Subject

Mathematics

Class

JEE Class 12

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 Multiple Choice QuestionsMultiple Choice Questions

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1.

The sum of all real values of x satisfying the equation
left parenthesis straight x squared minus 5 straight x space plus 5 right parenthesis to the power of straight x squared plus 4 straight x minus 60 space equals space 1 end exponent is:

  • 3

  • -4

  • 6

  • 6


A.

3

Given, 

left parenthesis straight x squared minus 5 straight x space plus 5 right parenthesis to the power of straight x squared plus 4 straight x minus 60 end exponent space equals space 1
clearly comma space this space is space possible space when
straight I. space straight x squared space plus space 4 straight x minus 60 space equals space 0 space and space straight x squared minus 5 straight x space plus 5 space not equal to space 0
II. space straight x squared minus 5 straight x plus 5 space equals space 1
III. space straight x squared minus 5 straight x space plus 5 space equals space minus 1 space and space straight x squared plus 4 straight x minus 60 space equals space Even

Case space straight I space when space straight x squared space plus space 4 space straight x minus 60 space equals space 0 comma space then
straight x squared space plus space 10 space straight x space minus 6 straight x space minus 60 equals 0
rightwards double arrow space straight x left parenthesis straight x plus 10 right parenthesis minus 6 left parenthesis straight x plus 10 right parenthesis space equals space 0
rightwards double arrow left parenthesis straight x plus 10 right parenthesis left parenthesis straight x plus 6 right parenthesis space equals space 0
rightwards double arrow straight x space equals space minus space 10 space or space straight x equals space 6
Note space that comma space for space these space two space vaues space of space straight x comma space straight x squared minus 5 straight x space plus space 5 space not equal to 0
Case space II space when space straight x squared space minus 5 straight x space plus 5 space equals space 1
straight x squared space minus 5 straight x space plus 4 space equals space 0
straight x squared minus 4 straight x minus straight x space plus 4 equals 0
straight x left parenthesis straight x minus 4 right parenthesis minus 1 left parenthesis straight x minus 4 right parenthesis space equals 0
left parenthesis straight x minus 4 right parenthesis left parenthesis straight x minus 1 right parenthesis space equals 0
straight x equals space 4 space or space straight x space equals 1

Case space III space when space straight x squared minus 5 straight x space plus 5 space equals space minus 1
straight x squared minus 5 straight x space plus 5 space equals space minus 1
straight x squared minus 5 straight x space plus space 6 space equals space 0
rightwards double arrow space straight x squared minus 2 straight x minus 3 straight x space plus 6 space equals space 0
straight x left parenthesis straight x minus 2 right parenthesis minus 3 left parenthesis straight x minus 2 right parenthesis space equals space 0
left parenthesis straight x minus 2 right parenthesis left parenthesis straight x minus 3 right parenthesis space equals 0
straight x equals 2 space or space straight x space equals space 3
Now comma space when space straight x space equals space 2 comma space straight x to the power of 2 space end exponent plus 4 straight x minus 60 space equals space 4 plus 8 minus 60
equals negative 48 comma space which space is space an space interger
when space straight x space equals space 3 comma space straight x squared plus space 4 straight x space minus 60 space equals space 9 space plus 12 minus 60 space equals space minus 39
which space is space not space an space even space interger.
Thus comma space in space this space case comma space we space get space straight x space equals space 2
Hence comma space the space sum space of space all space real space values space of space
straight x space equals space minus 10 plus 6 plus 4 plus 1 plus 2 space equals 3

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2.

If all the words (with or without meaning) having five letters, formed using the letters of the word SMALL and arranged as in a dictionary; then the position of the word SMALL is:

  • 46th

  • 59th

  • 52nd

  • 52nd

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3.

If the number of terms in the expansion of open parentheses 1 minus 2 over straight x space plus 4 over straight x squared close parentheses to the power of straight n comma straight x not equal to 0 comma is 28, then the sum of the coefficients of all the terms in this expansion is

  • 64

  • 2187

  • 243

  • 243

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4.

If the 2nd, 5th and 9th terms of a non-constant A.P. are in G.P., then the common ratio of this G.P. is:

  • 8/5

  • 4/3

  • 1

  • 1

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5.

If the sum of the first ten terms of the series,

open parentheses 1 3 over 5 close parentheses squared space plus open parentheses 2 2 over 5 close parentheses squared space plus open parentheses 3 1 fifth close parentheses squared space plus space 4 squared space plus space open parentheses 4 4 over 5 close parentheses squared space plus.....is 16/5 m, the m is equal to

  • 102

  • 101

  • 100

  • 100

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6.

Let straight p space equals space limit as straight x rightwards arrow 0 to the power of plus of left parenthesis 1 plus tan squared square root of straight x right parenthesis to the power of 1 divided by 2 straight x end exponent comma space then log p is equal to 

  • 2

  • 1

  • 1/2

  • 1/2

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7.

A man is walking towards a vertical pillar in a straight path, at a uniform speed. At a certain point A on the path, he observes that the angle of elevation of the top of the pillar is 30o. After walking for 10 minutes from A in the same direction, at a point B, he observes that the angle of elevation of the top of the pillar is 60o. Then the time taken (in minutes) by him, from B to reach the pillar, is:

  • 6

  • 10

  • 20

  • 20

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8.

If 
straight f left parenthesis straight x right parenthesis space plus space 2 straight f open parentheses 1 over straight x close parentheses space equals space 3 straight x comma
straight x space not equal to space 0 space and space straight S space equals space open curly brackets straight x space straight epsilon space straight R colon space straight f space left parenthesis straight x right parenthesis space equals space straight f left parenthesis negative straight x right parenthesis close curly brackets semicolon space then space straight S

  • is an empty set

  • contains exactly one element.

  • contains exactly two elements.

  • contains exactly two elements.

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9.

A value of θ for which fraction numerator 2 space plus space 3 straight i space sin space straight theta over denominator 1 minus 2 straight i space sin space straight theta end fraction is purely imaginary is

  • π/3

  • π/6

  • sin to the power of negative 1 end exponent open parentheses fraction numerator square root of 3 over denominator 4 end fraction close parentheses
  • sin to the power of negative 1 end exponent open parentheses fraction numerator square root of 3 over denominator 4 end fraction close parentheses
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10.

If A = open square brackets table row cell 5 straight a end cell cell negative straight b end cell row 3 2 end table close square brackets and A adj A = AAT, then 5a +b is equal to

  • -1

  • 5

  • 4

  • 4

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