List of top Mathematics Questions

If \( a \neq b \neq c \), then

\[ \Delta_1 = \begin{vmatrix} 1 & a^2 & bc \\ 1 & b^2 & ca \\ 1 & c^2 & ab \end{vmatrix}, \quad \Delta_2 = \begin{vmatrix} 1 & 1 & 1 \\ a^2 & b^2 & c^2 \\ a^3 & b^3 & c^3 \end{vmatrix} \]

and

\[ \frac{\Delta_1}{\Delta_2} = \frac{6}{11} \]

then what is \( 11(a + b + c) \)?

If \( A = \begin{pmatrix} 9 & 3 & 0 \\ 1 & 5 & 8 \\ 7 & 6 & 2 \end{pmatrix} \) and

\[ A^T A^{-2} = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix} \]

then

\[ \sum_{1 \leq i \leq 3} \sum_{1 \leq j \leq 3} a_{ij} \]

is:

Solving the System of Linear Equations

If (x,y,z) = (α,β,γ) is the unique solution of the system of simultaneous linear equations:

    3x - 4y + 2z + 7 = 0,    2x + 3y - z = 10,    x - 2y - 3z = 3,    

then α = ?

If \(AX = D\) represents the system of linear equations \[ 3x - 4y + 7z + 6=0,\quad 5x + 2y - 4z + 9=0,\quad 8x - 6y - z + 5=0, \] then AX = D ?

Matrix Inverse Sum Calculation

Given the matrix:

    A =    | 1  2  2 |    | 3  2  3 |    | 1  1  2 |    

The inverse matrix is represented as:

    A-1 =    | a11  a12  a13 |    | a21  a22  a23 |    | a31  a32  a33 |    

The sum of all elements in A-1 is:

∑ aij = 3

If the real-valued function

\[ f(x) = \sin^{-1}(x^2 - 1) - 3\log_3(3^x - 2) \]

is not defined for all \( x \in (-\infty, a] \cup (b, \infty) \), then what is \( 3^a + b^2 \)?

∫ √(2x2 - 5x + 2) dx = ∫ (41/60) dx,

and

-1/2 > α > 0, then α = ?

The number of common roots among the 12th and 30th roots of unity is ?

Calculate the determinant of the matrix:
 


 

S = (-1,1) is the focus, \( 2x - 3y + 1 = 0 \) is the directrix corresponding to S and \( \frac{1}{2} \) is the eccentricity of an ellipse. If \( (a,b) \) is the centre of the ellipse, then \( 3a + 2b \) is:

Calculate the variance of the data set: 1, 2, 3, 5, 8, 13, 17.
Evaluate the integral: \[ \int e^{-2x} \left( \tan 2x - 2\sec^2 2x \tan 2x \right) dx. \]
If \( Z_1, Z_2, Z_3 \) are three complex numbers with unit modulus such that \[ |Z_1 - Z_2|^2 + |Z_1 - Z_3|^2 = 4 \] then \[ Z_1 Z_2 + \overline{Z_1} Z_2 + Z_1 Z_3 + \overline{Z_1} Z_3 = \]

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

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 \( a_{22} = 2 \). 
 

The direction cosines of two lines are connected by the relations \[ l - m + n = 0, \quad 2lm - 3mn + nl = 0. \] If \( \theta \) is the angle between these two lines, then \( \cos \theta \) is:

The mean deviation about the mean for the following data is:
 


 

The position vectors of the vertices \( A, B, C \) of a triangle are given as: \[ \vec{A} = \hat{i} - 2\hat{j} + k, \quad \vec{B} = 2\hat{i} + \hat{j} - k, \quad \vec{C} = \hat{i} - \hat{j} - 2k \] If \( D \) and \( E \) are the midpoints of \( BC \) and \( CA \) respectively, then the unit vector along \( \overrightarrow{DE} \) is:
If \(\tan A<0\) and \(\tan 2A = -\frac{4}{3}\), then \(\cos 6A\) is:

A student has to write the words ABILITY, PROBABILITY, FACILITY, MOBILITY. He wrote one word and erased all the letters in it except two consecutive letters. If 'LI' is left after erasing then the probability that the boy wrote the word PROBABILITY is: \

One of the roots of the equation \( x^{14} + x^9 - x^5 - 1 = 0 \) is:
x and y are two complex numbers such that \(|x| = |y| = 1\). If \( \text{Arg}(x) = 2\alpha \), \( \text{Arg}(y) = 3\beta \) and \( \alpha + \beta = \frac{\pi}{36} \), then \( x^6 y^4 + \frac{1}{x^6 y^4} \) is:
The point P denotes the complex number \(z = x + iy\) in the Argand plane. If \( \frac{2z - i}{z - 2} \) is purely real number, then the equation of the locus of P is.

Let

\( A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \alpha & \beta \\ 0 & \beta & \alpha \end{bmatrix} \)

and \(|2A|^3 = 2^{21}\) where \(\alpha, \beta \in \mathbb{Z}\). Then a value of \(\alpha\) is:

If \(\vec{a},\,\vec{b},\,\vec{c}\) are non-coplanar vectors. If the three points \[ \lambda \vec{a} - 2\vec{b} + \vec{c},\quad 2 \vec{a} + \lambda\vec{b} - 2\vec{c},\quad 4\vec{a} + 7\vec{b} - 8\vec{c} \] are collinear, then \(\lambda = \,?\)