If a unit vector \(\vec r\) makes angles \(\frac{\pi }{3}\) with \(\hat i\), \(\frac{\pi }{4}\) with \(\hat j\) and \(\theta  \in \left( {0,\pi } \right)\) with  \(\hat k\), then a value of \(\theta\) is

Consider the function. \(f(x)=\left\{\begin{array}{cc} \frac{a\left(7 x-12-x^2\right)}{b\left|x^2-7 x+12\right|} & , x<3 \\ 2^{\frac{\sin (x-3)}{x-[x]}} & , x>3 \\ b & , x=3 \end{array}\right.\) Where \([\mathrm{x}]\) denotes the greatest integer less than or equal to \(x\). If \(S\) denotes the set of all ordered pairs \((a, b)\) such that \(f(x)\) is continuous at \(x=3\), then the number of elements in \(\mathrm{S}\) is :

The sum of all the real roots of the equation \(\left( e ^{2 x }-4\right)\left(6 e ^{2 x }-5 e ^{ x }+1\right)=0\) is

The value of \(\int \limits_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{(2+3 \sin x)}{\sin x(1+\cos x)} d x\) is equal to

An unbiased coin is tossed. If the outcome is a head then a pair of unbiased dice is rolled and the sum of the numbers obtained on the is noted. If the toss of the coin results in tail then a card from a well-shuffled pack of nine cards numbered \(1, 2, 3,….., 9\) is randomly picked and the number on the card is noted. The probability that the noted number is either \(7\) or \(8\) is

Define a relation R on the interval \(\left[0, \frac{\pi}{2}\right)\) by \(x \mathrm{R} y\) if and only if \(\sec ^2 x-\tan ^2 y=1\). Then R is :

Let \(\vec \alpha \, = \,3\hat i\, + \hat j\) and \(\vec \beta \, = \,2\hat i\, - \hat j + 3\hat k.\) If \(\vec \beta \, = \,{\vec \beta _1} - {\vec \beta _2},\) where \({\vec \beta _1}\) is parallel to \(\vec \alpha \) and \(\vec \beta_2 \) is perpendicular to \(\vec \alpha ,\)  then \({\vec \beta _1} \times {\vec \beta _2}\) is equal to

Let \(x(t)=2 \sqrt{2} \cos t \sqrt{\sin 2 t}\) and \(y ( t )=2 \sqrt{2} \sin t \sqrt{\sin 2 t }, t \in\left(0, \frac{\pi}{2}\right)\). Then \(\frac{1+\left(\frac{ dy }{ dx }\right)^{2}}{\frac{ d ^{2} y }{ dx ^{2}}}\) at \(t =\frac{\pi}{4}\) is equal to..

Let \(E ^{ C }\) denote the complement of an event \(E\). Let \(E _{1}, E _{2}\) and \(E _{3}\) be any pairwise independent events with \(P \left( E _{1}\right) > 0\) and \(P \left( E _{1} \cap E _{2} \cap E _{3}\right)=0\) Then \(P \left( E _{2}^{ C } \cap E _{3}^{ C } / E _{1}\right)\) is equal to

If \(y=y(x)\) is the solution curve of the differential equation \(x^{2} d y+\left(y-\frac{1}{x}\right) d x=0 \quad ; x>0\) and \(\mathrm{y}(1)=1\), then \(\mathrm{y}\left(\frac{1}{2}\right)\) is equal to :

Suppose the vectors \(x_{1}, x_{2}\) and \(x_{3}\) are the solutions of the system of linear equations, \(Ax = b\) when the vector \(b\) on the right side is equal to \(b _{1}, b _{2}\) and \(b _{3}\) respectively. If \(x =\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right], x _{2}=\left[\begin{array}{l}0 \\ 2 \\ 1\end{array}\right], x _{3}=\left[\begin{array}{l}0 \\ 0 \\ 1\end{array}\right], b _{1}=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]\) \(b _{2}=\left[\begin{array}{l}0 \\ 2 \\ 0\end{array}\right]\) and \(b _{3}=\left[\begin{array}{l}0 \\ 0 \\ 2\end{array}\right],\) then the determinant of \(A\) is equal to

Let \(A =\{0,1,2, \ldots 9)\). Let R be a relation on A defined by \(( x , y ) \in R\) if and only if \(| x - y |\) is a multiple of 3 .
Given below are two statements:
Statement I: \(n ( R )=36\)
Statement II: \(R\) is an equivalence relation.
In the light of the above statements, choose the correct answer from the options given below

Let \(C\) be a circle passing through the points \(A (2,-1)\) and \(B (3,4)\). The line segment \(AB\) is not a diameter of \(C\). If \(r\) is the radius of \(C\) and its centre lies on the circle \((x-5)^{2}+(y-1)^{2}=\frac{13}{2}\), then \(r^{2}\) is equal to