List of top Mathematics Questions

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 Rolle's Theorem is applicable for the function:

\[ f(x) = \begin{cases} x^p \log x, & x \neq 0 \\ 0, & x = 0 \end{cases} \]

on the interval \([0,1]\), then a possible value of \( p \) is:

If are the roots of the equation \[ 2x^3 \;-\; 5x^2 \;+\; 4x \;-\; 3 \;=\; 0, \] then \[ \sum \alpha\beta(\alpha+\beta) \;=\;? \]
The trigonometric equation \( \sin^{-1}x = 2\sin^{-1}a \) has a solution. Find the valid range for \( a \).
The number of normals that can be drawn through the point \((9,6)\) to the parabola \(y^2 = 4x\) is:
If the normal drawn at the point \( P(9, 9) \) on the parabola \( y^2 = 9x \) meets the parabola again at \( Q(a, b) \), then \( 2a + b = \text{?} \)
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:
S' is the focus of the ellipse \( \frac{x^2}{25} + \frac{y^2}{b^2} = 1, (b<5) \) lying on the negative X-axis and \( P(\theta) \) is a point on this ellipse. If the distance between the foci of this ellipse is 8 and \( S'P = 7 \), then \( \theta \) is:
If \(\vec{a},\vec{b}\) are two vectors such that \(\lvert \vec{a}\rvert =3,\;\lvert \vec{b}\rvert =4,\;\lvert \vec{a}+\vec{b}\rvert =\sqrt{37},\;\lvert \vec{a}-\vec{b}\rvert = k,\) and the angle between \(\vec{a}\) and \(\vec{b}\) is \(\theta,\) then \(\frac{4}{13}\,(k \sin \theta)^2 =\,?\)
If \( f(x) = ax^2 + bx + c \) is an even function and \( g(x) = px^3 + qx^2 + rx \) is an odd function, and if \( h(x) = f(x) + g(x) \) and \( h(-2) = 0 \), then \( 8p + 4q + 2r = \)?
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 \[ Z_1 Z_2 + \overline{Z_1} Z_2 + Z_1 Z_3 + \overline{Z_1} Z_3 = \]

If the function \( f(x) \) is given by \[ f(x) = \begin{cases} \frac{\tan(a(x-1))}{\frac{x-1}{x}}, & tif0<x<1 
\frac{x^3-125}{x^2 - 25} , & \text{if } 1 \leq x \leq 4 
\frac{b^x - 1}{x}, & \text{if } x>4 \end{cases} \] is continuous in its domain, then find \( 6a + 9b^4 \).

If the coefficients of three consecutive terms in the expansion of (1 + x)23 are in arithmetic progression, then those terms are ?
If \( T_4 \) represents the 4th term in the expansion of \( \left( 5x + \frac{7}{x} \right)^{-\frac{3}{2}} \) and \( x \not\in \left[ \frac{\sqrt{7}}{5}, \frac{\sqrt{7}}{5} \right] \), then \( \left( x^3 \cdot \sqrt{5x} \right) T_4 = \):}
Given \((\sin \theta - \csc \theta)^2 + (\cos \theta + \sec \theta)^2 = 5\) and \(\theta\) lies in the third quadrant, find \((\sin \theta + \cos \theta)^3\).
The sum of the maximum and minimum values of the function \[ f(x) = \frac{x^2 - x + 1}{x^2 + x + 1} \] is:
If \[ \frac{3x^4 - 2x^2 +1}{(x-2)^4} = A + \frac{B}{x-2} + \frac{C}{(x-2)^2} + \frac{D}{(x-2)^3} + \frac{E}{(x-2)^4}, \] then \(2A + 3B - C - D + E =\;?\)
If \(\frac{x^2}{2x^2 + 7x + 6} = \frac{Ax + B}{x^2 + a} + \frac{Cx + D}{ax^2 + 3}\), then \(A + B + C - 2D =\)
f is a real valued function satisfying the relation \(f\bigl(3x + \tfrac{1}{2x}\bigr) \;=\; 9x^2 + \tfrac{1}{4x^2}\). If \(f\bigl(x + \tfrac{1}{x}\bigr) = 1\) then \(x =\) ?

A bacteria sample of certain number of bacteria is observed to grow exponentially in a given amount of time. Using exponential growth model, the rate of growth of this sample of bacteria is calculated.


The differential equation representing the growth of bacteria is given as: \[ \frac{dP}{dt} = kP, \] where \( P \) is the population of bacteria at any time \( t \). bf{Based on the above information, answer the following questions:} 
[(i)] Obtain the general solution of the given differential equation and express it as an exponential function of \( t \). 
[(ii)] If the population of bacteria is 1000 at \( t = 0 \), and 2000 at \( t = 1 \), find the value of \( k \).

Using integration, find the area of the region enclosed between the circle \( x^2 + y^2 = 16 \) and the lines \( x = -2 \) and \( x = 2 \).

The traffic police has installed Over Speed Violation Detection (OSVD) system at various locations in a city. These cameras can capture a speeding vehicle from a distance of 300 m and even function in the dark. A camera is installed on a pole at the height of 5 m. It detects a car travelling away from the pole at the speed of 20 m/s. At any point, \(x\) m away from the base of the pole, the angle of elevation of the speed camera from the car C is \(\theta\). 
On the basis of the above information, answer the following questions: 
(i)Express \(\theta\) in terms of the height of the camera installed on the pole and x.
(ii) Find \(\frac{d\theta}{dx}\).
(iii) (a) Find the rate of change of angle of elevation with respect to time at an instant when the car is 50 m away from the pole.
(iii) (b) If the rate of change of angle of elevation with respect to time of another car at a distance of 50 m from the base of the pole is \(\frac{3}{101} \, \text{rad/s}\), then find the speed of the car.

A relation \( R \) on set \( A = \{-4, -3, -2, -1, 0, 1, 2, 3, 4\} \) is defined as: \[ R = \{(x, y) : x + y \text{ is an integer divisible by 2}\}. \]

Show that \( R \) is an equivalence relation. Also, write the equivalence class \([2]\).

Let $A = I_2 - MM^T$, where $M$ is a real matrix of order $2 \times 1$ such that the relation $M^T M = I_1$ holds. If $\lambda$ is a real number such that the relation $AX = \lambda X$ holds for some non-zero real matrix $X$ of order $2 \times 1$, then the sum of squares of all possible values of $\lambda$ is equal to:
Find the particular solution of the differential equation: \[ x^2 \frac{dy}{dx} - xy = x^2 \cos^2\left(\frac{y}{2x}\right), \] given that when \(x = 1\), \(y = \frac{\pi}{2}\).