MCQ 11 Mark
Directions: In the following questions, the Assertions $(A)$ and Reason$(s)\ (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion $(A)$ If $\text{y}=\log_7(\text{x}^2+7\text{x}+4),$ then $\frac{\text{dy}}{\text{dx}}=\frac{(2\text{x}+7)}{(\text{x}^2+7\text{x}+4),}$
Reason $(R) \log_\text{b}=\frac{\log_\text{e}}{\log_\text{e}\text{b}}$
- 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.
AnswerCorrect option: D. $A$ is false but $R$ is true.
View full question & answer→MCQ 21 Mark
Directions: In the following questions, the Assertions (A) and Reason(s) (R) have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion (A) If $x^2 + 2xy + y^3 = 42$, Then $\frac{\text{dy}}{\text{dx}}=\frac{2(\text{x+y})}{(2\text{x+3}\text{y}^2)}$
Reason(R) $\frac{\text{dy}^\text{n}}{\text{dx}}=\text{ny}^{(\text{n-1})}$
- 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
- ✓
View full question & answer→MCQ 31 Mark
Directions: In the following questions, the Assertions $(A)$ and Reason$(s)\ (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion $(A)$ $\frac{\text{dx}^{\sin\text{x}}}{\text{dx}}=\text{x}^{\sin\text{x}}[(\cos)\log\text{x}+\frac{\sin\text{x}}{\text{x}}]$
Reason $(R)$ if $y = x^{f(x)}$ then $\frac{\text{dy}}{\text{dx}}=\text{x}^\text{f(x)}[\text{f '(x)}\log\text{x}+\frac{\text{f(x)}}{\text{x}}]$
- ✓
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
- D
$A$ is false but $R$ is true
AnswerCorrect option: A. Both $A$ and $R$ are true and $R$ is the correct explanation of $A$
View full question & answer→MCQ 41 Mark
Directions : In the following questions, the Assertions $(A)$ and Reason $(s)\ (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following :
Assertion $(A) :$ Acontinuous funection is always differentiable.
Reason $(R) :$ Adifferentiable function is always continuous.
- A
$A$ is true, $R$ is true: $R$ is a correct explanation for $A.$
- B
$A$ is true $R$ is true; $R$ is not a correct explanation for $A.$
- C
$A$ is true: $R$ is false.
- ✓
$A$ is false: $R$ is true.
AnswerCorrect option: D. $A$ is false: $R$ is true.
View full question & answer→MCQ 51 Mark
Directions : In the following questions, the Assertions $(A)$ and Reason $(s) \ (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following :
Assertion $(A) : $ if $\text{y}=\sin^{-1}\frac{2\text{x}}{1+\text{x}^2}$ then $\frac{\text{dy}}{\text{dx}}=\frac{2}{1+\text{x}^2}$
Reason $(R) : \sin2\theta=\frac{2\tan\theta}{1+\tan^2\theta}$
- ✓
$A$ is true, $R$ is true: $R$ is a correct explanation for $A.$
- B
$A$ is true $R$ is true; $R$ is not a correct explanation for $A.$
- C
$A$ is true: $R$ is false.
- D
$A$ is false: $R$ is true.
AnswerCorrect option: A. $A$ is true, $R$ is true: $R$ is a correct explanation for $A.$
View full question & answer→MCQ 61 Mark
Directions : In the following questions, the Assertions $(A)$ and Reason $(s)\ (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following :
Assertion $(A) : f(x) = x - 1 + x - 2$ is continuous but not differentiable at $x = 1, 2.$
Reason $(R) : $ Every differentiable function is continuous
- A
$A$ is true, $R$ is true: $R$ is a correct explanation for $A.$
- ✓
$A$ is true $R$ is true; $R$ is not a correct explanation for $A.$
- C
$A$ is true: $R$ is false.
- D
$A$ is false: $R$ is true.
AnswerCorrect option: B. $A$ is true $R$ is true; $R$ is not a correct explanation for $A.$
View full question & answer→MCQ 71 Mark
Directions : In the following questions, the Assertions $(A)$ and Reason $(s)\ (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion : $\text{u}=\text{f}(\cot\text{x})\ \text{f}(1)=\sqrt2$ and $\text{g}(\sqrt{2})=2$ then $\Big(\frac{\text{du}}{\text{dv}}\Big)_{\text{x}=\frac{\text{x}}{4}}=1.$
Reason : If $u = f(x), v = g(x)$ then derivative of $\text{f w.r.t}.$ to $g$ is $\frac{\text{du}}{\text{dv}}=\frac{\frac{\text{du}}{\text{dx}}}{\frac{\text{dv}}{\text{dx}}}.$
- ✓
Both Assertion and Reason are correct and Reason is the correct explanation for Assertion.
- B
Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion.
- C
Assertion is correct but Reason is incorrect.
- D
Both Assertion and Reason are incorrect.
AnswerCorrect option: A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion.
View full question & answer→MCQ 81 Mark
Directions : In the following questions, the Assertions $(A)$ and Reason $(s)\ (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following :
Assertion $(A) $ The value of the constant $'k\ ’$ so that $\text{f(x)}=\begin{cases}\text{kx}^2,\text{if x}\leq2\\3,\text{if x}>2\end{cases}$ is continuous at $x = 2$ is $\text{k}=\frac{4}{3}$
Reason $(R)$ A function $f(x)$ is continuous at a point $x= a$ of its domain if $\lim\limits_{\text{x}\rightarrow 0}\text{f(x)}=\text{f(x)}$
- A
$A$ is true, $R$ is true: $R$ is a correct explanation for $A.$
- B
$A$ is true $R$ is true; $R$ is not a correct explanation for $A.$
- C
$A$ is true: $R$ is false.
- ✓
$A$ is false: $R$ is true.
AnswerCorrect option: D. $A$ is false: $R$ is true.
View full question & answer→MCQ 91 Mark
Directions : In the following questions, the Assertions $(A)$ and Reason $(s)\ (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion $(A): \text{f(x)}=\sin\text{x}$ is continuous $x = 0.$
Reason $(R) : \sin\text{x}$ is differentiable at $x = 0.$
- A
$A$ is true, $R$ is true: $R$ is a correct explanation for $A.$
- B
$A$ is true $R$ is true; $R$ is not a correct explanation for $A.$
- ✓
$A$ is true: $R$ is false.
- D
$A$ is false: $R$ is true.
AnswerCorrect option: C. $A$ is true: $R$ is false.
View full question & answer→MCQ 101 Mark
Directions : In the following questions, the Assertions $(A)$ and Reason $(s)\ (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following :
Assertion $(A) : \text{f(x)}=\sin\text{x}$ is continuous for all $\text{x }\in\text{ R}$
Reason $ (R) : \sin\text{x}$ and $\text{x}$ are continuous at on $R.$
- ✓
$A$ is true, $R$ is true: $R$ is a correct explanation for $A.$
- B
$A$ is true $R$ is true; $R$ is not a correct explanation for $A.$
- C
$A$ is true: $R$ is false.
- D
$A$ is false: $R$ is true.
AnswerCorrect option: A. $A$ is true, $R$ is true: $R$ is a correct explanation for $A.$
View full question & answer→MCQ 111 Mark
Directions : In the following questions, the Assertions $(A)$ and Reason $(s)\ (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following :
Assertion $(A) : f(x) = [x]$ greatest integer function is not differentiable at $x = 2$
Reason $(R) :$ The greatest integer function is not continuous at any integer
- ✓
$A$ is true, $R$ is true: $R$ is a correct explanation for $A.$
- B
$A$ is true $R$ is true; $R$ is not a correct explanation for $A.$
- C
$A$ is true: $R$ is false.
- D
$A$ is false: $R$ is true.
AnswerCorrect option: A. $A$ is true, $R$ is true: $R$ is a correct explanation for $A.$
View full question & answer→MCQ 121 Mark
Directions : In the following questions, the Assertions $(A)$ and Reason $(s)\ (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following :
Assertion $(A) : $ The function $\text{f(x)}=\begin{cases}12\text{x} -13 , \text{if x}\leq3\\2\text{x}^2+5,\text{if x} > 3\end{cases}$ is differentiable at $x = 3.$
Reason $(R) :$ The function $f(x)$ is differentiable at $x = c$ of its domain if Left hand derivative of $f$ at $c =$ Right hand derivative of $f$ at $c$.
- A
$A$ is true, $R$ is true: $R$ is a correct explanation for $A.$
- ✓
$A$ is true $R$ is true; $R$ is not a correct explanation for $A.$
- C
$A$ is true: $R$ is false.
- D
$A$ is false: $R$ is true.
AnswerCorrect option: B. $A$ is true $R$ is true; $R$ is not a correct explanation for $A.$
View full question & answer→MCQ 131 Mark
Directions : In the following questions, the Assertions $(A)$ and Reason $(s)\ (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion $(A) : \text{f(x)}=\tan^2\text{x}$ is continuous at $\text{x}=\frac{\pi}{2}$
Reason $(R) :\ ?^2$ is continuous at $\text{x}=\frac{\pi}{2}$
- 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 $\text{NOT}$ the correct explanation of $A$
- C
$A$ is true but $R$ is false.
- ✓
$A$ is false but $R$ is true.
AnswerCorrect option: D. $A$ is false but $R$ is true.
View full question & answer→MCQ 141 Mark
Directions : In the following questions, the Assertions $(A)$ and Reason $(s)\ (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following :
Assertion $(A) : $ if $\text{y}=\tan^{-1}\Big(\frac{\cos\text{x}+\sin\text{x}}{\sin\text{x}-\cos\text{x}}\Big) ,\frac{-\pi}{4} < \text{x} < \frac{\pi}{4},\text{then}\frac{\text{dy}}{\text{dx}}=-1$
Reason $(R) : \frac{\cos\text{x}+\sin\text{x}}{\sin\text{x}-\cos\text{x}}=\tan\Big(\text{x}+\frac{\pi}{4}\Big)$
- A
$A$ is true, $R$ is true: $R$ is a correct explanation for $A.$
- B
$A$ is true $R$ is true; $R$ is not a correct explanation for $A.$
- C
$A$ is true : $R$ is false.
- ✓
$A$ is false : $R$ is true.
AnswerCorrect option: D. $A$ is false : $R$ is true.
View full question & answer→MCQ 151 Mark
Directions : In the following questions, the Assertions $(A)$ and Reason $(s) \ (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following :
Assertion $(A)$ If $\text{f(x)}=\cos,$ then $\text{ f '}\Big(\frac{\pi}{4}\Big)=\frac{-1}{\sqrt{2}}$ and $\text{ f '}\Big(\frac{3\pi}{4}\Big)=\frac{1}{\sqrt{2}}$
Reason $(R) : \text{f(x)}=\cos=\begin{cases}\cos\text{x },0\leq\text{x} \leq\frac{\pi}{2}\\-\cos\text{x },\text{if }\frac{\pi}{2} < \text{x}\leq\pi\end{cases}$
- ✓
$A$ is true, $R$ is true: $R$ is a correct explanation for $A.$
- B
$A$ is true $R$ is true; $R$ is not a correct explanation for $A.$
- C
$A$ is true: $R$ is false.
- D
$A$ is false: $R$ is true.
AnswerCorrect option: A. $A$ is true, $R$ is true: $R$ is a correct explanation for $A.$
View full question & answer→MCQ 161 Mark
Directions : In the following questions, the Assertions $(A)$ and Reason $(s) \ (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following :
Assertion : $\text{f}(\text{x})=\begin{cases}\text{x}^2\sin\big(\frac{1}{\text{x}}\big), &\text{x}=0\\0, &\text{x}=0\end{cases}$ is continuous at $x = 0$.
Reason : Both $\text{h}(\text{x})=\text{x}^2,\text{g}(\text{x})=\begin{cases}\text{x}^2\sin\big(\frac{1}{\text{x}}\big), &\text{x}=0\\0, &\text{x}=0\end{cases}$ are continuous at $x = 0$.
- A
Both $A$ and $R$ are true and $R$ is the correct explanation of $A.$
- B
Both $A $ and $R$ are true and $R$ is not the correct explanation of $A$.
- ✓
$A$ is true but $R$ is false.
- D
$R$ is true but $A$ is false.
AnswerCorrect option: C. $A$ is true but $R$ is false.
Assertion : $\text{f}(0)=0\lim\text{x}^2\sin\big(\frac{1}{\text{x}}\big)=0^2\times(\text{finite value})=0$
$\therefore$ It is continuous at $x = 0$
Reason : $h(x) = x^2$ is continuous but $g(x)$ is not continuous
$\lim\limits_{\text{x}\rightarrow0}\sin\big(\frac{1}{\text{x}}\big) =$ not defined $($value oscillates$)$
$\lim\limits_{\text{x}\rightarrow0}\sin\big(\frac{1}{\text{x}}\big)=0$
$\therefore$ not continuous.
View full question & answer→MCQ 171 Mark
Directions: In the following questions, the Assertions (A) and Reason(s) (R) have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion (A) If $x = at^2$ and $y = 2$ at where ‘t’ is the parameter and ‘a’ is a constant, then $\frac{\text{d}^2\text{y}}{\text{dy}^2}= \frac{-1}{\text{t}^2}.$
Reason(R ) $\frac{\text{d}^2\text{y}}{\text{dx}^2}=\frac{\text{d}^2\text{y}}{\text{dt}^2}\div\frac{\text{d}^2\text{x}}{\text{dt}^2}$
- 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
- ✓
View full question & answer→MCQ 181 Mark
Directions : In the following questions, the Assertions $(A)$ and Reason $(s)\ (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion $(A): f(x) = [x]$ is not differentiableat $x = 2.$
Reason $(R) : f(x) = [x]$ is not continuous at $x = 2.$
- ✓
$A$ is true, $R$ is true: $R$ is a correct explanation for $A.$
- B
$A$ is true $R$ is true; $R$ is not a correct explanation for $A.$
- C
$A$ is true: $R$ is false.
- D
$A$ is false: $R$ is true.
AnswerCorrect option: A. $A$ is true, $R$ is true: $R$ is a correct explanation for $A.$
View full question & answer→MCQ 191 Mark
Directions : In the following questions, the Assertions $(A)$ and Reason $(s)\ (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following :
Assertion $(A) :$ If $\text{f(x)}=\text{x}+\begin{vmatrix}\text{x}+2&\text{ab}\\\text{ab}&\text{x}+\text{b}^2\end{vmatrix}$ then $\text{f'(x)}=2\text{x}+\text{a}^2+\text{b}^2$
Reason $(R) : $ If $\triangle=\begin{vmatrix}\text{f(x)}&\text{g(x)} \\ \text{u(x)}&\text{g(x)} \end{vmatrix},$ Then $\frac{\text{d}\triangle}{\text{dx}}=\begin{vmatrix}\text{f'(x)}&\text{g'(x)} \\ \text{u(x)}&\text{g(x)} \end{vmatrix}+\begin{vmatrix}\text{f(x)}&\text{g(x)} \\ \text{u'(x)}&\text{g'(x)} \end{vmatrix}$
- ✓
$A$ is true, $R$ is true: $R$ is a correct explanation for $A.$
- B
$A$ is true $R$ is true; $R$ is not a correct explanation for $A.$
- C
$A$ is true: $R$ is false.
- D
$A$ is false: $R$ is true.
AnswerCorrect option: A. $A$ is true, $R$ is true: $R$ is a correct explanation for $A.$
View full question & answer→MCQ 201 Mark
Directions : In the following questions, the Assertions $(A)$ and Reason $(s)\ (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion $(A) :$ The derivative of $\log\sin\text{x}\text{ w.r.t}\sqrt{\cos\text{x}}$ is $2\sqrt{\cos\text{x}} \cos\text{x } \text{cosec x}$
Reason $(R) :$ The derivative of $\text{u w.r.t. v}$ is $\frac{\frac{\text{du}}{\text{dx}}}{\frac{\text{dv}}{\text{dx}}}$
- ✓
$A$ is true, $R$ is true: $R$ is a correct explanation for $A.$
- B
$A$ is true $R$ is true; $R$ is not a correct explanation for $A.$
- C
$A$ is true: $R$ is false.
- D
$A$ is false: $R$ is true.
AnswerCorrect option: A. $A$ is true, $R$ is true: $R$ is a correct explanation for $A.$
View full question & answer→