NTA JEE Mains 1st feb 2023 Shift 2

The number of integral values of $$k$$, for which one root of the equation $$2x^2 - 8x + k = 0$$ lies in the interval $$(1, 2)$$ and its other root lies in the interval $$(2, 3)$$, is :

Let $$a, b$$ be two real numbers such that $$ab < 0$$. If the complex number $$\frac{1+ai}{b+i}$$ is of unit modulus and $$a + ib$$ lies on the circle $$|z - 1| = |2z|$$, then a possible value of $$\frac{1+[a]}{4b}$$, where $$[t]$$ is greatest integer function, is :

Number of integral solutions to the equation $$x + y + z = 21$$, where $$x \geq 1, y \geq 3, z \geq 4$$, is equal to _____.

The total number of six digit numbers, formed using the digits 4, 5, 9 only and divisible by 6, is _____.

The sum $$\sum_{n=1}^{\infty} \frac{2n^2+3n+4}{(2n)!}$$ is equal to :

The sum of the common terms of the following three arithmetic progressions.
$$3, 7, 11, 15, \ldots, 399$$
$$2, 5, 8, 11, \ldots, 359$$ and
$$2, 7, 12, 17, \ldots, 197$$, is equal to _____.

If the term without $$x$$ in the expansion of $$\left(x^{2/3} + \frac{\alpha}{x^3}\right)^{22}$$ is 7315, then $$|\alpha|$$ is equal to _____.

Let the sixth term in the binomial expansion of $$\left(\sqrt{2^{\log_2(10-3^x)}} + \sqrt[5]{2^{(x-2)\log_2 3}}\right)^m$$ powers of $$2^{(x-2)\log_2 3}$$ be 21. If the binomial coefficients of the second, third and fourth terms in the expansion are respectively the first, third and fifth terms of an A.P., then the sum of the squares of all possible values of $$x$$ is _____.

If the $$x$$-intercept of a focal chord of the parabola $$y^2 = 8x + 4y + 4$$ is 3, then the length of this chord is equal to _____.

The line $$x = 8$$ is the directrix of the ellipse $$E : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ with the corresponding focus $$(2, 0)$$. If the tangent to $$E$$ at the point $$P$$ in the first quadrant passes through the point $$(0, 4\sqrt{3})$$ and intersects the $$x$$-axis at $$Q$$, then $$(3PQ)^2$$ is equal to _____.