NTA JEE Main 9th January 2020 Shift 1

The number of distinct solutions of the equation, $$\log_{\frac{1}{2}}|\sin x| = 2 - \log_{\frac{1}{2}}|\cos x|$$ in the interval $$[0, 2\pi]$$, is ___________.

The coefficient of $$x^4$$ in the expansion of $$(1 + x + x^2)^{10}$$ is ___________.

If for $$x \ge 0$$, $$y = y(x)$$ is the solution of the differential equation,
$$(x + 1)dy = ((x + 1)^2 + y - 3)dx$$, $$y(2) = 0$$ then $$y(3)$$ is equal to ___________.

If the vectors, $$\vec{p} = (a+1)\hat{i} + a\hat{j} + a\hat{k}$$, $$\vec{q} = a\hat{i} + (a+1)\hat{j} + a\hat{k}$$ and $$\vec{r} = a\hat{i} + a\hat{j} + (a+1)\hat{k}$$ $$(a \in R)$$ are coplanar and $$3(\vec{p} \cdot \vec{q})^2 - \lambda|\vec{r} \times \vec{q}|^2 = 0$$, then the value of $$\lambda$$ is ___________.

The projection of the line segment joining the point $$(1, -1, 3)$$ and $$(2, -4, 11)$$ on the line joining the points $$(-1, 2, 3)$$ and $$(3, -2, 10)$$ is ___________.