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Maths Assertion (A) & Reason (B) MCQ

CONTINUITY AND DIFFERENTIABILITY · Gujarat Board (GSEB) STD 12 Science (GSEB - English Medium). 20 questions.

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Assertion (A) & Reason (B) MCQ

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20 questions · timed · auto-graded

Question 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) $\frac{\text{dx}^{\sin\text{x}}}{\text{dx}}=\text{x}^{\sin\text{x}}[(\cos)\log\text{x}+\frac{\sin\text{x}}{\text{x}}]$
Reason(R) if y = xf(x)  then $\frac{\text{dy}}{\text{dx}}=\text{x}^\text{f(x)}[\text{f '(x)}\log\text{x}+\frac{\text{f(x)}}{\text{x}}]$
  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
Answer
  1. Both A and R are true and R is the correct explanation of A
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Question 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) f(x) = x - 1 +  x - 2 is continuous but not differentiable at x = 1, 2.
Reason(R) Every differentiable function is continuous
  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
Answer
  1. Both A and R are true but R is NOT the correct explanation of A.
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Question 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): $\text{f(x)}=\sin\text{x}$ is continuous x = 0.
Reason(R): $\sin\text{x}$ is differentiable at x = 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.
Answer
  1. A is true but R is false.
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Question 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) f(x) = [x] greatest integer function is not differentiable at x = 2
Reason(R) The greatest integer function is not continuous at any integer
  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
Answer
  1. Both A and R are true and R is the correct explanation of A
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Question 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): $\text{f(x)}=\tan^2\text{x}$ is continuous at $\text{x}=\frac{\pi}{2}$
Reason(R): 𝑥2 is continuous at $\text{x}=\frac{\pi}{2}$
  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.
Answer
  1. A is false but R is true.
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Question 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)  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)  $\frac{\cos\text{x}+\sin\text{x}}{\sin\text{x}-\cos\text{x}}=\tan\Big(\text{x}+\frac{\pi}{4}\Big)$
  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
Answer
  1. A is false but R is true
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Question 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(A): Acontinuous funection is always differentiable.
Reason(R): Adifferentiable function is always continuous.
  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.
Answer
  1. A is false but R is true.
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Question 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) If $\text{f(x)}=\cos,\text{then}\text{ f '}\Big(\frac{\pi}{4}\Big)=\frac{-1}{\sqrt{2}} \text{ and} \text{ f '}\Big(\frac{3\pi}{4}\Big)=\frac{1}{\sqrt{2}}$
Reason(R) $\text{f(x)}=\cos=\begin{cases}\cos\text{x },0\leq\text{x} \leq\frac{\pi}{2}\\-\cos\text{x },\text{if }\frac{\pi}{2}<\text{x}\leq\pi\end{cases}$
  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
Answer
  1. Both A and R are true and R is the correct explanation of A
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Question 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) 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 $\frac{\text{d}\triangle}{\text{dx}}=\begin{vmatrix}\text{f'(x)}&\text{g'(x)} \\ \text{u(x)}&\text{g(x)} \end{vmatrix}+\begin{vmatrix}\text{f(x)}&\text{g(x)} \\ \text{u'(x)}&\text{g'(x)} \end{vmatrix}$
  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
Answer
  1. Both A and R are true and R is the correct explanation of A
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Question 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: $\text{u}=\text{f}(\cot\text{x})\&\text{f}(1)=\sqrt2$ 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 f w.r.t. to g is $\frac{\text{du}}{\text{dv}}=\frac{\frac{\text{du}}{\text{dx}}}{\frac{\text{dv}}{\text{dx}}}.$
  1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion.
  2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion.
  3. Assertion is correct but Reason is incorrect.
  4. Both Assertion and Reason are incorrect.
Answer
  1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion.
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Question 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) 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 u w.r.t. v is $\frac{\frac{\text{du}}{\text{dx}}}{\frac{\text{dv}}{\text{dx}}}$
  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
Answer
  1. Both A and R are true and R is the correct explanation of A
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Question 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) If x = at2 and y = 2at 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 ) $\frac{\text{d}^2\text{y}}{\text{dx}^2}=\frac{\text{d}^2\text{y}}{\text{dt}^2}\div\frac{\text{d}^2\text{x}}{\text{dt}^2}$
  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 fals
Answer
  1. Both A and R are fals
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Question 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) 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) $\sin2\theta=\frac{2\tan\theta}{1+\tan^2\theta}$

  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
Answer
  1. Both A and R are true and R is the correct explanation of A
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Question 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 x2 + 2xy + y3 = 42, Then $\frac{\text{dy}}{\text{dx}}=\frac{2(\text{x+y})}{(2\text{x+3}\text{y}^2)}$
Reason(R) $\frac{\text{dy}^\text{n}}{\text{dx}}=\text{ny}^{(\text{n-1})}$
  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 fals
Answer
  1. Both A and R are fals
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Question 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) 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.
  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
Answer
  1. Both A and R are true and R is the correct explanation of A
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Question 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.
  1. Both A and R are true and R is the correct explanation of A.
  2. Both A and R are true and R is not the correct explanation of A.
  3. A is true but R is false.
  4. R is true but A is false.
Answer
  1. A is true but R is false.

Solution:

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) = x2 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.

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Question 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): :f(x) = [x] is not differentiableat x = 2.
Reason(R): f(x) = [x] is not continuous at x = 2.
  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.
Answer
  1. Both A and R are true and R is the correct explanation of A
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Question 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): $\text{f(x)}=\sin\text{x}$ is continuous for all $\text{x }\in\text{ R}$
Reason (R): $\sin\text{x}$ and $\text{x}$ are continuous at on R.
  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.
Answer
  1. Both A and R are true and R is the correct explanation of A
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Question 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 $\text{y}=\log_7(\text{x}^2+7\text{x}+4),$ then $\frac{\text{dy}}{\text{dx}}=\frac{(2\text{x}+7)}{(\text{x}^2+7\text{x}+4),}$
Reason(R) $\log_\text{b}=\frac{\log_\text{e}}{\log_\text{e}\text{b}}$
  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
Answer
  1. A is false but R is true
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Question 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 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 $\lim\limits_{\text{x}\rightarrow 0}\text{f(x)}=\text{f(x)}$
  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 
Answer
  1. A is false but R is true
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