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Application of Integrals — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSApplication of Integrals4 Marks
Question
Graphs of two function $\text{f}(\text{x})=\text{sin}\text{ x}$ and $\text{(g)}\text{x}=\text{cos}\text{ x}$ is given below:

Based on the above information, answer the following questions.
  1. In $(0, \pi)$, the curves $\text{f}(\text{x})=\text{sin}\text{ x}$ and $\text{g}\text{ (x)}=\text{cos}\text{ x}$ at $\text{x}=$ 
    1. $\frac{\pi}{2}$
    2. $\frac{\pi}{3}$
    3. $\frac{\pi}{4}$
    4. ${\pi}$
  2. Value of $\int\limits_{0}^{\frac{\pi}{4}}\text{sin}\text{ x}\text{ dx}$ is.
    1. $1-\frac{1}{\sqrt{2}}$
    2. $1+\frac{1}{\sqrt{2}}$
    3. $2-\frac{1}{\sqrt{2}}$
    4. $2+\frac{1}{\sqrt{2}}$
  1. Value of $\int\limits_\frac{\pi}{4}^{\frac{\pi}{2}}\text{cos}\text{ x}\text{ dx}$ is.
    1. $1+\frac{1}{\sqrt{2}}$
    2. $1-\frac{1}{\sqrt{2}}$
    3. $2-\sqrt{2}$
    4. $2+\sqrt{2}$
  2. Value of $\int\limits_{0}^{\pi}\text{sin}\text{ x}\text{ dx}$ is.
  1. 0
  2. 1
  3. 2
  4. -2
  1. Value of $\int\limits_{0}^\frac{\pi}{2}\text{sin}\text{ x}\text{ dx}$ is.
  1. 0
  2. 1
  3. 3
  4. 4
✓

Answer

  1. (c) $\frac{\pi}{4}$
Solution:
for point of intersection, we have
$\text{sin}\text{ x}=\text{cos}\text{ x}$
$\Rightarrow\frac{\text{sin}\text{ x}}{\text{cos}\text{ x}}=1$
$\Rightarrow\text{tan}\text{ x}=1$
$\Rightarrow\text{x}=\frac{\pi}{4}$
  1. (a) $1-\frac{1}{\sqrt{2}}$
Solution:
$\int\limits_{0}^{\frac{\pi}{4}}\text{sin}\text{ x}\text{ dx}=\big[-\text{cos x}\big]^\frac{\pi}{4}_0=-\text{cos}\frac{\pi}{4}+\text{cos}0$
$=1-\frac{1}{\sqrt{2}}$
  1. (b) $1-\frac{1}{\sqrt{2}}$
Solution:
$\int\limits_{\frac{\pi}{2}}^{\frac{\pi}{4}}\text{cos}\text{ x}\text{ dx}=\big[\text{sin x}\big]^\frac{\pi}{2}_\frac{\pi}{4}=\text{sin}\frac{\pi}{2}-\text{sin}\frac{\pi}{4}$
$=1-\frac{1}{\sqrt{2}}$
  1. (c) 2
Solution:
$\int\limits_{0}^{\pi}\text{sin}\text{ x}\text{ dx}=\big[-\text{cos x}\big]^{\pi}_0=\big[-\text{cos}{\pi}+\text{cos}0\big]=2$
  1. (b) 1
Solution:
$\int\limits_{0}^\frac{\pi}{2}\text{sin}\text{ x}\text{ dx}=\big[-\text{cos x}\big]^\frac{\pi}{2}_0=\Big[-\text{cos}\frac{\pi}{2}+\text{cos}0\Big]$
$=0+1+1$

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Shobhit's father wants to construct a rectangular garden using a brick wall on one side of the garden and wire fencing for the other three sides as shown in figure. He has $200\  ft$ of wire fencing.

Based on the above information, answer the following questions.
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  1. $x + 2y = 150$
  2. $x + 2y = 50$
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  2. $x - 2x^2$
  3. $200x - 2x^2$
  4. $200 - x^2$
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  4. $\frac{\cos\text{x}}{1+\sin\text{x}},\ \frac{\text{x}}{1+\sin\text{x}}$
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  2. $\cos\text{x}$
  3. $1+\sin\text{x}$
  4. $1-\cos\text{x}$
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  3. $\text{y}(1-\sin\text{x})=\frac{\text{-x}^2}{2}\text{+c}$
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  4. $\frac{2+\text{x}^2}{2(1-\sin\text{x})}$
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  2. x + y  + 3z = 2
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