List of top Mathematics Questions

If the coefficients of \((5r+4)th\) term and \((r-1)th\) term in the expansion of \((1+x)^{25}\) are equal, then \(r\) is 
The remainder when 7103 is divided by 17, is
If matrix $A = \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix}$ and $A^2 = kA$, then the value of $k$ is:

The integral \(\int \left(\frac{x}{2}\right)^x + \left(\frac{2}{x}\right)^x \log x \, dx\) is equal to:

The integrating factor of the differential equation $(3x^2 + y) \frac{dx}{dy} = x$ is:
For the expression \((1-x)^{100}\). Then sum of coefficient of first 50 terms is:

\(\lim\limits_{x\rightarrow0}\left(\left(\frac{1-cos^2(3x)}{cos^3(4x)}\right)\left(\frac{sin^3(4x)}{(log_e(2x+1))^5}\right)\right)\)is equal to

In a group activity class, there are 10 students whose ages are 16, 17, 15, 14, 19, 17, 16, 19, 16 and 15 years. One student is selected at random such that each has equal chance of being chosen and age of the student is recorded.
On the basis of the above information, answer the following questions :
Ahousing society wants to commission a swimming pool for its residents. For this, they have to purchase a square piece of land and dig this to such a depth that its capacity is 250 cubic metres. Cost of land is 500 per square metre. The cost of digging increases with the depth and cost for the whole pool is 4000 (depth)2. Suppose the side of the square plot is x metres and depth is h metres.
On the basis of the above information, answer the following questions
Given:- Square plot side = x metres
- Depth = h metres
- Volume = x2h = 250 cubic metres
- Cost of land: 500 per sq.metre
- Cost of digging: 4000 ×h

In an agricultural institute, scientists conduct experiments with varieties of seeds to grow them in different environments for producing healthy plants and obtaining higher yields.
A scientist observed that a particular seed grew very fast after germination. He recorded the growth of the plant from the time of germination and modeled its growth with the function:
Given:
\( f(x) = \frac{1}{3}x^3 - 4x^2 + 15x + 2 \),   \( 0 \leq x \leq 10 \)
where \( x \) is the number of days the plant is exposed to sunlight.
On the basis of the above information, answer the following questions: 

Find the area of triangle ABC bounded by:
5x −2y−10 =0, x−y−9=0, 3x−4y−6=0, usingintegrationmethod.
Show that the relation \( S = \{ (a, b) : a \leq b^3,\ a, b \in \mathbb{R} \} \) is neither reflexive, nor symmetric, nor transitive.
Show that the lines
\[ \frac{x + 1}{3} = \frac{y + 3}{5} = \frac{z + 5}{7} \quad \text{and} \quad \frac{x - 2}{1} = \frac{y - 4}{3} = \frac{z - 6}{5} \] intersect and find their point of intersection.
Find the shortest distance between the pair of lines:
Line \( L_1 \) : \[ \frac{x - 1}{2} = \frac{y + 1}{3} = z, \] Line \( L_2 \) : \[ \frac{x + 1}{5} = \frac{y - 2}{1} = z = 2. \]
Relation \( R \) on \( A = \{1,2,3,4,5,6,7\} \):
\[ R = \{(a,b) : a \text{ and } b \text{ are both odd or both even} \} \] Show that \( R \) is an equivalence relation and find the equivalence class \([1]\).
50 people, 20 always tell truth.Two selected. Find probability distribution of number of truth-tellers.
Find the particular solution of: \[ (1 + x^2) \frac{dy}{dx} + 2xy = \frac{1}{1 + x^2}, y = 0 \text{ when } x = 1. \]
Find: \[ I = \int \frac{2x^2 + 1}{x^2(x^2 + 4)} \, dx \]
Given: \[ A = \begin{bmatrix} 3 & 2 & 1 \\ 4 & 1 & 3 \\ 1 & 1 & 1 \end{bmatrix} \]
Find \( A^{-1} \) and hence solve the following system of equations:
\[ \begin{cases} 3x + 2y + z = 2000 \\ 4x + y + 3z = 2500 \\ x + y + z = 900 \end{cases} \]
Two bags: A (2W, 3R), B (4W, 5R). One red drawn. Find probability it came from B.
Find the particular solution of the differential equation:
\[ \frac{dy}{dx} = \frac{xy}{x^2 + y^2} \] given that \( y = 1 \) when \( x = 0 \).
Find the integral:
\[ I = \int \frac{\cos \theta \, d\theta}{\sqrt{3 - 3\sin \theta - \cos^2 \theta}} \]
Evaluate: \[ \int_{1}^{3} (|x - 1| + |x - 2|) dx. \]
Minimise $z = 3x + 8y$ subject to: \[ 3x + 4y \geq 8, 5x + 2y \geq 11, x \geq 0,\; y \geq 0. \]
If a line makes angles \( \alpha, \beta, \gamma \) with the x-axis, y-axis, and z-axis respectively,
then prove that: \[ \sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma = 2 \]