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Most Important Selected Qs for JEE AdvancedMathematicsVector Algebra

Let f: R (0,1) be a continuous function, then which of the following pair of vectors are linearly dependent for some x (0,1) ?

Options

  1. Aa =f(x) i +2 j ; b =x^2 i +3 j
  2. Ba =f(x) i +3 j ; b =x^2 i +2 j
  3. Ca = ( ₀^ 1-x f(t) d t ) i +3 j ; b =x i +2 j
  4. Da = ( ₀^ 1-x f(t) d t ) i +2 j ; b =x i +3 j

Correct answer

D. a = ( ₀^ 1-x f(t) d t ) i +2 j ; b =x i +3 j

Step-by-step solution

(1) For linearly dependent f(x) x^2 = 2 3 3 f(x)-2 x^2=0 Let g(x)=3 f(x)-2 x^2 g(0)=3 f(0)>0 and g(1)=3 f(1)-2 , may be positive or negative (2) f(x) x^2 = 3 2 2 f(x)-3 x^2=0 Let g(x)=2 f(x)-3 x^2g(0)=2 f(0)>0 and g(1)=2 f(1)-3 0 and g(1)=-3 < 0 a and b are linearly dependent. Similarly for (4) also vectors are linearly dependent.

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