Lakshya Education MCQs

Question: What is the common difference of an AP which has its first term as 100 and the sum of its first 6 terms = 5 times the sum of its next six terms?
Options:
A.12
B.15
C.-10
D.-2
E.a < -2
Answer: Option C
: C

Option a and b can be eliminated directly, as from the statement sum of its first 6 terms = 5 times the sum of its next six terms, we can deduce that the Common Difference is negative (decreasing AP). Go from answer options : Take option c Sum to 1st 6 terms is given by the formula n2(2a+[n1]d) = 3(200-50)= 450 Sum to 2nd6 terms = Sum to 12 terms – sum to 6 terms = 6( 200-110) -450 = 540 – 450 = 90 5 × Sum to2nd6 terms = 450 Hence option c

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Question 1. In the given figure AB is a diameter of the circle, CD is a chord parallel to AB, and AC intersects BD at E. If the ratio of area of triangles AEB and DEC= 4: 1, then find out the value of (\theta\)

.
  1.    30
  2.    60
  3.    45
  4.    15
  5.    None of these
Answer: Option B
: B

Join AD. As AB is the diameter ADB = 90. And hence ΔADB is a right angle triangle. Triangles AEB and DEC are similar hence if the ratio of area is 4: 1, ratio of sides will be 2:1. So, AE: DE = 2: 1. In triangle ADE, hypotenuse AE = 2, DE = 1 AD2 = 4 – 1 = 3. AD = 3 So, AED = 60 .
Question 2. Find the value of g, given that 8x2+2x+g=0 and 2x2+gx+1=0 have a common root?
  1.    1
  2.    2
  3.    3
  4.    a and b
  5.    None of the above
Answer: Option E
: E

OPTION ETo find the common root, equate the two equations2x2+gx+1=8x2+2x+g-6x2+x(g-2)+(1-g)=0Now find the discriminant of this equation and equate that to 0(D=0 means that the roots are equal and there exists a single root, acording to the question also, there is a single root which is common, hence D=0)we will get it as : g2 - 28g +28 = 0Check with answer options at this stageNone of the values of x is being satisfied, hence option e.
Question 3.  Find the last two digits of  (148)1084
  1.    06
  2.    66 
  3.     56
  4.    16
  5.    a < -2
Answer: Option D
: D

(148)1084=(37)1084×(4)1084

=(374)271×(2)2168

=(372×372)271×....56

=(..69×..69)271×.....56

=(61)271×....56=....61×....56=......16

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