List of top Mathematics Questions

If \( \alpha, \beta \) are the roots of \( 2x^2 - 4x + 5 = 0 \), then \( (\alpha + 1)(\beta + 1) = \):

The value of $\dfrac{1+\cot^2 \theta}{1+\tan^2 \theta}$ will be:
 

The number of positive integers less than 10000 which contain the digit 5 at least once is
Ram and Syam are friends. Probability that both will have same birthday is:

Four students of class XII are given a problem to solve independently. Their respective chances of solving the problem are: \[ \frac{1}{2},\quad \frac{1}{3},\quad \frac{2}{3},\quad \frac{1}{5} \] Find the probability that at most one of them will solve the problem.

Consider a Cartesian coordinate system defined over a 3-dimensional vector space with orthogonal unit basis vectors \(\hat{i}, \hat{j}\), and \(\hat{k}\). Let vector \(\mathbf{a} = \sqrt{2}\hat{i} + \frac{1}{\sqrt{2}}\hat{k}\), and vector \(\mathbf{b} = \frac{1}{\sqrt{2}}\hat{i} + \sqrt{2}\hat{j} - \hat{k}\). The inner product of these vectors (\(\mathbf{a} \cdot \mathbf{b}\)) is:
Find the derivative of \[ \frac{d}{dx} \left( \lim_{n \to 1} \frac{x^n - 1}{n+1} \right). \]

$PQ$ is a chord of length $4\ \text{cm}$ of a circle of radius $2.5\ \text{cm}$. The tangents at $P$ and $Q$ intersect at a point $T$. Find the length of $TP$.
 

Find the angle between the pair of lines:
\( \vec{r} = 3\hat{i} + \hat{j} - 2\hat{k} + \lambda(\hat{i} - \hat{j} - 2\hat{k}) \) and
\( \vec{r} = 2\hat{i} - \hat{j} - 56\hat{k} + \mu(3\hat{i} - 5\hat{j} - 4\hat{k}) \).
If DE : BC = 3 : 5, then find A(\(\triangle\)ADE) : A(\(\square\)DBCE).
Consider a volume V enclosed by a closed surface S having unit surface normal \(\hat{n}\). For \(\mathbf{r} = x\hat{i} + y\hat{j} + z\hat{k}\), the value of the surface integral \(\frac{1}{9} \oint_{S} \mathbf{r} \cdot \hat{n} \,dS\) is

In the given figure, \( PQ \) and \( PR \) are tangents to the circle such that \( PQ = 7 \, \text{cm} \) and \( \angle RPQ = 60^\circ \).

The length of chord QR is: 
 

From the word ARRANGEMENT, how many distinct 9-letter words can be formed?
Prove that \(\int_{\pi/6}^{\pi/3} \frac{dx}{1+\sqrt{\tan x}} = \frac{\pi}{12}\).
The domain and range of a real valued function \( f(x) = \cos (x-3) \) are respectively.
If \( A = \begin{bmatrix} 0 & -\tan{\frac{\alpha}{2}} \\[4pt] \tan{\frac{\alpha}{2}} & 0 \end{bmatrix} \), then prove that \[ (I + A) = (I - A)\begin{bmatrix} \cos{\alpha} & -\sin{\alpha} \\[4pt] \sin{\alpha} & \cos{\alpha} \end{bmatrix}. \]
Solve the system of equations by matrix method:
\( 3x - 2y + 3z = 8 \)
\( 2x + y - z = 1 \)
\( 4x - 3y + 2z = 4 \)
If $LCM(35, 63) = 315$, the $HCF(35, 63)$ will be:
Find the inverse of the matrix \( A = \begin{bmatrix} 2 & 0 & -1 \\ 5 & 1 & 0 \\ 0 & 1 & 3 \end{bmatrix} \).
Let \( f, g : \mathbb{R} \to \mathbb{R} \) be two functions defined by \[ f(x) = \begin{cases} x |x| \sin \frac{1}{x} & \text{if } x \neq 0 \\ 0 & \text{if } x = 0 \end{cases} \] and \[ g(x) = \begin{cases} x^2 \sin \frac{1}{x} + x \cos \frac{1}{x} & \text{if } x \neq 0 \\ 0 & \text{if } x = 0 \end{cases} \] Then, which one of the following is TRUE?
Obtain the differential equation of the family of curves \(y = \frac{2ce^{2x}}{1+ce^{2x}}\).
Solve: \((\tan^{-1}y - x)dy = (1+y^2)dx\).
Suppose \( t_1, t_2, t_3, \dots, t_{55} \) are in AP such that \( \sum_{l=0}^{18} t_{3l+1} = 1197 \) and \( t_7 + 3t_{22} = 174 \). If \( \sum_{l=1}^{9} t_l^2 = 947b \), then the value of \( b \) is
If \(A\) is an identity matrix of order \(n\), then \(A (\text{Adj } A)\) is a/an:
\(\triangle\)ABC \(\sim\) \(\triangle\)PQR. In \(\triangle\)ABC, AB = 3.6 cm, BC = 4 cm and AC = 4.2 cm. The corresponding sides of \(\triangle\)ABC and \(\triangle\)PQR are in the ratio 2 : 3, construct \(\triangle\)ABC and \(\triangle\)PQR.