The equation of the plane through the intersection of the planes \(\mathrm{x}+\mathrm{y}+\mathrm{z}=1\) and \(2 \mathrm{x}+3 \mathrm{y}-\mathrm{x}+4=0\) and parallel to \(\mathrm{X}\)-axis is

The parametric equations of the circle \(x^2+y^2+2 x-4 y-4=0\) are

If the function \(\mathrm{f}\) defined by \(\mathrm{f}(\mathrm{x})= \begin{cases}\mathrm{K}\left(\mathrm{x}-\mathrm{x}^2\right) & \text { if } 0 < \mathrm{x} < 1 \ 0 & \text {, other wise }\end{cases}\) is the p.d.f. of a r.v.x, then the value of \(\mathrm{P}\left(\mathrm{X} < \frac{1}{2}\right)\) is

Consider the three statements -
\(\mathrm{p}: \forall \mathrm{n} \in \mathbb{N}, 10 \mathrm{n}-3\) is a prime number, when n is not divisible by 3 .
\(\mathrm{q}: \frac{2}{\sqrt{3}}, \frac{-2}{\sqrt{3}}, \frac{-1}{\sqrt{3}}\) are the direction cosines of a directed line.
\(\mathrm{r}: \sin x\) is an increasing function in the interval \(\left[\frac{-\pi}{2}, \frac{\pi}{2}\right]\).
Then which of the following statement pattern has truth value true?

The acute angle between the diagonals of a parallelogram whose vertices are \(\mathrm{A}(2,-1), \mathrm{B}(0,2), \mathrm{C}(2,3)\) and \(\mathrm{D}(4,0)\) is

The value of $\underset{-3}{\overset{3}{\int }}\left(a{x}^5+b{x}^3+cx+k\right)dx,$ where $a,b,c,k$ are constants, depends only on ________

The value of \(c\) of Lagrange's mean value theorem for \(\mathrm{f}(x)=\sqrt{25-x^2}\) on \([1,5]\) is

In LCR series resonant circuit, at resonance, voltage across ' \(\mathrm{L}\) ' and ' \(C\) ' will cancel each other because they are

\(\lim _{x \rightarrow 0} \frac{63^x-9^x-7^x+1}{\sqrt{2}-\sqrt{1+\cos x}}=\ldots\).

If \(\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}\) are non-coplanar vectors and \(\overline{\mathrm{p}}=\frac{\overline{\mathrm{b}} \times \overline{\mathrm{c}}}{[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]}, \overline{\mathrm{q}}=\frac{\overline{\mathrm{c}} \times \overline{\mathrm{a}}}{[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]}, \overline{\mathrm{r}}=\frac{\overline{\mathrm{a}} \times \overline{\mathrm{b}}}{[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]}, \quad\) then \(2 \overline{\mathrm{a}} \cdot \overline{\mathrm{p}}+\overline{\mathrm{b}} \cdot \overline{\mathrm{q}}+\overline{\mathrm{c}} \cdot \overline{\mathrm{r}}=\)

Calculate the molal elevation constant of solvent if boiling point of 0.12 m solution is 319.8 K (Boling point of solvent \(=319.5 \mathrm{~K}\) )

In a single throw of three dice, the probability of getting a sum at least 5 is

Which from following n mole molecules of carbohydrate contains 2 n mole molecules of galactose, n moles of glucose and n moles of fructose in it?