Subject

Mathematics

Class

JEE Class 12

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JEE Mathematics 2008 Exam Questions

Multiple Choice Questions

31.

The area of the plane region bounded by the curves x + 2y2= 0 and x + 3y2= 1 is equal to 

  • 5/3

  • 1/3

  • 2/3

  • 4/3


D.

4/3

Solving the equations we get the points of intersection (–2, 1) and (–2,  1)The bounded region is shown as shaded region



The space required space area space equals space 2 integral subscript 0 superscript 1 space left parenthesis 1 minus 3 straight y squared right parenthesis space minus space left parenthesis negative 2 straight y squared right parenthesis
space equals space 2 integral subscript 0 superscript 1 space left parenthesis 1 minus straight y squared right parenthesis space dy space equals space 2 space open square brackets straight y space minus space straight y cubed over 3 close square brackets subscript 0 superscript 1
space equals space 2 space straight x space 2 over 3 space equals space 4 over 3

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32.

How many real solutions does the equation x7+ 14x5+ 16x3+ 30x – 560 = 0 have?

  • 7

  • 1

  • 3

  • 5


B.

1

x7+ 14x5+ 16x3+ 30x – 560 = 0
Let f(x) = x7+ 14x5+ 16x3+ 30x
⇒ f′(x) = 7x6+ 70x4+ 48x2+ 30 > 0 ∀ x.
∴ f (x) is an increasing function ∀ x.

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33.

The value of square root of 2 space integral fraction numerator sin space straight x space dx over denominator sin space open parentheses straight x minus begin display style straight pi over 4 end style close parentheses end fraction space is

  • space straight x plus space log open vertical bar cos space open parentheses straight x minus straight pi over 4 close parentheses close vertical bar space plus space straight C
  • straight x space minus log space open vertical bar sin space open parentheses straight x minus straight pi over 4 close parentheses close vertical bar space plus straight c
  • straight x space plus space log space open vertical bar sin space open parentheses straight x minus straight pi over 4 close parentheses close vertical bar space plus straight c
  • straight x space plus space log space open vertical bar cos space open parentheses straight x minus straight pi over 4 close parentheses close vertical bar space plus straight c

C.

straight x space plus space log space open vertical bar sin space open parentheses straight x minus straight pi over 4 close parentheses close vertical bar space plus straight c
square root of 2 space integral fraction numerator sin space straight x space dx over denominator sin space open parentheses straight x minus begin display style straight pi over 4 end style close parentheses end fraction space equals space square root of 2 integral fraction numerator sin space open parentheses straight x minus begin display style straight pi over 4 end style plus begin display style straight pi over 4 end style close parentheses dx over denominator sin space open parentheses straight x minus space begin display style straight pi over 4 end style close parentheses end fraction
space equals space square root of 2 space integral open parentheses cos space straight pi over 4 space plus space cot space open parentheses straight x minus space straight pi over 4 close parentheses space sin space straight pi over 4 close parentheses space dx
space equals space integral space dx space plus space integral cot space open parentheses straight x space minus space straight pi over 4 close parentheses dx
equals space straight x space plus space In space open vertical bar sin space open parentheses straight x minus straight pi over 4 close parentheses close vertical bar space plus straight c
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34. Let space straight I space space equals space integral subscript 0 superscript 1 space fraction numerator sin space straight x over denominator square root of straight x end fraction dx space and space straight J space equals space integral subscript 0 superscript 1 space fraction numerator cos space straight x over denominator square root of straight x end fraction space dx then which one is true?
  • I > 2/ 3 and J > 2

  • I < 2/ 3 and J < 2

  • I < 2/ 3 and J > 2

  • I > 2/ 3 and J < 2


B.

I < 2/ 3 and J < 2

straight I equals space integral subscript 0 superscript 1 fraction numerator sin space straight x over denominator square root of straight x end fraction space dx space less than space integral subscript 0 superscript 1 space fraction numerator straight x over denominator square root of straight x end fraction dx space equals space integral subscript 0 superscript 1 square root of straight x dx space equals space 2 over 3 straight x to the power of 3 divided by 2 end exponent space left enclose space subscript 0 superscript 1 end enclose space equals space 2 over 3
rightwards double arrow straight l space less than space 2 over 3
straight J space equals space integral subscript 0 superscript 1 fraction numerator cos space straight x over denominator square root of straight x end fraction space dx space less than integral subscript 0 superscript 1 fraction numerator 1 over denominator square root of straight x end fraction space dx space equals space 2 square root of straight x vertical line subscript 0 superscript 1 space equals space 2
therefore space straight J less or equal than 2
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