Let  be three unit vectors such that  then the angle between

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 Multiple Choice QuestionsMultiple Choice Questions

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1.

Let   be three unit vectors such that   then the angle between   is

  • 3π/4

  • π/2

  • 2π/3

  • 5π/6


D.

5π/6

Given comma space vertical line straight a with hat on top vertical line space equals space vertical line stack straight b vertical line with hat on top space equals vertical line straight c with hat on top vertical line space equals space 1
and space space straight a with hat on top space straight x space left parenthesis straight b with hat on top space straight x space straight c with hat on top space right parenthesis equals space fraction numerator square root of 3 over denominator 2 end fraction space left parenthesis straight b with hat on top space plus space straight c with hat on top space right parenthesis
Now comma space consider space straight a with hat on top space straight x space left parenthesis straight b with hat on top space straight x space straight c with hat on top space right parenthesis space equals space fraction numerator square root of 3 over denominator 2 end fraction space left parenthesis straight b with hat on top space plus space straight c with hat on top space right parenthesis
rightwards double arrow space left parenthesis straight a with hat on top. straight c with hat on top right parenthesis straight b with hat on top space minus space left parenthesis straight a with hat on top. straight b with hat on top right parenthesis straight c with hat on top space equals space fraction numerator square root of 3 over denominator 2 end fraction space left parenthesis straight b with hat on top space plus fraction numerator square root of 3 over denominator 2 end fraction straight c with hat on top
On space comparing comma space we space get

straight a with hat on top. straight b with hat on top space equals space minus space fraction numerator square root of 3 over denominator 2 end fraction space rightwards double arrow space vertical line straight a with hat on top vertical line space vertical line straight b with hat on top space vertical line space cosθ space equals space minus fraction numerator square root of 3 over denominator 2 end fraction
rightwards double arrow space cos space straight theta space equals space minus fraction numerator square root of 3 over denominator 2 end fraction space vertical line because space vertical line straight a with hat on top vertical line space equals space vertical line straight b with hat on top vertical line space equals space 1
rightwards double arrow space cos space straight theta space equals space cos space open parentheses straight pi space minus straight pi over 6 close parentheses rightwards double arrow space straight theta space equals space fraction numerator 5 straight pi over denominator 6 end fraction space
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2.

If the standard deviation of the numbers 2, 3, a and 11 is 3.5, then which of the following is true?

  • 3a2−26a+55=0

  • 3a2−32a+84=0

  • 3a2−34a+91=0

  • 3a2−34a+91=0

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3.

Let y(x) be the solution of the differential equation left parenthesis straight x space log space straight x right parenthesis dy over dx space plus straight y space equals space 2 space straight x space log space straight x comma (x  ≥1). Then, y (e) is equal to 

  • e

  • 0

  • 2

  • 2

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4.

If left parenthesis 2 space sin space straight x right parenthesis space dy over dx space plus space left parenthesis space straight y space plus 1 right parenthesis space cos space straight x space equals space 0 space and space straight y left parenthesis 0 right parenthesis space equals 1 space then space straight y open parentheses straight pi over 2 close parentheses is equal to

  • 4/3

  • 1/3

  • -2/3

  • -2/3

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5.

Let In ∫tan x dx,(n> 1) . I4 + I6 = a tan5x + bx5 + C, where C is a constant of integration, then the ordered pair (a, b) is equal to

  • open parentheses negative 1 fifth comma 0 close parentheses
  • open parentheses negative 1 fifth comma 1 close parentheses
  • open parentheses 1 fifth comma 0 close parentheses
  • open parentheses 1 fifth comma 0 close parentheses
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6.

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

  • 7

  • 1

  • 3

  • 3

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7.

Let the orthocentre and centroid of a triangle be A(–3, 5) and B(3, 3) respectively. If C is the circumcentre of this triangle, then the radius of the circle having line segment AC as diameter, is

  • 352

  • 10

  • 210

  • 352


8.

Consider the non-constant differentiable function f of one variable which obeys the relation f(x)fy f(x - y ). If f'(0)= p and f(y) . f'(5) = q, then f'(- 5) is

  • p2q

  • qp

  • pq

  • q


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9.

If f(x)=log5(log3(x)), then f'(e) is equal to

  • e loge(5)

  • e loge(3)

  • 1eloge5

  • 1eloge3


10.

Let F(x)=ex, G(x)=e- x and H(x) = G(F(x)), where x is a real variable. Then, dHdx at x= 0 is

  • 1

  • - 1

  • - 1e

  • - e


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