List of top Mathematics Questions

If \( L \) is the line of intersection of two planes \[ x + 2y + 2z = 15 \quad \text{and} \quad x - y + z = 4 \] and the direction ratios of the line \( L \) are \( (a, b, c) \), then the value of \[ \frac{a^2 + b^2 + c^2}{b^2} \] is:
Evaluate the integral: \[ I = \int \frac{1}{x^m \sqrt[m]{x^m + 1}} dx. \]
In a \( \triangle ABC \), the sides \( b, c \) are fixed. In measuring angle \( A \), if there is an error of \( \delta A \), then the percentage error in measuring the length of the side \( a \) is:
If \( A \) is the solution set of the equation \( \cos^2 x = \cos^2 \frac{\pi}{6} \) and \( B \) is the solution set of the equation \( \cos^2 x = \log_{10} P \) where \( P + \frac{16}{P} = 10 \), then \( B - A = ? \)
Given the two parabolas: \[ S = y^2 - 4ax = 0, \quad S' = y^2 + ax = 0 \] where \( P(t) \) is a point on the parabola \( S' = 0 \). If \( A \) and \( B \) are the feet of the perpendiculars from \( P \) to the coordinate axes and \( AB \) is a tangent to the parabola \( S = 0 \) at the point \( Q(t_1) \), then the value of \( t_1 \) is:

The equation of a circle which passes through the points of intersection of the circles \[ 2x^2 + 2y^2 - 2x + 6y - 3 = 0, \quad x^2 + y^2 + 4x + 2y + 1 = 0 \] and whose centre lies on the common chord of these circles is: 

If \( 3^{2n+2} - 8n - 9 \) is divisible by \( 2^p \) \(\forall n \in \mathbb{N}\), then the maximum value of \( p \) is

Let \( A = [a_{ij}] \) be a \( 3 \times 3 \) matrix with positive integers as its elements. The elements of \( A \) are such that the sum of all the elements of each row is equal to 6, and \( a_{22} = 2 \). 
 

\[ \textbf{If } | \text{Adj} \ A | = x \text{ and } | \text{Adj} \ B | = y, \text{ then } \left( | \text{Adj}(AB) | \right)^{-1} \text{ is } \]

\[ D = \begin{vmatrix} -\frac{bc}{a^2} & \frac{c}{a} & \frac{b}{a} \\ \frac{c}{b} & -\frac{ac}{b^2} & \frac{a}{b} \\ \frac{b}{c} & \frac{a}{c} & -\frac{ab}{c^2} \end{vmatrix} \]

The roots of the equation \( x^3 - 3x^2 + 3x + 7 = 0 \) are \( \alpha, \beta, \gamma \) and \( w, w^2 \) are complex cube roots of unity. If the terms containing \( x^2 \) and \( x \) are missing in the transformed equation when each one of these roots is decreased by \( h \), then 

The equation of the straight line whose slope is \( -\frac{2}{3} \) and which divides the line segment joining \( (1,2) \) and \( (-3,5) \) in the ratio 4:3 externally is:
If \( m \) is the slope and \( P(\beta, \beta) \) is the midpoint of a chord of contact of the circle \[ x^2 + y^2 = 125, \] then the number of values of \( \beta \) such that \( \beta \) and \( m \) are integers is: \
By shifting the origin to the point \( (h,5) \) by the translation of coordinate axes, if the equation \[ y = x^2 - 9x^2 + cx - d \] transforms to \( Y = X^2 \), then \( \left( \frac{d - c}{h} \right) \) is: \
The sum of the series \( \frac{1}{1.5} + \frac{1}{5.9} + \frac{1}{9.13} + \cdots \) up to \( n \) terms is:
If \( \vec{a} = i\hat{i} + j\hat{j} + 3k\hat{k} \), \( \vec{b} = i\hat{i} + 2k\hat{k} \), \( \vec{c} = -3i\hat{i} + 2j\hat{j} + k\hat{k} \) are linearly dependent vectors and the magnitude of \( \vec{a} \) is \( \sqrt{14} \), then if \( \alpha, \beta \) are integers, find \( \alpha + \beta \):
If \( \vec{a} = \hat{i} - \hat{j} + \hat{k} \), \( \vec{b} = \hat{i} + \hat{j} - 2\hat{k} \), \( \vec{c} = 2\hat{i} - 3\hat{j} - 3\hat{k} \), and \( \vec{d} = 2\hat{i} + \hat{j} + \hat{k} \) are four vectors, then \( (\vec{a} \times \vec{c}) \times (\vec{b} \times \vec{d}) = \):
The rate of change of \( x^{\sin x} \) with respect to \( (\sin x)^x \) is:
A manufacturing company noticed that 1% of its products are defective. If a dealer orders 300 items from this company, then the probability that the number of defective items is at most one is:
In a class consisting of 40 boys and 30 girls, 30% of the boys and 40% of the girls are good at Mathematics. If a student selected at random from that class is found to be a girl, then the probability that she is not good at Mathematics is:
If \( (a, \beta) \) is the orthocenter of the triangle with the vertices \( A(2, 5), B(1, 5), C(1, 4) \), then \( a + \beta = \):

If the inverse point of the point \( (-1, 1) \) with respect to the circle \( x^2 + y^2 - 2x + 2y - 1 = 0 \) is \( (p, q) \), then \( p^2 + q^2 = \) 

The equations \( 2x - 3y + 1 = 0 \) and \( 4x - 5y - 1 = 0 \) are the equations of two diameters of the circle \( S = x^2 + y^2 + 2gx + 2fy - 11 = 0 \) and \( R \) are the points of contact of the tangents drawn from the point \( P(-2, -2) \) to this circle. If \( C \) is the centre of the circle, \( S = 0 \) is the equation of the circle, then the area (in square units) of the quadrilateral \( PQCR \) is:

Roots of the equation \( a(b - c)x^2 + b(c - a)x + c(a - b) = 0 \) are:
The number of 5-digit odd numbers greater than 40,000 that can be formed by using 3, 4, 5, 6, 7, 0 such that at least one of its digits must be repeated is: