A circle C passing through the point \((1,1)\) bisects the circumference of the circle \(x^2+y^2-2 x=0\). If C is orthogonal to the circle \(x^2+y^2+2 y-3=0\) then the centre of the circle C is

By using the non-zero digits, the number of 5 digit numbers that can be formed so that each number has largest digit in its middle place and the digits in the number are different is

If \(a, b, c, \neq 0\) and belong to the set to \(\{0,1,2\), \(3 \ldots \ldots, 9\}\), then \(\log _{10}\left(\frac{a+10 b+10^2 c}{10^{-4} a+10^{-3} b+10^{-2} c}\right)\) is equal to

If \(\cos x+\cos y=\frac{2}{3}\) and \(\sin x-\sin y=\frac{3}{4}\) then \(\sin (x-y)+\cos (x-y)=\)

The angle between the line of intersection of the two planes \(\mathbf{r} \cdot(2 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}-3 \hat{\mathbf{k}})=5\), \(\mathbf{r} \cdot(3 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}-5 \hat{\mathbf{k}})=3\) and the line \(\mathbf{r}=3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}}+t(5 \hat{\mathbf{i}}+5 \hat{\mathbf{j}}-7 \hat{\mathbf{k}})\) is

A bag contains \(n\) white and \(n\) black balls. Pairs of balls are drawn at random without replacement successively, until the bag is empty. If the number of ways in which each pair consists of one white and one black ball is 14400 , then \(n\) is equal to

The normal to a circle \(S=0\) at \(P(1,3)\) is \(x+2 y=7\) and it has another normal at \(Q(3,5)\) which is the polar of the point \(A\left(7,-\frac{1}{2}\right)\) with respect to the circle \(x^2+y^2-4 x+6 y-12=0\). Then, the equation of the circle \(S=0\) is

Identify the correct statements from the folluwing (A) At 298 K, the potential of hydrogen electrode placed in a solution of \(\mathrm{pH}=10\), is -0.59 V (B) The limiting molar conductivity of \(\mathrm{Ca}^{2+}\) and \(\mathrm{Cl}^{-}\)is 119 and \(76 \mathrm{~S} \mathrm{~cm}^2 \mathrm{~mol}^{-1}\) respectively. The limiting molar conductivity of \(\mathrm{CaCl}_2\) is \(195 \mathrm{~S} \mathrm{~cm}^2 \mathrm{~mol}^{-\frac{1}{2}}\) (C) The correct relationship between \(\mathrm{K}_{\mathrm{c}}\) and \(\mathrm{E}_{\text {cell }}^0\) is \(\mathrm{E}_{\mathrm{ccll}}^{-}=\frac{2.303 \mathrm{RT}}{\mathrm{nF}} \log \mathrm{~K}_{\mathrm{c}}\)

Let \(\mathrm{p}_1, \mathrm{p}_2, \mathrm{p}_3\) be the altitudes of a triangle ABC drawn through the vertices \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) respectively. If \(r_1=4, r_2=6, r_3=12\) are the ex-radii of triangle ABC then \(\frac{1}{\mathrm{p}_1^2}+\frac{1}{\mathrm{p}_2^2}+\frac{1}{\mathrm{p}_3^2}=\)

Fluorosis disease is caused due to the reaction of ...... with excess of fluoride in the body.

\(\int \frac{x^2 \operatorname{Tan}^{-1} x}{\left(1+x^2\right)^2} d x=\)

Which alkali metal emits blue colour light in flame test?

Identify an antioxidant, an antiseptic, and an antibiotic respectively from the following:

Equanil	Chloramphenicol	Bithionol
$\left(\mathrm{A}\right)$	$\left(\mathrm{B}\right)$	$\left(\mathrm{C}\right)$
Aspartame	Dimetapp	Butylated hydroxytoluene
$\left(\mathrm{D}\right)$	$\left(\mathrm{E}\right)$	$\left(\mathrm{F}\right)$