Download App

5. continuity and differentiation — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths5. continuity and differentiation1 Mark
MCQ
If $f(x) = \int_0^x {t(\sin \,\,x\, - \sin \,\,t)\,dt} $ then ?
  • ✓
    $f'''(x) + f'(x) = \cos \,x\, - 2x\,\,\sin \,x$
  • B
    $f'''(x) + f''(x) - f'(x) = \cos \,x\,$
  • C
    $f'''(x) - f''(x) = \cos \,x\,\, - \,2x\,\,\sin \,x$
  • D
    $f'''(x) + f''(x) = \,\sin \,x$
✓

Answer

Correct option: A.
$f'''(x) + f'(x) = \cos \,x\, - 2x\,\,\sin \,x$
a
$f\left( x \right) = \int_0^x {t\left( {\sin x - \sin t} \right)} .dt$

$ = \sin x\int_0^x {t.dt}  - \int_0^x {t\sin t.dt} $

$ = \frac{{{x^2}}}{2}\sin x + \left[ {t\cos t_0^x} \right] + \sin x$

$ \Rightarrow f\left( x \right) = \frac{{{x^2}}}{2}\sin x + x\cos x + \sin x$

$f'\left( x \right) = \frac{{{x^2}}}{2}\cos x + 2\,\cos x$

$f''\left( x \right) = x\cos x - \frac{{{x^2}}}{2}\sin x - 2\sin x$

$f'''\left( x \right) = \cos x - 2x\sin x - \frac{{{x^2}}}{2}\cos x - 2\cos x$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "M.C.Q (1 Marks)" from 5. continuity and differentiation→5. continuity and differentiation — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

The solution of the differential equation $x\,dy - y\,dx = (\sqrt {{x^2} + {y^2})} dx$is
→
$\int\limits^\sqrt{3}_1\frac{1}{1+\text{x}^2}\text{ dx}$ is equal to:
  1. $\frac{\pi}{12}$
  2. $\frac{\pi}{6}$
  3. $\frac{\pi}{4}$
  4. $\frac{\pi}{3}$
→
Let $f (x)$ be integrable over $(a, b) , b > a > 0$.

If $I_1 = \int\limits_{\frac{\pi }{6}}^{\frac{\pi }{3}} \, f (\tan\, \theta + \cot\, \theta )\cdot sec^2\, \theta\, d\, \theta$ &

$I_2 = \int\limits_{\frac{\pi }{6}}^{\frac{\pi }{3}} \, f (\tan\, \theta + \cot\, \theta )\cdot cosec^2\, \theta\, d \, \theta$ ,

then the ratio $\frac{{{I_1}}}{{{I_2}}}$ :

→
If $\vec{a}$ is a nonzero vector of magnitude 'a' and $\lambda$ a nonzero scalar, then $\lambda\ \vec{a}$ is unit vector if
  1. $\lambda=1$
  2. $\lambda=-1$
  3. $a=\big|\lambda\big|$
  4. $a=1/\big|\lambda\big|$
→
Degree and order of the differential equation $\frac{\text{d}^2\text{y}}{\text{dx}^2}=\Big(\frac{\text{dy}}{\text{dx}}\Big)^2$  are respectively:
  1. 1, 2
  2. 2, 1
  3. 2, 2
  4. 1, 1
→
Which of the following differentials equation has $\text{y}=\text{C}_{1}\text{e}^{\text{x}}+\text{C}_{2}\text{e}^{-\text{x}}$ as the general solution?
  1. $\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}+\text{y}=0$
  2. $\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}-\text{y}=0$
  3. $\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}+\text{1}=0$
  4. $\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}-\text{1}=0$ 
→
Let $P, Q, R$ and $S$ be the points on the plane with position vectors $-2 \hat{i}-\hat{j}, 4 \hat{i}, 3 \hat{i}+3 \hat{j}$ and $-3 \hat{i}+2 \hat{j}$ respectively. The quadrilateral $\mathrm{PQRS}$ must be a
→
Choose the correct answer from the given four options.
The area of the quadrilateral ABCD, where A(0, 4, 1), B(2, 3, -1), C(4, 5, 0) and D(2, 6, 2), is equal to:
  1. 9 sq. units.
  2. 18 sq. units.
  3. 27 sq. units.
  4. 81 sq. units.
→
A random variable has the following probability distribution:

X = xi 0 1 2 3 4 5 6 7
P(X = Xi) 0 2p 2p 3p p2 2p2 7p2 2p

  1. $\frac{1}{10}$

  2. $-1$

  3. $-\frac{1}{10}$

  4. $\frac{1}{5}$

→
Choose the correct answer in Exercise:
The value of $\int^{1}_{0}\tan^{-1}\bigg(\frac{2\text{x}-1}{1+\text{x}-\text{x}^{2}}\bigg)\text{dx}$ is
  1. 1
  2. 0
  3. -1
  4. $\frac{\pi}{4}$
→