NTA JEE Main 15th April 2023 Shift 1

The number of real roots of the equation $$x|x| - 5|x + 2| + 6 = 0$$, is

If the set $$\left\{Re\left(\frac{z - \bar{z} + z\bar{z}}{2 - 3z + 5\bar{z}}\right) : z \in \mathbb{C}, \ Re \ z = 3\right\}$$ is equal to the interval $$(\alpha, \beta]$$, then $$24(\beta - \alpha)$$ is equal to

The total number of three-digit numbers, divisible by 3, which can be formed using the digits 1, 3, 5, 8, if repetition of digits is allowed, is

Let $$A_1$$ and $$A_2$$ be two arithmetic means and $$G_1$$, $$G_2$$ and $$G_3$$ be three geometric means of two distinct positive numbers. Then $$G_1^4 + G_2^4 + G_3^4 + G_1^2 G_3^2$$ is equal to

Let $$(a + bx + cx^2)^{10} = \sum_{i=0}^{20} p_i x^i$$, $$a, b, c \in \mathbb{N}$$. If $$p_1 = 20$$ and $$p_2 = 210$$, then $$2(a + b + c)$$ is equal to

If $$(\alpha, \beta)$$ is the orthocenter of the triangle $$ABC$$ with vertices $$A(3, -7)$$, $$B(-1, 2)$$ and $$C(4, 5)$$, then $$9\alpha - 6\beta + 60$$ is equal to

The number of common tangents, to the circles $$x^2 + y^2 - 18x - 15y + 131 = 0$$ and $$x^2 + y^2 - 6x - 6y - 7 = 0$$, is

Negation of $$p \wedge (q \wedge \sim(p \wedge q))$$ is

The mean and standard deviation of 10 observations are 20 and 8 respectively. Later on, it was observed that one observation was recorded as 50 instead of 40. Then the correct variance is

Let the determinant of a square matrix $$A$$ of order $$m$$ be $$m - n$$, where $$m$$ and $$n$$ satisfy $$4m + n = 22$$ and $$17m + 4n = 93$$. If $$det(n \ adj(adj(mA))) = 3^a 5^b 6^c$$, then $$a + b + c$$ is equal to