List of practice Questions

Let \( (a_n) \) be a sequence of positive real numbers such that \[ a_1 = 1, \quad a_{n+1} = 2a_n a_{n+1} - a_n = 0 \text{ for all } n \geq 1. \] Then the sum of the series \[ \sum_{n=1}^{\infty} a_n \] lies in the interval

Let \( G \) be a nonabelian group, \( y \in G \), and let the maps \( f, g, h \) from \( G \) to itself be defined by \[ f(x) = yxy^{-1}, \quad g(x) = x^{-1} \quad \text{and} \quad h = g \circ f \circ g. \] Then

The function \[ f(x, y) = x^3 + 2xy + y^3 \] has a saddle point at

The area of the part of the surface of the paraboloid \[ x^2 + y^2 + z = 8 \] lying inside the cylinder \[ x^2 + y^2 = 4 \] is

Let \( C \) be the circle \( (x - 1)^2 + y^2 = 1 \), oriented counterclockwise. Then the value of the line integral \[ \int_C \left( \frac{4}{3} x y^3 \, dx + x^4 \, dy \right) \]

The tangent line to the curve of intersection of the surface \( x^2 + y^2 - z = 0 \) and the plane \( x + z = 3 \) at the point \( (1, 1, 2) \) passes through

For \( -1<x<1 \), the sum of the power series \[ 1 + \sum_{n=2}^{\infty} (-1)^{n-1} n^2 x^{n-1} \text{ is} \]

Let \( f(x) = (\ln x)^2, x>0 \). Then

Let \( f : \mathbb{R} \to \mathbb{R} \) be a differentiable function such that \( f'(x)>f(x) \) for all \( x \in \mathbb{R} \), and \( f(0) = 1 \). Then \( f(1) \) lies in the interval

For which one of the following values of \( k \), the equation \[ 2x^3 + 3x^2 - 12x - k = 0 \] has three distinct real roots?

Which one of the following series is divergent?

Let \( S \) be the family of orthogonal trajectories of the family of curves \[ 2x^2 + y^2 = k, \text{ for } k \in \mathbb{R} \text{ and } k>0. \] If \( \ell \in S \) and \( C \) passes through the point \( (1, 2) \), then \( C \) also passes through

Let \( x + e^x \) and \( 1 + x + e^x \) be solutions of a linear second-order ordinary differential equation with constant coefficients. If \( y(x) \) is the solution of the same equation satisfying \( y(0) = 3 \) and \( y'(0) = 4 \), then \( y(1) \) is equal to

The value of the integral \[ \int_0^1 \int_0^{1 - y^2} y \sin (\pi(1 - x^2)^2) \, dx \, dy \] is

The area of the surface generated by rotating the curve \( y = x^3 \), \( 0 \leq x \leq 1 \), about the y-axis, is

Let \( H \) and \( K \) be subgroups of \( \mathbb{Z}_{144} \). If the order of \( H \) is 24 and the order of \( K \) is 36, then the order of the subgroup \( H \cap K \) is

Let \( P \) be a \( 4 \times 4 \) matrix with entries from the set of rational numbers. If \( \sqrt{2} + i \), with \( i = \sqrt{-1} \), is a root of the characteristic polynomial of \( P \) and \( I \) is the \( 4 \times 4 \) identity matrix, then

The set \[ \left\{ \frac{x}{1+x^2} : -1<x<1 \right\}, \text{ as a subset of } \mathbb{R}, \text{ is} \]

The set \[ \left\{ \frac{1}{m} + \frac{1}{n} : m, n \in \mathbb{N} \right\} \cup \{0\}, \text{ as a subset of } \mathbb{R}, \text{ is} \]

Let \( g: \mathbb{R} \to \mathbb{R} \) be a twice differentiable function. Define \( f: \mathbb{R}^3 \to \mathbb{R} \) by \[ f(x, y, z) = g(x^2 + y^2 - 2z^2). \] Then \[ \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2} \] is equal to

Let \( \{a_n\}_{n=0}^{\infty} \) and \( \{b_n\}_{n=0}^{\infty} \) be sequences of positive real numbers such that \( n a_n<b_n<n^2 a_n \), for all \( n \geq 2 \). If the radius of convergence of the power series \[ \sum_{n=0}^{\infty} a_n x^n \] is 4, then the power series \[ \sum_{n=0}^{\infty} b_n x^n \] is

Let \( S \) be the set of all limit points of the set \[ \left\{ \frac{n}{\sqrt{2}} + \frac{\sqrt{2}}{n} : n \in \mathbb{N} \right\}. \] Let \( \mathbb{Q}_+ \) be the set of all positive rational numbers. Then

If \( x^h y^k \) is an integrating factor of the differential equation \[ y(1 + xy) \, dx + x(1 - xy) \, dy = 0, \] then the ordered pair \( (h, k) \) is equal to

The equation of the tangent plane to the surface \[ x^2 z + \sqrt{8 - x^2 - y^4} = 6 \text{ at the point } (2, 0, 1) \] is

Let \( a_1 = b_1 = 0 \), and for each \( n \geq 2 \), let \( a_n \) and \( b_n \) be real numbers given by \[ a_n = \sum_{m=2}^{n} (-1)^{m} m (\log(m))^m \] \[ b_n = \sum_{m=2}^{n} \frac{1}{(\log(m))^m}. \] Then which one of the following is TRUE about the sequences \( \{a_n\} \) and \( \{b_n\} \)?