Let \(S = \{ x \in R:x \ge 0\) and \(2\left| {\sqrt x - 3} \right| + \sqrt x \left( {\sqrt x - 6} \right) + 6 = 0\} \) then \(S:\) . . .

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 \(\mathrm{A}=\{1,2,3,4,5\}\). Let \(\mathrm{R}\) be a relation on \(\mathrm{A}\) defined by \(x R y\) if and only if \(4 x \leq 5 y\). Let \(m\) be the number of elements in \(\mathrm{R}\) and \(\mathrm{n}\) be the minimum number of elements from \(\mathrm{A} \times \mathrm{A}\) that are required to be added to \(\mathrm{R}\) to make it a symmetric relation. Then \(m+n\) is equal to :

A bag contains 6 blue and 6 green balls. Pairs of balls are drawn without replacement until the bag is empty. The probability that each drawn pair consists of one blue and one green ball is :

If \(\int_0^1 \frac{1}{\sqrt{3+x}+\sqrt{1+x}} d x=a+b \sqrt{2}+c \sqrt{3}\), where \(a, b, c\) are rational numbers, then \(2 a+3 b-4 c\) is equal to :

If the orthocentre of the triangle formed by the lines \(y=x+1, y=4 x-8\) and \(y=m x+c\) is at \((3,-1)\), then \(\mathrm{m}-\mathrm{c}\) is :

Let \(a_1, a_2, a_3, \ldots\) be a G.P. of increasing positive terms. If \(a_1 a_5=28\) and \(a_2+a_4=29\), then \(a_6\) is equal to:

The general solution of the differential equation \((y^2 -x^3) dx -xydy = 0\, (x \ne 0)\) is (where \(c\) is a constant of integration)

Let \(A(2,3,5)\) and \(C(-3,4,-2)\) be opposite vertices of a parallelogram \(A B C D\) if the diagonal \(\overrightarrow{B D}=\hat{i}+2 \hat{j}+3 \hat{k}\) then the area of the parallelogram is equal to

For \(z \in C\) if the minimum value of \((|z-3 \sqrt{2}|+|z-p \sqrt{2} i|)\) is \(5 \sqrt{2}\), then a value of \(P\) is \(.......\)

The sum of all values of \(x\) in \([0,2 \pi]\), for which \(\sin x+\sin 2 x+\sin 3 x+\sin 4 x=0\), is equal to:

Let \(P\) be the plane passing through the intersection of the planes
\(\overrightarrow{ r } \cdot(\hat{ i }+3 \hat{ j }-\hat{ k })=5\) and \(\overrightarrow{ r } \cdot(2 \hat{ i }-\hat{ j }+\hat{ k })=3\), and the point \((2,1,-2)\). Let the position vectors of the points \(X\) and \(Y\) be \(\hat{ i }-2 \hat{ j }+4 \hat{ k }\) and \(5 \hat{ i }-\hat{ j }+2 \hat{ k }\) respectively. Then the points
 

Let the mean and the variance of seven observations \(2, 4, \alpha, 8, \beta, 12, 14\), \(\alpha < \beta\), be \(8\) and \(16\) respectively. Then the quadratic equation whose roots are \(3\alpha + 2\) and \(2\beta + 1\) is :