NTA JEE Mains 6th April 2024 Shift 1

Let $$x_1, x_2, x_3, x_4$$ be the solution of the equation $$4x^4 + 8x^3 - 17x^2 - 12x + 9 = 0$$ and $$(4 + x_1^2)(4 + x_2^2)(4 + x_3^2)(4 + x_4^2) = \frac{125}{16}m$$. Then the value of $$m$$ is

Let the first term of a series be $$T_1 = 6$$ and its $$r^{th}$$ term $$T_r = 3T_{r-1} + 6^r$$, $$r = 2, 3, \ldots, n$$. If the sum of the first $$n$$ terms of this series is $$\frac{1}{5}(n^2 - 12n + 39)(4 \cdot 6^n - 5 \cdot 3^n + 1)$$, then $$n$$ is equal to ______

If the second, third and fourth terms in the expansion of $$(x + y)^n$$ are 135, 30 and $$\frac{10}{3}$$, respectively, then $$6(n^3 + x^2 + y)$$ is equal to _______

Let a conic $$C$$ pass through the point $$(4, -2)$$ and $$P(x, y), x \geq 3$$, be any point on $$C$$. Let the slope of the line touching the conic $$C$$ only at a single point $$P$$ be half the slope of the line joining the points $$P$$ and $$(3, -5)$$. If the focal distance of the point $$(7, 1)$$ on $$C$$ is $$d$$, then $$12d$$ equals ______

Let $$L_1, L_2$$ be the lines passing through the point $$P(0, 1)$$ and touching the parabola $$9x^2 + 12x + 18y - 14 = 0$$. Let $$Q$$ and $$R$$ be the points on the lines $$L_1$$ and $$L_2$$ such that the $$\triangle PQR$$ is an isosceles triangle with base $$QR$$. If the slopes of the lines $$QR$$ are $$m_1$$ and $$m_2$$, then $$16(m_1^2 + m_2^2)$$ is equal to _______

Let $$\alpha\beta\gamma = 45$$; $$\alpha, \beta, \gamma \in \mathbb{R}$$. If $$x(\alpha, 1, 2) + y(1, \beta, 2) + z(2, 3, \gamma) = (0, 0, 0)$$ for some $$x, y, z \in \mathbb{R}, xyz \neq 0$$, then $$6\alpha + 4\beta + \gamma$$ is equal to _______

For $$n \in \mathbb{N}$$, if $$\cot^{-1}3 + \cot^{-1}4 + \cot^{-1}5 + \cot^{-1}n = \frac{\pi}{4}$$, then $$n$$ is equal to _____

Let $$r_k = \frac{\int_0^1 (1-x^7)^k dx}{\int_0^1 (1-x^7)^{k+1} dx}$$, $$k \in \mathbb{N}$$. Then the value of $$\sum_{k=1}^{10} \frac{1}{7(r_k - 1)}$$ is equal to ________

Let $$\vec{a} = 2\hat{i} - 3\hat{j} + 4\hat{k}$$, $$\vec{b} = 3\hat{i} + 4\hat{j} - 5\hat{k}$$ and a vector $$\vec{c}$$ be such that $$\vec{a} \times (\vec{b} + \vec{c}) + \vec{b} \times \vec{c} = \hat{i} + 8\hat{j} + 13\hat{k}$$. If $$\vec{a} \cdot \vec{c} = 13$$, then $$(24 - \vec{b} \cdot \vec{c})$$ is equal to _______

Let $$P$$ be the point $$(10, -2, -1)$$ and $$Q$$ be the foot of the perpendicular drawn from the point $$R(1, 7, 6)$$ on the line passing through the points $$(2, -5, 11)$$ and $$(-6, 7, -5)$$. Then the length of the line segment $$PQ$$ is equal to ________