Let $$\vec{a}=\alpha \hat{i}+\hat{j}+\beta \hat{k}$$ and $$\vec{b}=3 \hat{i}-5 \hat{j}+4 \hat{k}$$ be two vectors, such that $$\vec{a} \times \vec{b}=-\hat{i}+9 \hat{j}+12 \hat{k}$$. Then the projection of $$\vec{b}-2 \vec{a}$$ on $$\vec{b}+\vec{a}$$ is equal to :

Let $$S$$ be the sample space of all five digit numbers. It $$p$$ is the probability that a randomly selected number from $$S$$, is a multiple of 7 but not divisible by 5 , then $$9 p$$ is equal to :

Let $$A(1,1), B(-4,3), C(-2,-5)$$ be vertices of a triangle $$A B C, P$$ be a point on side $$B C$$, and $$\Delta_{1}$$ and $$\Delta_{2}$$ be the areas of triangles $$A P B$$ and $$A B C$$, respectively. If $$\Delta_{1}: \Delta_{2}=4: 7$$, then the area enclosed by the lines $$A P, A C$$ and the $$x$$-axis is :

If the circle $$x^{2}+y^{2}-2 g x+6 y-19 c=0, g, c \in \mathbb{R}$$ passes through the point $$(6,1)$$ and its centre lies on the line $$x-2 c y=8$$, then the length of intercept made by the circle on $$x$$-axis is :