Let \(S\) be the set of all the natural numbers, for which the line \(\frac{x}{a}+\frac{y}{b}=2\) is a tangent to the curve \(\left(\frac{ x }{ a }\right)^{ n }+\left(\frac{ y }{ b }\right)^{ n }=2\) at the point \(( a , b ), ab \neq 0\). Then

Let \(a-2 b+c=1\) If \(f(x)=\left|\begin{array}{lll}{x+a} & {x+2} & {x+1} \\ {x+b} & {x+3} & {x+2} \\ {x+c} & {x+4} & {x+3}\end{array}\right|,\) then

If the area of the region \(\{(x, y):|x-5| \leq y \leq 4 \sqrt{x}\}\) is A , then 3 A is equal to _____ .

If one of the diameter s of the circle, given by the equation, \({x^2} + {y^2} - 4x + 6y - 12 = 0\) is a chord of a circle \(S,\) whose centre is at \((-3, 2),\) then the radius of \(S\) is:

If \(\alpha\) denotes the number of solutions of \(|1-i|^x=2^x\) and \(\beta=\left(\frac{|z|}{\arg (z)}\right)\), where \(z=\frac{\pi}{4}(1+i)^4\left(\frac{1-\sqrt{\pi i}}{\sqrt{\pi}+i}+\frac{\sqrt{\pi}-i}{1+\sqrt{\pi} \mathrm{i}}\right), i=\sqrt{-1}\), then the distance of the point \((\alpha, \beta)\) from the line \(4 x-3 y=7\) is

Let \(\vec{a}, \vec{b}, \vec{c}\) be three vectors mutually perpendicular to each other and have same magnitude. If a vector \(\overrightarrow{\mathrm{r}}\) satisfies. \(\overrightarrow{\mathrm{a}} \times\{(\overrightarrow{\mathrm{r}}-\overrightarrow{\mathrm{b}}) \times \overrightarrow{\mathrm{a}}\}+\overrightarrow{\mathrm{b}} \times\{(\overrightarrow{\mathrm{r}}-\overrightarrow{\mathrm{c}}) \times \overrightarrow{\mathrm{b}}\}+\overrightarrow{\mathrm{c}} \times\{(\overrightarrow{\mathrm{r}}-\overrightarrow{\mathrm{a}}) \times \overrightarrow{\mathrm{c}}\}=\overrightarrow{0}\) then \(\overrightarrow{\mathrm{r}}\) is equal to:

The number of elements in the set \(\{ z = a + ib \in C : a , b \in Z\) and \(1<| z -3+2 i |<4\}\) is.......

A multiple choice examination has \(5\) questions. Each question has three alternative answers of which exactly one is correct. The probability that a student will get \(4\) or more correct answers just by guessing is :

Let the tangent drawn to the parabola \(y ^{2}=24 x\) at the point \((\alpha, \beta)\) is perpendicular to the line \(2 x\) \(+2 y=5\). Then the normal to the hyperbola \(\frac{x^{2}}{\alpha^{2}}-\frac{y^{2}}{\beta^{2}}=1\) at the point \((\alpha+4, \beta+4)\) does \(NOT\) pass through the point.

The square of the distance of the image of the point \((6,1,5)\) in the line \(\frac{x-1}{3}=\frac{y}{2}=\frac{z-2}{4}\), from the origin is .............

If the area of an equilateral triangle inscribed in the circle, \(x^2 + y^2 + 10x + 12y + c = 0\) is \(27\sqrt 3 \,sq.\,units\) then \(c\) is equal to

Let the function \(f(x)=2 x^3+(2 p-7) x^2+3(2 p-9) x-6\) have a maxima for some value of \(x < 0\) and a minima for some value of \(x > 0\). Then, the set of all values of \(p\) is \(......\)

The integral \(\int {\frac{{{{\sin }^2}\,x\,{{\cos }^2}\,x}}{{({{\sin }^3}\,x\, + {{\cos }^3}\,x)^2}}} dx\) is equal to