The number of arrangements of the letters of the word "INDEPENDENCE" in which all the vowels always occur together is :

The number of ways, in which 5 girls and 7 boys can be seated at a round table so that no two girls sit together, is :

Let the number of elements in sets $$A$$ and $$B$$ be five and two respectively. Then the number of subsets of $$A\times B$$ each having at least 3 and at most 6 elements is :

All the letters of the word PUBLIC are written in all possible orders and these words are written as in a dictionary with serial numbers. Then the serial number of the word PUBLIC is :

The number of integral values of k, for which one root of the equation $$2x^2-8x+k=0$$ lies in the interval (1, 2) and its other root lies in the interval (2, 3), is :

The equation ${\mathrm{e}}^{4x}+8{\mathrm{e}}^{3x}+13{\mathrm{e}}^{2x}-8{\mathrm{e}}^x+1=0,x\in \mathbb{R}$ has :

Let $$S=\left\{x:x\in \mathbb{R}\mathrm{and}{(\sqrt{3}+\sqrt{2})}^{x^2-4}+{(\sqrt{3}-\sqrt{2})}^{x^2-4}=10\right\}$$. Then $$n(S)$$ is equal to

The number of real roots of the equation $$\sqrt{x^2-4x+3}+\sqrt{x^2-9}=\sqrt{4x^2-14x+6}$$, is :

Let $$\lambda \ne 0$$ be a real number. Let $$\alpha ,\beta$$ be the roots of the equation $$14x^2-31x+3\lambda =0$$ and $$\alpha ,\gamma$$ be the roots of the equation $$35x^2-53x+4\lambda =0$$. Then {{3\alpha } \over \beta } and {{4\alpha } \over \gamma } are the roots of the equation

The number of real solutions of the equation 3\left( {{x^2} + {1 \over {{x^2}}}} \right) - 2\left( {x + {1 \over x}} \right) + 5 = 0, is

The equation $$x^2-4x+[x]+3=x[x]$$, where $$[x]$$ denotes the greatest integer function, has :

The number of real roots of the equation $x|x|-5|x+2|+6=0$, is :

Let $$\alpha ,\beta$$ be the roots of the equation $$x^2-\sqrt{2}x+2=0$$. Then $${\alpha }^{14}+{\beta }^{14}$$ is equal to

The set of all $$a\in \mathbb{R}$$ for which the equation $$x|x-1|+|x+2|+a=0$$ has exactly one real root, is :

Let $$\alpha ,\beta$$ be the roots of the quadratic equation $$x^2+\sqrt{6}x+3=0$$. Then $$\frac{{\alpha }^{23}+{\beta }^{23}+{\alpha }^{14}+{\beta }^{14}}{{\alpha }^{15}+{\beta }^{15}+{\alpha }^{10}+{\beta }^{10}}$$ is equal to :

Let $$\alpha ,\beta ,\gamma$$ be the three roots of the equation $$x^3+bx+c=0$$. If $$\beta \gamma =1=-\alpha$$, then $$b^3+2c^3-3{\alpha }^3-6{\beta }^3-8{\gamma }^3$$ is equal to :

Let $$A=\{x\in R:[x+3]+[x+4]\le 3\},$$ $$B = \left\{ {x \in R:{3^x}{{\left( {\sum\limits_{r = 1}^$\infty$ {{3 \over {{{10}^r}}}} } \right)}^{x - 3}}

The sum of all the roots of the equation $$\left|x^2-8x+15\right|-2x+7=0$$ is :

Let $$\mathrm{P}\left(x_0,y_0\right)$$ be the point on the hyperbola $$3x^2-4y^2=36$$, which is nearest to the line $$3x+2y=1$$. Then $$\sqrt{2}\left(y_0-x_0\right)$$ is equal to :

Let $\mathrm{H}$ be the hyperbola, whose foci are $(1\pm \sqrt{2},0)$ and eccentricity is $\sqrt{2}$. Then the length of its latus rectum is :