NTA JEE Main 20th July 2021 Shift 1 - Mathematics

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:

Let $$[x]$$ denote the greatest integer $$\le x$$, where $$x \in R$$. If the domain of the real valued function $$f(x) = \sqrt{\frac{|x|-2}{|x|-3}}$$ is $$(-\infty, a) \cup [b, c) \cup [4, \infty)$$, $$a < b < c$$, then the value of $$a + b + c$$ is:

Let a function $$f : R \to R$$ be defined as,
$$$f(x) = \begin{cases} \sin x - e^x & \text{if } x \le 0 \\ a + [-x] & \text{if } 0 < x < 1 \\ 2x - b & \text{if } x \ge 1 \end{cases}$$$
Where $$[x]$$ is the greatest integer less than or equal to $$x$$. If $$f$$ is continuous on $$R$$, then $$(a + b)$$ is equal to:

Let $$A = [a_{ij}]$$ be a $$3 \times 3$$ matrix, where $$a_{ij} = \begin{cases} 1, & \text{if } i = j \\ -x, & \text{if } |i-j| = 1 \\ 2x+1, & \text{otherwise} \end{cases}$$
Let a function $$f : R \to R$$ be defined as $$f(x) = \det(A)$$. Then the sum of maximum and minimum values of $$f$$ on $$R$$ is equal to:

Let $$a$$ be a real number such that the function $$f(x) = ax^2 + 6x - 15$$, $$x \in R$$ is increasing in $$\left(-\infty, \frac{3}{4}\right)$$ and decreasing in $$\left(\frac{3}{4}, \infty\right)$$. Then the function $$g(x) = ax^2 - 6x + 15$$, $$x \in R$$ has a

Let $$a$$ be a positive real number such that $$\int_0^a e^{x-[x]} dx = 10e - 9$$ where $$[x]$$ is the greatest integer less than or equal to $$x$$. Then, $$a$$ is equal to:

The value of the integral $$\int_{-1}^{1} \log_e\left(\sqrt{1-x} + \sqrt{1+x}\right)dx$$ is equal to:

Let $$y = y(x)$$ be the solution of the differential equation $$x \tan\left(\frac{y}{x}\right) dy = \left(y \tan\left(\frac{y}{x}\right) - x\right) dx$$, $$-1 \le x \le 1$$, $$y\left(\frac{1}{2}\right) = \frac{\pi}{6}$$. Then the area of the region bounded by the curves $$x = 0$$, $$x = \frac{1}{\sqrt{2}}$$ and $$y = y(x)$$ in the upper half plane is:

Let $$y = y(x)$$ be the solution of the differential equation $$e^x\sqrt{1-y^2}dx + \left(\frac{y}{x}\right)dy = 0$$, $$y(1) = -1$$. Then the value of $$(y(3))^2$$ is equal to:

Let $$\vec{a} = 2\hat{i} + \hat{j} - 2\hat{k}$$ and $$\vec{b} = \hat{i} + \hat{j}$$. If $$\vec{c}$$ is a vector such that $$\vec{a} \cdot \vec{c} = |\vec{c}|$$, $$|\vec{c} - \vec{a}| = 2\sqrt{2}$$ and the angle between $$(\vec{a} \times \vec{b})$$ and $$\vec{c}$$ is $$\frac{\pi}{6}$$, then the value of $$|(\vec{a} \times \vec{b}) \times \vec{c}|$$ is:

Words with or without meaning are to be formed using all the letters of the word EXAMINATION. The probability that the letter M appears at the fourth position in any such word is:

There are 15 players in a cricket team, out of which 6 are bowlers, 7 are batsmen and 2 are wicketkeepers. The number of ways, a team of 11 players be selected from them so as to include at least 4 bowlers, 5 batsmen and 1 wicketkeeper, is ___.

The number of rational terms in the binomial expansion of $$\left(4^{1/4} + 5^{1/6}\right)^{120}$$ is ___.

Let $$y = mx + c$$, $$m > 0$$ be the focal chord of $$y^2 = -64x$$, which is tangent to $$(x+10)^2 + y^2 = 4$$. Then, the value of $$4\sqrt{2}(m+c)$$ is equal to ___.

If the value of $$\lim_{x \to 0}\left(2 - \cos x\sqrt{\cos 2x}\right)^{\left(\frac{x+2}{x^2}\right)}$$ is equal to $$e^a$$, then $$a$$ is equal to ___.

Let $$A = \begin{bmatrix} 1 & -1 & 0 \\ 0 & 1 & -1 \\ 0 & 0 & 1 \end{bmatrix}$$ and $$B = 7A^{20} - 20A^7 + 2I$$, where $$I$$ is an identity matrix of order $$3 \times 3$$. If $$B = [b_{ij}]$$, then $$b_{13}$$ is equal to ___.

Let $$a, b, c, d$$ be in arithmetic progression with common difference $$\lambda$$. If
$$\begin{vmatrix} x+a-c & x+b & x+a \\ x-1 & x+c & x+b \\ x-b+d & x+d & x+c \end{vmatrix} = 2$$,
then value of $$\lambda^2$$ is equal to ___.

Let $$T$$ be the tangent to the ellipse $$E : x^2 + 4y^2 = 5$$ at the point $$P(1, 1)$$. If the area of the region bounded by the tangent $$T$$, ellipse $$E$$, lines $$x = 1$$ and $$x = \sqrt{5}$$ is $$\alpha\sqrt{5} + \beta + \gamma\cos^{-1}\left(\frac{1}{\sqrt{5}}\right)$$, then $$|\alpha + \beta + \gamma|$$ is equal to ___.

Let $$\vec{a}, \vec{b}, \vec{c}$$ be three mutually perpendicular vectors of the same magnitude and equally inclined at an angle $$\theta$$, with the vector $$\vec{a} + \vec{b} + \vec{c}$$. Then $$36\cos^2 2\theta$$ is equal to ___.

Let $$P$$ be a plane passing through the points $$(1, 0, 1)$$, $$(1, -2, 1)$$ and $$(0, 1, -2)$$. Let a vector $$\vec{a} = \alpha\hat{i} + \beta\hat{j} + \gamma\hat{k}$$ be such that $$\vec{a}$$ is parallel to the plane $$P$$, perpendicular to $$(\hat{i} + 2\hat{j} + 3\hat{k})$$ and $$\vec{a} \cdot (\hat{i} + \hat{j} + 2\hat{k}) = 2$$, then $$(\alpha - \beta + \gamma)^2$$ equals ___.

If the shortest distance between the lines $$\vec{r_1} = \alpha\hat{i} + 2\hat{j} + 2\hat{k} + \lambda(\hat{i} - 2\hat{j} + 2\hat{k})$$, $$\lambda \in R$$, $$\alpha > 0$$ and $$\vec{r_2} = -4\hat{i} - \hat{k} + \mu(3\hat{i} - 2\hat{j} - 2\hat{k})$$, $$\mu \in R$$ is 9, then $$\alpha$$ is equal to ___.