If the circles \(x^2+y^2-8 x-8 y+28=0\) and \(x^2+y^2-8 x-6 y\) \(+25-\alpha^2=0\) have only one common tangent, then \(\alpha=\)

If the line joining the points \(\overline{\mathrm{i}}+2 \overline{\mathrm{j}}\) and \(\overline{\mathrm{j}}-2 \overline{\mathrm{k}}\) intersects the plane passing through the points \(2 \overline{\mathrm{i}}-\overline{\mathrm{j}}, 2 \overline{\mathrm{j}}+3 \overline{\mathrm{k}}\) and \(\overline{\mathrm{k}}-2 \overline{\mathrm{i}}\) at \(\overline{\mathrm{r}}\), then \(\overline{\mathrm{r}} \cdot(\overline{\mathrm{i}}+\overline{\mathrm{j}}+\overline{\mathrm{k}})=\)

If \(x>0, y>0, z>0, x y+y z+z x < 1\) and if \(\tan ^{-1} x+\tan ^{-1} y+\tan ^{-1} z=\pi\), then \(x+y+z\) equals to

In a triangle ABC , if \((a-b)(s-c)=(b-c)(s-a)\), then \(r_1+r_3=\)

For \(|x| < 1\), the constant term in the expansion of \(\frac{1}{(x-1)^2(x-2)}\) is

In \(\triangle A B C, a=6 \mathrm{~cm}, b=10 \mathrm{~cm}\) and \(c=14 \mathrm{~cm}\). Then, the sum of the acute angles of the triangle is

If \(z=x+i y\) is a complex number satisfying \(\left|z+\frac{i}{2}\right|^2=\left|z-\frac{i}{2}\right|^2\), then the locus of \(z\) is

A student is at a distance \(16 \mathrm{~m}\) from a bus when the bus begins to move with a constant acceleration of \(9 \mathrm{~m} \mathrm{~s}^{-2}\). The minimum velocity with which the student should run towards the bus so as the catch it is \(\alpha \sqrt{2} \mathrm{~ms}^{-1}\). The value of \(\alpha\) is

If the circles \(\mathrm{S} \equiv x^2+y^2-14 x+6 y+33=0\) and \(\mathrm{S}^{\prime} \equiv x^2+\) \(y^2-a^2=0(a \in \mathrm{~N})\) have 4 common tangents then possible number of values of \(a\) is

A prism \((\mu=1.5)\) has the refracting angle of \(30^{\circ}\). The deviation of a monochromatic ray incident normally on its one surface will be (given, \(\sin 48^{\circ} 36^{\prime}=0.75\))

The formation of oxide ion \(\mathrm{O}^{2-}\) from oxygen atom requires an exothermic reaction, followed by an endothermic step as shown below. The process of formation of \(\mathrm{O}^{2-}\) in gas phase is unfavorable \(\left(\Delta H^{\ominus}=+\right.\) ve \()\), even though it has stable configuration of the nearest noble gas neon. This is because
\(\begin{aligned} & \mathrm{O}(g)+e^{-} \longrightarrow \mathrm{O}^{-}(g) ; \Delta H^{\ominus}=-141 \mathrm{~kJ} \mathrm{~mol}^{-1} \\ & \mathrm{O}^{-}(g)+e^{-} \longrightarrow \mathrm{O}^{2-}(g) ; \Delta H^{\ominus}=+760 \mathrm{~kJ} \mathrm{~mol}^{-1}\end{aligned}\)

For a binomial variate \(X\) with parameters \(n=5\) and \(p=\frac{3}{4}\), if \(\alpha=\frac{1}{9} P(X \geq 3)\) and \(\beta=P(X \leq 2)\), then \(256(\beta-\alpha)=\)

Four resistors \(A, B, C\) and \(D\) form a Wheatstone bridge as shown in the figure. The bridge is balanced when \(C=100 \Omega\). If \(A\) and \(B\) are interchanged, the bridge balances for \(C=121 \Omega\). The value of \(D\) is