Let \(d\) be the distance of the point of intersection of the lines \(\frac{x+6}{3}=\frac{y}{2}=\frac{z+1}{1} \quad\) and \(\frac{x-7}{4}=\frac{y-9}{3}=\frac{z-4}{2}\) from the point \((7,8,9)\). Then \(\mathrm{d}^2+6\) is equal to :

For the system of linear equations : \(x-2 y=1, x-y+k z=-2, k y+4 z=6, k \in R\) consider the following statements : \((A)\) The system has unique solution if \(k \neq 2\), \(k \neq-2\) \((B)\) The system has unique solution if \(k =-2\). \((C)\) The system has unique solution if \(k =2\). \((D)\) The system has no-solution if \(k =2\). \((E)\) The system has infinite number of solutions if \(k \neq-2\) Which of the following statements are correct?

Let \({a_1},{a_2}...,{a_{10}}\) be a \(G.P.\) If \(\frac{{{a_3}}}{{{a_1}}} = 25,\) then \(\frac {{{a_9}}}{{{a_{  5}}}}\) equal

Let \(\theta=\frac{\pi}{5}\) and \(A=\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta\end{array}\right] \cdot\) If \(B=A + A ^{4},\) then \(\operatorname{det}( B )\)

Consider the quadratic equation \(\left( {c - 5} \right)\,{x^2} - 2cs + \left( {c - 4} \right) = 0\), \(c \ne 5\). Let \(S\) be the set of all integral values of \(c\) for which one root of the equation lies in the interval \((0, 2)\) and its other root lies in the interval \((2, 3)\). Then the number of elements in \(S\) is

The system of linear equation \(x + y + z = 2, 2x + 3y + 2z = 5\), \(2x + 3y + (a^2 -1)\,z = a + 1\) then

The mean and standard deviation of \(15\) observations are found to be \(8\) and \(3\) respectively. On rechecking it was found that, in the observations, \(20\) was misread as \(5\) . Then, the correct variance is equal to......

If the constant term in the expansion of \(\left(\frac{\sqrt[5]{3}}{x}+\frac{2 x}{\sqrt[3]{5}}\right)^{12}, x \neq 0\), is \(\alpha \times 2^8 \times \sqrt[5]{3}\), then \(25 \alpha\) is equal to :

All five letter words are made using all the letters \(\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}, \mathrm{E}\) and arranged as in an English dictionary with serial numbers. Let the word at serial number \(n\) be denoted by \(W_n\). Let the probability \(\mathrm{P}\left(\mathrm{W}_{\mathrm{n}}\right)\) of choosing the word \(\mathrm{W}_{\mathrm{n}}\) satisfy \(\mathrm{P}\left(\mathrm{W}_{\mathrm{n}}\right)=2 \mathrm{P}\left(\mathrm{W}_{\mathrm{n}-1}\right), \mathrm{n} \gt 1\).
If \(\mathrm{P}(\mathrm{CDBEA})=\frac{2^\alpha}{2^\beta-1}, \alpha, \beta \in \mathbb{N}\), then \(\alpha+\beta\) is equal to : _______

Let the system of linear equations \(4 x+\lambda y+2 z=0\) ;  \(2 x-y+z=0\) ;  \(\mu x +2 y +3 z =0, \lambda, \mu \in R\) has a non-trivial solution. Then which of the following is true?

If \( lim_{x\rightarrow0}\frac{e^{(a-1)x}+2~cos~bx+(c-2)e^{-x}}{x~cos~x-log_{e}(1+x)}=2, \) then \( a^{2}+b^{2}+c^{2} \) is equal to:

Let \(P_{1}: \vec{r} \cdot(2 \hat{ i }+\hat{ j }-3 \hat{ k })=4\) be a plane. Let \(P_{2}\) be another plane which passes through the points \((2,-\) \(3,2)(2,-2,-3)\) and \((1,-4,2)\). If the direction ratios of the line of intersection of \(P_{1}\) and \(P_{2}\) be \(16\) , \(\alpha, \beta\), then the value of \(\alpha+\beta\) is equal to

Let \(E\) and \(F\) be two independent events. The probability that both \(E\) and \(F\) happen is \(\frac{1}{12}\) and the probability that neither \(E\) nor \(F\) happens is  \(\frac{1}{2}\) , then a value of  \(\frac{{P(E)}}{{P\left( F \right)}}\) is