Let J _ n, m = ∫ _ 0 ^ 1 / 2 x ^ n x ^ m - 1 dx, n > m and n, m ∈ N . Consider a matrix A = [ a _ ij ] _ 3 × 3 where a _ ij = \ cases J _ 6 + i, 3 - J _ i + 3, 3 & , i ≤ j 0 & , i > j cases ). Then |adj A ^ - 1 | is :

Correct answer: D. ( 105 ) ^ 2 × 2 ^ 38 ∵ A = [ cases a _ 11 & a _ 12 & a _ 13 0 & a _ 22 & a _ 23 0 & 0 & a _ 33 cases ] ⇒ |A| = a _ 11 a _ 22 a _ 33 ⇒ | A | = ( J _ 7, 3 - J _ 4, 3 ) · ( J _ 8, 3 - J _ 5, 3 ) · ( J _ 9, 3 - J _ 6, 3 ) = ∫ _ 0 ^ 1 / 2 ( x ^ 7 - x ^ 4 x ^ 3 - 1 )dx · ∫ _ 0 ^ 1 / 2 ( x ^ 8 - x ^ 5 x ^ 3 - 1 )dx · ∫ _ 0 ^ 1 / 2 ( x ^ 9 - x ^ 6 x ^ 3 - 1 )dx = ∫ _ 0 ^ 1 / 2 x ^ 4 dx · ∫ _ 0 ^ 1 / 2 x ^ 5 dx · ∫ _ 0 ^ 1 / 2 x ^ 6 dx = 1 ( 210 ) · ( 2 ) ^ 18 Now |adj A ^ - 1 | = | A ^ - 1 | ^ 2 = 1 | A | ^ 2 = (( 210 ) · ( 2 ^ 18 )) ^ 2 = ( 105 ) ^ 2 · ( 2 ) ^ 38

JEE Main 20211 SeptemberEvening Shift

Exam: JEE Main
Year: 2021
Date: 1 September
Shift: Evening Shift
Subject: Mathematics
Chapter: Definite Integration

JEE Main 2021 · 1 September · Evening Shift · Mathematics · Definite Integration

Let J _ n, m = ∫ _ 0 ^ 1 / 2 x ^ n x ^ m - 1 dx, n > m and n, m ∈ N...

Let

{J}_{n, m} = {∫}_{0}^{1 / 2}\dfrac{{x}^{n}}{{x}^{m} - 1}dx, \forall n > m and

n, m \in N .

Consider a matrix

A = {\left[{a}_{ij}\right]}_{3 \times 3}

where

{a}_{ij} = \left\{

\begin{cases}{J}_{6 + i, 3} - {J}_{i + 3, 3} & , i \le j \\ 0 & , i > j\end{cases}

\right). Then

\left|adj {A}^{- 1}\right|

is :

Options

A $( 15{)}^{2} \times {2}^{34}$
B $( 15{)}^{2} \times {2}^{42}$
C $( 105{)}^{2} \times {2}^{36}$
D $( 105{)}^{2} \times {2}^{38}$

Correct answer

D. $( 105{)}^{2} \times {2}^{38}$

Step-by-step solution

∵ A = \left[ $\begin{cases}{a}_{11} & {a}_{12} & {a}_{13} \\ 0 & {a}_{22} & {a}_{23} \\ 0 & 0 & {a}_{33}\end{cases}$ \right] ⇒ \left|A\right| = {a}_{11}{a}_{22}{a}_{33} $$ ⇒ | A | = \left({J}_{7, 3} - {J}_{4, 3}\right) \cdot \left({J}_{8, 3} - {J}_{5, 3}\right) \cdot \left({J}_{9, 3} - {J}_{6, 3}\right) $$ = {∫}_{0}^{1 / 2}\left(\dfrac{{x}^{7} - {x}^{4}}{{x}^{3} - 1}\right)dx \cdot {∫}_{0}^{1 / 2}\left(\dfrac{{x}^{8} - {x}^{5}}{{x}^{3} - 1}\right)dx \cdot {∫}_{0}^{1 / 2}\left(\dfrac{{x}^{9} - {x}^{6}}{{x}^{3} - 1}\right)dx $$ = {∫}_{0}^{1 / 2}{x}^{4}dx \cdot {∫}_{0}^{1 / 2}{x}^{5}dx \cdot {∫}_{0}^{1 / 2}{x}^{6}dx $$ = \dfrac{1}{( 210 ) \cdot ( 2{)}^{18}} $Now\left|adj {A}^{- 1}\right| = {\left|{A}^{- 1}\right|}^{2} = \dfrac{1}{| A{|}^{2}} = {\left(( 210 ) \cdot \left({2}^{18}\right)\right)}^{2} = ( 105{)}^{2} \cdot ( 2{)}^{38}$

Practice Definite Integration on Quantrex Academy →

More from Definite Integration

The value of the definite integral ∫_ 0 ^ 2 1 3^ (x+3)/(2) \, dx is JEE Advanced 2026 The value of the integral ∫_ 0 ^ 2 x(x^2+x+1) ( x+1 )( x^4+x^2+1 ) \, dx is equal to: JEE Main 2026 If ∫_ π/6 ^ π/4 ( (x-(π)/(3)) (x+(π)/(3))+1)dx = α _e( 3 -1) , then 9α^2 is equal to… JEE Main 2026 Let A = bmatrix 1 & 3 & -1 2 & 1 & α 0 & 1 & -1 bmatrix be a singular matrix. Let f(x) =… JEE Main 2026 Let f: ℝ arrow ℝ be such that f(xy) = f(x)f(y) , for all x, y ∈ ℝ and f(0) ≠ 0 . Let g… JEE Main 2026 The value of the integral ∫ _ -1 ^ 1 ((x^3 + |x| + 1)/(x^2 + 2|x| + 1)) dx is equal to : JEE Main 2026 The value of the integral ∫_ -π/4 ^ π/4 ( 32 ^4 x 1 + e^ x )dx is: JEE Main 2026 Let (2^ 1-a + 2^ 1+a ) , f(a) , (3^a + 3^ -a ) be in A.P. and α be the minimum value of… JEE Main 2026 Let f: [1, ∞) arrow R be a differentiable function defined as f(x) = ∫_1^x f(t)\,dt +… JEE Main 2026 The value of the integral ∫_0^∞ ( _e(x))/(x^2 + 4)\,dx is: JEE Main 2026 The value of the integral ∫_ π/6 ^ π/3 ((4 - ^2 x)/( ^4 x)) dx is: JEE Main 2026 The integral ∫_ 0 ^ 1 ^ -1 (1+x+x^2)dx is equal to: JEE Main 2026 Let f be a twice differentiable function such that f(x)=∫_ 0 ^ x (t-x)dt-∫_ 0 ^ x f(t)… JEE Main 2026 Let ∫_ -2 ^ 2 (| x| + [x x])\,dx = 2(3 - 2) + β , where [·] is the greatest integer… JEE Main 2026 The value of ∫_0^ 20π ( ^4 x + ^4 x) \, dx is equal to: JEE Main 2026 Let [·] denote the greatest integer function. Then the value of ∫_0^3 ( e^x + e^ -x [x]!… JEE Main 2026 Full Definite Integration list Definite Integration JEE Maths PYQs JEE Mathematics PYQs All JEE Main PYQs