NTA JEE Main 12th April 2023 Shift 1

Let $$\alpha$$, $$\beta$$ be the roots of the quadratic equation $$x^2 + \sqrt{6}x + 3 = 0$$. Then $$\frac{\alpha^{23} + \beta^{23} + \alpha^{14} + \beta^{14}}{\alpha^{15} + \beta^{15} + \alpha^{10} + \beta^{10}}$$ is equal to

Let $$C$$ be the circle in the complex plane with centre $$z_0 = \frac{1}{2}(1 + 3i)$$ and radius $$r = 1$$. Let $$z_1 = 1 + i$$ and the complex number $$z_2$$ be outside circle $$C$$ such that $$|z_1 - z_0||z_2 - z_0| = 1$$. If $$z_0$$, $$z_1$$ and $$z_2$$ are collinear, then the smaller value of $$|z_2|^2$$ is equal to

The number of five-digit numbers, greater than 40000 and divisible by 5, which can be formed using the digits 0, 1, 3, 5, 7 and 9 without repetition, is equal to

Let $$\langle a_n \rangle$$ be a sequence such that $$a_1 + a_2 + \ldots + a_n = \frac{n^2 + 3n}{(n+1)(n+2)}$$. If $$28 \sum_{k=1}^{10} \frac{1}{a_k} = p_1 p_2 p_3 \ldots p_m$$, where $$p_1, p_2, \ldots p_m$$ are the first $$m$$ prime numbers, then $$m$$ is equal to

If $$\frac{1}{n+1}$$ $$^nC_n + \frac{1}{n}$$ $$^nC_{n-1} + \ldots + \frac{1}{2}$$ $$^nC_1 + ^nC_0 = \frac{1023}{10}$$, then $$n$$ is equal to

The sum, of the coefficients of the first 50 terms in the binomial expansion of $$(1 - x)^{100}$$, is equal to

If the point $$\left(\alpha, \frac{7\sqrt{3}}{3}\right)$$ lies on the curve traced by the mid-points of the line segments of the lines $$x \cos\theta + y \sin\theta = 7$$, $$\theta \in \left(0, \frac{\pi}{2}\right)$$ between the co-ordinates axes, then $$\alpha$$ is equal to

In a triangle $$ABC$$, if $$\cos A + 2\cos B + \cos C = 2$$ and the lengths of the sides opposite to the angles $$A$$ and $$C$$ are 3 and 7 respectively, then $$\cos A - \cos C$$ is equal to

Let $$P\left(\frac{2\sqrt{3}}{\sqrt{7}}, \frac{6}{\sqrt{7}}\right)$$, Q, R and S be four points on the ellipse $$9x^2 + 4y^2 = 36$$. Let PQ and RS be mutually perpendicular and pass through the origin. If $$\frac{1}{(PQ)^2} + \frac{1}{(RS)^2} = \frac{p}{q}$$, where $$p$$ and $$q$$ are coprime, then $$p + q$$ is equal to

Among the two statements
$$(S_1) : (p \Rightarrow q) \wedge (p \wedge (\sim q))$$ is a contradiction and
$$(S_2) : (p \wedge q) \vee ((\sim p) \wedge q) \vee (p \wedge (\sim q)) \vee ((\sim p) \wedge (\sim q))$$ is a tautology