NTA JEE Main 27th August 2021 Shift 2

Let $$z_1$$ and $$z_2$$ be two complex numbers such that $$\arg(z_1 - z_2) = \frac{\pi}{4}$$ and $$z_1, z_2$$ satisfy the equation $$|z - 3| = \text{Re}(z)$$. Then the imaginary part $$z_1 + z_2$$ is equal to _________.

Let $$S = \{1, 2, 3, 4, 5, 6, 9\}$$. Then the number of elements in the set $$T = \{A \subseteq S : A \neq \phi$$ and the sum of all the elements of $$A$$ is not a multiple of 3$$\}$$ is _________.

$$3 \times 7^{22} + 2 \times 10^{22} - 44$$ when divided by 18 leaves the remainder _________.

Let $$S$$ be the sum of all solutions (in radians) of the equation $$\sin^4\theta + \cos^4\theta - \sin\theta\cos\theta = 0$$ in $$[0, 4\pi]$$ then $$\frac{8S}{\pi}$$ is equal to _________.

Two circles each of radius 5 units touch each other at the point $$(1, 2)$$. If the equation of their common tangent is $$4x + 3y = 10$$, and $$C_1(\alpha, \beta)$$ and $$C_2(\gamma, \delta)$$, $$C_1 \neq C_2$$ are their centres, then $$|(\alpha + \beta)(\gamma + \delta)|$$ is equal to _________.

Let $$P(a\sec\theta, b\tan\theta)$$ and $$Q(a\sec\phi, b\tan\phi)$$ where $$\theta + \phi = \frac{\pi}{2}$$, be two points on the hyperbola $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$. If the ordinate of the point of intersection of normals at $$P$$ and $$Q$$ is $$-k\left(\frac{a^2+b^2}{2b}\right)$$, then $$k$$ is equal to _________.

An online exam is attempted by 50 candidates out of which 20 are boys. The average marks obtained by boys is 12 with a variance 2. The variance of marks obtained by 30 girls is also 2. The average marks of all 50 candidates is 15. If $$\mu$$ is the average marks of girls and $$\sigma^2$$ is the variance of marks of 50 candidates, then $$\mu + \sigma^2$$ is equal to _________.

$$\int \frac{2e^x+3e^{-x}}{4e^x+7e^{-x}} dx = \frac{1}{14}(ux + v\log_e(4e^x + 7e^{-x})) + C$$, where $$C$$ is a constant of integration, then $$u + v$$ is equal to _________.

Let $$S$$ be the mirror image of the point $$Q(1, 3, 4)$$ with respect to the plane $$2x - y + z + 3 = 0$$ and let $$R(3, 5, \gamma)$$ be a point of this plane. Then the square of the length of the line segment $$SR$$ is _________.

The probability distribution of random variable $$X$$ is given by

Let $$p = P(1 < X < 4 | X < 3)$$. If $$5p = \lambda K$$, then $$\lambda$$ is equal to _________.