The locus of a point on the argand plane represented by the complex number \(z\), when \(z\) satisfies the condition \(\left|\frac{z-1+i}{z+1-i}\right|=\left|\operatorname{Re}\left(\frac{z-1+i}{z+1-i}\right)\right|\) is

\(\int \frac{3^x(x \log 3-1)}{x^2} d x=\)

If the lines \(L_1 \equiv x-2 y+3=0, L_2 \equiv 2 x+y+1=0\) and \(L_3 \equiv 3 x+y+c=0\) are concurrent and \(\theta\) is the acute angle between the lines \(L_1=0\) and \(L_3=0\), then \(\tan \theta=\)

If the coefficient of \(x\) is in expansion of \(\left(x^2+\frac{k}{x}\right)^5\) is 270 , then \(k\) is equal to

The angle between the curves $2x^2+y^2=20$ and $4y^2-x^2=8$ at a point where they intersect in the $4^{\mathrm{th}}$ quadrant is

If $M_1$ and $M_2$ are the mean deviations from mean and median of the first $15$ even integers then $M_1+M_2=$

\(\int_{-\pi / 2}^{\pi / 2} \sin |x| d x\) is equal to

If two equal sides of an isosceles triangle are given by the equations \(7 x-y+3=0\) and \(x+y-3=0\), then the equation of its third side passing through the point \((2,-5)\) is

Let \(A=(2,3), B=(3,-5)\) be two vertices of \(\triangle A B C\) such that \(C\) is a point on the line \(L \equiv 3 x+4 y-5=0\). Then the locus of the centroid of \(\triangle A B C\) is a line parallel to

The indicator used during the estimation of $\mathrm{COD}$ of a water sample is

If five dice are thrown simultaneously, then the probability that atleast three of them show the same numbered face is

Consider the following statements: Assertion (A): The direction ratios of a line \(\mathrm{L}_1\) are \(2,5,7\) and the direction ratios of another line \(\mathrm{L}_2\) are \(\frac{4}{\sqrt{19}}, \frac{10}{\sqrt{19}}, \frac{14}{\sqrt{19}}\). Then the lines \(\mathrm{L}_1, \mathrm{~L}_2\) are parallel Reason (R): If the direction ratios of a line \(L_1\) are \(a_1, b_1, c_1\), the direction ratios of a line \(\mathrm{L}_2\) are \(\mathrm{a}_2, \mathrm{~b}_2, \mathrm{c}_2\) and \(\mathrm{a}_1 \mathrm{a}_2+\mathrm{b}_1 \mathrm{~b}_2\) \(+c_1 c_2=0\), then the lines of \(L_1, L_2\) are parallel Which one of the following is True?

The equation of a given straight line is \(\frac{x-x_1}{\cos \theta}=\frac{y-y_1}{\sin \theta}=\gamma\). If the equation of the line perpendicular to the given line and passing through \((\alpha, \beta)\) is \(\frac{x}{a}+\frac{y}{b}=1\), then \(\frac{b}{a}\) is equal to