Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
NDA2018MathematicsDefinite IntegrationActual

If _a^b x^3 d x=0 and _a^b x^2 d x= 2 3 , then what are the values of a and b respectively?

Options

  1. A–1, 1
  2. B1, 1
  3. C0, 0
  4. D2, –2

Correct answer

A. –1, 1

Step-by-step solution

aligned & _a^b x^3 d x=0 & ( x ^4 4 )_ a ^ b =0 & ~b ^4- a ^4=0 & ( ~b ^2- a ^2 ) ( b ^2+ a ^2 )=0 & ~b ^2- a ^2=0 (or) b ^2+ a ^2=0 & ( ~b + a )( b - a )=0( or ) b ^2+ a ^2=0 aligned b =- a ( or ) a = b ( or ) b ^2+ a ^2=0 or aligned & _a^b x^2 d x= 2 3 & ( x^3 3 )_a^b= 2 3 & b^3-a^3=2 aligned Since, b=-a . aligned & b ^3- (- b ^3 )=2 & 2 ~b ^3=2 & ~b ^3=1 & ~b =1, a =-1 aligned

Practice Definite Integration on Quantrex Academy →

More from Definite Integration

Passage: Let f(x)= x and g(x)-f(x)=f(4-x) . Question: What is ₀^4 f(x) g(x) ,dx equal to ? 2026Passage: Let ₀^ 2 a x+b x (a+b)( x+ x) ,dx=k . Question: What is the value of k ? 2026Passage: Let f(x)= x and g(x)-f(x)=f(4-x) . Question: What is ₀^4 f(4-x) g(4-x) ,dx equal to ? 2026Passage: Let ₀^ 2 a x+b x (a+b)( x+ x) ,dx=k . Question: What is ₀^ 2 a x+b x x+ x ,dx equal to ? 2026Consider the following for the two (02) items that follow: Let f(x) = [x^2] , where [ ] is the greatest integer function. What is _ 2 ² f(x) , dx equal to? 2025What is _n^ n+1 (x - [x]) , dx , where [ ] is the greatest integer function and n is a natural number? 2025If I₁ = _e^ e^2 dx x and I₂ = ₁^2 e^x x , dx , then which one of the following is correct? 2025Paragraph: Let I = ₀^ 2 f(x) g(x) d x , where f(x)= x and g(x)= x+ x+1 . Question: What is ₀^ 2 d x g(x) equal to? 2024 Full Definite Integration list All NDA PYQs