List of top Mathematics Questions

Consider the parabola \(25[(x-2)^2 + (y+5)^2] = (3x+4y-1)^2\), match the characteristic of this parabola given in List-I with its corresponding item in List-II. 

The length of the normal drawn at \( t = \frac{\pi}{4} \) on the curve \( x = 2(\cos 2t + t \sin 2t) \), \( y = 4(\sin 2t + t \cos 2t) \) is: 

Evaluate the integral: \[ \int \frac{x^2 - 1}{x^4 + 3x^2 + 1} \, dx \]
Consider the function
\[ f(x) = x(x - 1)(x - 2) \cdots (x - 100). \]
Which one of the following is correct?
If \( x - 2y - 6 = 0 \) is a normal to the circle \( x^2 + y^2 + 2gx + 2fy - 8 = 0 \) and the line \( y = 2 \) touches this circle, then the radius of the circle can be:
If \( \tan \left( \frac{\pi}{12} + 2x \right) = \cot 3x \), where \( 0<x<\frac{\pi}{2} \), then the value of \( x \) is:
If \(U_n (n = 1, 2)\) denotes the \(n\)-th derivative (\(n = 1, 2\)) of \(U(x) = \frac{Lx + M}{x^2 - 2Bx + C}\) (\(L, M, B, C\) are constants), then \(PU_2 + QU_1 + RU = 0\) holds for:

If \[ \int e^x (x^3 + x^2 - x + 4) \, dx = e^x f(x) + C, \] then \( f(1) \) is:

If \( (1,1) \) is the vertex and \( x + y + 1 = 0 \) is the directrix of a parabola. If \( (a, b) \) is its focus and \( (c, d) \) is the point of intersection of the directrix and the axis of the parabola, then \( a + b + c + d \) is:
The numbers \(1, 2, 3, \ldots, m\) are arranged in random order. The number of ways this can be done, so that the numbers \(1, 2, \ldots, r \, (r < m)\) appear as neighbours is:

Let \( I = \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{\tan^2 x}{1+5^x} \, dx \). Then:

If \( z = x + iy \) and the point \( P \) represents \( z \) in the Argand plane. If the amplitude of \( \frac{2z-i}{z+2i} \) is \( \frac{\pi}{4} \), then the equation of the locus of \( P \) is:
In \( \triangle ABC \), if \( (r_2 - r_1)(r_3 - r_1) = 2r_2r_3 \), then \( 2(r + R) = \):
If 35 is removed from the data 30, 34, 35, 36, 37, 38, 39, 40, then the median increases by
If \(\cos x + \cos y = \tfrac{2}{3}\) and \(\sin x - \sin y = \tfrac{3}{4}\), then \[ \sin(x - y) \;+\; \cos(x - y) \;=\;? \]
The common ratio of a G.P. is \( \frac{1}{2} \). If the product of the first three terms is 64, then the sum of the first 10 terms is:

The focus of the parabola \(y^2 + 4y - 8x + 20 = 0\) is at the point:

If three numbers are randomly selected from the set \( \{1,2,3,\dots,50\} \), then the probability that they are in arithmetic progression is: 

If 1000! = 3n × m, where m is an integer not divisible by 3, then n =?
The symmetric equation of the straight line passing through the points \( (-1, 4, 2) \) and \( (-3, 0, 5) \) is
Chords AB CD of a circle intersect at right angle at the point P. If the lengths of AP, PB, CP, PD are 2, 6, 3, 4 units respectively, then the radius of the circle is:
The pair of linear equations \(a_1x+b_1y+c_1=0\) and \(a_2x+b_2y+c_2=0\) has a unique solution, then
If m and M denote the mean deviations about mean and about median respectively of the data 20, 5, 15, 2, 7, 3, 11, then the mean deviation about the mean of m and M is:

The general solution of the differential equation: \[ (6x^2 - 2xy - 18x + 3y) dx - (x^2 - 3x) dy = 0 \] 

Let
\[ I(R) = \int_0^R e^{-R \sin x} \, dx, \quad R > 0. \]
Which of the following is correct?