NTA JEE Main 30th January 2023 Shift 1

Let the system of linear equations
$$x + y + kz = 2$$
$$2x + 3y - z = 1$$
$$3x + 4y + 2z = k$$
have infinitely many solutions. Then the system
$$(k+1)x + (2k-1)y = 7$$
$$(2k+1)x + (k+5)y = 10$$ has:

Suppose $$f: R \to (0, \infty)$$ be a differentiable function such that $$5f(x+y) = f(x) \cdot f(y), \forall x, y \in R$$. If $$f(3) = 320$$, then $$\sum_{n=0}^{5} f(n)$$ is equal to:

The number of points on the curve $$y = 54x^5 - 135x^4 - 70x^3 + 180x^2 + 210x$$ at which the normal lines are parallel to $$x + 90y + 2 = 0$$ is:

If $$[t]$$ denotes the greatest integer $$\leq 1$$, then the value of $$\frac{3e-1}{e} \int_1^2 x^2 e^{x+x^3} dx$$ is:

Let the solution curve $$y = y(x)$$ of the differential equation $$\frac{dy}{dx} - \frac{3x^5\tan^{-1}x^3}{1+x^{6\cdot 7}} y = 2x \exp\frac{x^3 - \tan^{-1}x^3}{(1+x)^7}$$ pass through the origin. Then $$y(1)$$ is equal to:

If $$\vec{a}$$, $$\vec{b}$$, $$\vec{c}$$ are three non-zero vectors and $$\hat{n}$$ is a unit vector perpendicular to $$\vec{c}$$ such that $$\vec{a} = \alpha\vec{b} - \hat{n}$$, $$\alpha \neq 0$$ and $$\vec{b} \cdot \vec{c} = 12$$, then $$\vec{c} \times \vec{a} \times \vec{b}$$ is equal to:

The line $$l_1$$ passes through the point $$(2, 6, 2)$$ and is perpendicular to the plane $$2x + y - 2z = 10$$. Then the shortest distance between the line $$l_1$$ and the line $$\frac{x+1}{2} = \frac{y+4}{-3} = \frac{z}{2}$$ is:

Let a unit vector $$\vec{OP}$$ make angle $$\alpha, \beta, \gamma$$ with the positive directions of the co-ordinate axes OX, OY, OZ respectively, where $$\beta \in (0, \frac{\pi}{2})$$. $$\vec{OP}$$ is perpendicular to the plane through points $$(1, 2, 3)$$, $$(2, 3, 4)$$ and $$(1, 5, 7)$$, then which one is true?

If an unbiased die, marked with $$-2, -1, 0, 1, 2, 3$$ on its faces is thrown five times, then the probability that the product of the outcomes is positive, is:

A straight line cuts off the intercepts OA = a and OB = b on the positive directions of x-axis and y-axis respectively. If the perpendicular from origin O to this line makes an angle of $$\frac{\pi}{6}$$ with positive direction of y-axis and the area of $$\triangle OAB$$ is $$\frac{98}{3}\sqrt{3}$$, then $$a^2 - b^2$$ is equal to: