Let \(\vec a = 2\hat i - \hat j + \hat k\), \(\vec b = \hat i + 2\hat j - \hat k\) and \(\vec c = \hat i + \hat j - 2\hat k\) be three vectors. A vector of the type \(\vec b + \lambda \vec c\) for some scalar \(\lambda \),  whose projection on \(\vec a\) is of magnitude \(\sqrt {\frac{2}{3}} \) is

The area (in sq. units) of the region bounded by the curve \(x^2 = 4y\) and the straight line \(x = 4y - 2\) is

Let \(x=4\) be a directrix to an ellipse whose centre is at the origin and its eccentricity is \(\frac{1}{2}\) If \(P (1, \beta), \beta>0\) is a point on this ellipse, then the equation of the normal to it at \(P\) is

Let \(N\) denote the set of all natural numbers. Define two binary relations on \(N\) as \(R_1 = \{(x,y) \in  N \times  N : 2x + y= 10\}\) and \(R_2 = \{(x,y) \in  N\times  N : x+ 2y= 10\} \). Then

The length of the latus rectum of a parabola, whose vertex and focus are on the positive \(x\)-axis at a distance \(\mathrm{R}\) and \(\mathrm{S}(\,>\,\mathrm{R})\) respectively from the origin, is:

Let \(3, a, b, c\) be in \(A.P.\) and \(3, a-1, b+1, c+9\) be in \(G.P.\) Then, the arithmetic mean of \(a, b\) and \(c\) is :

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 ...........

A circle in inscribed in an equilateral triangle of side of length \(12\). If the area and perimeter of any square inscribed in this circle are \(\mathrm{m}\) and \(\mathrm{n}\), respectively, then \(m+n^2\) is equal to

If \(A\) and \(B\) are the points of intersection of the circle \(x^2+y^2-8 x=0\) and the hyperbola \(\frac{x^2}{9}-\frac{y^2}{4}=1\) and a point P moves on the line \(2 x-3 y+4=0\), then the centroid of \(\triangle \mathrm{PAB}\) lies on the line :

\(\mathop {\lim }\limits_{x \to 0} \frac{{x\,\tan \,2x - 2x\,\tan \,x}}{{{{\left( {1 - \cos \,2x} \right)}^2}}}\) equals

Let \(\alpha\) and \(\beta\) be two real roots of the equation \((\mathrm{k}+1) \tan ^{2} \mathrm{x}-\sqrt{2} \cdot \lambda \tan \mathrm{x}=(1-\mathrm{k})\) where \(\mathrm{k}(\neq-1)\) and \(\lambda\) are real numbers. If \(\tan ^{2}(\alpha+\beta)=50,\) then a value of \(\lambda\) is :

Let \(I(x)=\int \frac{x^2\left(x \sec ^2 x+\tan x\right)}{(x \tan x+1)^2} d x\). If \(I(0)=0\) the \(I\) \(\left(\frac{\pi}{4}\right)\) is equal to

Let \(w\) \((Im\, w \neq 0)\) be a complex number. Then the set of all complex number \(z\) satisfying the equation \(w - \overline {w}z  = k\left( {1 - z} \right)\) , for some real number \(k\), is