The equation of the plane passing through the line of intersection of planes \(\pi_1=2 x+6 y+4 z-7=0\), \(\pi_2=x-y-2 z-2=03\) and perpendicular to the plane \(x+y+2 z-5=0\) is

Let \(A, B, C\) be three points on \(\overline{O X}, \overline{O Y}, \overline{O Z}\) respectively at the distances 3, 6, 9 from origin. Let \(Q\) be the point \((2,5,8)\) and \(P\) be the point equidistant from \(O, A, B, C\). Then, the coordinates of the point \(R\) which divides \(P Q\) in the ratio \(3: 2\) is

If a line \(l\) passes through \((k, 2 k),(3 k, 3 k)\) and \((3,1), k \neq 0\), then the distance from the origin to the line \(l\) is

If \(A=\left[\begin{array}{ll}0 & 3 \\ 0 & 0\end{array}\right]\) and \(f(x)=x+x^2+x^3+\ldots . .+x^{2023}\) then \(\mathrm{f}(\mathrm{A})+\mathrm{I}=\)

There are two boxes each containing 10 balls. In each box, few of them are black balls and rest are white. A ball is drawn at random from one of the boxes and found that it is black. If the probability that the black ball drawn is from the second box is \(\frac{1}{5}\), then number of black balls in the first box is

A ladder of length \(13 \mathrm{mts}\) has one end resting against a vertical wall and the other on the ground. If the lower end moves away from the wall at a speed of \(2 \mathrm{mts} /\) minute, then the speed (in \(\mathrm{mts} / \mathrm{min}\) ) at which upper end falls when the bottom is \(5 \mathrm{mts}\) away from the wall is

If \(\{x\}=x-[x]\) where \([x]\) is the greatest integer \(\leq x\) and \(\lim _{x \rightarrow 0^{-}} \frac{\operatorname{Cos}^{-1}\left(1-\{x\}^2\right) \operatorname{Sin}^{-1}(1-\{x\})}{\{x\}-\{x\}^4}=\theta\), then \(\tan \theta=\)

For a wire, as shown in the figure, carrying a current of \(10 \mathrm{~A}\), then the magnetic induction field at the point \(O\) is [talse, \(\mu_0=4 \pi \times 10^{-7} \mathrm{H} / \mathrm{m}\) ]

Statement (I) : The slope of kinetic energy-displacement curve of a body in motion will be directly proportional to its acceleration. Statement (II) : From a height of \(15 \mathrm{~m}\) a ball is projected vertically upwards with a velocity of \(30 \mathrm{~m} / \mathrm{s}\). If the ball rises to the same height after hitting the ground, the loss of its energy on hitting the ground is \(30 \%\). Statement (III) : The velocity acquired by a body of mass ' \(m\) ' after travelling a fixed distance from rest under the action of a constant force is directly proportional to mass ' \(\mathrm{m}\) '. Which of the following is correct?

A cube of edge length $1 \mathrm{cm}$ is divided into smaller cubes of uniform size of length $1 \mathrm{nm}$. Assuming that no voids are present, the ratio of total surface area of all the cubes of $1 \mathrm{nm}$ edge length to the surface area of the initial cube is

The combined equation of the two diameters of a circle which divide the circle into $4$ sectors is $a{x}^2+2hxy+b{y}^2=0$. If the area of the bigger sector is $5$ times the area of the smaller sector, then $\frac{\left|a+b\right|}{\sqrt{{\left(a-b\right)}^2+4h^2}}=$

Consider the following sequence of reactions

The incorrect statement about \(\mathrm{R}\) is

The line among the following which touches the parabola \(y^2=4 a x\), is