NTA JEE Main 26th July 2022 Shift 1

If for some $$p, q, r \in \mathbb{R}$$, all have positive sign, one of the roots of the equation $$(p^2 + q^2)x^2 - 2q(p + r)x + q^2 + r^2 = 0$$ is also a root of the equation $$x^2 + 2x - 8 = 0$$, then $$\dfrac{q^2 + r^2}{p^2}$$ is equal to ______.

The number of 5-digit natural numbers, such that the product of their digits is 36, is ______.

The series of positive multiples of 3 is divided into sets: $$\{3\}, \{6, 9, 12\}, \{15, 18, 21, 24, 27\}, \ldots$$ Then the sum of the elements in the $$11^{th}$$ set is equal to ______.

If the coefficients of $$x$$ and $$x^2$$ in the expansion of $$(1 + x)^p(1 - x)^q$$, $$p, q \le 15$$, are $$-3$$ and $$-5$$ respectively, then the coefficient of $$x^3$$ is equal to ______.

The equations of the sides $$AB$$, $$BC$$ and $$CA$$ of a triangle $$ABC$$ are $$2x + y = 0$$, $$x + py = 15a$$ and $$x - y = 3$$ respectively. If its orthocentre is $$(2, a)$$, $$-\dfrac{1}{2} < a < 2$$, then $$p$$ is equal to ______.

The number of distinct real roots of the equation $$x^5(x^3 - x^2 - x + 1) + x(3x^3 - 4x^2 - 2x + 4) - 1 = 0$$ is ______.

Let the function $$f(x) = 2x^2 - \log_e x$$, $$x > 0$$, be decreasing in $$(0, a)$$ and increasing in $$(a, 4)$$. A tangent to the parabola $$y^2 = 4ax$$ at a point $$P$$ on it passes through the point $$(8a, 8a - 1)$$ but does not pass through the point $$\left(-\dfrac{1}{a}, 0\right)$$. If the equation of the normal at $$P$$ is $$\dfrac{x}{\alpha} + \dfrac{y}{\beta} = 1$$, then $$\alpha + \beta$$ is equal to ______.

If $$n(2n + 1) \displaystyle\int_0^1 (1 - x^n)^{2n} dx = 1177 \int_0^1 (1 - x^n)^{2n+1} dx$$, $$n \in \mathbb{N}$$, then $$n$$ is equal to ______.

Let a curve $$y = y(x)$$ pass through the point $$(3, 3)$$ and the area of the region under this curve, above the $$x$$-axis and between the abscissae $$3$$ and $$x (> 3)$$ be $$\left(\dfrac{y}{x}\right)^3$$. If this curve also passes through the point $$(\alpha, 6\sqrt{10})$$ in the first quadrant, then $$\alpha$$ is equal to ______.

Let $$Q$$ and $$R$$ be two points on the line $$\dfrac{x+1}{2} = \dfrac{y+2}{3} = \dfrac{z-1}{2}$$ at a distance $$\sqrt{26}$$ from the point $$P(4, 2, 7)$$. Then the square of the area of the triangle $$PQR$$ is ______.