A chord is drawn through the focus of the parabola \(y^2\, = 6x\) such that its distance from the vertex of this parabola is \(\frac{{\sqrt 5 }}{2}\) , then its slope can be:

Let \(P = \{\theta \in [0, 4\pi] : \tan^2\theta \neq 1\}\) and \(S = \{a \in \mathbb{Z} : 2(\cos^8\theta - \sin^8\theta)\sec 2\theta = a^2, \theta \in P\}\). Then \(n(S)\) is:

Let \(A\) be a point on the line \(\vec r = \left( {1 - 3\mu } \right)\hat i + \left( {\mu  - 1} \right)\hat j + \left( {2 + 5\mu } \right)\hat k\) and \(B(3, 2, 6)\) be a point in the space. Then the value of \(\mu \) for which the vector \(\overrightarrow {AB} \) is parallel to the plane \(x -4y +3z = 1\) is

Consider the system of linear equations in \(x, y, z\):
\(x + 2y + tz = 0\),
\(6x + y + 5tz = 0\),
\(3x + t^2 y + f(t) z = 0\),
where \(f: \mathbb{R} \rightarrow \mathbb{R}\) is a differentiable function. If this system has infinitely many solutions for all \(t \in \mathbb{R}\), then \(f\)

If \(\mathrm{n}\) is the number of solutions of the equation \(2 \cos x\left(4 \sin \left(\frac{\pi}{4}+x\right) \sin \left(\frac{\pi}{4}-x\right)-1\right)=1, x \in[0, \pi]\) and \(S\) is the sum of all these solutions, then the ordered pair \((\mathrm{n}, \mathrm{S})\) is :

All the pairs \((x, y)\) that satisfy the inequality \({2^{\sqrt {{{\sin }^2}{\kern 1pt} x - 2\sin {\kern 1pt} x + 5} }}.\frac{1}{{{4^{{{\sin }^2}\,y}}}} \leq 1\) also Satisfy the equation

Let the tangent to the circle \(x^{2}+y^{2}=25\) at the point \(R (3,4)\) meet \(x\) -axis and \(y\) -axis at point \(P\) and \(Q\), respectively. If \(r\) is the radius of the circle passing through the origin \(O\) and having centre at the incentre of the triangle \(OPQ ,\) then \(r ^{2}\) is equal to

If a point \(R (4, y, z)\) lies on the line segment joining the points \(P (2, -3, 4)\) and \(Q (8, 0, 10)\), then the distance of \(R\) from the origin is

Let a random variable X take values \(0,1,2,3\) with \(\mathrm{P}(\mathrm{X}=0)=\mathrm{P}(\mathrm{X}=1)=\mathrm{p}, \mathrm{P}(\mathrm{X}=2)=\mathrm{P}(\mathrm{X}=3)\) and \(\mathrm{E}\left(\mathrm{X}^2\right)=2 \mathrm{E}(\mathrm{X})\). Then the value of \(8 \mathrm{p}-1\) is :

The number of strictly increasing functions f from the set {1, 2, 3, 4, 5, 6} to the set {1, 2, 3,...,9} such that \( f(i)\ne i \) for \( 1\le i\le6, \) is equal to:

Let a unit vector \(\hat{\mathrm{u}}=\mathrm{x} \hat{\mathrm{i}}+\mathrm{y} \hat{\mathrm{j}}+\mathrm{z} \hat{\mathrm{k}}\) make angles \(\frac{\pi}{2}, \frac{\pi}{3}\) and \(\frac{2 \pi}{3}\) with the vectors \(\frac{1}{\sqrt{2}} \hat{\mathrm{i}}+\frac{1}{\sqrt{2}} \hat{\mathrm{k}}, \frac{1}{\sqrt{2}} \hat{\mathrm{j}}+\frac{1}{\sqrt{2}} \hat{\mathrm{k}}\) and \(\frac{1}{\sqrt{2}} \hat{\mathrm{i}}+\frac{1}{\sqrt{2}} \hat{\mathrm{j}}\) respectively. If \(\overrightarrow{\mathrm{v}}=\frac{1}{\sqrt{2}} \hat{\mathrm{i}}+\frac{1}{\sqrt{2}} \hat{\mathrm{j}}+\frac{1}{\sqrt{2}} \hat{\mathrm{k}}\), then \(|\hat{\mathrm{u}}-\overrightarrow{\mathrm{v}}|^2\) is equal to

The value of the integral \(\int_0^{\frac{\pi}{4}} \frac{x d x}{\sin ^4(2 x)+\cos ^4(2 x)}\) equals :

In an examination, there are \(10\) true-false type questions. Out of \(10\) , a student can guess the answer of \(4\) questions correctly with probability \(\frac{3}{4}\) and the remaining \(6\) questions correctly with probability \(\frac{1}{4}\). If the probability that the student guesses the answers of exactly \(8\) questions correctly out of \(10\) is \(\frac{27 k }{4^{10}}\), then \(k\) is equal to