List of top Mathematics Questions asked in KEAM

Simplify the following expression: \[ \cos 18^\circ \cos 42^\circ \cos 78^\circ. \] The simplified form is:

The integral

\[ \int \frac{\sec x}{(\sec x + \tan x)^2} \, dx \] is:

If \[ \sin^{-1} x + \sin^{-1} y + \sin^{-1} z = \frac{3\pi}{2}, \] then \( x + y + z = \):

Real part of \[ \left( \frac{1+i}{1-i} \right) \left( \frac{2+i}{2-i} \right) \] is:

Let \( \overrightarrow{a} = 2i + 3j - 4k, \quad \overrightarrow{b} = i + j - k, \quad \overrightarrow{c} = -i + 2j + 3k, \quad \overrightarrow{d} = i + j + k. \) Then \[ (\overrightarrow{a} \times \overrightarrow{b}) \cdot (\overrightarrow{c} \times \overrightarrow{d}) = \]

The angle between the two straight lines \[ \overrightarrow{r_1} = (4i - k) + t(2i + j - 2k), \quad t \in \mathbb{R}, \quad {and} \quad \overrightarrow{r_2} = (i - j + 2k) + s(2i - 2j + k), \quad s \in \mathbb{R} \] is:

The common point of the two straight lines \[ \overrightarrow{r_1} = (i - 2j + 3k) + s(2i + j + k) \quad {and} \quad \overrightarrow{r_2} = (-i + 2j + 7k) + t(i + j + k), \quad t, s \in \mathbb{R} \] is:

If \( z \) is a complex number of unit modulus, then \[ \left| \frac{1+z}{1+ \overline{z}} \right| \] equals:

The volume of the parallelepiped whose coterminous vectors are given by the vectors \[ \overrightarrow{a} = i - j + k, \quad \overrightarrow{b} = 3i + j - k, \quad \overrightarrow{c} = 5i + 2j - 7k \] is (in cubic units):

The integral

\[ \int \frac{dx}{x^8 \left( 1 + x^7 \right)^{2/3}} \] is equal to:

The area enclosed by the curve \[ x = 3 \cos \theta, \quad y = 5 \sin \theta, \quad 0 \leq \theta \leq 2\pi, \] is equal to:

Let \( z \) be a non-zero complex number such that \[ z = \frac{16}{\bar{z}}. \] Then the locus of \( z \) is:

If \( \left\lfloor x^2 \right\rfloor \) is the greatest integer less than or equal to \( x^2 \), then \[ \int_0^{\sqrt{2}} \left\lfloor x^2 \right\rfloor \, dx = \]

The solution of \( \frac{dy}{\cos y} = dx \) is:

Let \(a,b,c,d \) be an increasing sequence of real numbers,which are in geometric progression .If \(a+d=112\) and \(b+c=48\) ,then the value of \(\dfrac{a+c+8}{b}\) is 

A thin particle moves from \((0,1)\) and gets reflected upon hitting the x-axis at \((√3,0)\). Then the slope of the reflected line is ?

A six faced fair die is rolled for a large number of times. Then, the mean value of the outcome is 

In a linear programming problem ,the restrictions under which the objective function is to be optimized are called as?

Suppose the point \(P(1,1)\) is translated to \(Q\) in the direction of y = 2x. If \(PQ = 1\),then Qis

If\( \dfrac{1+sinx}{1-sinx}=\dfrac{(1+siny)^3}{(1-siny)^3}\) for some real values \(x\) and \(y\) ,then \(\dfrac{sinx}{siny}\)=