When two dice are rolled, what is the probability that at least one face shows a six?

If $\left(a,b,c\right)$ are the direction ratios of a line joining the points $\left(4,3,-5\right)$ and $\left(-2,1,-8\right)$ then the point $P\left(a,3 b,2c\right)$ lies on the plane

Let \(0(\overrightarrow{0}), \mathrm{A}(\hat{i}+2 \hat{j}+\hat{k}), \mathrm{B}(-2 \hat{i}+3 \hat{k}), \mathrm{C}(-2 \hat{i}+\hat{j}), \mathrm{D}(4 \hat{k})\) are position vectors of the points \(\mathrm{O}, \mathrm{A}, \mathrm{B}, \mathrm{C}\) and D , If a line passing through A and B intersects the plane passing through \(\mathrm{O}, \mathrm{C}\) and D at the point R . then position vector of R is

Equation of the locus of the centroid of the triangle whose vertices are \((a \cos k, a \sin k),(b \sin k,-b \cos k)\) and \((1,0)\), where \(k\) is a parameter, is

If \(f(x)=\frac{\cos ^2 x+\sin ^4 x}{\sin ^2 x+\cos ^4 x} \forall x \in R\), then \(f(2023)=\)

The variance of the random variable \(X\) having the following distribution

If \((2 k-1) x^2-2(3 k-2) x+4 k>0\) for every \(x \in R\), then the sum of all possible integral values of \(k\) is

The angle between the lines represented by $\cos \theta (\cos \theta +1)x^2-\left(2\cos \theta +{\sin }^2\theta \right)xy+$$(1-\cos \theta )y^2=0$ is

When ' \(\mathrm{X}\) ' \(\mathrm{g}\) of graphite is completely burnt in a bomb calorimeter in excess of \(\mathrm{O}_2\) at \(298 \mathrm{~K}\) and \(1 \mathrm{~atm}\) pressure as given in the equation
\(\mathrm{C}\) (graphite) \(+\mathrm{O}_2(\mathrm{~g}) \rightarrow \mathrm{CO}_2(\mathrm{~g})\)
The temperature of calorimeter raised from \(298 \mathrm{~K}\) to \(302 \mathrm{~K}\). If the heat capacity of the calorimeter and molar enthalpy change for the reaction at \(1 \mathrm{~atm}\) and \(298 \mathrm{~K}\) are \(20.7 \mathrm{~kJ} \mathrm{~K}^{-1}\) and \(-248.4 \mathrm{~kJ} \mathrm{~mol}^{-1}\), ' \(\mathrm{X}^{\prime}\) in \(\mathrm{g}\) is

The order of basic strength of methyl substituted amines and \(\mathrm{NH}_3\) in aqueous solution is

Let \(\mathrm{P}(\mathrm{h}, \mathrm{k})\) be the point of contact of the tangent to the hyperbola \(5 x^2-7 y^2-35=0\) which is parallel to the line \(\sqrt{2} x-y+\lambda=0\). If \(P\) lies in the third quadrant then \(3 h^2-2 k=\)

A soccer ball of mass 250 g is moving horizontally to the left with a speed \(22 \mathrm{~ms}^{-1}\). This ball is kicked towards right with a velocity \(30 \mathrm{~ms}^{-1}\) at an angle \(53^{\circ}\) with the horizontal in upward direction. Assuming that it took 0.01 s for the collision to take place, the average force acting is \(\left(\cos 53^{\circ}=\frac{3}{5} ; \sin 53^{\circ}=\frac{4}{5}\right)\)

Assertion (A) The aggregate of carboxylate ions of fatty acids form a spherical shape.
Reason (R) Soap helps in emulsification and washing away of oils.