NTA JEE Main 17th March 2021 Shift 2

If $$1, \log_{10}(4^x - 2)$$ and $$\log_{10}\left(4^x + \frac{18}{5}\right)$$ are in arithmetic progression for a real number $$x$$ then the value of the determinant $$\begin{vmatrix} 2(x-\frac{1}{2}) & x-1 & x^2 \\ 1 & 0 & x \\ x & 1 & 0 \end{vmatrix}$$ is equal to ________.

Let the coefficients of third, fourth and fifth terms in the expansion of $$\left(x + \frac{a}{x^2}\right)^n$$, $$x \neq 0$$, be in the ratio 12 : 8 : 3. Then the term independent of $$x$$ in the expansion, is equal to ________.

Let $$\tan\alpha$$, $$\tan\beta$$ and $$\tan\gamma$$; $$\alpha, \beta, \gamma \neq \frac{(2n-1)\pi}{2}$$, $$n \in N$$ be the slopes of the three line segments $$OA$$, $$OB$$ and $$OC$$, respectively, where $$O$$ is origin. If circumcentre of $$\triangle ABC$$ coincides with origin and its orthocentre lies on $$y$$-axis, then the value of $$\left(\frac{\cos 3\alpha + \cos 3\beta + \cos 3\gamma}{\cos\alpha \cdot \cos\beta \cdot \cos\gamma}\right)^2$$ is equal to ________.

Consider a set of $$3n$$ numbers having variance 4. In this set, the mean of first $$2n$$ numbers is 6 and the mean of the remaining $$n$$ numbers is 3. A new set is constructed by adding 1 into each of the first $$2n$$ numbers, and subtracting 1 from each of the remaining $$n$$ numbers. If the variance of the new set is $$k$$, then $$9k$$ is equal to ________.

Let $$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$$ and $$B = \begin{bmatrix} \alpha \\ \beta \end{bmatrix} \neq \begin{bmatrix} 0 \\ 0 \end{bmatrix}$$ such that $$AB = B$$ and $$a + d = 2021$$, then the value of $$ad - bc$$ is equal to ________.

Let $$f : [-1, 1] \to R$$ be defined as $$f(x) = ax^2 + bx + c$$ for all $$x \in [-1, 1]$$, where $$a, b, c \in R$$ such that $$f(-1) = 2$$, $$f'(-1) = 1$$ and for $$x \in (-1, 1)$$ the maximum value of $$f''(x)$$ is $$\frac{1}{2}$$. If $$f(x) \leq \alpha$$, $$x \in [-1, 1]$$, then the least value of $$\alpha$$ is equal to ________.

Let $$I_n = \int_1^e x^{19}(\log|x|)^n dx$$, where $$n \in N$$. If $$(20)I_{10} = \alpha I_9 + \beta I_8$$, for natural numbers $$\alpha$$ and $$\beta$$, then $$\alpha - \beta$$ is equal to ________.

Let $$f : [-3, 1] \to R$$ be given as $$f(x) = \begin{cases} \min\{(x+6), x^2\}, & -3 \leq x \leq 0 \\ \max\{\sqrt{x}, x^2\}, & 0 \leq x \leq 1 \end{cases}$$. If the area bounded by $$y = f(x)$$ and $$x$$-axis is $$A$$ sq units, then the value of $$6A$$ is equal to ________.

Let $$\vec{x}$$ be a vector in the plane containing vectors $$\vec{a} = 2\hat{i} - \hat{j} + \hat{k}$$ and $$\vec{b} = \hat{i} + 2\hat{j} - \hat{k}$$. If the vector $$\vec{x}$$ is perpendicular to $$(3\hat{i} + 2\hat{j} - \hat{k})$$ and its projection on $$\vec{a}$$ is $$\frac{17\sqrt{6}}{2}$$, then the value of $$|\vec{x}|^2$$ is equal to ________.

Let $$P$$ be an arbitrary point having sum of the squares of the distances from the planes $$x + y + z = 0$$, $$lx - nz = 0$$ and $$x - 2y + z = 0$$ equal to 9 units. If the locus of the point $$P$$ is $$x^2 + y^2 + z^2 = 9$$, then the value of $$l - n$$ is equal to ________.