Download App

Application of Derivatives — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsApplication of Derivatives1 Mark
MCQ
Assertion (A): If the function $f(x)=\frac{a e^x+b e^{-x}}{c e^x+d e^{-x}}$ is increasing function of $x$, then $b c>a d$.
Reason (R): A function $f(x)$ is increasing if $f^{\prime}(x)>0$ for all $x$.
  • A
    Both (A) and (R) are true and (R) is the correct explanation of (A).
  • B
    Both (A) and (R) are true but (R) is not the correct explanation of (A).
  • C
    (A) is true but (R) is false.
  • ✓
    (A) is false but (R) is true.
✓

Answer

Correct option: D.
(A) is false but (R) is true.
(d) : $f^{\prime}(x)=\frac{2(a d-b c)}{\left(c e^x+d e^{-x}\right)^2}$and $f(x)$ is an increasing function.
$
\begin{array}{l}
\therefore \quad f^{\prime}(x)>0 \Rightarrow \frac{2(a d-b c)}{\left(c e^x+d e^{-x}\right)^2}>0 \\
\Rightarrow \quad 2(a d-b c)>0 \Rightarrow a d>b c \Rightarrow b c < ad
\end{array}
$

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 "Assertion (A) & Reason (B) MCQ" from Application of Derivatives→Application of Derivatives — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Directions: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Consider the function f : R → R defined as
$\text{f}(\text{x})=\frac{\text{x}}{\text{x}^{2}+1}.$
Assertion: f(x) is not one - one.
Reason: f(x) is not onto.
  1. Both A and R are true and R is the correct explanation of A.
  2. Both A and R are true but R is not the correct explanation of A.
  3. A is true but R is false.
  4. A is false and R is true.
→
Directions: In these questions, a statement of Assertion is followed by a statement of Reason is given.Choose the correct answer out of the following choices:
Assertion: The magnitude of the resultant of vectors $\overline{\text{a}}=2\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}$ and $\hat{\text{b}}=\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}}$
Reason: The magnitude of a vector can never be negative.
  1. Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
  2. Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
  3. Assertion is correct statement but Reason is wrong statement.
  4. Assertion is wrong statement but Reason is correct statement.
→
Let $\vec{a}$ and $\vec{b}$ be two non-zero vectors and $\theta$ be the angle between then.
Assertion (A) : $(\vec{a} \times \vec{b})^2+(\vec{a} \cdot \vec{b})^2 \neq|\vec{a}|^2|\vec{b}|^2$
Reason (R) : $\sin ^2 \theta+\cos ^2 \theta=1$
→
Let $F(x)$ be an indefinite integral of $\sin ^2 x$.
Assertion (A) : The function $F(x)$ satisfies $F(x+\pi)=F(x)$ for all real $x$.
Reason (R) : $\sin ^2(x+\pi)=\sin ^2 x$ for all real $x$.
→
Directions: In these questions, a statement of Assertion is followed by a statement of Reason is given.Choose the correct answer out of the following choices:
Assertion: If $ (\vec{\text{a}}\times\vec{\text{b}})+(\vec{\text{a}}.\vec{\text{b}})=400$ and $|\vec{\text{a}}|=4,$ then $|\vec{\text{b}}|=9.$
Reason: If $\vec{\text{a}}$ and $\vec{\text{b}}$ are any two vectors, then $(\vec{\text{a}}\times\vec{\text{b}})^2$ is equal to $(\vec{\text{a}})^2(\vec{\text{b}})^2-(\vec{\text{a}}.\vec{\text{b}})^2.$
  1. Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion. 
  2. Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion. 
  3. Assertion is correct statement but Reason is wrong statement. 
  4. Assertion is wrong statement but Reason is correct statement.
→
Assertion (A) : Range of $f(x)=\sin ^{-1} x$ $+\tan ^{-1} x+\sec ^{-1} x$ is $\left\{\frac{\pi}{4}, \frac{3 \pi}{4}\right\}$.
Reason (R) : $f(x)=\sin ^{-1} x+\tan ^{-1} x+\sec ^{-1} x$ is defined for all $x \in[-1,1]$.
→
Assertion (A) : If the cartesian equation of a line is $\frac{x-5}{3}=\frac{y+4}{7}=\frac{z-6}{2}$, then its vector form is $\vec{r}=5 \hat{i}-4 \hat{j}+6 \hat{k}+\lambda(3 \hat{i}+7 \hat{j}+2 \hat{k})$.
Reason (R): The vector equation of line passing through the points $A(\vec{a})$ and parallel to vector $\vec{b}$ is $\vec{r}=\vec{a}-\lambda(\vec{a}-\vec{b})$, where $\lambda \in R$ is a parameter.
→
Directions: In these questions, a statement of Assertion is followed by a statement of Reason is given.Choose the correct answer out of the following choices:
Assertion: If $\frac{\text{dy}}{\text{dy}}+\text{xy}=\text{x}^3\text{y}^3,\text{x}>0,\text{y}\geq0$ and $\text{y}(0)=1,$ then $\text{y}(1)=\frac{1}{\sqrt{2}}$
Reason: The differential equation is linear with integrating factor ex
  1. Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
  2. Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
  3. Assertion is correct statement but Reason is wrong statement.
  4. Assertion is wrong statement but Reason is correct statement.
→
Directions: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion: Value of x for which the matrix $\begin{bmatrix}1&2&0\\0&1&2\\-1&2&\text{x}\end{bmatrix}$ is singular is 5.
Reason: A square matrix is singular if $\mid\text{A}\mid=0.$
  1. Both A and R are true and R is the correct explanation of A.
  2. Both A and R are true but R is not the correct explanation of A.
  3. A is true but R is false.
  4. A is false but R is true.
  5. Both A and R are false.
→
Directions: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion: $\tan^{-1}\big(\frac{3}{4}\big)+\tan^{-1}\big(\frac{1}{7}\big)=\frac{\pi}{4}.$
Reason: For x > 0, y > 0, xy < 1, $\tan^{-1}\text{x}+\tan^{-1}\text{y}=\tan^{-1}\Big(\frac{\text{x}+\text{y}}{1-\text{xy}}\Big).$
  1. Both A and R are true and R is the correct explanation of A.
  2. Both A and R are true but R is not the correct explanation of A.
  3. A is true but R is false.
  4. A is false but R is true.
  5. Both A and R are false.
→