NTA JEE Main 29th June 2022 Shift 2 - Mathematics

NTA JEE Main 29th June 2022 Shift 2 - Mathematics - Question 61

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Let $$\alpha$$ be a root of the equation $$1 + x^2 + x^4 = 0$$. Then the value of $$\alpha^{1011} + \alpha^{2022} - \alpha^{3033}$$ is equal to:

A

1

B

$$\alpha$$

C

$$1 + \alpha$$

D

$$1 + 2\alpha$$

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NTA JEE Main 29th June 2022 Shift 2 - Mathematics - Question 62

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Let $$(z)$$ represent the principal argument of the complex number $$z$$. The, $$|z| = 3$$ and $$\arg(z-1) - \arg(z+1) = \frac{\pi}{4}$$ intersect:

A

Exactly at one point

B

Exactly at two points

C

Nowhere

D

At infinitely many points.

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NTA JEE Main 29th June 2022 Shift 2 - Mathematics - Question 63

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The sum of the infinite series $$1 + \frac{5}{6} + \frac{12}{6^2} + \frac{22}{6^3} + \frac{35}{6^4} + \frac{51}{6^5} + \frac{70}{6^6} + \ldots$$ is equal to:

A

$$\frac{425}{216}$$

B

$$\frac{429}{216}$$

C

$$\frac{288}{125}$$

D

$$\frac{280}{125}$$

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NTA JEE Main 29th June 2022 Shift 2 - Mathematics - Question 64

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Let $$n \geq 5$$ be an integer. If $$9^n - 8n - 1 = 64\alpha$$ and $$6^n - 5n - 1 = 25\beta$$, then $$\alpha - \beta$$ is equal to:

A

$$1 + {}^nC_2(8-5) + {}^nC_3(8^2-5^2) + \ldots + {}^nC_n(8^{n-2}-5^{n-2})$$

B

$$1 + {}^nC_3(8-5) + {}^nC_4(8^2-5^2) + \ldots + {}^nC_n(8^{n-2}-5^{n-2})$$

C

$$\binom{n}{3}(8-5)+\binom{n}{4}(8^2-5^2)+\cdots+\binom{n}{n}(8^{\,n-2}-5^{\,n-2})$$

D

$${}^nC_4(8-5) + {}^nC_5(8^2-5^2) + \ldots + {}^nC_n(8^{n-3}-5^{n-3})$$

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NTA JEE Main 29th June 2022 Shift 2 - Mathematics - Question 65

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The distance of the origin from the centroid of the triangle whose two sides have the equations $$x - 2y + 1 = 0$$ and $$2x - y - 1 = 0$$ and whose orthocenter is $$\left(\frac{7}{3}, \frac{7}{3}\right)$$ is:

A

$$\sqrt{2}$$

B

2

C

$$2\sqrt{2}$$

D

4

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NTA JEE Main 29th June 2022 Shift 2 - Mathematics - Question 66

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Let a triangle $$ABC$$ be inscribed in the circle $$x^2 - \sqrt{2}(x+y) + y^2 = 0$$ such that $$\angle BAC = \frac{\pi}{2}$$. If the length of side $$AB$$ is $$\sqrt{2}$$, then the area of the $$\triangle ABC$$ is equal to:

A

1

B

$$\left(\frac{\sqrt{6}+\sqrt{3}}{2}\right)$$

C

$$\left(\frac{\sqrt{3}+\sqrt{3}}{2}\right)$$

D

$$\left(\frac{\sqrt{6}+2\sqrt{3}}{4}\right)$$

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NTA JEE Main 29th June 2022 Shift 2 - Mathematics - Question 67

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Let $$P : y^2 = 4ax$$, $$a > 0$$ be a parabola with focus $$S$$. Let the tangents to the parabola $$P$$ make an angle of $$\frac{\pi}{4}$$ with the line $$y = 3x + 5$$ touch the parabola $$P$$ at $$A$$ and $$B$$. Then the value of $$a$$ for which $$A$$, $$B$$ and $$S$$ are collinear is:

A

8 only

B

2 only

C

$$\frac{1}{4}$$ only

D

any $$a > 0$$

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NTA JEE Main 29th June 2022 Shift 2 - Mathematics - Question 68

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The value of $$\lim_{x \to 1} \frac{(x^2-1)\sin^2(\pi x)}{x^4-2x^3+2x-1}$$ is equal to:

A

$$\frac{\pi^2}{6}$$

B

$$\frac{\pi^2}{3}$$

C

$$\frac{\pi^2}{2}$$

D

$$\pi^2$$

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NTA JEE Main 29th June 2022 Shift 2 - Mathematics - Question 69

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Negation of the Boolean statement $$(p \vee q) \Rightarrow ((\sim r) \vee p)$$ is equivalent to:

A

$$p \wedge (\sim q) \wedge r$$

B

$$(\sim p) \wedge (\sim q) \wedge r$$

C

$$(\sim p) \wedge q \wedge r$$

D

$$p \wedge q \wedge (\sim r)$$

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NTA JEE Main 29th June 2022 Shift 2 - Mathematics - Question 70

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The number of values of $$a \in N$$ such that the variance of 3, 7, 12, $$a$$, 43 $$- a$$ is a natural number is:

A

0

B

2

C

5

D

infinite

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NTA JEE Main 29th June 2022 Shift 2 - Mathematics - Question 71

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From the base of a pole of height 20 meter, the angle of elevation of the top of a tower is 60°. The pole subtends an angle 30° at the top of the tower. Then the height of the tower is:

A

$$15\sqrt{3}$$

B

$$20\sqrt{3}$$

C

$$20 + 10\sqrt{3}$$

D

30

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NTA JEE Main 29th June 2022 Shift 2 - Mathematics - Question 72

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Let $$A = \begin{pmatrix} 2 & -1 \\ 0 & 2 \end{pmatrix}$$. If $$B = I - {}^5C_1(\text{adj } A) + {}^5C_2(\text{adj } A)^2 - \ldots - {}^5C_5(\text{adj } A)^5$$, then the sum of all elements of the matrix $$B$$ is:

A

$$-5$$

B

$$-6$$

C

$$-7$$

D

$$-8$$

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NTA JEE Main 29th June 2022 Shift 2 - Mathematics - Question 73

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Let $$f : R \to R$$ be a function defined by $$f(x) = (x-3)^{n_1}(x-5)^{n_2}$$, $$n_1, n_2 \in N$$. The, which of the following is NOT true?

A

For $$n_1 = 3$$, $$n_2 = 4$$, there exists $$\alpha \in (3, 5)$$ where $$f$$ attains local maxima.

B

For $$n_1 = 4$$, $$n_2 = 3$$, there exists $$\alpha \in (3, 5)$$ where $$f$$ attains local maxima.

C

For $$n_1 = 3$$, $$n_2 = 5$$, there exists $$\alpha \in (3, 5)$$ where $$f$$ attains local maxima.

D

For $$n_1 = 4$$, $$n_2 = 6$$, there exists $$\alpha \in (3, 5)$$ where $$f$$ attains local maxima.

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NTA JEE Main 29th June 2022 Shift 2 - Mathematics - Question 74

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Let $$f$$ be a real valued continuous function on $$[0, 1]$$ and $$f(x) = x + \int_0^1 (x-t)f(t)dt$$. Then which of the following points $$(x, y)$$ lies on the curve $$y = f(x)$$?

A

$$(2, 4)$$

B

$$(1, 2)$$

C

$$(4, 17)$$

D

$$(6, 8)$$

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NTA JEE Main 29th June 2022 Shift 2 - Mathematics - Question 75

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If $$\int_0^2 \left(\sqrt{2x} - \sqrt{2x - x^2}\right)dx = \int_0^1 \left(1 - \sqrt{1-y^2}-\frac{y^2}{2}\right)dy + \int_1^2 \left(2 - \frac{y^2}{2}\right)dy + I$$, then $$I$$ equal to

A

$$\int_0^1 (1 + \sqrt{1-y^2}) dy$$

B

$$\int_0^1 \left(\frac{y^2}{2} - \sqrt{1-y^2} + 1\right) dy$$

C

$$\int_0^1 (1 - \sqrt{1-y^2}) dy$$

D

$$\int_0^1 \left(\frac{y^2}{2} + \sqrt{1-y^2} + 1\right) dy$$

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NTA JEE Main 29th June 2022 Shift 2 - Mathematics - Question 76

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If $$y = y(x)$$ is the solution of the differential equation $$(1+e^{2x})\frac{dy}{dx} + 2(1+y^2)e^x = 0$$ and $$y(0) = 0$$, then $$6\left(y'(0) + \left(y\left(\log_e \sqrt{3}\right)\right)^2\right)$$ is equal to:

A

2

B

$$-2$$

C

$$-4$$

D

$$-1$$

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NTA JEE Main 29th June 2022 Shift 2 - Mathematics - Question 77

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Let $$A$$, $$B$$, $$C$$ be three points whose position vectors respectively are:
$$\vec{a} = \hat{i} + 4\hat{j} + 3\hat{k}$$
$$\vec{b} = 2\hat{i} + \alpha\hat{j} + 4\hat{k}$$, $$\alpha \in R$$
$$\vec{c} = 3\hat{i} - 2\hat{j} + 5\hat{k}$$
If $$\alpha$$ is the smallest positive integer for which $$\vec{a}$$, $$\vec{b}$$, $$\vec{c}$$ are non-collinear, then the length of the median, through $$A$$, of $$\triangle ABC$$ is:

A

$$\frac{\sqrt{82}}{2}$$

B

$$\frac{\sqrt{62}}{2}$$

C

$$\frac{\sqrt{69}}{2}$$

D

$$\frac{\sqrt{66}}{2}$$

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NTA JEE Main 29th June 2022 Shift 2 - Mathematics - Question 78

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Let $$\frac{x-2}{3} = \frac{y+1}{-2} = \frac{z+3}{-1}$$ lie on the plane $$px - qy + z = 5$$, for some $$p, q \in R$$. The shortest distance of the plane from the origin is:

A

$$\sqrt{\frac{3}{109}}$$

B

$$\sqrt{\frac{5}{142}}$$

C

$$\sqrt{\frac{5}{71}}$$

D

$$\sqrt{\frac{1}{142}}$$

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NTA JEE Main 29th June 2022 Shift 2 - Mathematics - Question 79

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Let $$Q$$ be the mirror image of the point $$P(1, 2, 1)$$ with respect to the plane $$x + 2y + 2z = 16$$. Let $$T$$ be a plane passing through the point $$Q$$ and contains the line $$\vec{r} = -\hat{k} + \lambda(\hat{i} + \hat{j} + 2\hat{k})$$, $$\lambda \in R$$. Then, which of the following points lies on $$T$$?

A

$$(2, 1, 0)$$

B

$$(1, 2, 1)$$

C

$$(1, 2, 2)$$

D

$$(1, 3, 2)$$

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NTA JEE Main 29th June 2022 Shift 2 - Mathematics - Question 80

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The probability that a relation $$R$$ from $$\{x, y\}$$ to $$\{x, y\}$$ is both symmetric and transitive, is equal to:

A

$$\f\frac{5}{16}$$

B

$$\f\frac{9}{16}$$

C

$$\f\frac{11}{16}$$

D

$$\f\frac{13}{16}$$

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NTA JEE Main 29th June 2022 Shift 2 - Mathematics - Question 81

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The total number of four digit numbers such that each of the first three digits is divisible by the last digit, is equal to ______.

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NTA JEE Main 29th June 2022 Shift 2 - Mathematics - Question 82

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Let 3, 6, 9, 12, ... upto 78 terms and 5, 9, 13, 17, ... upto 59 terms be two series. Then, the sum of the terms common to both the series is equal to ______.

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NTA JEE Main 29th June 2022 Shift 2 - Mathematics - Question 83

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Let the coefficients of $$x^{-1}$$ and $$x^{-3}$$ in the expansion of $$\left(2x^{1/5} - \frac{1}{x^{1/5}}\right)^{15}$$, $$x > 0$$, be $$m$$ and $$n$$ respectively. If $$r$$ is a positive integer such $$mn^2 = {}^{15}C_r \cdot 2^r$$, then the value of $$r$$ is equal to ______.

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NTA JEE Main 29th June 2022 Shift 2 - Mathematics - Question 84

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The number of solutions of the equation $$\sin x = \cos^2 x$$ in the interval $$(0, 10)$$ is ______.

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NTA JEE Main 29th June 2022 Shift 2 - Mathematics - Question 85

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Let $$M = \begin{bmatrix} 0 & -\alpha \\ \alpha & 0 \end{bmatrix}$$, where $$\alpha$$ is a non-zero real number and $$N = \sum_{k=1}^{49} M^{2k}$$. If $$(I - M^2)N = -2I$$, then the positive integral value of $$\alpha$$ is ______.

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NTA JEE Main 29th June 2022 Shift 2 - Mathematics - Question 86

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Let $$f(x)$$ and $$g(x)$$ be two real polynomials of degree 2 and 1 respectively. If $$f(g(x)) = 8x^2 - 2x$$, and $$g(f(x)) = 4x^2 + 6x + 1$$, then the value of $$f(2) + g(2)$$ is ______.

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NTA JEE Main 29th June 2022 Shift 2 - Mathematics - Question 87

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Let $$f$$ and $$g$$ be twice differentiable even functions on $$(-2, 2)$$ such that $$f\left(\frac{1}{4}\right) = 0$$, $$f\left(\frac{1}{2}\right) = 0$$, $$f(1) = 1$$ and $$g\left(\frac{3}{4}\right) = 0$$, $$g(1) = 2$$. Then, the minimum number of solutions of $$f(x)g''(x) + f'(x)g'(x) = 0$$ in $$(-2, 2)$$ is equal to ______.

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NTA JEE Main 29th June 2022 Shift 2 - Mathematics - Question 88

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For real numbers $$a$$, $$b$$ ($$a > b > 0$$), let
Area $$\{(x, y) : x^2 + y^2 \leq a^2$$ and $$\frac{x^2}{a^2} + \frac{y^2}{b^2} \geq 1\} = 30\pi$$
and
Area $$\{(x, y) : x^2 + y^2 \geq b^2$$ and $$\frac{x^2}{a^2} + \frac{y^2}{b^2} \leq 1\} = 18\pi$$
Then the value of $$(a-b)^2$$ is equal to ______.

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NTA JEE Main 29th June 2022 Shift 2 - Mathematics - Question 89

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Let $$y = y(x)$$, $$x > 1$$, be the solution of the differential equation $$(x-1)\frac{dy}{dx} + 2xy = \frac{1}{x-1}$$, with $$y(2) = \frac{1+e^4}{2e^4}$$. If $$y(3) = \frac{e^{\alpha}+1}{\beta e^{\alpha}}$$, then the value of $$\alpha + \beta$$ is equal to ______.

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NTA JEE Main 29th June 2022 Shift 2 - Mathematics - Question 90

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Let $$\vec{a} = \hat{i} - 2\hat{j} + 3\hat{k}$$, $$\vec{b} = \hat{i} + \hat{j} + \hat{k}$$ and $$\vec{c}$$ be a vector such that $$\vec{a}  \times (\vec{b}+ \vec{c}) = \vec{0}$$, then the value of $$3(\vec{c} \cdot \vec{a})$$ is equal to ______.

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