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APPLICATION OF DERIVATIVES — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsAPPLICATION OF DERIVATIVES4 Marks
Question
Show that $\text{f(x)}=\tan^{-1}(\sin\text{x}+\cos\text{x})$ is an increasing function in $\Big(0,\frac{\pi}{4}\Big).$
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Answer

We have, $\text{f(x)}=\tan^{-1}(\sin\text{x}+\cos\text{x})$
$\therefore\ \text{f}'(\text{x})=\frac{1}{1+(\sin\text{x}+\cos\text{x})^2}\cdot(\cos\text{x}-\sin\text{x})$
$=\frac{1}{1+\sin^2\text{x}+\cos^2\text{x}+2\sin\text{x}.\cos\text{x}}(\cos\text{x}-\sin\text{x})$
$=\frac{1}{(2+\sin2\text{x})}(\cos\text{x}-\sin\text{x})$
$\big[\because\sin2\text{x}=2\sin\text{x}\cos\text{x and }\sin^2\text{x}+\cos^2\text{x}=1\big]$
For $\text{f}'(\text{x})\geq0.$
$\frac{1}{(2+\sin2\text{x})}\cdot(\cos\text{x}-\sin\text{x})\geq0$
$\Rightarrow\ \cos\text{x}-\sin\text{x}\geq0$ $\Big[\because(2+\sin2\text{x})\geq0\text{ in }\Big(0,\frac{\pi}{4}\Big)\Big]$
$\Rightarrow\ \cos\text{x}\geq\sin\text{x}$
Which is true, if $\text{x}\in\Big(0,\frac{\pi}{4}\Big)$
Hence, f(x) is an increasing function in $\Big(0,\frac{\pi}{4}\Big).$

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Based on the above information, answer the following questions.

  1. Probability that defective bolt drawn is manufactured by machine A, is:

  1. $\frac{4}{13}$

  2. $\frac{5}{13}$

  3. $\frac{6}{13}$

  4. $\frac{9}{13}$
  1. Probability that defective bolt drawn is manufactured by machine B, is:
  1. 0.3
  2. 0.1
  3. 0.2
  4. 0.4
  1. Probability that defective bolt drawn is manufactured by machine C, is:

  1. $\frac{16}{39}$

  2. $\frac{17}{39}$

  3. $\frac{20}{39}$

  4. $\frac{15}{39}$

  1. Probability that defective bolt is not manufactured by machine B, is:

  1. $\frac{35}{39}$

  2. $\frac{61}{39}$

  3. $\frac{41}{39}$

  4. None of these.

  1. Probability that defective bolt is not manufactured by machine C, is:
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  2. 0.09
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On the basis of above information, answer the following questions.
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Image

(i) Find the rate of growth of the plant with respect to sunlight.

(ii) What is the number of days it will take for the plant to grow to the maximum height?

(iii) Verify that height of the plant is maximum after four days by second derivative test and find the maximum height of plant.

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What will be the height of the plant after 2 days?

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Based on the above information, answer the following questions.

  1. If the line joining the positions of the helicopter and boat is perpendicular to the plane in which boat moves, then equation of plane is:
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  1. $\sqrt{5}\text{m}$

  2. $\sqrt{8}\text{m}$

  3. $\sqrt{10}\text{m}$

  4. $\sqrt{11}\text{m}$

  1. If the speed of bullet is 30m/ sec, then how much time will the bullet take to hit the boat after the shot is fired?
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  3. $\frac{1}{2}\text{second}$

  4. $\frac{\sqrt{11}}{30}\text{seconds}$

  1. At the given instant of time, the equation of line passing through the positions of helicopter and boat is:

  1. $\frac{\text{x}}{1}=\frac{\text{y}}{-1}=\frac{\text{z}}{3}$

  2. $\frac{\text{x}-1}{1}=\frac{\text{y}-4}{-1}=\frac{\text{z}-2}{3}$

  3. $\frac{\text{x}}{1}=\frac{\text{y}}{1}=\frac{\text{z}}{-3}$

  4. $\frac{\text{x}-1}{1}=\frac{\text{y}-4}{1}=\frac{\text{z}-2}{-3}$

  1. At a different instant of time, the boat moves to a different position along the planar surface. What should be the coordinates of the location of the boat for the bullet to hit the boat if soldier shoots the bullet along the line whose equation is $\frac{\text{x}-1}{1}=\frac{\text{y}-1}{-2}=\frac{\text{z}-2}{3}?$

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  2. $\Big(\frac{3}{4},\frac{3}{2},\frac{5}{4}\Big)$

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(i) Find the probability that Sophia processed the form and committed an error.
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  3. $-\vec{\text{p}}$

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  2. $2\overline{\text{AB}}$

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  4. $2\overline{\text{BD}}$

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  3. $2\vec{\text{b}}$

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  1. $\overline{\text{AC}}+\overline{\text{DB}}$

  2.  $\overline{\text{AC}}+\overline{\text{BD}}$

  3. $\overline{\text{BC}}+\overline{\text{AD}}$

  4. $\overline{\text{BD}}+\overline{\text{CA}}$

  1. If T is the mid point of side YZ of $\triangle\text{XYZ},$ then $\overline{\text{XY}}+\overline{\text{XZ}}=$

  1. $2\overline{\text{YT}}$

  2. $2\overline{\text{XT}}$

  3. $2\overline{\text{TZ}}$

  4. None of these
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