left parenthesis 3 right parenthesis squared space minus space left parenthesis 1 right parenthesis squared end root
space space space space space space equals space square root of 9 space minus space 1 end root space equals space square root of 8 space equals space 2 square root of 2 space cm
therefore space space PQ space equals space 2 space cross times space 2 square root of 2 space equals space 4 square root of 2 space cm.

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102.

If x + y + z = 6  and x2 + y2 + z2 = 20,  then the value of x3 + y3 + z3 - 3xyz is

  • 64

  • 70

  • 72

  • 76


C.

72

x + y  + z = 6
S.B.S.
(x + y + z)2 = x2 + y2 + z2 + 2(xy + yz + zx)
⟹      (6)2 = 20 + 2(xy + yz + zx)
⟹        36 = 20 + 2(xy + yz + zx)
⟹         36 - 20 = 2(xy + yz + zx)
⟹               16 = 2(xy + yz + zx)
⟹             (xy + yz + zx) = 8
∴        x3 + y3 + z3 - 3xyz
       = (x + y + z) (x2 + y2 + z2 - (xy + yz + zx)
       =  6 (20 - 8) = 6 x 12 = 72

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110.

AB is the chord of a circle with centre O and DOC is a line segment originating from a point D on the circle and intersecting AB produced at C such that BC = OD. If angle BCD space equals space 20 degree comma then angle AOD is equal to

  • 20°

  • 30°

  • 40°

  • 60°


D.

60°



Given that, OD = BC   ...(i)
              and OD = OB  ...(ii) [Radii of a circle]
           From (i) and (ii),
                  OB  =  BC
      therefore space space space angle BCO space equals space angle BOC space space left square bracket Angle space opposite space to space equal space sides space are space equal right square bracket
Then,
       angle OBC space equals space 180 degree space minus space left parenthesis angle BCO space plus space angle BOC right parenthesis
space space space space space space space space space space space equals space 180 degree space minus space 20 degree space minus space 20 degree space equals space 140 degree
space space space
   therefore space space angle OBA space equals space 180 degree space minus space 140 degree space equals space 40 degree
As,  OB = OA  (Radii of a circle)
    therefore space space space space angle OBA space equals space angle OAB
Now,
       angle AOB space equals space 180 degree space minus space left parenthesis angle OBA space plus space angle OAB right parenthesis
space space space space space space space space space space space equals space 180 degree space minus space left parenthesis 40 degree space plus space 40 degree right parenthesis space equals space 180 degree space minus space 80 degree space equals space 100 degree
therefore space space angle AOD space equals space 180 degree space minus space left parenthesis angle AOB space plus space angle BOC right parenthesis space equals space 180 degree space minus space left parenthesis 100 degree space plus space 20 degree right parenthesis space equals space 180 degree space minus space 120 degree space equals space 60 degree


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101.

The minimum value of (x - 2) (x - 9) is

  • negative 11 over 4

  • 49 over 4

  • 0

  • negative 49 over 4


D.

negative 49 over 4

(x - 2) (x - 9)
      = (x2 - 2x - 9x + 18) = (x2 - 11x + 18)
      As the co-efficient of x2 is greater than 0, 
∴ Minimum value of (x - 2) (x - 9) is given by,
      = (-D/4a)
      =  - (b2 - 4ac)/4a
      = - (121 - 4*1*18)/4*1
      = - (121 - 72)/4
     =   - 49/4
     

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109.

ABC is a triangle. The bisectors of the internal angle angle straight B and external angle angle straight C intersect at D. If angle BDC space equals space 50 degree,  then angle straight A is

  • 100°

  • 90°

  • 120°

  • 60°


A.

100°



In Δ BDC,
       y + 180° - 2x + x + 50° = 180°
                   y - x + 50° = 0
                        y - x = -50°    ...(i)
In Δ ABC,
         2 straight y space plus space 180 degree minus 2 straight x space plus space angle straight A space equals space 180 degree
space space space space space 2 left parenthesis straight y space minus space straight x right parenthesis space plus space angle straight A space equals space 0
space space space space space space space 2 left parenthesis negative 50 right parenthesis space plus space angle straight A space equals space 0
space space space space space space space space space space space space space space space space space space space space space space space angle straight A space equals space 100 degree
space space space space space space space space space

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106.

ABCD is a cyclic quadrilateral. AB and DC are produced to meet at P. If angle ADC space equals space 60 degree comma then the angle PBC space plus space angle PCB is

  • 130°

  • 150°

  • 155°

  • 180°


A.

130°


As ABCD is a cyclic quadrilateral,
    therefore space space space angle ABC space equals space 180 degree space minus space 70 degree space equals space 110 degree space and space angle BCD space equals space 180 degree space minus space 60 degree space equals space 120 degree
Now,
angle PBC space plus space angle PCB space equals space left parenthesis 180 degree space minus space 110 degree right parenthesis space plus space left parenthesis 180 degree space minus space 120 degree right parenthesis
space space space space space space space space space space space space space space space space space space space space space space space space space equals space 70 degree space plus space 60 degree space equals space 130 degree

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103.

The third proportional to
       open parentheses straight x over straight y space plus space straight y over straight x close parentheses space space and space square root of straight x squared space plus space straight y squared end root space is

  • xy

  • square root of xy

  • cube root of xy

  • fourth root of xy


A.

xy

Third proportion is given by,
             straight c space equals space straight b squared over straight a
            space straight c space equals space fraction numerator open parentheses square root of straight x squared space plus space straight y squared end root close parentheses squared over denominator begin display style straight x over straight y end style space plus space begin display style straight y over straight x end style end fraction
straight c space equals space fraction numerator straight x squared space plus space straight y squared over denominator begin display style fraction numerator straight x squared space plus space straight y squared over denominator xy end fraction end style end fraction space space rightwards double arrow space space straight c space equals space xy

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104.

In a ΔABC, the side BC is extended upto D. Such that CD = AC, if angle BAD space equals space 109 degree and angle ACB space equals space 72 degree comma then the value of angle BAC is

  • 35°

  • 60°

  • 40°

  • 45°


A.

35°



angle ACD space equals space 180 degree space minus space 72 degree space equals space 108 degree
As AC = CD  (given),
angle CAD space equals space angle CDA space equals space fraction numerator 180 degree space minus space 108 degree over denominator 2 end fraction space equals space fraction numerator 72 degree over denominator 2 end fraction space equals space 36 degree
angle BAC space equals space 109 degree space minus space angle CAD space equals space 109 degree minus space 36 degree space equals space 73 degree
Now, in Δ BAC,
   angle ABC space equals space 180 degree space minus space angle BAC space minus space angle ACB
space space space space space space space space space space equals space 180 degree space minus space 73 degree space minus space 72 degree space equals space 35 degree

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107.

From a point P which is at a distance of 13 cm from centre O of a circle of radius 5 cm, in the same plane, a pair of tangents PQ and PR are drawn to the circle. Area of quadrilateral PQOR is

  • 65 cm2

  • 60 cm2

  • 30 cm2

  • 90 cm2


B.

60 cm2



angle OQP space equals space angle ORP space equals space 90 degree space left parenthesis Angle space at space point space of space contact space of space tangent right parenthesis
Now, 
  In Δ OQP,  
             QP space equals space square root of OP squared space minus space OQ squared end root space equals space square root of left parenthesis 13 right parenthesis squared space minus space left parenthesis 5 right parenthesis squared end root space equals space square root of 169 space minus space 25 end root space equals space square root of 144 space equals space 12 space cm
∴  Area of Δ OQP
                      equals space 1 half space cross times space 12 space cross times space 5 space equals space 30 space cm squared
∴  Area of PQOR
              = 2 x 30 = 60 cm2
    



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