List of top Mathematics Questions on Algebra asked in CUET (UG)

Let \(y = y(x)\) satisfy 

\[ \cos x\,(\log(\cos x))^2\,dy + \left( \sin x - 3y\sin x\log(\cos x) \right)dx = 0, \quad x \in \left(0,\frac{\pi}{2}\right). \]

If \[ y\!\left(\frac{\pi}{4}\right) = -\frac{1}{\log 2}, \] then \[ y\!\left(\frac{\pi}{6}\right) \] equals:

Let \(a \in \mathbb{R}\) and \(A\) be a matrix of order \(3 \times 3\) such that \(\det(A) = -4\) and 

\[ A + I = \begin{bmatrix} 1 & a & 1 \\ 2 & 1 & 0 \\ a & 1 & 2 \end{bmatrix}. \]

If \[ \det\!\left((a+1)\,\operatorname{adj}\!\big((a-1)A\big)\right) \] is \(2^m3^n\), then \(m+n\) equals:

Let \[ A= \begin{bmatrix} \log_5 128 & \log_4 5\\ \log_5 8 & \log_4 25 \end{bmatrix}. \] If \(A_{ij}\) is cofactor of \(a_{ij}\), \(C_{ij}=\sum_{k=1}^2 a_{ik}A_{jk}\), and \(C=[C_{ij}]\), then \(8|C|\) equals:
If the system \[ 2x+\lambda y+3z=5,\quad 3x+2y-z=7,\quad 4x+5y+\mu z=9 \] has infinitely many solutions, then \(\lambda^2+\mu^2\) equals:
Let \( f: \mathbf{R} \rightarrow \mathbf{R} \) be a thrice differentiable odd function satisfying \( f'(x) \ge 0 \), \( f''(x)=f(x) \), \( f(0)=0 \), \( f'(0)=3 \). Then \( 9f(\ln 3) \) equals:
If \[ f(x)=\int \frac{1}{x^{1/4}(1+x^{1/4})}dx,\quad f(0)=-6, \] then \(f(1)\) equals:
The number of singular matrices of order \(2\), whose elements are from the set \(\{2,3,6,9\}\), is:
If \[ y(x)=\det\begin{pmatrix} \sin x & \cos x & \sin x+\cos x+1\\ 27 & 28 & 27\\ 1 & 1 & 1 \end{pmatrix} \] for \(x\in\mathbb{R}\), then \(\frac{d^2y}{dx^2}+y\) equals:
Let \(y=f(x)\) satisfy \[ \frac{dy}{dx}+\frac{xy}{x^2-1}=\frac{x^6+4x}{\sqrt{1-x^2}},\ -1<x<1 \] with \(f(0)=0\). If \[ 6\int_{-1/2}^{1/2} f(x)\,dx = 2\pi-\alpha, \] then \(\alpha^2\) equals:
The number of 5-digit numbers that can be formed using the digits \(1,2,3,4,5,6\) without repetition and divisible by \(5\) is:
If \[ z=\frac{1+i}{1-i}, \] then \(z^8\) equals:
If the system of equations \[ x-3y+5z=3 \] \[ x-2y+4z=4 \] \[ 2x-7y+\lambda z=5 \] has infinite number of solutions, then the value of \(\lambda\) is: