Download App

STD 12 - 7.2 definite integral — Mathematics STD 12 Science — Question

CBSE BoardEnglish MediumSTD 12 ScienceMathematicsSTD 12 - 7.2 definite integral1 Mark
MCQ
Let $f$ be a differentiable function in $\left(0, \frac{\pi}{2}\right)$ If $\int\limits_{\cos x}^{1} t^{2} f(t) d t=\sin ^{3} x+\cos x-1$ then $\frac{1}{\sqrt{3}} f^{\prime}\left(\frac{1}{\sqrt{3}}\right)$ is equal to
  • A
    $6-9 \sqrt{2}$
  • B
    $\frac{9}{\sqrt{2}}-6$
  • C
    $\frac{9}{2}-6 \sqrt{2}$
  • ✓
    $6-\frac{9}{\sqrt{2}}$
✓

Answer

Correct option: D.
$6-\frac{9}{\sqrt{2}}$
d
$\int\limits_{\cos x}^{1} t^{2} f(t) d t=\sin ^{3} x+\cos x-1$

Calculation for option

differentiating both sides

$-\cos ^{2} x f(\cos x) \cdot(-\sin x)=3 \sin ^{2} x \cdot \cos x-\sin x$

$\Rightarrow f (\cos x )=3 \tan x -\sec ^{2} x$

$\Rightarrow f^{\prime}(\cos x)(-\sin x)=3 \sec ^{2} x-2 \sec ^{2} x \tan x$

$\Rightarrow f^{\prime}(\cos x) \cos x=\frac{2}{\cos ^{2} x}-\frac{3}{\sin x \cdot \cos x}$

When $\cos x=\frac{1}{\sqrt{3}} ; \sin x=\frac{\sqrt{2}}{\sqrt{3}}$

$\therefore f^{\prime}\left(\frac{1}{\sqrt{3}}\right) \frac{1}{\sqrt{3}}=6-\frac{9}{\sqrt{2}}$

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 STD 12 - 7.2 definite integral→STD 12 - 7.2 definite integral — all question types→Mathematics STD 12 Science question papers→CBSE Board STD 12 Science question papers→CBSE Board question paper generator→

Similar questions

The moment of a force represented by $\overrightarrow F = i + 2j + 3k$ about the point $2\,i - j + k = $
→
If $y = a{e^{mx}} + b{e^{ - mx}}$, then ${{{d^2}y} \over {d{x^2}}} - {m^2}y = $
→
If $A$ is a singular matrix, then $\text{adj A}$ is:
→
The function $\text{f(x)}=\begin{cases}\frac{\text{x}^2}{\text{a}},&0\leq\text{x}<1\\\text{a},&1\leq\text{x}<\sqrt{2}\\\frac{2\text{b}^2-4\text{b}}{\text{x}^2},&\sqrt{2}\leq\text{x}<\infty\end{cases}$ is continuous for $0\leq\text{x}<\infty,$ then the most suitable values of $a$ and $b$ are:
→
Consider the system of equations : $a_1 x+b_1 y+c_1 z=0 , a_2 x+b_2 y+c_2 z=0, a_3 x+b_3 y+c_3 z=0,$ if $\begin{vmatrix}\text{a}_1&\text{b}_1&\text{c}_1\\\text{a}_2&\text{b}_2&\text{c}_2\\\text{a}_3&\text{b}_3&\text{c}_3\end{vmatrix}=0,$ then the system has
→
The probability that a randomly chosen one-one function from the set $\{a, b, c, d\}$ to the set $\{1,2,3,4,5\}$ satisfies $f(a)+2 f(b)-f(c)=f(d)$ is
→
Let $A=\left(a_{i j}\right)_{1 \leq i, j \leq 3}$ be a $3 \times 3$ invertible matrix where each $a_{i j}$ is a real number. Denote the inverse of the matrix $A$ by $A^{-1}$. If $\Sigma_{j=1}^3 a_{i j}=1$ for $1 \leq i \leq 3$, then
→
If $\int_{}^{} {x\sin xdx = - x\cos x + A} $, then $A = $
→
The solution of the differential equation $\frac{\text{dy}}{\text{dx}}=1+\text{x}+\text{y}^{2}+\text{xy}^{2}, \text{y}=(0)$ is:
→
If $\left[ {\begin{array}{*{20}{c}}x&0\\1&y\end{array}} \right] + \left[ {\begin{array}{*{20}{c}}{ - 2}&1\\3&4\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}3&5\\6&3\end{array}} \right] - \left[ {\begin{array}{*{20}{c}}2&4\\2&1\end{array}} \right]$, then
→