NTA JEE Main 28th June 2022 Shift 2 - Mathematics

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

The probability that a randomly chosen one-one function from the set $$\{a, b, c, d\}$$ to the set $$\{1, 2, 3, 4, 5\}$$ satisfies $$f(a) + 2f(b) - f(c) = f(d)$$ is

Let $$f, g : \mathbf{R} \to \mathbf{R}$$ be functions defined by
$$f(x) = \begin{cases} [x] & , x < 0 \\ |1-x| & , x \geq 0 \end{cases}$$ and
$$g(x) = \begin{cases} e^x - x & , x < 0 \\ (x-1)^2 - 1 & , x \geq 0 \end{cases}$$
where $$[x]$$ denote the greatest integer less than or equal to $$x$$. Then, the function fog is discontinuous at exactly

Let $$f : \mathbf{R} \to \mathbf{R}$$ be a differentiable function such that $$f\left(\frac{\pi}{4}\right) = \sqrt{2}$$, $$f\left(\frac{\pi}{2}\right) = 0$$ and $$f'\left(\frac{\pi}{2}\right) = 1$$ and let $$g(x) = \int_x^{\pi} (f'(t) \sec t + \tan t \sec t \, f(t)) dt$$ for $$x \in \left[\frac{\pi}{4}, \frac{\pi}{2}\right)$$. Then $$\lim_{x \to \left(\frac{\pi}{2}\right)^-} g(x)$$ is equal to

Let $$f : \mathbf{R} \to \mathbf{R}$$ be continuous function satisfying $$f(x) + f(x+k) = n$$, for all $$x \in \mathbf{R}$$ where $$k > 0$$ and $$n$$ is a positive integer. If $$I_1 = \int_0^{4nk} f(x) dx$$ and $$I_2 = \int_{-k}^{3k} f(x) dx$$, then

The area of the bounded region enclosed by the curve $$y = 3 - \left|x - \frac{1}{2}\right| - |x + 1|$$ and the $$x$$-axis is

Let $$x = x(y)$$ be the solution of the differential equation $$2ye^{x/y^2} dx + \left(y^2 - 4xe^{x/y^2}\right) dy = 0$$ such that $$x(1) = 0$$. Then, $$x(e)$$ is equal to

Let the slope of the tangent to a curve $$y = f(x)$$ at $$(x, y)$$ be given by $$2 \tan x(\cos x - y)$$. If the curve passes through the point $$\left(\frac{\pi}{4}, 0\right)$$, then the value of $$\int_0^{\pi/2} y \, dx$$ is equal to

Let $$\vec{a} = \alpha \hat{i} + 2\hat{j} - \hat{k}$$ and $$\vec{b} = -2\hat{i} + \alpha \hat{j} + \hat{k}$$, where $$\alpha \in \mathbf{R}$$. If the area of the parallelogram whose adjacent sides are represented by the vectors $$\vec{a}$$ and $$\vec{b}$$ is $$\sqrt{15(\alpha^2 + 4)}$$, then the value of $$2|\vec{a}|^2 + (\vec{a} \cdot \vec{b})|\vec{b}|^2$$ is equal to

Let $$\vec{a}$$ be a vector which is perpendicular to the vector $$3\hat{i} + \frac{1}{2}\hat{j} + 2\hat{k}$$. If $$\vec{a} \times (2\hat{i} + \hat{k}) = 2\hat{i} - 13\hat{j} - 4\hat{k}$$, then the projection of the vector $$\vec{a}$$ on the vector $$2\hat{i} + 2\hat{j} + \hat{k}$$ is

Let the plane $$ax + by + cz = d$$ pass through $$(2, 3, -5)$$ and is perpendicular to the planes $$2x + y - 5z = 10$$ and $$3x + 5y - 7z = 12$$.
If $$a, b, c, d$$ are integers $$d > 0$$ and $$\gcd(|a|, |b|, |c|, d) = 1$$ then the value of $$a + 7b + c + 20d$$ is equal to

Sum of squares of modulus of all the complex numbers $$z$$ satisfying $$\bar{z} = iz^2 + z^2 - z$$ is equal to

Let for $$n = 1, 2, \ldots, 50$$, $$S_n$$ be the sum of the infinite geometric progression whose first term is $$n^2$$ and whose common ratio is $$\frac{1}{(n+1)^2}$$. Then the value of $$\frac{1}{26} + \sum_{n=1}^{50} \left(S_n + \frac{2}{n+1} - n - 1\right)$$ is equal to

If one of the diameters of the circle $$x^2 + y^2 - 2\sqrt{2}x - 6\sqrt{2}y + 14 = 0$$ is a chord of the circle $$(x - 2\sqrt{2})^2 + (y - 2\sqrt{2})^2 = r^2$$, then the value of $$r^2$$ is equal to

If $$\lim_{x \to 1} \left(\frac{\sin(3x^2 - 4x + 1) - x^2 + 1}{2x^3 - 7x^2 + ax + b}\right) = -2$$, then the value of $$(a - b)$$ is equal to

The maximum number of compound propositions, out of $$p \vee r \vee s$$, $$p \vee r \vee \sim s$$, $$p \vee \sim q \vee s$$, $$\sim p \vee \sim r \vee s$$, $$\sim p \vee \sim r \vee \sim s$$, $$\sim p \vee q \vee \sim s$$, $$q \vee r \vee \sim s$$, $$q \vee \sim r \vee \sim s$$, $$\sim p \vee \sim q \vee \sim s$$ that can be made simultaneously true by an assignment of the truth values to $$p, q, r$$ and $$s$$, is equal to

Suppose a class has 7 students. The average marks of these students in the mathematics examination is 62, and their variance is 20. A student fails in the examination if he/she gets less than 50 marks, then in worst case, the number of students can fail is

Let $$A = \begin{pmatrix} 1+i & 1 \\ -i & 0 \end{pmatrix}$$ where $$i = \sqrt{-1}$$. Then, the number of elements in the set $$\{n \in \{1, 2, \ldots, 100\} : A^n = A\}$$ is

If the system of linear equations
$$2x - 3y = \gamma + 5$$
$$\alpha x + 5y = \beta + 1$$,
where $$\alpha, \beta, \gamma \in \mathbf{R}$$ has infinitely many solutions, then the value of $$|9\alpha + 3\beta + 5\gamma|$$ is equal to

Let $$S = \{1, 2, 3, 4\}$$. Then the number of elements in the set $$\{f : S \times S \to S : f$$ is onto and $$f(a,b) = f(b,a) \geq a \forall (a,b) \in S \times S\}$$ is

Let the image of the point $$P(1, 2, 3)$$ in the line $$L : \frac{x-6}{3} = \frac{y-1}{2} = \frac{z-2}{3}$$ be $$Q$$. Let $$R(\alpha, \beta, \gamma)$$ be a point that divides internally the line segment $$PQ$$ in the ratio 1 : 3. Then the value of $$22(\alpha + \beta + \gamma)$$ is equal to