NTA JEE Main 25th January 2023 Shift 2

If the function $$f(x) = \begin{cases} (1+|\cos x|)^{\frac{\lambda}{|\cos x|}}, & 0 < x < \frac{\pi}{2} \\ \mu, & x = \frac{\pi}{2} \\ e^{\frac{\cot 6x}{\cot 4x}}, & \frac{\pi}{2} < x < \pi \end{cases}$$ is continuous at $$x = \frac{\pi}{2}$$, then $$9\lambda + 6\log_e \mu + \mu^6 - e^{6\lambda}$$ is equal to

Let the function $$f(x) = 2x^3 + (2p-7)x^2 + 3(2p-9)x - 6$$ have a maxima for some value of $$x < 0$$ and a minima for some value of $$x > 0$$. Then, the set of all values of $$p$$ is

The integral $$16\int_1^2 \frac{dx}{x^3(x^2+2)^2}$$ is equal to

Let T and C respectively, be the transverse and conjugate axes of the hyperbola $$16x^2 - y^2 + 64x + 4y + 44 = 0$$. Then the area of the region above the parabola $$x^2 = y + 4$$, below the transverse axis T and on the right of the conjugate axis C is:

Let $$y = y(t)$$ be a solution of the differential equation $$\frac{dy}{dt} + \alpha y = \gamma e^{-\beta t}$$ Where, $$\alpha > 0, \beta > 0$$ and $$\gamma > 0$$. Then $$\lim_{t \to \infty} y(t)$$

If the four points, whose position vectors are $$3\hat{i} - 4\hat{j} + 2\hat{k}$$, $$\hat{i} + 2\hat{j} - \hat{k}$$, $$-2\hat{i} - \hat{j} + 3\hat{k}$$ and $$5\hat{i} - 2\alpha\hat{j} + 4\hat{k}$$ are coplanar, then $$\alpha$$ is equal to

Let $$\vec{a} = -\hat{i} - \hat{j} + \hat{k}$$, $$\vec{a} \cdot \vec{b} = 1$$ and $$\vec{a} \times \vec{b} = \hat{i} - \hat{j}$$. Then $$\vec{a} - 6\vec{b}$$ is equal to

The foot of perpendicular of the point $$(2, 0, 5)$$ on the line $$\frac{x+1}{2} = \frac{y-1}{5} = \frac{z+1}{-1}$$ is $$(\alpha, \beta, \gamma)$$. Then. Which of the following is NOT correct?

Let $$a \in R$$ and let $$\alpha, \beta$$ be the roots of the equation $$x^2 + 60^{1/4}x + a = 0$$. If $$\alpha^4 + \beta^4 = -30$$, then the product of all possible values of $$a$$ is _____.

Suppose Anil's mother wants to give 5 whole fruits to Anil from a basket of 7 red apples, 5 white apples and 8 oranges. If in the selected 5 fruits, at least 2 orange, at least one red apple and at least one white apple must be given, then the number of ways, Anil's mother can offer 5 fruits to Anil is _____.