If the chord joining the points \(P_{1}(x_{1},y_{1})\) and \(P_{2}(x_{2},y_{2})\) on the parabola \(y^{2}=12x\) subtends a right angle at the vertex of the parabola, then \(x_{1}x_{2}-y_{1}y_{2}\) is equal to

Let \([\bullet]\) be the greatest integer function. If \( \alpha=\int_{0}^{64}(x^{1/3}-[x^{1/3}])dx, \) then \( \frac{1}{\pi}\int_{0}^{\alpha\pi}(\frac{sin^{2}\theta}{sin^{6}\theta+cos^{6}\theta})d\theta \) is equal to ___ .

Consider the following system of questions \(\alpha x+2 y+z=1\)  ;  \(2 \alpha x+3 y+z=1\)  ;  \(3 x+\alpha y+2 z=\beta\) . For some \(\alpha, \beta \in R\). Then which of the following is NOT correct.

If the equations \(x^2 + bx - 1 = 0\) and  \(x^2 + x + b= 0\) have a common root different from \(-1,\)  then \(\left| b \right|\)  is equal to

For \(x \in R\), two real valued functions \(f(x)\) and \(g(x)\) are such that, \(g(x)=\sqrt{x}+1\) and \(f o g(x)=x+3-\sqrt{x}\). Then \(f(0)\) is equal to

If the line \(x-1=0\), is a directrix of the hyperbola \(kx ^{2}- y ^{2}=6\), then the hyperbola passes through the point.

Let \(a, b, c, d\) and \(p\) be any non zero distinct real numbers such that  \(\left(a^{2}+b^{2}+c^{2}\right) p^{2}-2(a b+b c+ cd ) p +\left( b ^{2}+ c ^{2}+ d ^{2}\right)=0 .\) Then

In a survey of \(220\) students of a higher secondary school, it was found that at least \(125\) and at most \(130\) students studied Mathematics; at least \(85\) and at most \(95\) studied Physics; at least \(75\) and at most \(90\) studied Chemistry; \(30\) studied both Physics and Chemistry; \(50\) studied both Chemistry and Mathematics; \(40\) studied both Mathematics and Physics and \(10\) studied none of these subjects. Let \(\mathrm{m}\) and \(\mathrm{n}\) respectively be the least and the most number of students who studied all the three subjects. Then \(\mathrm{m}+\mathrm{n}\) is equal to ...........

Let \(y=f(x)\) represent a parabola with focus \(\left(-\frac{1}{2}, 0\right)\) and directrix \(y =-\frac{1}{2}\). Then \(S=\left\{x \in R : \tan ^{-1}\left(\sqrt{f(x)}+\sin ^{-1}(\sqrt{f(x)+1})\right)=\frac{\pi}{2}\right\}:\)

If \(12\) identical balls are to be placed in \(3\) identical boxes, then the probability that one of the boxes contains exactly \(3\) balls is :

If the probability of hitting a target by a shooter, in any shot, is \(\frac{1}{3}\), then the minimum number of independent shots at the target required by him so that the probability of hitting the target at least once is greater than \(\frac{5}{6}\), is

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?

The probability of selecting integers \(a \in[-5,30]\) such that \(x^{2}+2(a+4) x-5 a+64>0\), for all \(x \in R\), is: