CSIR NET Maths previous year question papers

If the function $s : [0, 4] \to \mathbb{R}$ defined by $s(x) = a(x - 2)^{2} + b(x - 1)^{2}$ for $0 \le x \le 1, s(x) = (x - 2)^{2}$ for 1 < x ≤ 3, and $s(x) = 2c(x - 2)^{2} + (x - 3)^{3}$ for 3 < x ≤ 4, is a cubic spline, then the value of 2a + b + 2c is

If p(x) is the interpolating polynomial for the data x = −2,−1,0,1,2 with y = −1,3,1,−1,3, then the value of p(1/2) is

If $\alpha, \beta \in \mathbb{R}$ are such that the equation $\int_{0}^{3} f(x)$ dx $= (3/2)[f(\alpha) + f(\alpha + \beta)]$ holds for all polynomials f(x) of degree less than or equal to 2, then which of the following statements are true?

Consider the initial value problem (IVP) y′ + y = 0, y(0) = 1. Let $(y_{n})$ be the iterates of forward Euler method, applied to the IVP, with step size h where 0 < h < 1. Then which of the following statements are true?

Let S be the set of all 2×2 matrices A such that the iterative sequence generated by the Gauss-Seidel method converges for every initial guess, when employed to solve the system of equations $A(x_{1},x_{2})^{T}=(1,2)^{T}$. Then which of the following statements are true?

Let {$x_{n}$} be a convergent iterative sequence generated by Newton-Raphson method for solving the equation sin x−1=0 such that $x_{n}\to\pi/2$ as $n\to\infty$. For $n\in\mathbb{N}$, let $e_{n}=x_{n}-\pi/2$. Let p>0 be such that $\lim(n\to\infty)$|$e_{n}$₊$_{1}$|/|$e_{n}$|^p exists and is non-zero. Then which of the following statements are true?

Consider the quadrature formula $\int$₋$_{1}^{1}$ |x|f(x)dx ≈ (1/2)(f(−1) + f(1)). Then the degree of precision (also known as order of exactness) of the quadrature formula is

Let $f : \mathbb{R} \to \mathbb{R}$ be such that sup(x≠y) |f(x) − f(y)|/|x − y| = L, where $1 < L < \infty$. Let $h : \mathbb{R} \to \mathbb{R}$ be a differentiable function satisfying |h′(x)| ≤ 3/4 for all $x \in \mathbb{R}$. For $\alpha > 0$, define $g(x) = \alpha{}f(x) + h(x)$ for $x \in \mathbb{R}$. Consider the sequence {$x_{k}$}(k≥0) defined by $x_(k+1) = g(x_{k}), k = 0, 1, \dots$, where $x_{0} \in \mathbb{R}$. The sequence {$x_{k}$}(k≥0) converges to the solution of the equation x = g(x) if

If the value of the approximate solution of the initial value problem $y'(x) = x(y(x) + 1), x \in \mathbb{R}, y(0) = \beta$ at x = 0.2 using the forward Euler method with step size 0.1 is 1.02, then the value of $\beta$ is

Consider the following statements: $S_{1}$: There exists an $\varepsilon > 0$ such that $\forall{}x_{0} \in (2 - \varepsilon, 2 + \varepsilon)$, the iterative sequence defined by $x(n+1) = 6/(5 - x_{n}), n = 0, 1, 2, 3, \dots$, converges to $2. S_{2}$: There exists an $\varepsilon > 0$ such that $\forall{}x_{0} \in (3 - \varepsilon, 3 + \varepsilon)$, the iterative sequence defined by $x(n+1) = (x_{n}^{2} + 6)/5, n = 0, 1, 2, 3, \dots$, converges to $3. S_{3}$: There exists an $\varepsilon > 0$ such that $\forall{}x_{0} \in (3 - \varepsilon, 3 + \varepsilon)$, the iterative sequence defined by $x(n+1) = (5x_{n} - 6)/x_{n}, n = 0, 1, 2, 3, \dots$, converges to 3. Then which of the following statements are true?

Let $f : [0, 1] \times \mathbb{R} \to \mathbb{R}$ be a function such that f and its partial derivatives of orders less than or equal to 3 are continuous and bounded. Let $y : [0, 1] \to \mathbb{R}$ be the solution of dy/dt $= f(t, y(t)), 0 \le t \le 1, y(0) = y_{0}$, where $y_{0} \in \mathbb{R}$. For h > 0, denote t_j = jh. Let Y_j be an approximation of y(t_j), defined by Y_(j+1) = Y_j + ahf(t_j, Y_j) + bhf(t_(j+1), Y_j + chf$(t_j, Y_j)), 0 < (j+1)h < 1, Y_{0} = y_{0}$, where $a, b, c \in \mathbb{R}$. If there exists an M > 0 such that |y(t_(j+1)) − y(t_j) − ahf(t_j, y(t_j)) − bhf(t_(j+1), y(t_j) + chf(t_j, y(t_j)))| ≤ Mh$^{3}$ for every h > 0, and every j with 0 ≤ (j+1)h < 1, then

If $\lambda \in \mathbb{R}$ and $p \in \mathbb{R}$ are such that the quadrature formula $\int_(x_{0})^(x_{0}+h) f(x)$dx $\approx \lambda{}h(f(x_{0}) + f(x_{0} + h)) +$ ph$^{3}(f''(x_{0}) + f''(x_{0} + h))$ is exact for all polynomials of degree as high as possible, then

Let f(x) be the polynomial of degree at most 2 that interpolates the data (−1, 2), (0, 1), and (1, 2). If g(x) is a polynomial of degree at most 3 such that f(x) + g(x) interpolates the data (−1, 2), (0, 1), (1, 2), and (2, 17), then

Let S denote the set of all 2 × 2 matrices A such that the iterative sequence generated by the Gauss-Seidel method applied to the system of linear equations $A(x_{1}, x_{2})^{T} = (2, 3)^{T}$ converges for every initial guess. Then which of the following statements are true?

Let g(x) be the polynomial of degree at most 4 that interpolates the data (x, y) = (−1, −30), (0, 1), (2, c), (3, 10), (6, 19). If g(4) = 5, then which of the following statements are true?

Using Euler's method with step size 0.05, the approximate value of the solution of dy/dx $= \sqrt{3x + 2y + 1}, y(1) = 1$, at x = 1.1 (rounded to two decimal places) is

The coefficient of $x^{3}$ in the interpolating polynomial for the data x = 0, 1, 2, 3, 4 with y = 1, 2, 1, 3, 5 is

For the IVP dy/dx $= f(x, y), y(x_{0}) = y_{0}$, with step size h, let the first iterate of a second-order scheme be $y_{1} = y_{0} +$ Pk$_{1} +$ Qk$_{2}$, where $k_{1} = h f(x_{0}, y_{0}), k_{2} = h f(x_{0} + \alpha_{0}h, y_{0} + \beta_{0}k_{1})$ and $P, Q, \alpha_{0}, \beta_{0} \in \mathbb{R}$. Which of the following statements are correct?