Answer

Verified

402.6k+ views

**Hint:**First we have to draw the figure and understand the area $S$. Then the first two options can be checked simply using the values of $x,y$. Using B, C can be checked. Certain rearrangements are needed to check D. The concept used here is that the area under a curve is equal to the integral value. For applying this we have to find the limits as well.

**Formula used:**

Area enclosed by a curve $y = f(x)$ within the limits $x = a$ to $x = b$ is simply the integral of the curve with same limits.

$ \Rightarrow S = \int\limits_a^b {f(x)dx} $

**Complete step by step answer:**

Given that $S$ is the area enclosed by $y = {e^{ - {x^2}}},y = 0,x = 0,x = 1.$

$y = 0,x = 0$ represent the $x,y$ axis respectively.

Let the curve drawn represent $y = {e^{ - {x^2}}}$.

In the figure, the shaded portion represents the area $S$.

When $x = 1 \Rightarrow y = {e^{ - {x^2}}} = {e^{ - 1}} = \dfrac{1}{e}$.

Since $S$ contains a rectangle OSDC with vertices $(0,0),(1,0),(1,\dfrac{1}{e}),(0,\dfrac{1}{e})$, where S(1,0).

The area of that rectangle will be $A = length \times breadth $

$\Rightarrow A= 1 \times \dfrac{1}{e} = \dfrac{1}{e}$.

Therefore clearly $S \geqslant \dfrac{1}{e}$.

This implies option A is right.

Also, since $S$ is the area enclosed by the curve $y = {e^{ - {x^2}}},x = 0,y = 0,x = 1$

We can consider $S = \int\limits_0^1 {{e^{ - {x^2}}}} dx$.

$\Rightarrow x \in [0,1] \Rightarrow {x^2} \leqslant x$

Multiplying both sides by $ - 1$ and taking exponents we have,

$\Rightarrow - {x^2} \geqslant - x \Rightarrow {e^{ - {x^2}}} \geqslant {e^{ - x}}$

Therefore, $S = \int\limits_0^1 {{e^{ - {x^2}}}} dx \geqslant \int\limits_0^1 {{e^{ - x}}} dx$

Integrating we get, $S \geqslant [ - {e^{ - x}}]_0^1 = {e^0} - {e^{ - 1}} = 1 - \dfrac{1}{e}$

$ \Rightarrow S \geqslant 1 - \dfrac{1}{e}$

Therefore option B is also right.

To show that option C is incorrect,

$\Rightarrow e > \sqrt e \Rightarrow \dfrac{1}{e} < \dfrac{1}{{\sqrt e }}$

$ \Rightarrow - \dfrac{1}{e} < \dfrac{1}{{\sqrt e }} \Rightarrow 1 - \dfrac{1}{e} < 1 + \dfrac{1}{{\sqrt e }} < \dfrac{1}{4}(1 + \dfrac{1}{{\sqrt e }})$

This implies $1 - \dfrac{1}{e} < \dfrac{1}{4}(1 + \dfrac{1}{{\sqrt e }})$

From B, we have $S \geqslant 1 - \dfrac{1}{e}$

Using these two equations we can say C is incorrect.

Moving on, consider two rectangles in the figure OAPQ and QBRS where $S(1,0)$,

Clearly $S \leqslant $ Area of OAPQ$ + $ Area of QBRS

PQ is drawn parallel to Y axis at the point $x = \dfrac{1}{{\sqrt 2 }}$.

Then $y = {e^{ - {x^2}}} = {e^{ - \dfrac{1}{2}}} = \dfrac{1}{{\sqrt e }}$.

Therefore, B is the point $(\dfrac{1}{{\sqrt 2 }},\dfrac{1}{{\sqrt e }})$.

Area of OAPQ $ = lenth \times breadth = 1 \times \dfrac{1}{{\sqrt 2 }} = \dfrac{1}{{\sqrt 2 }}$

In QBRS, $QB = \dfrac{1}{{\sqrt e }},BR = 1 - \dfrac{1}{{\sqrt 2 }}$

Therefore, Area of QBRS $ = length \times breadth = \dfrac{1}{{\sqrt e }} \times 1 - \dfrac{1}{{\sqrt 2 }}$

$ \Rightarrow S \leqslant AreaOAPQ + AreaQBRS = \dfrac{1}{{\sqrt 2 }} + \dfrac{1}{{\sqrt e }}(1 - \dfrac{1}{{\sqrt 2 }})$

So, option D is correct.

**Therefore, Option (A), (B) and (D) are correct.**

**Note:**

It is important to identify the graph of the given function and mark the given region. If there is some portion under the X-axis, this area will be negative. So be careful in those cases to split the region if necessary and find the area separately.

Recently Updated Pages

The base of a right prism is a pentagon whose sides class 10 maths CBSE

A die is thrown Find the probability that the number class 10 maths CBSE

A mans age is six times the age of his son In six years class 10 maths CBSE

A started a business with Rs 21000 and is joined afterwards class 10 maths CBSE

Aasifbhai bought a refrigerator at Rs 10000 After some class 10 maths CBSE

Give a brief history of the mathematician Pythagoras class 10 maths CBSE

Trending doubts

Difference Between Plant Cell and Animal Cell

Give 10 examples for herbs , shrubs , climbers , creepers

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Name 10 Living and Non living things class 9 biology CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE

Select the word that is correctly spelled a Twelveth class 10 english CBSE

Fill the blanks with proper collective nouns 1 A of class 10 english CBSE