If \(C_1\) and \(C_2\) are the centres of similitude with respect to the circles \(x^2+y^2+6 x+8 y+24=0\) and \(x^2+y^2-6 x-8 y\) \(+9=0\) then \(C_1 C_2=\)

The number of admissible values of $C$ obtained when the Lagrange's mean value theorem is applied for the function $f\left(x\right)=x$ on $\left[2,5\right]$ is

\(\text { If } \tan ^{-1} x+\tan ^{-1} y+\tan ^{-1} z=\frac{\pi}{2}\)
then \(1-x y-y z-z x\) is equal to

In any triangle \(A B C\), if \(a: b: c=2: 3: 4\), then \(R: r=\)

Two dice $A$ and $B$ are rolled. If it is known that the number on $B$ is $5,$ then the probability that the sum of the numbers on the two dice will be greater than $9$ is

If Rolle's Theorem is applicable for the function \(f(x)=\left\{\begin{array}{cc}x^p \log x, & x \neq 0 \ 0, & x=0\end{array}\right.\) on the interval \([0,1]\), then a possible value of \(p\) is

If \(m\) and \(\mathrm{M}\) are respectively the smallest and greatest rational roots of the equation \(6 x^6-25 x^5+31 x^4-31 x^2+25 x-6=0\), then \(\mathrm{M}-m=\)

Two similar coils which are separated by \(2 \mathrm{~m}\) having radius of \(1 \mathrm{~m}\) and number of turns 80 have a common axis. Calculate the strength of magnetic field in microtesla at a point midway between them on their common axis when a current is \(0.2\)

The ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(b>a)\) and the parabola \(y^2=8 a x\) cut at right angles. If \(e\) is the eccentricity of the ellipse, then \(e^4\) is equal to

What is the approximate volume (in \(\mathrm{mL}\) ) of 10 vol \(\mathrm{H}_2 \mathrm{O}_2\) solution that will react completely with \(1 \mathrm{~L}\) of \(0.02 \mathrm{M} \mathrm{KMnO}_4\) solution in acidic medium?

If \(\cosh \beta=\sec \alpha \cos \theta, \sinh \beta=\operatorname{cosec} \alpha \sin \theta\), then \(\sinh ^2 \beta=\)

Four identical masses of $m$ are kept at corners of a square. If the gravitational force exerted on one of masses by the other masses is $\left(\frac{2\sqrt{2}+1}{32}\right)\frac{G{m}^2}{{ L}^2},$ then the length of the side of the square is

Let \(\overrightarrow{\mathbf{a}}=a_1 \hat{\mathbf{i}}+a_2 \hat{\mathbf{j}}+a_3 \hat{\mathbf{k}}\) Assertion (A) : The identity \(|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{i}}|^2+|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{j}}|^2+|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{k}}|^2=2|\overrightarrow{\mathbf{a}}|^2\) holds for \(\vec{a}\). Reason (R) : \(\overrightarrow{\mathbf{a}} \times \hat{\mathbf{i}}=a_3 \hat{\mathbf{j}}-a_2 \hat{\mathbf{k}}\), \(\overrightarrow{\mathbf{a}} \times \hat{\mathbf{j}}=a_1 \hat{\mathbf{k}}-a_3 \hat{\mathbf{i}}, \overrightarrow{\mathbf{a}} \times \hat{\mathbf{k}}=a_2 \hat{\mathbf{i}}-a_1 \hat{\mathbf{j}}\) Which of the following is correct?