A' can run 1 km in 5 minutes 20 seconds and 'B' can run the same distance in 6 minutes. How many meters start can 'A' give 'B' in a kilometer race so that they finish the race together?

If \(\hat{a}\) is a unit vector perpendicular to both the vectors \(\vec{b}=\hat{j}+2 \hat{k}\) and \(\vec{c}=\hat{i}+2 \hat{j}\), then \(\hat{a}\) is equal to :

Let \(A=\left[\begin{array}{ccc}1 & 2 & 1 \\ -1 & 3 & 2 \\ 2 & 4 & 1\end{array}\right]\) and \(M_{i j}, A_{i j}\) respectively denote the minor, co-factor of an element \(a_{i j}\) of matrix \(A\), then which of the following are true?
(A) \(M_{22}=-1\)
(B) \(A_{23}=0\)
(C) \(A_{32}=3\)
(D) \(M_{23}=1\)
(E) \(M_{32}=-3\)
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In context of Alligation and mixture the ratio of quantity of cheaper ingredient to the quantity of dearer ingredient =

If \(a, b, c\) are in A.P., then the value of determinant \(\left|\begin{array}{lll}x+2 & x+3 & x+2 a \\ x+3 & x+4 & x+2 b \\ x+4 & x+5 & x+2 c\end{array}\right|\) is:

A value of \(p\) for which the points \((1,1, p)\) and \((-3,0,1)\) are equidistant from the plane \(\vec{r} \cdot(3 \hat{i}+4 \hat{j}-12 \hat{k})+13=0\) is:

If \(\int_0^a 3 x^2 d x=8\), then the value of \(a\) is:

Which of the following term refers to the confinement of the species to a specific area and also the same species are not found anywhere else?

Arrange the following alkyl halides in increasing order of reactivity towards \(SN ^2\) reaction?
(A)\(CH _3- Cl\)
(B)\(CH _3- Br\)
(C)\(CH _3- F\)
(D)\(CH _3- I\)
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Bioactive molecule, cyclosporin A, that is used as an immunosuppressive agent in organ-transplant patients, is produced by the fungus -

A bag contains 4 red and 6 black balls. Two balls are drawn in succession without replacement. The probability that the first is red and the second is black is

If \(A\) is any event associated with sample space and If \(E_1, E_2, E_3\) are mutually exclusive and exhaustive events. Then which of the following are true?
(A) \(P(A)=P\left(E_1\right) P\left(E_1 \mid A\right)+P\left(E_2\right) P\left(E_2 \mid A\right)+P\left(E_3\right) P\left(E_3 \mid A\right)\)
(B) \(P(A)=P\left(A \mid E_1\right) P\left(E_1\right)+P\left(A \mid E_2\right) P\left(E_2\right)+P\left(A \mid E_3\right) P\left(E_3\right)\)
(C) \(P\left(E_i \mid A\right)=\frac{P\left(A \mid E_i\right) P\left(E_i\right)}{\sum_{i=1}^3 P\left(A \mid E_i\right) P\left(E_i\right)}, i=1,2,3\)
(D) \(P\left(A \mid E_i\right)=\frac{P\left(E_i \mid A\right) P\left(E_i\right)}{\sum_{i=1}^3 P\left(E_i \mid A\right) P\left(E_i\right)}, i=1,2,3\)
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Which one of the following is also called 'Terror of Bengal'?