NTA JEE Main 15th April 2023 Shift 1

Let the system of linear equations
$$-x + 2y - 9z = 7$$
$$-x + 3y + 7z = 9$$
$$-2x + y + 5z = 8$$
$$-3x + y + 13z = \lambda$$
has a unique solution $$x = \alpha, y = \beta, z = \gamma$$. Then the distance of the point $$(\alpha, \beta, \gamma)$$ from the plane $$2x - 2y + z = \lambda$$ is

If the domain of the function $$f(x) = \log_e(4x^2 + 11x + 6) + \sin^{-1}(4x + 3) + \cos^{-1}\left(\frac{10x + 6}{3}\right)$$ is $$(\alpha, \beta]$$, then $$36|\alpha + \beta|$$ is equal to

Let $$[x]$$ denote the greatest integer function and $$f(x) = \max\{1 + x + [x], 2 + x, x + 2[x]\}$$, $$0 \leq x \leq 2$$, where $$f$$ is not continuous and $$n$$ be the number of points in $$(0, 2)$$, where $$f$$ is not differentiable. Then $$(m + n)^2 + 2$$ is equal to

If $$\int_0^1 \frac{1}{(5+2x-2x^2)(1+e^{(2-4x)})} dx = \frac{1}{\alpha} \log_e\left(\frac{\alpha+1}{\beta}\right)$$, $$\alpha, \beta > 0$$, then $$\alpha^4 - \beta^4$$ is equal to

Let $$x = x(y)$$ be the solution of the differential equation
$$2(y+2)\log_e(y+2)dx + (x + 4 - 2\log_e(y+2))dy = 0$$, $$y > -1$$ with $$x(e^4 - 2) = 1$$. Then $$x(e^9 - 2)$$ is equal to

Let $$S$$ be the set of all $$(\lambda, \mu)$$ for which the vectors $$\lambda\hat{i} - \hat{j} + \hat{k}$$, $$\hat{j} + 2\hat{j} + \mu\hat{k}$$ and $$3\hat{i} - 4\hat{j} + 5\hat{k}$$, where $$\lambda - \mu = 5$$, are coplanar, then $$\sum_{(\lambda,\mu) \in S} 80(\lambda^2 + \mu^2)$$ is equal to

Let $$ABCD$$ be a quadrilateral. If $$E$$ and $$F$$ are the mid points of the diagonals $$AC$$ and $$BD$$ respectively and $$\left(\vec{AB} - \vec{BC}\right) + \left(\vec{AD} - \vec{DC}\right) = k\vec{FE}$$, then $$k$$ is equal to

Let the foot of perpendicular of the point $$P(3, -2, -9)$$ on the plane passing through the points $$(-1, -2, -3)$$, $$(9, 3, 4)$$, $$(9, -2, 1)$$ be $$Q(\alpha, \beta, \gamma)$$. Then the distance $$Q$$ from the origin is

Let $$S$$ be the set of all values of $$\lambda$$, for which the shortest distance between the lines $$\frac{x-\lambda}{0} = \frac{y-3}{-4} = \frac{z+6}{1}$$ and $$\frac{x+\lambda}{3} = \frac{y}{-4} = \frac{z-6}{0}$$ is 13. Then $$8|\sum_{\lambda \in S} \lambda|$$ is equal to

A bag contains 6 white and 4 black balls. A die is rolled once and the number of balls equal to the number obtained on the die are drawn from the bag at random. The probability that all the balls drawn are white is