List of top Mathematics Questions

Let \( x_1 = 1 \). For \( n \in \mathbb{N} \), define \[ x_{n+1} = \left( \frac{1}{2} + \frac{\sin^2 n}{n} \right) x_n. \] Then, which one of the following is TRUE?

The number of elements in the set \[ \{ x \in \mathbb{R} : 8x^2 + x^4 + x^8 = \cos x \} \] is equal to:

Let \( \Omega \) be the bounded region in \( \mathbb{R}^3 \) lying in the first octant \( (x \geq 0, y \geq 0, z \geq 0) \), and bounded by the surfaces \( z = x^2 + y^2 \), \( z = 4 \), \( x = 0 \) and \( y = 0 \). Then, the volume of \( \Omega \) is equal to:

Let \( x_1 = 1 \). For \( n \in \mathbb{N} \), define \[ x_{n+1} = \left( \frac{1}{2} + \frac{\sin^2 n}{n} \right) x_n. \] Then, which one of the following is TRUE?

Let \( x_1>0 \). For \( n \in \mathbb{N} \), define \[ x_{n+1} = x_n + 4. \] If \[ \lim_{n \to \infty} \left( \frac{1}{x_1 x_2 x_3} + \frac{1}{x_2 x_3 x_4} + \cdots + \frac{1}{x_{n+1} x_{n+2} x_{n+3}} \right) = \frac{1}{24}, \] then the value of \( x_1 \) is equal to:

Let \( f : \mathbb{R}^2 \to \mathbb{R} \) be defined by \[ f(x, y) = e^{y}(x^2 + y^2) \quad \text{for all } (x, y) \in \mathbb{R}^2. \] Then, which one of the following is TRUE?

Let \( \mathbb{R}/\mathbb{Z} \) denote the quotient group, where \( \mathbb{Z} \) is considered as a subgroup of the additive group of real numbers \( \mathbb{R} \). Let \( m \) denote the number of injective (one-one) group homomorphisms from \( \mathbb{Z}_3 \) to \( \mathbb{R}/\mathbb{Z} \) and \( n \) denote the number of group homomorphisms from \( \mathbb{R}/\mathbb{Z} \) to \( \mathbb{Z}_3 \). Then, which one of the following is TRUE?

Let \( f_1, f_2, f_3 \) be nonzero linear transformations from \( \mathbb{R}^4 \) to \( \mathbb{R} \) and \[ \ker(f_1) \subset \ker(f_2) \cap \ker(f_3). \] Let \( T : \mathbb{R}^4 \to \mathbb{R}^3 \) be the linear transformation defined by \[ T(v) = (f_1(v), f_2(v), f_3(v)) \quad \text{for all } v \in \mathbb{R}^4. \] Then, the nullity of \( T \) is equal to:

Let \( \varphi : \mathbb{R} \to \mathbb{R} \) be the solution of the differential equation \[ x \frac{dy}{dx} = (y - 1)(y - 3), \] satisfying \( \varphi(0) = 2 \). Then, which one of the following is TRUE?

Let \( \varphi : (0, \infty) \to \mathbb{R} \) be the solution of the differential equation \[ x \frac{dy}{dx} = \left( \ln y - \ln x \right) y, \] satisfying \( \varphi(1) = e^2 \). Then, the value of \( \varphi(2) \) is equal to:

Let \( C \) denote the family of curves described by \( yx^2 = \lambda \), for \( \lambda \in (0, \infty) \) and lying in the first quadrant of the \( xy \)-plane. Let \( O \) denote the family of orthogonal trajectories of \( C \). Which one of the following curves is a member of \( O \), and passes through the point \( (2, 1) \)?

Let \( G \) be a finite abelian group of order 10. Let \( x_0 \) be an element of order 2 in \( G \). If \( X = \{ x \in G : x^3 = x_0 \} \), then which one of the following is TRUE?

Let \( T : P_2(\mathbb{R}) \to P_2(\mathbb{R}) \) be the linear transformation defined by \[ T(p(x)) = p(x + 1), \quad \text{for all } p(x) \in P_2(\mathbb{R}) \] If \( M \) is the matrix representation of \( T \) with respect to the ordered basis \( \{1, x, x^2\} \) of \( P_2(\mathbb{R}) \), then which one of the following is TRUE?

Which one of the following is the general solution of the differential equation \[ \frac{d^2 y}{dx^2} - 8 \frac{dy}{dx} + 16y = 2e^{4x} ? \]

Define \( T : \mathbb{R}^3 \to \mathbb{R}^3 \) by \[ T(x, y, z) = (x + z, 2x + 3y + 5z, 2y + 2z), \quad \text{for all } (x, y, z) \in \mathbb{R}^3 \] Then, which one of the following is TRUE?

For which one of the following choices of \( N(x, y) \), is the equation \[ (e^x \sin y - 2y \sin x) \, dx + N(x, y) \, dy = 0 \] an exact differential equation?

If \(A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}\), then show that \(A^2 - 5A + 7I = O\). Using this, obtain \(A^{-1}\).

If \( y = \sin^{-1} x \), then prove that \( (1 - x^2) \frac{d^2y}{dx^2} - x \frac{dy}{dx} = 0 \).

If \(A = \begin{bmatrix} 1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4 \end{bmatrix}\), then find \(A^{-1}\).

Solve: \( (1 + x^2)\frac{dy}{dx} + 2xy - 4x^2 = 0 \).

Prove that \(\int_0^\pi \sqrt{\frac{1+\cos 2x}{2}} \, dx = 2\).

At \( t = 2 \), the slope of the vector function \( \vec{f}(t) = 2\hat{i} + 3\hat{j} + 5t^2\hat{k} \) is

If the relation \( R \) is given by \[ R = \{(4, 5), (1, 4), (4, 6), (7, 6), (3, 7)\}, \] then find \( R^{-1} \circ R^{-1} \).

If \(A = \begin{bmatrix} 3 & \sqrt{3} & 2 \\ 4 & 2 & 0 \end{bmatrix}\) and \(B = \begin{bmatrix} 0 & 1/4 \\ 0 & 0 \\ 1/2 & 1/8 \end{bmatrix}\), then prove that \(|C| = 1\), where \(C = (A')' B\).

If \( A \) and \( B \) are two matrices of order \( n \) which are invertible, then prove that \( (AB)^{-1} = B^{-1}A^{-1} \).