Numerical differentiation and error, Numerical integration - Numerical Analysis - Mathematics Previous Year Questions

1

2008 · Mathematics · Numerical Analysis · Numerical differentiation and error, Numerical integration

Mathematics (MA) 2008

If a quadrature formula \(\frac{3}{2} f\left(-\frac{1}{3}\right) + K f\left(\frac{1}{3}\right) + \frac{1}{2} f(1)\), that approximates \(\int_{-1}^{1} f(x) dx\), is found to be exact for quadratic polynomials, then the value of \(K\) is

A

2

B

1

C

0

D

-1

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2

2009 · Mathematics · Numerical Analysis · Numerical differentiation and error, Numerical integration

Mathematics (MA) 2009

For what values of α and β, the quadrature formula ∫_–1¹ f(x) dx = α f(–1) + f(β) is exact for all polynomials of degree ≤ 1?

A

α = 1, β = 1

B

α = –1, β = 1

C

α = 1, β = –1

D

α = –1, β = –1

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3

2009 · Mathematics · Numerical Analysis · Numerical differentiation and error, Numerical integration

Mathematics (MA) 2009

Let \(f : [0, 2] \to \mathbb{R}\) be a twice continuously differentiable function. If \(\int_0^2 f(x) dx = 2 f(1)\), then the error in the approximation is

A

\(\frac{f''(\xi)}{12}\) for some \(\xi \in (0, 2)\)

B

\(\frac{f''(\xi)}{2}\) for some \(\xi \in (0, 2)\)

C

\(\frac{f''(\xi)}{3}\) for some \(\xi \in (0, 2)\)

D

\(\frac{f''(\xi)}{6}\) for some \(\xi \in (0, 2)\)

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4

2010 · Mathematics · Numerical Analysis · Numerical differentiation and error, Numerical integration

Mathematics (MA) 2010

The numerical value obtained by applying the two-point trapezoidal rule to the integral \( \int_{0}^{1} \frac{\ln(1 + x)}{x} dx \) is

A

\(\frac{1}{2}(\ln 2 + 1) \)

B

\(\frac{1}{2} \)

C

\(\frac{1}{2}(\ln 2 - 1) \)

D

\(\frac{1}{2} \ln 2 \)

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5

2011 · Mathematics · Numerical Analysis · Numerical differentiation and error, Numerical integration

Mathematics (MA) 2011

The value of the integral \( I = \int_{-1}^1 \exp(x^2) dx \) using a rectangular rule is approximated as 2. Then, the approximation error \( |I - 2| \) lies in the interval

A

(2e, 3e]

B

(2/3, 2e]

C

(e/8, 2/3]

D

(0, e/8]

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6

2012 · Mathematics · Numerical Analysis · Numerical differentiation and error, Numerical integration

Mathematics (MA) 2012

Given the data:

\(x\)	1	2	3	4	5
\(y\)	-1	2	-3	4	-5

If the derivative of \(y(x)\) is approximated as: \(y'(x_k)\approx\frac{1}{h}(\Delta y_k+\frac{1}{2}\Delta^2 y_k-\frac{1}{4}\Delta^3 y_k)\), then the value of \(y'(2)\) is

A

4

B

8

C

12

D

16

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7

2012 · Mathematics · Numerical Analysis · Numerical differentiation and error, Numerical integration

Mathematics (MA) 2012

The data given in the following table summarizes the monthly budget of an average household.

Category	Amount (Rs.)
Food	4000
Clothing	1200
Rent	2000
Savings	1500
Other expenses	1800

The approximate percentage of the monthly budget NOT spent on savings is

A

10%

B

14%

C

81%

D

86%

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8

2014 · Mathematics · Numerical Analysis · Numerical differentiation and error, Numerical integration

Mathematics (MA) 2014

Let the following discrete data be obtained from a curve \(y = y(x)\):

x	0	0.25	0.5	0.75	1.0
y	1	0.9896	0.9589	0.9089	0.8415

Let \(S\) be the solid of revolution obtained by rotating the above curve about the x-axis between \(x = 0\) and \(x = 1\) and let \(V\) denote its volume. The approximate value of \(V\), obtained using Simpson's \(\frac{1}{3}\) rule, is ______________

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9

2015 · Mathematics · Numerical Analysis · Numerical differentiation and error, Numerical integration

Mathematics (MA) 2015

If, for some \(\alpha, \beta \in \mathbb{R}\), the integration formula \[\int_0^2 p(x)dx = p(\alpha) + p(\beta)\] holds for all polynomials \(p(x)\) of degree at most 3, then the value of \(3(\alpha - \beta)^2\) is equal to ________

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10

2016 · Mathematics · Numerical Analysis · Numerical differentiation and error, Numerical integration

Mathematics (MA) 2016

Let the integral \( I = \int_0^4 f(x)dx \), where \( f(x) = \begin{cases} x & 0 \le x \le 2 \\ 4-x & 2 \le x \le 4 \end{cases} \). Consider the following statements P and Q: (P) : If \( I_2 \) is the value of the integral obtained by the composite trapezoidal rule with two equal sub-intervals, then \( I_2 \) is exact. (Q) : If \( I_3 \) is the value of the integral obtained by the composite trapezoidal rule with three equal sub-intervals, then \( I_3 \) is exact. Which of the above statements hold TRUE?

A

both P and Q

B

only P

C

only Q

D

Neither P nor Q

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11

2017 · Mathematics · Numerical Analysis · Numerical differentiation and error, Numerical integration

Mathematics (MA) 2017

If the quadrature rule \(\int_0^2 f(x) dx \approx c_1 f(0) + 3 f(c_2)\), where \(c_1, c_2 \in \mathbb{R}\), is exact for all polynomials of degree \(\le 1\), then \(c_1 + 3c_2\) equals ________.

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12

2018 · Mathematics · Numerical Analysis · Numerical differentiation and error, Numerical integration

Mathematics (MA) 2018

Let \(a \in (-1, 1)\) be such that the quadrature rule \[ \int_{-1}^1 f(x) \, dx \approx f(-a) + f(a) \] is exact for all polynomials of degree less than or equal to 3. Then \(3a^2 = \) ________.

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13

2019 · Mathematics · Numerical Analysis · Numerical differentiation and error, Numerical integration

Mathematics (MA) 2019

The maximum value of the error term of the composite Trapezoidal rule when it is used to evaluate the definite integral \(\int_{0.2}^{1.4} (\sin x - \log_e x) dx\) with 12 sub-intervals of equal length, is equal to ______ (round off to 3 places of decimal).

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14

2020 · Mathematics · Numerical Analysis · Numerical differentiation and error, Numerical integration

Mathematics (MA) 2020

Let \(a, b, c \in \mathbb{R}\) be such that the quadrature rule
\[\int_{-1}^1 f(x)dx \approx a f(-1) + b f(0) + c f'(1)\]
is exact for all polynomials of degree less than or equal to 2. Then \(b\) is equal to __________ (rounded off to two decimal places)

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15

2021 · Mathematics · Numerical Analysis · Numerical differentiation and error, Numerical integration

Mathematics (MA) 2021

The quadrature formula
\[\int_0^2 x f(x) dx \approx \alpha f(0) + \beta f(1) + \gamma f(2)\]
is exact for all polynomials of degree \(\leq 2\). Then \(2\beta - \gamma = \) ________.

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16

2023 · Mathematics · Numerical Analysis · Numerical differentiation and error, Numerical integration

Mathematics (MA) 2023

For a fixed \( c \in \mathbb{R} \), let \( \alpha = \int_0^2 (9x^2 - 5cx^4) dx \).
If the value of \( \int_0^2 (9x^2 - 5cx^4) dx \) obtained by using the Trapezoidal rule is equal to \( \alpha \), then the value of \( c \) is __________ (rounded off to 2 decimal places).

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17

2023 · Mathematics · Numerical Analysis · Numerical differentiation and error, Numerical integration

Mathematics (MA) 2023

For \(h > 0\), and \(\alpha\), \(\beta\), \(\gamma \in \mathbb{R}\), let
\(D_h f(a) = \frac{\alpha f(a-h) + \beta f(a) + \gamma f(a+2h)}{6h}\)
be a three-point formula to approximate \(f'(a)\) for any differentiable function \(f : \mathbb{R} \to \mathbb{R}\) and \(a \in \mathbb{R}\).
If \(D_h f(a) = f'(a)\) for every polynomial \(f\) of degree less than or equal to 2 and for all \(a \in \mathbb{R}\), then

A

\(\alpha + 2\gamma = -2\)

B

\(\alpha + 2\beta - 2\gamma = 0\)

C

\(\alpha + 2\gamma = 2\)

D

\(\alpha + 2\beta - 2\gamma = 1\)

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18

2024 · Mathematics · Numerical Analysis · Numerical differentiation and error, Numerical integration

Mathematics (MA) 2024

Let \(f(x) = |x| + |x - 1| + |x - 2|, \; x \in [-1, 2]\). Which one of the following numerical integration rules gives the exact value of the integral \[\int_{-1}^2 f(x) dx\]

A

The Simpson's rule

B

The trapezoidal rule

C

The composite Simpson's rule by dividing \([-1, 2]\) into 4 equal subintervals

D

The composite trapezoidal rule by dividing \([-1, 2]\) into 3 equal subintervals

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19

2025 · Mathematics · Numerical Analysis · Numerical differentiation and error, Numerical integration

Mathematics (MA) 2025

Let \(\alpha, \beta, \gamma, \delta \in \mathbb{R}\) be such that the quadrature formula \(\int_{-1}^1 f(x) \, dx = \alpha f(-1) + \beta f(1) + \gamma f'(-1) + \delta f'(1)\) is exact for all polynomials of degree less than or equal to 3. Then, \(9(\alpha^2 + \beta^2 + \gamma^2 + \delta^2)\) is equal to ____ (in integer)

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20

2007 · Mathematics · Numerical Analysis · Numerical differentiation and error, Numerical integration

Mathematics (MA) 2007

Consider the quadrature formula \[\int_{-1}^1 |x| f(x) dx \approx \frac{1}{2} [f(x_0) + f(x_1)],\] where \(x_0\) and \(x_1\) are quadrature points. Then the highest degree of the polynomial, for which the above formula is exact, equals

A

1

B

2

C

3

D

4

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