If f 2a-x -f x and -0af x dx- then -02af x dx is

The correct option is A 2λ We know that: ∫_0^2af(x)dx=2∫_0^af(x)dx #160; when, f(2a x)=f(x) Hence, #160; ∫_0^2af(x)dx=2∫_0^af(x)dx ...

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Standard XII

Mathematics

De Morgan's Law

If f 2a-x =...

Question

If $f (2 a - x) = f (x)$ and $\int_{0}^{a} f (x) d x = λ$ , then $\int_{0}^{2 a} f (x) d x$ is

2 λ

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none of these

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Solution

The correct option is A $2 λ$
We know that:-

$\int_{0}^{2 a} f (x) d x = 2 \int_{0}^{a} f (x) d x$ when, $f (2 a - x) = f (x)$

Hence, $\int_{0}^{2 a} f (x) d x = 2 \int_{0}^{a} f (x) d x$

$= 2 λ$

Hence, answer is option-(A).

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Similar questions

Q. Assertion :The value of

\int_{0}^{π} x f (sin x) d x

\frac{π}{2} \int_{0}^{π} f (sin x) d x

π \int_{0}^{π / 2} f (sin x) d x

Reason:

$\int_{0}^{a} f (x) d x = \int_{0}^{a} f (a - x) d x$ and $\int_{0}^{2 a} f (x) d x = 2 \int_{0}^{a} f (x) d x$ If $f (2 a - x) = f (x)$

Q. Let

\int_{0}^{a} f (x) d x = λ

and

\int_{0}^{a} f (2 a - x) d x = μ .

Then

\int_{0}^{2 a} f (x) d x

is equal to?

Q. If A =

[\begin{matrix} 2 & λ & - 3 \\ 0 & 2 & 5 \\ 1 & 1 & 3 \end{matrix}]

, then A^-1 exists if
(a) λ=2 (b) λ≠2 (c) λ≠-2 (d) none of these

Q. Observe the following Lists

List-I	List-II
$A) \int_{0}^{a} f (x) d x$	1) $\int_{0}^{a} f (a + x) d x$
B) $\int_{- a}^{a} f (x) d x$	2) $\int_{0}^{a} f (a - x) d x$
C) $\int_{0}^{2 a} f (x) d x$	3) 0, if $f (x)$ is odd
D) $\int_{0}^{n a} f (x) d x$	4) $0$ , if $f (2 a - x)$ $= - f (x)$
	5) $(n - 1) \int_{0}^{a} f (x) d x$ , if period of f(x) is $a$ .
	6) $\int_{0}^{a} f (x) d x$ , if period of $f (x)$ is $a$ .

$(a) P r o v e t h a t \int_{0}^{2 a} f (x) d x = 2 \int_{0}^{a} f (X) d x, i f f (2 a - x) = f (x) a n d e v a l u a t e \int_{0}^{2 π} c o s^{5} x d x$
$i f f (2 a - x) = - f (x)$

(b) Find the values of a and b such that the function defined by

$f (x) = ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} 5, & i f x \leq 2 a x + b & i f 2 < x < 10 i s a c o n t i n o u s f u n c t i o n 21, & i f x \geq 10 \end{matrix}$

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If f(2a−x)=f(x) and ∫a0f(x)dx=λ, then ∫2a0f(x)dx is

The correct option is A 2λWe know that:-∫2a0f(x)dx=2∫a0f(x)dx when, f(2a−x)=f(x)Hence, ∫2a0f(x)dx=2∫a0f(x)dx=2λHence, answer is option-(A).

If $f (2 a - x) = f (x)$ and $\int_{0}^{a} f (x) d x = λ$ , then $\int_{0}^{2 a} f (x) d x$ is

The correct option is A $2 λ$
We know that:-

$\int_{0}^{2 a} f (x) d x = 2 \int_{0}^{a} f (x) d x$ when, $f (2 a - x) = f (x)$

Hence, $\int_{0}^{2 a} f (x) d x = 2 \int_{0}^{a} f (x) d x$

$= 2 λ$

Hence, answer is option-(A).