Recent questions tagged goclasses2024-calculus-1 / GATE Overflow for GATE CSE

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GO Classes Test Series 2023 | Calculus | Test 1 | Question: 1
Evaluate the limit $ \lim _{x \rightarrow 0} \frac{1-\cos x}{\sin ^{2} x} $ $1$ $\frac{1}{2}$ $2$ $0$

Evaluate the limit$$\lim _{x \rightarrow 0} \frac{1-\cos x}{\sin ^{2} x}$$$1$$\frac{1}{2}$$2$$0$

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3 votes

1 answer

2

GO Classes Test Series 2023 | Calculus | Test 1 | Question: 2
Determine the value of following limit $ \lim _{x \rightarrow \infty}\left(\sqrt{x^{2}+4 x+1}-x\right) $ $2$ $4$ $\frac{1}{2}$ $3$

Determine the value of following limit$$\lim _{x \rightarrow \infty}\left(\sqrt{x^{2}+4 x+1}-x\right)$$$2$$4$$\frac{1}{2}$$3$

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GO Classes Test Series 2023 | Calculus | Test 1 | Question: 3
The function $f(x)=x^{4}-6 x^{2}$ is increasing on the intervals $(0, \sqrt{3})$ only $(\sqrt{3}, \infty)$ only $(-\infty,-\sqrt{3})$ and $(0, \sqrt{3})$ only $(-\sqrt{3}, 0)$ and $(\sqrt{3}, \infty)$ only

The function $f(x)=x^{4}-6 x^{2}$ is increasing on the intervals$(0, \sqrt{3})$ only$(\sqrt{3}, \infty)$ only$(-\infty,-\sqrt{3})$ and $(0, \sqrt{3})$ only$(-\sqrt{3}, 0)...

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5 votes

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4

GO Classes Test Series 2023 | Calculus | Test 1 | Question: 4
The function $f(x)=\cos x-x$ is an even function is an odd function is neither an even nor an odd function None of these

The function $f(x)=\cos x-x$is an even functionis an odd functionis neither an even nor an odd functionNone of these

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8 votes

2 answers

5

GO Classes Test Series 2023 | Calculus | Test 1 | Question: 5
Which of the following functions satisfy the conditions of Rolle's Theorem on the interval $[-1,1]?$ $ \begin{aligned} &f(x)=1-x^{2 / 3}\\ &g(x)=x^{3}-2 x^{2}-x+2\\ &h(x)=\cos \left(\frac{\pi}{4}(x+1)\right) \end{aligned} $ Rolle's Theorem applies to: both $f$ and $g$ both $g$ and $h$ $g$ only $h$ only

Which of the following functions satisfy the conditions of Rolle's Theorem on the interval $[-1,1]?$ $$\begin{aligned}&f(x)=1-x^{2 / 3}\\&g(x)=x^{3}-2 x^{2}-x+2\\&h(x)=\c...

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4 votes

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GO Classes Test Series 2023 | Calculus | Test 1 | Question: 6
Suppose that the derivative of a function $h$ is given by: $ h^{\prime}(x)=x(x-1)^{2}(x-2) $ On what interval(s) is $h$ increasing? $(-\infty, 0)$ $(-\infty, 0)$ and $(2, \infty)$ $(0,2)$ $(0,1)$ and $(2, \infty)$

Suppose that the derivative of a function $h$ is given by:$$h^{\prime}(x)=x(x-1)^{2}(x-2)$$On what interval(s) is $h$ increasing?$(-\infty, 0)$$(-\infty, 0)$ and $(2, \in...

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7 votes

3 answers

7

GO Classes Test Series 2023 | Calculus | Test 1 | Question: 7
Let $q(x)$ be a continuous function which is defined for all real numbers. A portion of the graph of $q^{\prime}(x)$, the derivative of $q(x)$, is shown below. On which of the following interval(s) is $q(x)$ increasing? $(0,2)$ $(2,4)$ $(7,9)$ None of these

Let $q(x)$ be a continuous function which is defined for all real numbers. A portion of the graph of $q^{\prime}(x)$, the derivative of $q(x)$, is shown below.On which of...

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7 votes

2 answers

8

GO Classes Test Series 2023 | Calculus | Test 1 | Question: 8
Choose the CORRECT statement - The function $f(x)=\exp \left(-x^{2}\right)-1$ has the root $x=0$. If a function $f$ is differentiable on $[-1,1]$, then there is a point $x$ in that interval where $f^{\prime}(x)=0$. If $1$ is a ... at $1 .$ If $f^{\prime \prime}(0)0$ then there is a point in $(0,1)$, where $f$ has an inflection point.

Choose the CORRECT statement -The function $f(x)=\exp \left(-x^{2}\right)-1$ has the root $x=0$.If a function $f$ is differentiable on $[-1,1]$, then there is a point $x$...

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GO Classes asked Aug 28, 2022

3 votes

1 answer

9

GO Classes Test Series 2023 | Calculus | Test 1 | Question: 9
Evaluate $y^{\prime \prime}(1)$ where $y=e^{x}+x^{e}$. $0$ $1$ $e^{2}$ $e$

Evaluate $y^{\prime \prime}(1)$ where $y=e^{x}+x^{e}$.$0$$1$$e^{2}$$e$

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6 votes

1 answer

10

GO Classes Test Series 2023 | Calculus | Test 1 | Question: 10
Consider the following statements: $f(x)$ is continuous on $[a, b]$ $f(x)$ is differentiable on $(a, b)$ $f(a)=f(b)$ Which of the above statements are required in order to guarantee a $c \in(a, b)$ such that $f^{\prime}(c)(b-a)=f(b)-f(a) ?$ I only I and II only I, II, and III I and III only

Consider the following statements:$f(x)$ is continuous on $[a, b]$$f(x)$ is differentiable on $(a, b)$$f(a)=f(b)$Which of the above statements are required in order to gu...

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7 votes

1 answer

11

GO Classes Test Series 2023 | Calculus | Test 1 | Question: 11
Let $I=(a, b)$ be an open interval and let $f$ be a function which is differentiable on $I$. Which of the followings is/are true statements - If $f^{\prime}(x)=0$ for all $x \in I$, then there is a constant $r$ such that $f(x)=r$ ... decreasing on $I$. If $f^{\prime}(x)>0$ for all $x \in I$, then $f(x)$ is strictly decreasing on $I$.

Let $I=(a, b)$ be an open interval and let $f$ be a function which is differentiable on $I$. Which of the followings is/are true statements -If $f^{\prime}(x)=0$ for all ...

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8 votes

1 answer

12

GO Classes Test Series 2023 | Calculus | Test 1 | Question: 12
Which of the following is/are FALSE? The absolute maximum value of $f(x)=\dfrac{1}{x}$ on the interval $[2,4]$ is $2.$ If $f(x)$ is a continuous function and $f(3)=2$ and $f(5)=-1$, then $f(x)$ has a root between $3$ and $5 .$ ... $h(x)$ is a continuous function and $h(1)=4$ and $h(2)=5$, then $h(x)$ has no roots between $1$ and $2.$

Which of the following is/are FALSE?The absolute maximum value of $f(x)=\dfrac{1}{x}$ on the interval $[2,4]$ is $2.$If $f(x)$ is a continuous function and $f(3)=2$ and $...

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6 votes

1 answer

13

GO Classes Test Series 2023 | Calculus | Test 1 | Question: 13
Suppose $g(x)$ is a polynomial function such that $g(-1)=4$ and $g(2)=7$. Then there is a number $c$ between $-1$ and $2$ such that $g(c)=1$ $g^{\prime}(c)=1$ $g(c)=0$ $g^{\prime}(c)=0$

Suppose $g(x)$ is a polynomial function such that $g(-1)=4$ and $g(2)=7$. Then there is a number $c$ between $-1$ and $2$ such that$g(c)=1$$g^{\prime}(c)=1$$g(c)=0$$g^{\p...

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2 votes

2 answers

14

GO Classes Test Series 2023 | Calculus | Test | Question: 14
Which of the following limit is/are correct? $\displaystyle{}\lim _{x \rightarrow \infty} \sqrt[x]{x}=1$ $\displaystyle{}\lim _{x \rightarrow \infty} \sqrt[x]{x}=e$ $\displaystyle{}\lim _{x \rightarrow \infty}\left(1+\frac{2}{x}\right)^{x}=e^{2}$ $\displaystyle{}\lim _{x \rightarrow \infty}\left(1+\frac{2}{x}\right)^{x}=e$

Which of the following limit is/are correct?$\displaystyle{}\lim _{x \rightarrow \infty} \sqrt[x]{x}=1$$\displaystyle{}\lim _{x \rightarrow \infty} \sqrt[x]{x}=e$$\displa...

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GO Classes asked Aug 28, 2022

6 votes

1 answer

15

GO Classes Test Series 2023 | Calculus | Test 1 | Question: 15
Suppose $f$ is twice differentiable with $ f^{\prime \prime}(x)=7 x-2, \quad f^{\prime}(-2)=0, \quad \text { and } \quad f(-2)=-2 . $ Find $f(0)$. $-337 / 6$ $-74 / 3$ $23 / 9$ $37 / 4$

Suppose $f$ is twice differentiable with$$f^{\prime \prime}(x)=7 x-2, \quad f^{\prime}(-2)=0, \quad \text { and } \quad f(-2)=-2 .$$Find $f(0)$.$-337 / 6$$-74 / 3$$23 / 9...

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GO Classes asked Aug 28, 2022

4 votes

1 answer

16

GO Classes Test Series 2023 | Calculus | Test 1 | Question: 16
The sum of three positive numbers is $12$ and two of them are equal. Find the largest possible product. $86$ $64$ $48$ $72$

The sum of three positive numbers is $12$ and two of them are equal. Find the largest possible product.$86$$64$$48$$72$

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GO Classes asked Aug 28, 2022

3 votes

2 answers

17

GO Classes Test Series 2023 | Calculus | Test 1 | Question: 17
If $f(x)=e^{x} g(x), g(0)=2$ and $g^{\prime}(0)=1$, then $f^{\prime}(0)$ is $1$ $3$ $2$ $0$

If $f(x)=e^{x} g(x), g(0)=2$ and $g^{\prime}(0)=1$, then $f^{\prime}(0)$ is$1$$3$$2$$0$

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GO Classes asked Aug 28, 2022

6 votes

1 answer

18

GO Classes Test Series 2023 | Calculus | Test 1 | Question: 18
Let $f$ be differentiable for all $x$. If $f(1)=-2$ and $f^{\prime}(x) \geq 2$ for $x \in[1,6]$, then $f(6) \geq 8$ $f(6)<8$ $f(6)<5$ $f(6)=5$

Let $f$ be differentiable for all $x$. If $f(1)=-2$ and $f^{\prime}(x) \geq 2$ for $x \in[1,6]$, then$f(6) \geq 8$$f(6)<8$$f(6)<5$$f(6)=5$

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8 votes

1 answer

19

GO Classes Test Series 2023 | Calculus | Test 1 | Question: 19
The equation $x^{5}+x+1=0$ has a solution in the interval $[0,1]$ $[-1,0]$ $[-2,-1]$ $[1,2]$

The equation $x^{5}+x+1=0$ has a solution in the interval$[0,1]$$[-1,0]$$[-2,-1]$$[1,2]$

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2 votes

1 answer

20

GO Classes Test Series 2023 | Calculus | Test 1 | Question: 20
Which of the following expression evaluates to given integral $ \int \frac{\ln (\ln x)}{x \ln x} d x $ $\dfrac{\ln x}{x}+C$ $\frac{1}{2}(\ln \ln x)^{2}+C$ $(\ln x)^{2}+C$ $(\ln \ln x)+C$

Which of the following expression evaluates to given integral$$\int \frac{\ln (\ln x)}{x \ln x} d x$$$\dfrac{\ln x}{x}+C$$\frac{1}{2}(\ln \ln x)^{2}+C$$(\ln x)^{2}+C$$(\l...

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