Single-variable Calculus and Integration - Calculus - Statistics Previous Year Questions

1

2019 · Statistics · Calculus · Single-variable Calculus and Integration

Statistics (ST) 2019

\[\lim_{n\to\infty}\sum_{k=1}^{n}\frac{n}{n^2+k^2}\] is equal to

A

\(\frac{\pi}{3}\)

B

\(\frac{\pi}{6}\)

C

\(\frac{3}{4}\)

D

\(\frac{\pi}{4}\)

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2

2019 · Statistics · Calculus · Single-variable Calculus and Integration

Statistics (ST) 2019

Let \(\vec{F} = (x-y+z)(\vec{i}+\vec{j})\) be a vector field on \(\mathbb{R}^3\). The line integral \(\int_C \vec{F} \cdot d\vec{r}\), where \(C\) is the triangle with vertices \((0,0,0), (5,0,0)\) and \((5,5,0)\) traversed in that order is

A

-25

B

25

C

50

D

5

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3

2019 · Statistics · Calculus · Single-variable Calculus and Integration

Statistics (ST) 2019

Let \( f \) be a continuous and positive real valued function on \( [0, 1] \). Then \( \int_0^\pi f(\sin x) \cos x \, dx \) is equal to ...

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4

2019 · Statistics · Calculus · Single-variable Calculus and Integration

Statistics (ST) 2019

Let \( S \) be the solid whose base is the region in the \( xy \)-plane bounded by the curves \( y = x^2 \) and \( y = 8 - x^2 \), and whose cross-sections perpendicular to the \( x \)-axis are squares. Then the volume of \( S \) (rounded off to two decimal places) is ...

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5

2019 · Statistics · Calculus · Single-variable Calculus and Integration

Statistics (ST) 2019

Let \(f: \mathbb{R} \to \mathbb{R}\) be defined by \(f(x) = (3x^2 + 4) \cos x\). Then \(\lim_{h \to 0} \frac{f(h)+f(-h)-8}{h^2}\) is equal to ...

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6

2020 · Statistics · Calculus · Single-variable Calculus and Integration

Statistics (ST) 2020

Which of the following functions is uniformly continuous on the specified domain?

A

\(f_1(x) = e^{x^2}, -\infty < x < \infty \)

B

\(f_2(x) = \begin{cases} \frac{1}{x}, & 0 < x \le 1 \\ 0, & x = 0 \end{cases} \)

C

\(f_3(x) = \begin{cases} x^2, & |x| \le 1 \\ \frac{2}{1+x^2}, & |x| > 1 \end{cases} \)

D

\(f_4(x) = \begin{cases} x, & |x| \le 1 \\ x^2, & |x| > 1 \end{cases} \)

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7

2020 · Statistics · Calculus · Single-variable Calculus and Integration

Statistics (ST) 2020

Let \(D = \{(x, y, z) \in \mathbb{R} \times \mathbb{R} \times \mathbb{R} : 0 \le x, y, z \le 1, x + y + z \le 1\}\), where \(\mathbb{R}\) denotes the set of all real numbers. If \[ I = \iiint_D (x + y) \, dx \, dy \, dz, \] then \(84 \times I\) = ____________________

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8

2021 · Statistics · Calculus · Single-variable Calculus and Integration

Statistics (ST) 2021

\(\lim_{n \to \infty} \left( 2^n + n 2^n \sin^2 \frac{n}{2} \right)^{\frac{1}{2n - n \cos \frac{1}{n}}}\) equals ________

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9

2021 · Statistics · Calculus · Single-variable Calculus and Integration

Statistics (ST) 2021

Let \(I = 4 \int_{\frac{1}{\sqrt{2}}}^{1} \int_{\sqrt{1-x^2}}^{x} \frac{1}{\sqrt{x^2 + y^2}} \, dy \, dx\) Then the value of $e^{I+\pi}$ equals ________ (round off to 2 decimal places).

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10

2021 · Statistics · Calculus · Single-variable Calculus and Integration

Statistics (ST) 2021

Let \( f: \mathbb{R} \to \mathbb{R} \) be defined by \[ f(x) = \begin{cases} x^3 \sin x, & x = 0 \text{ or } x \text{ is irrational}, \\ \frac{1}{q^3}, & x = \frac{p}{q} \in \mathbb{Z} \setminus \{0\}, q \in \mathbb{N} \text{ and } \gcd(p,q) = 1, \end{cases} \) where \( \mathbb{R} \) denotes the set of all real numbers, \( \mathbb{Z} \) denotes the set of all integers, \( \mathbb{N} \) denotes the set of all positive integers and \( \gcd(p,q) \) denotes the greatest common divisor of \( p \) and \( q \). Then which one of the following statements is true?

A

\(f \) is not continuous at 0

B

\(f \) is not differentiable at 0

C

\(f \) is differentiable at 0 and the derivative of \(f \) at 0 equals 0

D

\(f \) is differentiable at 0 and the derivative of \(f \) at 0 equals 1

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11

2021 · Statistics · Calculus · Single-variable Calculus and Integration

Statistics (ST) 2021

Let \( f: [0, \infty) \to \mathbb{R} \) be a function, where \( \mathbb{R} \) denotes the set of all real numbers. Then which one of the following statements is true?

A

If \(f \) is bounded and continuous, then \(f \) is uniformly continuous

B

If \(f \) is uniformly continuous, then \(\lim_{x \to \infty} f(x) \) exists

C

If \(f \) is uniformly continuous, then the function \(g(x) = f(x) \sin x \) is also uniformly continuous

D

If \(f \) is continuous and \(\lim_{x \to \infty} f(x) \) is finite, then \(f \) is uniformly continuous

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12

2021 · Statistics · Calculus · Single-variable Calculus and Integration

Statistics (ST) 2021

Let \( f: \mathbb{R} \to \mathbb{R} \) be a differentiable function such that \( f(0) = 0 \) and \( f'(x) + 2f(x) > 0 \) for all \( x \in \mathbb{R} \), where \( f' \) denotes the derivative of \( f \) and \( \mathbb{R} \) denotes the set of all real numbers. Then which one of the following statements is true?

A

\(f(x) > 0 \), for all \(x > 0 \) and \(f(x) < 0 \), for all \(x < 0 \)

B

\(f(x) < 0 \), for all \(x \neq 0 \)

C

\(f(x) > 0 \), for all \(x \neq 0 \)

D

\(f(x) < 0 \), for all \(x > 0 \) and \(f(x) > 0 \), for all \(x < 0 \)

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13

2022 · Statistics · Calculus · Single-variable Calculus and Integration

Statistics (ST) 2022

If the line \( y = \alpha x, \alpha \geq \sqrt{2} \), divides the area of the region
\( R := \{(x, y) \in \mathbb{R}^2 \mid 0 \leq x \leq \sqrt{y}, 0 \leq y \leq 2\} \)
into two equal parts, then the value of \( \alpha \) is equal to

A

\(\frac{3}{\sqrt{2}} \)

B

\(2\sqrt{2} \)

C

\(\sqrt{2} \)

D

\(\frac{5}{2\sqrt{2}} \)

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14

2022 · Statistics · Calculus · Single-variable Calculus and Integration

Statistics (ST) 2022

Which of the following real valued functions is/are uniformly continuous on \( [0, \infty) \)?

A

\(\sin^2 x \)

B

\(x \sin x \)

C

\(\sin(\sin x) \)

D

\(\sin(x \sin x) \)

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15

2023 · Statistics · Calculus · Single-variable Calculus and Integration

Statistics (ST) 2023

The area of the region bounded by the parabola \( x = -y^2 \) and the line \( y = x + 2 \) equals

A

\(\frac{3}{2} \)

B

\(\frac{7}{2} \)

C

\(\frac{9}{2} \)

D

9

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16

2023 · Statistics · Calculus · Single-variable Calculus and Integration

Statistics (ST) 2023

Let \(g(x) = f(x) + f(2 - x)\) for all \(x \in [0, 2]\), where \(f: [0, 2] \to \mathbb{R}\) is continuous on \([0, 2]\) and twice differentiable on \((0, 2)\). If \(g'\) denotes the derivative of \(g\) and \(f''\) denotes the second derivative of \(f\), then which one of the following statements is NOT true?

A

There exists \(c \in (0, 2)\) such that \(g'(c) = 0\)

B

If \(f'' > 0\) on \((0, 2)\), then \(g\) is strictly decreasing on \((0, 1)\)

C

If \(f'' < 0\) on \((0, 2)\), then \(g\) is strictly increasing on \((1, 2)\)

D

If \(f'' = 0\) on \((0, 2)\), then \(g\) is a constant function

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17

2023 · Statistics · Calculus · Single-variable Calculus and Integration

Statistics (ST) 2023

Let \( f \) be a continuous function from \( [0,1] \) to the set of all real numbers. Then which one of the following statements is NOT true?

A

For any sequence \(\{x_n\}_{n \geq 1} \) in \([0,1] \), \(\sum_{n=1}^{\infty} \frac{f(x_n)}{n^2} \) is absolutely convergent

B

If \(|f(x)| = 1 \) for all \(x \in [0,1] \), then \(\int_0^1 f(x)dx = 1 \)

C

If \(\{x_n\}_{n \geq 1} \) is a sequence in \([0,1] \) such that \(\{f(x_n)\}_{n \geq 1} \) is convergent, then \(\{x_n\}_{n \geq 1} \) is convergent

D

If \(f \) is also monotonically increasing, then the image of \(f \) is given by \([f(0), f(1)] \)

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18

2024 · Statistics · Calculus · Single-variable Calculus and Integration

Statistics (ST) 2024

Let \( D \) be the region bounded by the line \( y = x \) and the parabola \( y = 4x - x^2 \). Then \( \iint_D x \, dx \, dy \) equals

A

\(\frac{27}{4} \)

B

\(\frac{29}{4} \)

C

7

D

6

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19

2024 · Statistics · Calculus · Single-variable Calculus and Integration

Statistics (ST) 2024

If \( f: [-2, 2] \to \mathbb{R} \) is a continuous function, then which of the following statements is/are true?

A

\(F: [0,2] \to \mathbb{R} \) defined by \(F(x) = \int_0^x f(t) \, dt \) is differentiable on \((0,2) \)

B

For any \(x_1, x_2, \ldots, x_{10} \in [-2,2] \), there exists a point \(x_0 \in [-2,2] \) such that \(f(x_0) = \frac{1}{10} (f(x_1) + f(x_2) + \cdots + f(x_{10})) \)

C

\(f \) is bounded on \([-2,2] \)

D

If \(f \) is differentiable at 0 and \(f(0)=0 \), then \(\lim_{x \to 0} \frac{1}{x} \left( f(x) + f\left(\frac{x}{2}\right) + \cdots + f\left(\frac{x}{10}\right) \right) = 10 f'(0) \), where \(f'(0) \) is the derivative of \(f \) at 0

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20

2025 · Statistics · Calculus · Single-variable Calculus and Integration

Statistics (ST) 2025

Let \( f: [0, \infty) \to [0, \infty) \) be a differentiable function with \( f(x) > 0 \) for all \( x > 0 \), and \( f(0) = 0 \). Further, \( f \) satisfies
\[ (f(x))^2 = \int_{0}^{x} \left( (f(t))^2 + f(t) \right) dt \, \, , \, \, x > 0. \]
Then which one of the following options is correct?

A

\(0 < f(2) \le 1 \)

B

\(1 < f(2) \le 2 \)

C

\(2 < f(2) \le 3 \)

D

\(3 < f(2) \le 4 \)

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