NTA JEE Main 28th July 2022 Shift 1

The sum of all real values of $$x$$ for which $$\frac{3x^2 - 9x + 17}{x^2 + 3x + 10} = \frac{5x^2 - 7x + 19}{3x^2 + 5x + 12}$$ is equal to

Let S be the set of all passwords which are six to eight characters long, where each character is either an alphabet from {A, B, C, D, E} or a number from {1, 2, 3, 4, 5} with the repetition of characters allowed. If the number of passwords in S whose at least one character is a number from {1, 2, 3, 4, 5} is $$\alpha \times 5^6$$, then $$\alpha$$ is equal to

For $$p, q \in \mathbb{R}$$, consider the real valued function $$f(x) = (x - p)^2 - q$$, $$x \in \mathbb{R}$$ and $$q > 0$$. Let $$a_1, a_2, a_3$$ and $$a_4$$ be in an arithmetic progression with mean $$p$$ and positive common difference. If $$|f(a_i)| = 500$$ for all $$i = 1, 2, 3, 4$$, then the absolute difference between the roots of $$f(x) = 0$$ is

Let $$x_1, x_2, x_3, \ldots, x_{20}$$ be in geometric progression with $$x_1 = 3$$ and the common ratio $$\frac{1}{2}$$. A new data is constructed replacing each $$x_i$$ by $$(x_i - i)^2$$. If $$\bar{x}$$ is the mean of new data, then the greatest integer less than or equal to $$\bar{x}$$ is

For the hyperbola $$H: x^2 - y^2 = 1$$ and the ellipse $$E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$, $$a > b > 0$$, let the
(1) eccentricity of E be reciprocal of the eccentricity of H, and
(2) the line $$y = \sqrt{\frac{5}{2}}x + K$$ be a common tangent of E and H.
Then $$4(a^2 + b^2)$$ is equal to

$$\lim_{x \to 0} \left(\frac{(x+2\cos x)^3 + 2(x+2\cos x)^2 + 3\sin(x+2\cos x)}{(x+2)^3 + 2(x+2)^2 + 3\sin(x+2)}\right)^{\frac{100}{x}}$$ is equal to

Let $$A = \begin{pmatrix} 1 & -1 \\ 2 & \alpha \end{pmatrix}$$ and $$B = \begin{pmatrix} \beta & 1 \\ 1 & 0 \end{pmatrix}$$, $$\alpha, \beta \in \mathbb{R}$$. Let $$\alpha_1$$ be the value of $$\alpha$$ which satisfies $$(A + B)^2 = A^2 + \begin{pmatrix} 2 & 2 \\ 2 & 2 \end{pmatrix}$$ and $$\alpha_2$$ be the value of $$\alpha$$ which satisfies $$(A + B)^2 = B^2$$. Then $$|\alpha_1 - \alpha_2|$$ is equal to

Let $$f: [0, 1] \to \mathbb{R}$$ be a twice differentiable function in (0, 1) such that $$f(0) = 3$$ and $$f(1) = 5$$. If the line $$y = 2x + 3$$ intersects the graph of $$f$$ at only two distinct points in (0, 1), then the least number of points $$x \in (0, 1)$$, at which $$f''(x) = 0$$, is

If $$\int_0^{\sqrt{3}} \frac{15x^3}{\sqrt{(1+x^2)} + \sqrt{(1+x^2)^3}} dx = \alpha\sqrt{2} + \beta\sqrt{3}$$, where $$\alpha, \beta$$ are integers, then $$\alpha + \beta$$ is equal to

Let $$P(-2, -1, 1)$$ and $$Q\left(\frac{56}{17}, \frac{43}{17}, \frac{111}{17}\right)$$ be the vertices of the rhombus PRQS. If the direction ratios of the diagonal RS are $$\alpha, -1, \beta$$, where both $$\alpha$$ and $$\beta$$ are integers of minimum absolute values, then $$\alpha^2 + \beta^2$$ is equal to