NTA JEE Main 28th June 2022 Shift 2

Let $$f(x)$$ be a quadratic polynomial such that $$f(-2) + f(3) = 0$$. If one of the roots of $$f(x) = 0$$ is $$-1$$, then the sum of the roots of $$f(x) = 0$$ is equal to

The number of ways to distribute 30 identical candies among four children $$C_1, C_2, C_3$$ and $$C_4$$ so that $$C_2$$ receives atleast 4 and atmost 7 candies, $$C_3$$ receives atleast 2 and atmost 6 candies, is equal to

If $$n$$ arithmetic means are inserted between a and 100 such that the ratio of the first mean to the last mean is 1 : 7 and $$a + n = 33$$, then the value of $$n$$ is

The term independent of $$x$$ in the expression of $$(1 - x^2 + 3x^3)\left(\frac{5}{2}x^3 - \frac{1}{5x^2}\right)^{11}$$, $$x \neq 0$$ is

If $$\cot \alpha = 1$$ and $$\sec \beta = -\frac{5}{3}$$, where $$\pi < \alpha < \frac{3\pi}{2}$$ and $$\frac{\pi}{2} < \beta < \pi$$, then the value of $$\tan(\alpha + \beta)$$ and the quadrant in which $$\alpha + \beta$$ lies, respectively are

Let a triangle be bounded by the lines $$L_1: 2x + 5y = 10$$; $$L_2: -4x + 3y = 12$$ and the line $$L_3$$, which passes through the point $$P(2,3)$$, intersect $$L_2$$ at $$A$$ and $$L_1$$ at $$B$$. If the point $$P$$ divides the line-segment $$AB$$, internally in the ratio 1 : 3, then the area of the triangle is equal to

If vertex of parabola is $$(2, -1)$$ and equation of its directrix is $$4x - 3y = 21$$, then the length of latus rectum is

Let $$a > 0$$, $$b > 0$$. Let $$e$$ and $$l$$ respectively be the eccentricity and length of the latus rectum of the hyperbola $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$. Let $$e'$$ and $$l'$$ respectively the eccentricity and length of the latus rectum of its conjugate hyperbola. If $$e^2 = \frac{11}{14}l$$ and $$(e')^2 = \frac{11}{8}l'$$, then the value of $$77a + 44b$$ is equal to

Let $$R_1 = \{(a,b) \in N \times N : |a - b| \leq 13\}$$ and $$R_2 = \{(a,b) \in N \times N : |a - b| \neq 13\}$$. Then on $$N$$:

The value of $$\lim_{n \to \infty} 6 \tan\left\{\sum_{r=1}^{n} \tan^{-1}\left(\frac{1}{r^2 + 3r + 3}\right)\right\}$$ is equal to