NTA JEE Main 20th July 2021 Shift 1

If $$\alpha$$ and $$\beta$$ are the distinct roots of the equation $$x^2 + (3)^{1/4}x + 3^{1/2} = 0$$, then the value of $$\alpha^{96}(\alpha^{12} - 1) + \beta^{96}(\beta^{12} - 1)$$ is equal to:

The probability of selecting integers $$a \in [-5, 30]$$ such that $$x^2 + 2(a+4)x - 5a + 64 > 0$$, for all $$x \in R$$, is:

If $$z$$ and $$\omega$$ are two complex numbers such that $$|z\omega| = 1$$ and $$\arg(z) - \arg(\omega) = \frac{3\pi}{2}$$, then $$\arg\left(\frac{1 - 2\bar{z}\omega}{1 + 3\bar{z}\omega}\right)$$ is:
(Here $$\arg(z)$$ denotes the principal argument of complex number $$z$$)

The coefficient of $$x^{256}$$ in the expansion of $$(1-x)^{101}(x^2 + x + 1)^{100}$$ is:

Let the tangent to the parabola $$S : y^2 = 2x$$ at the point $$P(2, 2)$$ meet the $$x$$-axis at $$Q$$ and normal at it meet the parabola $$S$$ at the point $$R$$. Then the area (in sq. units) of the triangle $$PQR$$ is equal to:

The Boolean expression $$(p \wedge \sim q) \Rightarrow (q \vee \sim p)$$ is equivalent to:

The mean of 6 distinct observations is 6.5 and their variance is 10.25. If 4 out of 6 observations are 2, 4, 5 and 7, then the remaining two observations are:

If in a triangle $$ABC$$, $$AB = 5$$ units, $$\angle B = \cos^{-1}\left(\frac{3}{5}\right)$$ and radius of circumcircle of $$\triangle ABC$$ is 5 units, then the area (in sq. units) of $$\triangle ABC$$ is:

Let $$A = \begin{bmatrix} 2 & 3 \\ a & 0 \end{bmatrix}$$, $$a \in R$$ be written as $$P + Q$$ where $$P$$ is a symmetric matrix and $$Q$$ is skew symmetric matrix. If det$$(Q) = 9$$, then the modulus of the sum of all possible values of determinant of $$P$$ is equal to:

The number of real roots of the equation $$\tan^{-1}\sqrt{x(x+1)} + \sin^{-1}\sqrt{x^2 + x + 1} = \frac{\pi}{4}$$ is: