CSIR NET Maths previous year question papers

If y(x) is the solution of the integral equation $y(x) = x^{2} + 2\int_{0}^{1}$ xt y(t) dt, then which of the following statements is true?

Suppose u(x) is the solution of the integral equation $u(x) = 3 + \int_{0}$ˣ (x−t)u(t) dt. Then which of the following statements is true?

Let u(x) be the solution to the Volterra integral equation $u(x) = x^{2} + 4\int_{0}$ˣ $(t - x)^{2}u(t)$ dt. Then which of the following statements are true?

Let f and K be such that the solution of the initial value problem y″ − 3y′ + 2y = 4sin(x), y(0) = 1, y′(0) = −2 satisfies the Volterra integral equation $y(x) = f(x) + \int_{0}$ˣ K(x, t)y(t) dt. Then which of the following statements are true?

Let y(x) be the solution of the integral equation (IE) y(x) = eˣ $+ e + 1 + (1/3)\int_{0}^{1}y(t)$dt, and R(x,t,1/3) be the resolvent kernel associated to IE. Then which of the following statements are true?

Let $\lambda\in\mathbb{R}$ be such that the integral equation $y(x) = \lambda\int$₋$_{1}^{1}(5$xt$^{3}+4x^{2}t+3$xt)y(t)dt admits a non-trivial solution y(x) such that y(1)=5/2. Then which of the following statements are true?

Let u be the solution of the Volterra integral equation $\int_{0}$ᵗ [½ $+ \sin(t - \tau)] u(\tau) d\tau = \sin t$. Then the value of u(1) is

The integral equation $u(x) = f(x) + (2/\pi)\int_{0}^\pi \cos(x + t)u(t)$dt has infinitely many solutions if

Let R(x, t) and u(x) denote the resolvent kernel and the solution, respectively, of the Volterra integral equation u(x) = eˣ $+ 2\int_(\ln 6)^x e^(2(t-x))u(t)$dt. Then which of the following statements are true?

The integral equation $u(x) = f(x) + (2/\pi)\int_{0}^\pi \sin(x - t)u(t)$dt has a unique solution if

If u is the solution of the Volterra integral equation $u(x) = 3 + \sin x + \int_{0}$ˣ [(3 + sin x)/(3 + sin t)]u(t)dt, then

For $c \in \mathbb{R}$, consider the following Fredholm integral equation y(x) = 1 + x + cx$^{2} + 2\int_{0}^{1} (1 - 3$xt)y(t)dt. Then the values of c for which the integral equation admits a solution are

For $\lambda \in \mathbb{R}$ such that |$\lambda$| < 5/32, let $R(x, t, \lambda)$ and u denote the resolvent kernel and the solution, respectively, of the Fredholm integral equation $u(x) = x + (\lambda/2)\int$₋$_{2}^{2} ($xt $+ x^{2}t^{2})u(t)$dt. Then which of the following statements are true?

For the unknown $y : [0,1] \to \mathbb{R}$, consider the boundary value problem y″(x) + 2y(x) = 0 for x ∈ (0,1), y(0) = y(1) = 0. It is given that it corresponds to the integral equation $y(x) = 2\int_{0}^{1} K(x, t) y(t)$ dt. Which of the following is the kernel K(x, t)?

The value of $\lambda$ for which the integral equation $y(x) = \lambda\int_{0}^{1} x^{2}$ $e^{x+t}$ y(t) dt has a non-zero solution is

Let y be the solution to the Volterra integral equation y(x) = eˣ $+ \int_{0}$ˣ $((1 + x^{2})/(1 + t^{2})) y(t)$ dt. Which of the following statements are true?

Consider the Fredholm integral equation $y(x) - 3\int_{0}^{1} t x y(t)$ dt = f(x), with f continuous on [0, 1]. Which of the following choices for f(x) have the property that the equation admits at least one solution?