NTA JEE Main 29th January 2023 Shift 1

If all the six digit numbers $$x_1x_2x_3x_4x_5x_6$$ with $$0 < x_1 < x_2 < x_3 < x_4 < x_5 < x_6$$ are arranged in the increasing order, then the sum of the digits in the $$72^{th}$$ number is ______.

Five digit numbers are formed using the digits 1, 2, 3, 5, 7 with repetitions and are written in descending order with serial numbers. For example, the number 77777 has serial number 1. Then the serial number of 35337 is

Let $$a_1, a_2, a_3, \ldots$$ be a GP of increasing positive numbers. If the product of fourth and sixth terms is $$9$$ and the sum of fifth and seventh terms is $$24$$, then $$a_1a_9 + a_2a_4a_9 + a_5 + a_7$$ is equal to

Let the coefficients of three consecutive terms in the binomial expansion of $$(1 + 2x)^n$$ be in the ratio $$2 : 5 : 8$$. Then the coefficient of the term, which is in the middle of these three terms, is

If the co-efficient of $$x^9$$ in $$\left(\alpha x^3 + \frac{1}{\beta x}\right)^{11}$$ and the co-efficient of $$x^{-9}$$ in $$\left(\alpha x - \frac{1}{\beta x^3}\right)^{11}$$ are equal, then $$(\alpha\beta)^2$$ is equal to

Let $$f : \mathbb{R} \to \mathbb{R}$$ be a differentiable function that satisfies the relation $$f(x+y) = f(x) + f(y) - 1, \forall x, y \in \mathbb{R}$$. If $$f'(0) = 2$$, then $$|f(-2)|$$ is equal to

Suppose f is a function satisfying $$f(x + y) = f(x) + f(y)$$ for all $$x, y \in \mathbb{N}$$ and $$f(1) = \frac{1}{5}$$. If $$\sum_{n=1}^{m} \frac{f(n)}{n(n+1)(n+2)} = \frac{1}{12}$$ then m is equal to ______.

Let $$\vec{a}, \vec{b}$$ and $$\vec{c}$$ be three non-zero non-coplanar vectors. Let the position vectors of four points $$A$$, $$B$$, $$C$$ and $$D$$ be $$\vec{a} - \vec{b} + \vec{c}$$, $$\lambda\vec{a} - 3\vec{b} + 4\vec{c}$$, $$-\vec{a} + 2\vec{b} - 3\vec{c}$$ and $$2\vec{a} - 4\vec{b} + 6\vec{c}$$ respectively. If $$\vec{AB}$$, $$\vec{AC}$$ and $$\vec{AD}$$ are coplanar, then $$\lambda$$ is:

Let the equation of the plane P containing the line $$x + 10 = \frac{8-y}{2} = z$$ be $$ax + by + 3z = 2(a+b)$$ and the distance of the plane P from the point $$(1, 27, 7)$$ be $$c$$. Then $$a^2 + b^2 + c^2$$ is equal to

Let the co-ordinates of one vertex of $$\triangle ABC$$ be $$A(0, 2, \alpha)$$ and the other two vertices lie on the line $$\frac{x+\alpha}{5} = \frac{y-1}{2} = \frac{z+4}{3}$$. For $$\alpha \in \mathbb{Z}$$, if the area of $$\triangle ABC$$ is $$21$$ sq. units and the line segment $$BC$$ has length $$2\sqrt{21}$$ units, then $$\alpha^2$$ is equal to ______.