List of top Mathematics Questions

lসমতল \( 2x + 2y - 2z + 1 = 0 \) বিন্দু \( (2, 1, 5) \) এবং \( (3, 4, 3) \) এর সংযোগকারী রেখাংশকে কী অনুপাতে বিভক্ত করে তা নির্ণয় কর।

যদি সরলরেখা দুটি \[ \frac{x-2}{3} = \frac{y+1}{-2\lambda} = \frac{z-2}{0} \quad \text{এবং} \quad \frac{x-1}{1} = \frac{2y+3}{3\lambda} = \frac{z+5}{2} \] পরস্পর লম্ব হয়, তবে \( \lambda \)-এর মান নির্ণয় কর।

যদি \( \vec{a} = 4\hat{i} - \hat{j} + \hat{k} \) এবং \( \vec{b} = 2\hat{i} - 2\hat{j} + \hat{k} \) হয়, তবে \( \vec{a} + \vec{b} \) ভেক্টরের সমান্তরাল একটি একক ভেক্টর নির্ণয় কর।

যদি ভেক্টর \( \vec{\alpha} = a\hat{i} + a\hat{j} + c\hat{k}, \quad \vec{\beta} = \hat{i} + \hat{k}, \quad \vec{\gamma} = c\hat{i} + c\hat{j} + b\hat{k} \) একই সমতলে অবস্থিত (coplanar) হয়, তবে প্রমাণ কর যে \( c^2 = ab \)।

The value of \[ \frac{{}^{100}C_{50}}{51} + \frac{{}^{100}C_{51}}{52} + \cdots + \frac{{}^{100}C_{100}}{101} \] is:
Let \( P \) be a \(6 \times 4\) matrix and \( Q \) be a \(4 \times 6\) matrix such that \( PQ = 0 \). Which of the following statements is correct?
Given that the solution of \[ \frac{d^2y}{dx^2} + \alpha \frac{dy}{dx} + \beta y = -e^{-x} \] is \[ y(x) = C_1 e^{-x} + C_2 e^{2x} + x e^{-x}, \] find the values of $\alpha$ and $\beta$.
Let \[ A = \begin{pmatrix} 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 \end{pmatrix}. \] Which of the following statements is correct?
There are four different types of bananas. In how many ways can 12 children select bananas so that at least one child selects different types of bananas?
If the function \( f(x) \) satisfies \[ f'(x) = f(x) - \pi x + \pi, \] then the possible value of \( f(1) \) is
Given \[ y = -3x - 3 + m e^{2x} \] Find the Orthogonal Trajectories (O.T.).
Find the radius of convergence of the series
\[ \sum_{n=0}^{\infty} \frac{\binom{n}{6}^2}{(2n)!} x^n \]
Evaluate: \[ {}^{5}C_{0} + {}^{6}C_{1} + {}^{7}C_{2} + {}^{8}C_{3} + {}^{9}C_{4} + {}^{10}C_{5} + {}^{11}C_{6}. \]
Which of the following statements are false?
Let $P$ be a $5 \times 5$ matrix such that $\det(P) = 2$. If $Q$ is the cofactor matrix of $P$, then find $\det(Q)$.
If \[ f(x) = \big( f(x) - \pi x \big) + \pi, \] then the possible value(s) of \( f(3) - f(2) \) is/are:
Find the number of automorphisms of the cyclic group $\mathbb{Z}_n$ for $n = 30$.
Solve the system: \[ x + 2y + 2z = 1 \] \[ 2x + 3y + 2z = 2 \] \[ ax + 5y + bz = b \] Find $a + b$ for infinite solutions.
Let $G = P(N)$, where the operation is
\[ A \Delta B = A \cup B - A \cap B \] Which of the following is true?
Determine whether the sequence \[ a_n = 1 - (-1)^n + \frac{1}{n} \] is convergent or divergent.
Find the radius of convergence of the series \[ \sum_{n=0}^{\infty} \frac{(n!)^2}{(2n)!}\, x^n. \]

If \(S=\{1,2,....,50\}\), two numbers \(\alpha\) and \(\beta\) are selected at random find the probability that product is divisible by 3 :

Find the number of matrices $A$ of order $3\times 2$ whose elements are from the set $\{\pm2,\pm1,0\}$, if $\mathrm{Tr}(A^TA)=5$.
A complex number 'z' satisfy both \(|z-6|=5\) & \(|z+2-6i|=5\) simultaneously. Find the value of \(z^3 + 3z^2 - 15z + 141\).
Which method is commonly used to find roots of nonlinear equations?