Let \(\hat{a}\) and \(\hat{b}\) be two unit vectors such that the angle between them is \(\frac{\pi}{4}\). If \(\theta\) is the angle between the vectors \((\hat{a}+\hat{b})\) and \((\hat{a}+2 \hat{b}+2(\hat{a} \times \hat{b}))\) then the value of \(164 \cos ^{2} \theta\) is equal to.

For two non-zero complex number \(z_1\) and \(z_2\), if \(\operatorname{Re}\left(z_1 z_2\right)=0\) and \(\operatorname{Re}\left(z_1+z_2\right)=0\), then which of the following are possible ? \((A)\) \(\operatorname{Im}\left(z_1\right) > 0\) and \(\operatorname{Im}\left(z_2\right) > 0\) \((B)\) \(\operatorname{Im}\left(z_1\right) < 0\) and \(\operatorname{Im}\left(z_2\right) > 0\) \((C)\) \(\operatorname{Im}\left(z_1\right) > 0\) and \(\operatorname{Im}\left(z_2\right) < 0\) \((D)\) \(\operatorname{Im}\left( z _1\right) < 0\) and \(\operatorname{Im}\left( z _2\right) < 0\) Choose the correct answer from the options given below :

The number of \(5-\)digit natural numbers, such that the product of their digits is \(36\), is

Let
\(\mathrm{f}(x)=\left\{\begin{array}{lc}3 x, & x \lt 0 \\ \min \{1+x+[x], x+2[x]\}, & 0 \leq x \leq 2 \\ 5, & x\gt2,\end{array}\right.\)
where [.] denotes greatest integer function. If \(\alpha\) and \(\beta\) are the number of points, where f is not continuous and is not differentiable, respectively, then \(\alpha+\beta\) equals __________

The equation of the planes parallel to the plane \(x\) \(-2 y +2 z -3=0\) which are at unit distance from the point \((1,2,3)\) is \(a x+b y+c z+d=0\). If \((b-d)=K(c-a),\) then the positive value of \(K\) is

Each of the angles \(\beta\) and \(\gamma\) that a given line makes with the positive y - and z -axes, respectively, is half of the angle that this line makes with the positive x -axes. Then the sum of all possible values of the angle \(\beta\) is

A value of \(x\) for which \(\sin \,\left( {{{\cot }^{ - 1}}\,\left( {1 + x} \right)} \right) = \cos \,\left( {{{\tan }^{ - 1}}\,x} \right)\), is

A line passes through the origin and makes equal angles with the positive coordinate axes. It intersects the lines
\(\mathrm{L}_1: 2 \mathrm{x}+\mathrm{y}+6=0\) and \(\mathrm{L}_2: 4 \mathrm{x}+2 \mathrm{y}-\mathrm{p}=0, \mathrm{p} \gt 0\), at the points \(A\) and \(B\), respectively. If \(A B=\frac{9}{\sqrt{2}}\) and the foot of the perpendicular from the point A on the line \(L_2\) is \(M\), then \(\frac{A M}{B M}\) is equal to

Let \(f :(0,1) \rightarrow R\) be a function defined by \(f(x)=\frac{1}{1-e^{-x}}\), and \(g(x)=(f(-x)-f(x))\). Consider two statement: \((I)\) \(g\) is an increasing function in \((0,1)\) \((II)\) \(g\) is one-one in \((0,1)\) Then,

If the point on the curve \(y^{2}=6 x\), nearest to the point \(\left(3, \frac{3}{2}\right)\) is \((\alpha, \beta)\), then \(2(\alpha+\beta)\) is equal to \(.....\)

If the function \(f(x)=\left\{\begin{array}{cc}\frac{72^x-9^x-8^x+1}{\sqrt{2}-\sqrt{1+\cos x}} & , x \neq 0 \\ a \log _e 2 \log _e 3 & , x=0\end{array}\right.\) is continuous at \(x=0\), then the value of \(a^2\) is equal to

If \(\alpha \neq \mathrm{a}, \beta \neq \mathrm{b}, \gamma \neq \mathrm{c}\) and \(\left|\begin{array}{lll}\alpha & \mathrm{b} & \mathrm{c} \\ \mathrm{a} & \beta & \mathrm{c} \\ \mathrm{a} & \mathrm{b} & \gamma\end{array}\right|=0\),then \(\frac{a}{\alpha-a}+\frac{b}{\beta-b}+\frac{\gamma}{\gamma-c}\) is equal to :

If the system of linear equations \(2 x+3 y-z=-2\)  ; \(x+y+z=4\)  ; \(x-y+|\lambda| z=4 \lambda-4\)  (where \(\lambda \in R\)), has no solution, then