Interpolation - Numerical Analysis - Mathematics Previous Year Questions

1

2008 · Mathematics · Numerical Analysis · Interpolation

Mathematics (MA) 2008

Using the least squares method, if a curve \(y = ax^2 + bx + c\) is fitted to the collinear data points \((-1, -3), (1, 1), (3, 5)\) and \((7, 13)\), then the triplet \((a, b, c)\) is equal to

A

\((-1, 2, 0)\)

B

\((0, 2, -1)\)

C

\((2, -1, 0)\)

D

\((0, -1, 2)\)

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2

2008 · Mathematics · Numerical Analysis · Interpolation

Mathematics (MA) 2008

A quadratic polynomial \(p(x)\) is constructed by interpolating the data points \((0, 1), (1, e)\) and \((2, e^2)\). If \(\sqrt{e}\) is approximated by using \(p(x)\), then its approximate value is

A

\(\frac{1}{8}(3 + 6e - e^2)\)

B

\(\frac{1}{8}(3 - 6e + 2e^2)\)

C

\(\frac{1}{8}(3 - 6e - e^2)\)

D

\(\frac{1}{8}(3 + 6e - 2e^2)\)

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3

2009 · Mathematics · Numerical Analysis · Interpolation

Mathematics (MA) 2009

Let f : [0,4] → ℝ be a three times continuously differentiable function. Then the value of f[1,2,3,4] is

A

f'''(ξ)/3 for some ξ ∈ (0,4)

B

f'''(ξ)/6 for some ξ ∈ (0,4)

C

f'''(ξ)/3 for some ξ ∈ (0,4)

D

f'''(ξ)/6 for some ξ ∈ (0,4)

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4

2010 · Mathematics · Numerical Analysis · Interpolation

Mathematics (MA) 2010

Let \( l_k(x), k = 0, 1, ..., n \) denote the Lagrange's fundamental polynomials of degree \( n \) for the nodes \( x_0, x_1, ..., x_n \). Then the value of \( \sum_{k=0}^{n} l_k(x) \) is

A

0

B

1

C

\(x^n + 1 \)

D

\(x^n - 1 \)

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5

2014 · Mathematics · Numerical Analysis · Interpolation

Mathematics (MA) 2014

Let \(\alpha \in \mathbb{R}\). If \(\alpha x\) is the polynomial which interpolates the function \(f(x) = \sin \pi x\) on \([-1,1]\) at all the zeroes of the polynomial \(4x^3 - 3x\), then \(\alpha\) is ______________

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6

2015 · Mathematics · Numerical Analysis · Interpolation

Mathematics (MA) 2015

Let \(p(x)\) be the polynomial of degree at most 3 that passes through the points \((-2, 12), (-1, 1), (0, 2)\) and \((2, -8)\). Then the coefficient of \(x^3\) in \(p(x)\) is equal to ________

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7

2016 · Mathematics · Numerical Analysis · Interpolation

Mathematics (MA) 2016

Let the polynomial x⁴ be approximated by a polynomial of degree ≤ 2, which interpolates x⁴ at x = −1, 0 and 1. Then, the maximum absolute interpolation error over the interval [−1, 1] is equal to __________

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8

2017 · Mathematics · Numerical Analysis · Interpolation

Mathematics (MA) 2017

If the fourth order divided difference of \(f(x) = \alpha x^4 + 5x^3 + 3x + 2\), \(\alpha \in \mathbb{R}\), at the points 0.1, 0.2, 0.3, 0.4, 0.5 is 5, then \(\alpha\) equals ________.

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9

2017 · Mathematics · Numerical Analysis · Interpolation

Mathematics (MA) 2017

Let \( p(x) \) be the polynomial of degree at most \( 2 \) that interpolates the data \( (-1,2) \), \( (0,1) \) and \( (1,2) \). If \( q(x) \) is a polynomial of degree at most \( 3 \) such that \( p(x)+q(x) \) interpolates the data \( (-1,2) \), \( (0,1) \), \( (1,2) \) and \( (2,11) \), then \( q(3) \) equals ________.

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10

2018 · Mathematics · Numerical Analysis · Interpolation

Mathematics (MA) 2018

If \(p(x) = 2 - (x+1) + x(x+1) - \beta x(x+1)(x-\alpha)\) interpolates the points \((x, y)\) in the table \[ \[\begin{array}{c|cccc}\] x & -1 & 0 & 1 & 2 \\ \hline y & 2 & 1 & 2 & -7 \end{array} \] then \(\alpha + \beta = \) __________ .

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11

2020 · Mathematics · Numerical Analysis · Interpolation

Mathematics (MA) 2020

Let \(f(x) = x^4\) and let \(p(x)\) be the interpolating polynomial of \(f\) at nodes 1, 2 and 3. Then \(p(0)\) is equal to __________

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12

2021 · Mathematics · Numerical Analysis · Interpolation

Mathematics (MA) 2021

If the polynomial \( p(x) = \alpha + \beta (x+2) + \gamma (x+2)(x+1) + \delta (x+2)(x+1)x \) interpolates the data

\( x \)	\( -2 \)	\( -1 \)	\( 0 \)	\( 1 \)	\( 2 \)
\( f(x) \)	\( 2 \)	\( -1 \)	\( 8 \)	\( 5 \)	\( -34 \)

then \( \alpha + \beta + \gamma + \delta = \) ________.

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13

2022 · Mathematics · Numerical Analysis · Interpolation

Mathematics (MA) 2022

If \( P(x) \) is a polynomial of degree 5 and \[ \alpha = \sum_{i=0}^{6} P(x_i) \left( \prod_{j=0, j \neq i}^{6} (x_i - x_j)^{-1} \right), \] where \( x_0, x_1, \cdots, x_6 \) are distinct points in the interval \( [2,3] \), then the value of \( \alpha^2 - \alpha + 1 \) is __________.

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14

2022 · Mathematics · Numerical Analysis · Interpolation

Mathematics (MA) 2022

The first derivative of a function \( f \in C^\infty(-3, 3) \) is approximated by an interpolating polynomial of degree 2, using the data
\( (-1, f(-1)), \ (0, f(0)) \) and \( (2, f(2)) \).
It is found that
\( f'(0) \approx -\frac{2}{3} f(-1) + \alpha f(0) + \beta f(2) \).
Then, the value of \( \frac{1}{\alpha \beta} \) is

A

3

B

6

C

9

D

12

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15

2023 · Mathematics · Numerical Analysis · Interpolation

Mathematics (MA) 2023

Let \( p(x) = x^3 - 2x + 2 \). If \( q(x) \) is the interpolating polynomial of degree less than or equal to 4 for the data

\( x \)	\( -2 \)	\( -1 \)	\( 0 \)	\( 1 \)	\( 3 \)
\( q(x) \)	\( p(-2) \)	\( p(-1) \)	\( 2.5 \)	\( p(1) \)	\( p(3) \)

,
then the value of \( \frac{d^4 q}{dx^4} \) at \( x = 0 \) is __________.

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16

2024 · Mathematics · Numerical Analysis · Interpolation

Mathematics (MA) 2024

Let \( a \in \mathbb{R} \) and \( h \) be a positive real number. For any twice-differentiable function \( f : \mathbb{R} \to \mathbb{R} \), let \( P_f(x) \) be the interpolating polynomial of degree at most two that interpolates \( f \) at the points \( a - h, a, a + h \). Define \( d \) to be the largest integer such that any polynomial \( g \) of degree \( d \) satisfies \[ g'''(a) = P_g'''(a). \] The value of \( d \) is equal to ________ (answer in integer)

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17

2024 · Mathematics · Numerical Analysis · Interpolation

Mathematics (MA) 2024

Let \( P_f(x) \) be the interpolating polynomial of degree at most two that interpolates the function \( f(x) = x^2|x| \) at the points \( x = -1, 0, 1 \). Then \[ \sup_{x \in [-1,1]} |f(x) - P_f(x)| = \] ________ (round off to TWO decimal places)

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18

2007 · Mathematics · Numerical Analysis · Interpolation

Mathematics (MA) 2007

Let \(f(x) = x^{10} + x - 1, \; x \in \mathbb{R}\) and let \(x_k = k, \; k = 0, 1, 2, ..., 10.\) Then the value of the divided difference \(f[x_0, x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8, x_9, x_{10}]\) is

A

-1

B

0

C

1

D

10

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19

2007 · Mathematics · Numerical Analysis · Interpolation

Mathematics (MA) 2007

The smallest degree of the polynomial that interpolates the data

x	-2	-1	0	1	2	3
f(x)	-58	-21	-12	-13	-6	27

is

A

3

B

4

C

5

D

6

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