NTA JEE Main 16th March 2021 Shift 2

Let $$\frac{1}{16}$$, $$a$$ and $$b$$ be in G.P. and $$\frac{1}{a}$$, $$\frac{1}{b}$$, 6 be in A.P., where $$a, b > 0$$. Then $$72(a+b)$$ is equal to ________.

Let $$S_n(x) = \log_{a^{1/2}} x + \log_{a^{1/3}} x + \log_{a^{1/6}} x + \log_{a^{1/11}} x + \log_{a^{1/18}} x + \log_{a^{1/27}} x + \ldots$$ up to $$n$$-terms, where $$a > 1$$. If $$S_{24}(x) = 1093$$ and $$S_{12}(2x) = 265$$, then value of $$a$$ is equal to ________.

Let $$n$$ be a positive integer. Let $$A = \sum_{k=0}^{n} (-1)^k \cdot {^nC_k}\left[\left(\frac{1}{2}\right)^k + \left(\frac{3}{4}\right)^k + \left(\frac{7}{8}\right)^k + \left(\frac{15}{16}\right)^k + \left(\frac{31}{32}\right)^k\right]$$. If $$63A = 1 - \frac{1}{2^{30}}$$, then $$n$$ is equal to ________.

Consider the statistics of two sets of observations as follows:

If the variance of the combined set of these two observations is $$\frac{17}{9}$$, then the value of $$n$$ is equal to ________.

In $$\triangle ABC$$, the lengths of sides $$AC$$ and $$AB$$ are 12 cm and 5 cm, respectively. If the area of $$\triangle ABC$$ is 30 cm$$^2$$ and $$R$$ and $$r$$ are respectively the radii of circumcircle and incircle of $$\triangle ABC$$, then the value of $$2R + r$$ (in cm) is equal to ________.

Let $$A = \begin{bmatrix} a_1 \\ a_2 \end{bmatrix}$$ and $$B = \begin{bmatrix} b_1 \\ b_2 \end{bmatrix}$$ be two $$2 \times 1$$ matrices with real entries such that $$A = XB$$, where $$X = \frac{1}{\sqrt{3}}\begin{bmatrix} 1 & -1 \\ 1 & k \end{bmatrix}$$, and $$k \in R$$. If $$a_1^2 + a_2^2 = \frac{2}{3}(b_1^2 + b_2^2)$$ and $$(k^2 + 1)b_2^2 \neq -2b_1 b_2$$, then the value of $$k$$ is ________.

Let $$f : R \to R$$ and $$g : R \to R$$ be defined as $$f(x) = \begin{cases} x+a, & x < 0 \\ |x-1|, & x \geq 0 \end{cases}$$ and $$g(x) = \begin{cases} x+1, & x < 0 \\ (x-1)^2 + b, & x \geq 0 \end{cases}$$, where $$a, b$$ are non-negative real numbers. If $$g \circ f(x)$$ is continuous for all $$x \in R$$, then $$a + b$$ is equal to ________.

For real numbers $$\alpha, \beta, \gamma$$ and $$\delta$$, if $$\int \frac{(x^2-1)+\tan^{-1}\left(\frac{x^2+1}{x}\right)}{(x^4+3x^2+1)\tan^{-1}\left(\frac{x^2+1}{x}\right)}dx = \alpha\log_e\left(\tan^{-1}\left(\frac{x^2+1}{x}\right)\right) + \beta\tan^{-1}\left(\frac{\gamma(x^2-1)}{x}\right) + \delta\tan^{-1}\left(\frac{x^2+1}{x}\right) + C$$ where $$C$$ is an arbitrary constant, then the value of $$10(\alpha + \beta\gamma + \delta)$$ is equal to ________.

Let $$\vec{c}$$ be a vector perpendicular to the vectors $$\vec{a} = \hat{i} + \hat{j} - \hat{k}$$ and $$\vec{b} = \hat{i} + 2\hat{j} + \hat{k}$$. If $$\vec{c} \cdot (\hat{i} + \hat{j} + 3\hat{k}) = 8$$, then the value of $$\vec{c} \cdot (\vec{a} \times \vec{b})$$ is equal to ________.

If the distance of the point $$(1, -2, 3)$$ from the plane $$x + 2y - 3z + 10 = 0$$ measured parallel to the line, $$\frac{x-1}{3} = \frac{2-y}{m} = \frac{z+3}{1}$$ is $$\sqrt{\frac{7}{2}}$$, then the value of $$|m|$$ is equal to ________.