Vector Algebra

Question
CBSEENMA12036083

For x ∈, (0, 5π/2) define f(x). Then f (x) = integral subscript 0 superscript straight x space square root of straight t space sin space straight t space dt has

  • local maximum at π and 2π.

  • local minimum at π and 2π

  • local minimum at π and the local maximum at 2π.

  • local maximum at π and local minimum at 2π.

Solution

D.

local maximum at π and local minimum at 2π.

straight f left parenthesis straight x right parenthesis space equals space integral subscript 0 superscript straight x square root of straight t space sin space straight t space dt
straight f apostrophe left parenthesis straight x right parenthesis space equals space square root of straight x space sin space straight x

local maximum at π
and local minimum at 2π

Question
CBSEENMA12036084

The area of the region enclosed by the curves y = x, x = e, y =1/x and the positive x-axis is

  • 1/2 square units

  • 1 square units

  • 3/2 square units

  • 5/2 square units

Solution

C.

3/2 square units

Required area = OAB + ACDB

1 half space straight x space 1 space straight x space 1 space plus space integral subscript 1 superscript straight e 1 over straight x space dx
space equals space 1 half space plus space left parenthesis l n straight x right parenthesis subscript 1 superscript straight e
space equals space 3 over 2 space square space unit

Question
CBSEENMA12036092

If space straight a with rightwards arrow on top space equals space fraction numerator 1 over denominator square root of 10 end fraction space left parenthesis 3 straight i with hat on top space plus straight k with hat on top right parenthesis space and space straight b with rightwards arrow on top space equals space 1 over 7 left parenthesis 2 straight i with hat on top space plus space 3 straight j with hat on top space minus 6 straight k with hat on top right parenthesis comma space then space the space value space of space left parenthesis 2 straight a with rightwards arrow on top minus straight b with rightwards arrow on top right parenthesis. left square bracket left parenthesis straight a with rightwards arrow on top space straight x space straight b with rightwards arrow on top right parenthesis space straight x left parenthesis straight a with rightwards arrow on top plus 2 straight b with rightwards arrow on top right parenthesis right square bracket space is
  • -5

  • -3

  • 5

  • 3

Solution

A.

-5

left parenthesis 2 straight a with rightwards arrow on top minus straight b with rightwards arrow on top right parenthesis. left square bracket left parenthesis straight a with rightwards arrow on top space straight x straight b with rightwards arrow on top right parenthesis space straight x space left parenthesis straight a with rightwards arrow on top space plus 2 straight b with rightwards arrow on top right parenthesis right square bracket
equals negative space left parenthesis 2 straight a with rightwards arrow on top minus straight b with rightwards arrow on top right parenthesis. left square bracket space left parenthesis straight a with rightwards arrow on top space plus 2 straight b with rightwards arrow on top right parenthesis straight x space left parenthesis straight a with rightwards arrow on top space straight x straight b with rightwards arrow on top right parenthesis space right square bracket
equals space minus space left parenthesis 2 straight a with rightwards arrow on top minus straight b with rightwards arrow on top right parenthesis. left square bracket left parenthesis straight a with rightwards arrow on top plus 2 straight b with rightwards arrow on top right parenthesis. straight b with rightwards arrow on top right parenthesis straight a with rightwards arrow on top minus left parenthesis straight a with rightwards arrow on top plus 2 straight b with rightwards arrow on top right parenthesis. straight a with rightwards arrow on top right parenthesis straight b with rightwards arrow on top right square bracket
space equals space minus space left parenthesis 2 straight a with rightwards arrow on top minus straight b with rightwards arrow on top right parenthesis. left square bracket left parenthesis straight a with rightwards arrow on top. straight b with rightwards arrow on top right parenthesis plus 2. straight b with rightwards arrow on top. straight b with rightwards arrow on top right parenthesis straight a with rightwards arrow on top minus left parenthesis straight a with rightwards arrow on top. straight a with rightwards arrow on top plus 2 straight b with rightwards arrow on top. straight a with rightwards arrow on top right parenthesis straight b with rightwards arrow on top right parenthesis right square bracket
equals space minus space left parenthesis 2 straight a with rightwards arrow on top minus straight b with rightwards arrow on top right parenthesis. left square bracket 0 space plus 2 straight a with rightwards arrow on top minus left parenthesis 0 space plus straight b with rightwards arrow on top right parenthesis right square bracket
equals negative left parenthesis 2 straight a with rightwards arrow on top space minus straight b with rightwards arrow on top right parenthesis. space left parenthesis 2 straight a with rightwards arrow on top minus straight b with rightwards arrow on top right parenthesis
equals negative space left parenthesis 2 straight a with rightwards arrow on top minus straight b with rightwards arrow on top right parenthesis squared space equals space minus 4 straight a with rightwards arrow on top squared space plus 4 straight a with rightwards arrow on top. straight b with rightwards arrow on top minus straight b with rightwards arrow on top squared
space equals space minus space 4 plus 0 minus 1 space equals space minus space 5

Question
CBSEENMA12036098

The vector straight a with rightwards arrow on top space and space straight b with rightwards arrow on top are not perpendicular  and straight c with rightwards arrow on top space and space straight d with rightwards arrow on top are two vectors satisfying: straight b with rightwards arrow on top space straight x straight c with rightwards arrow on top space equals space straight b with rightwards arrow on top space straight x straight d with rightwards arrow on top space and space straight a with rightwards arrow on top. straight b with rightwards arrow on top space equals space 0. The vector straight d with rightwards arrow on top is equal to

  • straight b with rightwards arrow on top space minus space open parentheses fraction numerator straight b. straight c over denominator straight a with rightwards arrow on top. straight d with rightwards arrow on top end fraction close parentheses straight c with rightwards arrow on top
  • straight c with rightwards arrow on top space plus open parentheses fraction numerator straight a with rightwards arrow on top. straight c with rightwards arrow on top over denominator straight a with rightwards arrow on top. straight b with rightwards arrow on top end fraction close parentheses straight b with rightwards harpoon with barb upwards on top
  • straight b with rightwards arrow on top space plus space open parentheses fraction numerator straight b with rightwards arrow on top. straight c with rightwards arrow on top over denominator straight a with rightwards arrow on top. straight b with rightwards arrow on top end fraction close parentheses straight c with rightwards arrow on top
  • straight c with rightwards arrow on top minus open parentheses fraction numerator straight a with rightwards arrow on top. straight c with rightwards arrow on top over denominator straight a with rightwards arrow on top. straight b with rightwards arrow on top end fraction close parentheses straight b with rightwards arrow on top

Solution

D.

straight c with rightwards arrow on top minus open parentheses fraction numerator straight a with rightwards arrow on top. straight c with rightwards arrow on top over denominator straight a with rightwards arrow on top. straight b with rightwards arrow on top end fraction close parentheses straight b with rightwards arrow on top straight a with rightwards arrow on top. straight b with rightwards arrow on top space not equal to space 0 comma space straight b with rightwards arrow on top space straight x straight d with rightwards arrow on top space comma straight a with rightwards arrow on top. straight d with rightwards arrow on top space equals space 0
left parenthesis straight b with rightwards arrow on top space straight x space straight c with rightwards arrow on top right parenthesis space space straight x space straight a with rightwards arrow on top space equals space left parenthesis straight b with rightwards arrow on top space straight x space straight d with rightwards arrow on top space right parenthesis space space straight x space straight a with rightwards arrow on top
left parenthesis straight b with rightwards arrow on top. straight a with rightwards arrow on top right parenthesis straight c with rightwards arrow on top space minus space left parenthesis straight c with rightwards arrow on top. straight a with rightwards arrow on top right parenthesis straight b with rightwards arrow on top space equals space straight b with rightwards arrow on top. straight a with rightwards arrow on top right parenthesis straight b with rightwards arrow on top space minus left parenthesis straight d with rightwards arrow on top. straight a with rightwards arrow on top right parenthesis straight b with rightwards arrow on top
straight d with rightwards arrow on top space equals straight c with rightwards arrow on top space minus space open parentheses fraction numerator straight a with rightwards arrow on top. straight c with rightwards arrow on top over denominator straight a with rightwards arrow on top. straight b with rightwards arrow on top end fraction close parentheses straight b with rightwards arrow on top