NTA JEE Main 31st August 2021 Shift 1 - Mathematics

The sum of 10 terms of the series $$\frac{3}{1^2 \times 2^2} + \frac{5}{2^2 \times 3^2} + \frac{7}{3^2 \times 4^2} + \ldots$$ is:

Three numbers are in an increasing geometric progression with common ratio $$r$$. If the middle number is doubled, then the new numbers are in an arithmetic progression with common difference $$d$$. If the fourth term of GP is $$3r^2$$, then $$r^2 - d$$ is equal to:

$$\text{cosec } 18°$$ is a root of the equation:

If $$p$$ and $$q$$ are the lengths of the perpendiculars from the origin on the lines, $$x\text{cosec}\alpha - y\sec\alpha = k\cot 2\alpha$$ and $$x\sin\alpha + y\cos\alpha = k\sin 2\alpha$$ respectively, then $$k^2$$ is equal to:

The length of the latus rectum of a parabola, whose vertex and focus are on the positive $$x$$-axis at a distance $$R$$ and $$S (> R)$$ respectively from the origin, is:

The line $$12x\cos\theta + 5y\sin\theta = 60$$ is tangent to which of the following curves?

$$\lim_{x \to 0} \frac{\sin^2(\pi\cos^4 x)}{x^4}$$ is equal to:

Let $$*, \square \in \{\wedge, \vee\}$$ be such that the Boolean expression $$(p * \sim q) \Rightarrow (p \square q)$$ is a tautology. Then:

A vertical pole fixed to the horizontal ground is divided in the ratio 3 : 7 by a mark on it with lower part shorter than the upper part. If the two parts subtend equal angles at a point on the ground 18 m away from the base of the pole, then the height of the pole (in meters) is:

Which of the following is not correct for relation $$R$$ on the set of real numbers?

If $$a_r = \cos\frac{2r\pi}{9} + i\sin\frac{2r\pi}{9}$$, $$r = 1, 2, 3, \ldots$$, $$i = \sqrt{-1}$$, then the determinant $$\begin{vmatrix} a_1 & a_2 & a_3 \\ a_4 & a_5 & a_6 \\ a_7 & a_8 & a_9 \end{vmatrix}$$ is equal to:

If the following system of linear equations
$$2x + y + z = 5$$
$$x - y + z = 3$$
$$x + y + az = b$$
has no solution, then:

The function $$f(x) = |x^2 - 2x - 3| \cdot e^{9x^2-12x+4}$$ is not differentiable at exactly:

If the function $$f(x) = \begin{cases} \frac{1}{x}\log_e\left(\frac{1+\frac{x}{b}}{1-\frac{x}{b}}\right), & x < 0 \\ k, & x = 0 \\ \frac{\cos^2 x - \sin^2 x - 1}{\sqrt{x^2+1}-1}, & x > 0 \end{cases}$$ is continuous at $$x = 0$$, then $$\frac{1}{a} + \frac{1}{b} + \frac{4}{k}$$ is equal to:

The number of real roots of the equation $$e^{4x} + 2e^{3x} - e^x - 6 = 0$$ is:

The integral $$\int \frac{1}{\sqrt[4]{(x-1)^3(x+2)^5}} dx$$ is equal to: (where $$C$$ is a constant of integration)

Let $$f$$ be a non-negative function in $$[0, 1]$$ and twice differentiable in $$(0, 1)$$. If
$$\int_0^x \sqrt{1 - (f'(t))^2} dt = \int_0^x f(t) dt$$, $$0 \leq x \leq 1$$ and $$f(0) = 0$$, then $$\lim_{x \to 0} \frac{1}{x^2} \int_0^x f(t) dt$$:

If $$\frac{dy}{dx} = \frac{2^{x+y} - 2^x}{2^y}$$, $$y(0) = 1$$, then $$y(1)$$ is equal to:

Let $$\vec{a}$$ and $$\vec{b}$$ be two vectors such that $$|2\vec{a} + 3\vec{b}| = |3\vec{a} + \vec{b}|$$ and the angle between $$\vec{a}$$ and $$\vec{b}$$ is 60°. If $$\frac{1}{8}\vec{a}$$ is a unit vector, then $$|\vec{b}|$$ is equal to:

Let the equation of the plane, that passes through the point $$(1, 4, -3)$$ and contains the line of intersection of the planes $$3x - 2y + 4z - 7 = 0$$ and $$x + 5y - 2z + 9 = 0$$, be $$\alpha x + \beta y + \gamma z + 3 = 0$$, then $$\alpha + \beta + \gamma$$ is equal to:

A point $$z$$ moves in the complex plane such that $$\arg\left(\frac{z-2}{z+2}\right) = \frac{\pi}{4}$$, then the minimum value of $$|z - 9\sqrt{2} - 2i|^2$$ is equal to _________.

The number of six letter words (with or without meaning), formed using all the letters of the word 'VOWELS', so that all the consonants never come together, is _________.

If $$\left(\frac{x^6}{4^4}\right) k$$ is the term, independent of $$x$$, in the binomial expansion of $$\left(\frac{x}{4} - \frac{12}{x^2}\right)^{12}$$, then $$k$$ is equal to _________.

If the variable line $$3x + 4y = \alpha$$ lies between the two circles $$(x-1)^2 + (y-1)^2 = 1$$ and $$(x-9)^2 + (y-1)^2 = 4$$, without intercepting a chord on either circle, then the sum of all the integral values of $$\alpha$$ is _________.

The mean of 10 numbers
$$7 \times 8, 10 \times 10, 13 \times 12, 16 \times 14, \ldots$$ is _________.

If $$R$$ is the least value of $$a$$ such that the function $$f(x) = x^2 + ax + 1$$ is increasing on $$[1, 2]$$ and $$S$$ is the greatest value of $$a$$ such that the function $$f(x) = x^2 + ax + 1$$ is decreasing on $$[1, 2]$$, then the value of $$|R - S|$$ is _________.

Let $$[t]$$ denote the greatest integer $$\leq t$$. Then the value of $$8 \cdot \int_{-\frac{1}{2}}^{1} \left([2x] + |x|\right) dx$$ is _________.

If $$x\phi(x) = \int_5^x (3t^2 - 2\phi'(t)) dt$$, $$x > -2$$, $$\phi(0) = 4$$, then $$\phi(2)$$ is _________.

The square of the distance of the point of intersection of the line $$\frac{x-1}{2} = \frac{y-2}{3} = \frac{z+1}{6}$$ and the plane $$2x - y + z = 6$$ from the point $$(-1, -1, 2)$$ is _________.

An electric instrument consists of two units. Each unit must function independently for the instrument to operate. The probability that the first unit functions is 0.9 and that of the second unit is 0.8. The instrument is switched on and it fails to operate. If the probability that only the first unit failed and second unit is functioning is $$p$$, then $$98p$$ is equal to _________.