In \(\triangle A B C\), the circle that touches the sides \(B C\) internally and other two sides \(A B\) and \(A C\) externally, is called

Let \(\overline{\mathrm{i}}-2 \overline{\mathrm{j}}+\overline{\mathrm{k}}, \overline{\mathrm{i}}+\overline{\mathrm{j}}-2 \overline{\mathrm{k}}, 2 \overline{\mathrm{i}}-\overline{\mathrm{j}}-\overline{\mathrm{k}}\) and \(\overline{\mathrm{i}}+\overline{\mathrm{j}}+\overline{\mathrm{k}}\) be the position vectors of four points \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) and D respectively. If a point P divides AB in the ratio \(2: 1\) internally and a point \(Q\) divides \(C D\) in the ratio \(1: 2\) externally, then the ratio in which the point with position vector \(5 \overline{\mathrm{i}}-6 \overline{\mathrm{j}}-5 \overline{\mathrm{k}}\) divides PQ is

\(\int x^3(\log x)^2 d x=\)

The angle between the circles \(x^2+y^2+4 x-14 y+28=0\) and \(x^2+y^2-12 x-6 y-4=0\) is

If each of the coefficients \(a, b, c\) in the equation \(a x^2+b x\) \(+c=0\) is determined by throwing a die, then the probability that the equation will have equal roots, is

The equation of the pair of straight lines perpendicular to the pair $2x^2+3xy+2y^2+10x+$$5y=0$ and passing though the origin is______

If the points whose position vectors are \(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, 6 \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}\) and \(14 \hat{\mathbf{i}}-5 \hat{\mathbf{j}}+p \hat{\mathbf{k}}\) are collinear, then the value of \(p\) is
are collinear, then the value of \(p\) is

For any natural number \(n,\left(15 \times 5^{2 n}\right)+\left(2 \times 2^{3 n}\right)\) is divisible by

The rms speed of oxygen molecule at a certain absolute temperature is ' \(v\) '. If the absolute temperature is doubled and the oxygen molecules dissociate into atomic oxygen, then the rms speed would be

The value of \(c\) in the Lagrange's mean value theorem for \(f(x)=\sqrt{x-2}\) in the interval \([2,6]\) is

From a bag containing 4 white and 5 red balls, if 3 balls are drawn at random, then the mean of the number of red balls among the balls drawn, is

The random variable takes the values \(1,2,3\), \(\ldots, m\). If \(P(X=n)=\frac{1}{m}\) to each \(n\), then the variance of \(X\) is

Assertion (A) : \(\mathrm{Na}^{+}\)and \(\mathrm{Mg}^{2+}\) ions are isoelectronic but the ionic radius of \(\mathrm{Na}^{+}\)is greater than that of \(\mathrm{Mg}^{2+}\).
Reason (R) : The effective nuclear charge of \(\mathrm{Na}^{+}\)ion is less than that of \(\mathrm{Mg}^{2+}\) ion.