(A) Given $f(x) = \begin{vmatrix} x & x^2 & x^3 \\ 1 & 2x & 3x^2 \\ 0 & 2 & 6x \end{vmatrix}$.Expanding along the first column:$f(x) = x(12x^2 - 6x^2)

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

(A) Given $f(x) = \begin{vmatrix} x & x^2 & x^3 \\ 1 & 2x & 3x^2 \\ 0 & 2 & 6x \end{vmatrix}$.Expanding along the first column:$f(x) = x(12x^2 - 6x^2) - 1(6x^3 - 2x^3) + 0$$f(x) = x(6x^2) - 1(4x^3) = 6x^3 - 4x^3 = 2x^3$.Now,find the derivatives:$f'(x) = \frac{d}{dx}(2x^3) = 6x^2$.$f''(x) = \frac{d}{dx}(6x^2) = 12x$.Therefore,the ratio is $\frac{f''(x)}{f'(x)} = \frac{12x}{6x^2} = \frac{2}{x}$ or $2 : x$.

Class 12 Mathematics 3-4. Matrices and Determinants Rank of Matrices , Some special determinants, differentiation and integration of determinants

Difficult

If $f(x) = \begin{vmatrix} x & x^2 & x^3 \\ 1 & 2x & 3x^2 \\ 0 & 2 & 6x \end{vmatrix}$,then the ratio $f^{\prime \prime}(x) : f^{\prime}(x) =$

AP EAMCET 2021 All AP EAMCET

A
$2 : x$
B
$x^2 : x$
C
$3x : 2$
D
$6 : x$

Explore More

Browse All

Question Bank

All Subjects

Class 12 Questions

Class 12

Mathematics Questions

Chapter

3-4. Matrices and Determinants

Subtopic

Rank of Matrices , Some special determinants, differentiation and integration of determinants

Previous Year

AP EAMCET 2021 Paper All AP EAMCET years →

The rank of the matrix $\begin{bmatrix} -1 & 2 & 5 \\ 2 & -4 & a - 4 \\ 1 & -2 & a + 1 \end{bmatrix}$ is

Medium

View Solution

If $\left|\begin{array}{ccc}x^2+3x & x+1 & x-3 \\ x-1 & 2-x & x+4 \\ x-3 & x-3 & 3x\end{array}\right|=a_0+a_1x+a_2x^2+a_3x^3+a_4x^4$, then find the value of $(a_1+a_3)+2(a_0+a_2+a_4)$.

MediumTS EAMCET 2019

View Solution

If $f(x) = \left| \begin{array}{ccc} 2 \cos x & 1 & 0 \\ x - \frac{\pi}{2} & 2 \cos x & 1 \\ 0 & 1 & 2 \cos x \end{array} \right|$, then $f^{\prime}(\pi)$ is equal to

MediumTS EAMCET 2010

View Solution

The rank of the matrix $A = \begin{bmatrix} 2 & 1 & 2 \\ 1 & 0 & 1 \\ 4 & 1 & 4 \end{bmatrix}$ is

EasyAP EAMCET 2022

View Solution

Which of the following matrices has rank $3$?

EasyAP EAMCET 2021

View Solution

Vedclass Products

For Students

Vedclass Test Series

Mock tests in real JEE/NEET style with performance analysis. 5-day free trial.

Start Free Trial

For Teachers

Exam Paper Generator

Generate Set A/B/C/D exam papers from 7.5L+ questions in 2 minutes. 3 chapters free.

Try Free

For Institutes

Online Exam Module

Live online exams with unlimited students, 360° analytics & white-label branding.

See Demo

If $f(x) = \begin{vmatrix} x & x^2 & x^3 \\ 1 & 2x & 3x^2 \\ 0 & 2 & 6x \end{vmatrix}$,then the ratio $f^{\prime \prime}(x) : f^{\prime}(x) =$

Similar Questions

The rank of the matrix $\begin{bmatrix} -1 & 2 & 5 \\ 2 & -4 & a - 4 \\ 1 & -2 & a + 1 \end{bmatrix}$ is

If $\left|\begin{array}{ccc}x^2+3x & x+1 & x-3 \\ x-1 & 2-x & x+4 \\ x-3 & x-3 & 3x\end{array}\right|=a_0+a_1x+a_2x^2+a_3x^3+a_4x^4$, then find the value of $(a_1+a_3)+2(a_0+a_2+a_4)$.

If $f(x) = \left| \begin{array}{ccc} 2 \cos x & 1 & 0 \\ x - \frac{\pi}{2} & 2 \cos x & 1 \\ 0 & 1 & 2 \cos x \end{array} \right|$, then $f^{\prime}(\pi)$ is equal to

The rank of the matrix $A = \begin{bmatrix} 2 & 1 & 2 \\ 1 & 0 & 1 \\ 4 & 1 & 4 \end{bmatrix}$ is

Which of the following matrices has rank $3$?

Vedclass Test Series

Exam Paper Generator

Online Exam Module