NTA JEE Main 20th July 2021 Shift 1

There are 15 players in a cricket team, out of which 6 are bowlers, 7 are batsmen and 2 are wicketkeepers. The number of ways, a team of 11 players be selected from them so as to include at least 4 bowlers, 5 batsmen and 1 wicketkeeper, is ___.

The number of rational terms in the binomial expansion of $$\left(4^{1/4} + 5^{1/6}\right)^{120}$$ is ___.

Let $$y = mx + c$$, $$m > 0$$ be the focal chord of $$y^2 = -64x$$, which is tangent to $$(x+10)^2 + y^2 = 4$$. Then, the value of $$4\sqrt{2}(m+c)$$ is equal to ___.

If the value of $$\lim_{x \to 0}\left(2 - \cos x\sqrt{\cos 2x}\right)^{\left(\frac{x+2}{x^2}\right)}$$ is equal to $$e^a$$, then $$a$$ is equal to ___.

Let $$A = \begin{bmatrix} 1 & -1 & 0 \\ 0 & 1 & -1 \\ 0 & 0 & 1 \end{bmatrix}$$ and $$B = 7A^{20} - 20A^7 + 2I$$, where $$I$$ is an identity matrix of order $$3 \times 3$$. If $$B = [b_{ij}]$$, then $$b_{13}$$ is equal to ___.

Let $$a, b, c, d$$ be in arithmetic progression with common difference $$\lambda$$. If
$$\begin{vmatrix} x+a-c & x+b & x+a \\ x-1 & x+c & x+b \\ x-b+d & x+d & x+c \end{vmatrix} = 2$$,
then value of $$\lambda^2$$ is equal to ___.

Let $$T$$ be the tangent to the ellipse $$E : x^2 + 4y^2 = 5$$ at the point $$P(1, 1)$$. If the area of the region bounded by the tangent $$T$$, ellipse $$E$$, lines $$x = 1$$ and $$x = \sqrt{5}$$ is $$\alpha\sqrt{5} + \beta + \gamma\cos^{-1}\left(\frac{1}{\sqrt{5}}\right)$$, then $$|\alpha + \beta + \gamma|$$ is equal to ___.

Let $$\vec{a}, \vec{b}, \vec{c}$$ be three mutually perpendicular vectors of the same magnitude and equally inclined at an angle $$\theta$$, with the vector $$\vec{a} + \vec{b} + \vec{c}$$. Then $$36\cos^2 2\theta$$ is equal to ___.

Let $$P$$ be a plane passing through the points $$(1, 0, 1)$$, $$(1, -2, 1)$$ and $$(0, 1, -2)$$. Let a vector $$\vec{a} = \alpha\hat{i} + \beta\hat{j} + \gamma\hat{k}$$ be such that $$\vec{a}$$ is parallel to the plane $$P$$, perpendicular to $$(\hat{i} + 2\hat{j} + 3\hat{k})$$ and $$\vec{a} \cdot (\hat{i} + \hat{j} + 2\hat{k}) = 2$$, then $$(\alpha - \beta + \gamma)^2$$ equals ___.

If the shortest distance between the lines $$\vec{r_1} = \alpha\hat{i} + 2\hat{j} + 2\hat{k} + \lambda(\hat{i} - 2\hat{j} + 2\hat{k})$$, $$\lambda \in R$$, $$\alpha > 0$$ and $$\vec{r_2} = -4\hat{i} - \hat{k} + \mu(3\hat{i} - 2\hat{j} - 2\hat{k})$$, $$\mu \in R$$ is 9, then $$\alpha$$ is equal to ___.