List of top Mathematics Questions

Let $f: R \rightarrow R$ be defined as, $f(x) = \begin{cases} -55x, & \text{if } x < -5 \\ 2x^3 -3x^2 -120x, & \text{if } 5 \le x \le 4 \\ 2x^3 -3x^2-36x-336& \text{if } x > 4, \end{cases} $ Let $A=\{ x \in R : f$ is increasing $\} .$ Then $A$ is equal to

Let $R =\{( P , Q ) | P$ and $Q$ are at the same distance from the origin $\}$ be a relation, then the equivalence class of (1,-1) is the set:

The pitch of the screw gauge is $1 mm$ and there are $100$ divisions on the circular scale. When nothing is put in between the jaws, the zero of the circular scale lies $ 8$ divisions below the reference line. When a wire is placed between the jaws, the first linear scale division is clearly visible while $72^{\text {nd }}$ division on circular scale coincides with the reference line. The radius of the wire is

The value of $-{ }^{15} C _{1}+2 .{ }^{15} C _{2}-3 .{ }^{15} C _{3}+\ldots $ $-15 .{ }^{15} C _{15}+{ }^{14} C _{1}+{ }^{14} C _{3}+{ }^{14} C _{5}+\ldots .+{ }^{14} C _{11}$ is

If $P$ is a point on the parabola $y=x^{2}+4$ which is closest to the straight line $y =4 x -1$, then the co-ordinates of $P$ are

Let $f$ be a twice differentiable function defined on $R$ such that $f (0)=1, f'(0)=2$ and $f'(x)\neq 0$ for all $x \in R$. If $\left|\begin{array}{cc}f(x) & f'(x) \\ f'(x) & f''(x)\end{array}\right|=0,$ for all $x \in R,$ then the value of $f (1)$ lies in the interval:

The value of the integral, \[ \int_{1}^{3} \left[ x^{2} - 2x - 2 \right] \, dx, \] where \( [x] \) denotes the greatest integer less than or equal to \( x \), is:

The total number of positive integral solutions $( x , y , z )$ such that $xyz =24$ is :

The negative of the statement $\sim p \wedge( p \vee q )$ is

If the curve $y=a x^{2}+b x+c, x \in R,$ passes through the point (1,2) and the tangent line to this curve at origin is $y = x ,$ then the possible values of $a , b , c$ are :

the angle of intersection of the curves y=x² and x=y² at (1,1) is

In the figure \(∠BAO 30°, ∠BCO=40°\) then \(∠AOC = ?\)

One of the roots of the equation, \( z^6 - 3z^4 - 16 = 0 \) is given by \( z_1 = 2 \). The value of the product of the other five roots is ...........

The radial component of acceleration in plane polar coordinates is given by:

In the Fourier series expansion of two functions \( f_1(t) = 4t^2 + 3 \) and \( f_2(t) = 6t^3 + 7t \) in the interval \( -\frac{T}{2} \) to \( +\frac{T}{2} \), the Fourier coefficients \( a_n \) and \( b_n \) (\( a_n \) and \( b_n \) are coefficients of \( \cos(n\omega t) \) and \( \sin(n\omega t) \), respectively) satisfy:

The solution \( y(x) \) of the differential equation \( y\frac{dy}{dx} + 3x = 0 \), \( y(1) = 0 \), is described by:

Let \( M \) be a \( 2 \times 2 \) matrix. Its trace is 6 and its determinant has value 8. Its eigenvalues are:

The function \( e^{\cos x} \) is Taylor expanded about \( x = 0 \). The coefficient of \( x^2 \) is:

Let \( \alpha, \beta \) and \( \gamma \) be the eigenvalues of \[ M = \begin{bmatrix} 0 & 1 & 0 \\ 1 & 3 & 3 \\ -1 & 2 & 2 \end{bmatrix}. \] If \( \gamma = 1 \) and \( \alpha > \beta \), then the value of \( 2\alpha + 3\beta \) is .............

Let \[ M = \begin{bmatrix} 5 & -6 \\ 3 & -4 \end{bmatrix} \] be a \(2 \times 2\) matrix. If \(\alpha = \det(M^4 - 6I_2)\), then the value of \(\alpha^2\) is ............

Let \( M \) be a \(3 \times 3\) real matrix. Let \[ \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}, \quad \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}, \quad \begin{pmatrix} 0 \\ -1 \\ \alpha \end{pmatrix} \] be eigenvectors of \(M\) corresponding to three distinct eigenvalues. Then, which of the following is NOT a possible value of \(\alpha\)?

For \( a \in \mathbb{R} \), consider the system of linear equations \[ \begin{cases} a x + a y = a + 2,
x + a y + (a - 1)z = a - 4,
a x + a y + (a - 2)z = -8, \end{cases} \] in the unknowns \(x, y, z\). Then, which of the following statements is TRUE?