NTA JEE Main 25th July 2021 Shift 1 - Mathematics

Let $$S_n$$ be the sum of the first $$n$$ terms of an arithmetic progression. If $$S_{3n} = 3S_{2n}$$, then the value of $$\frac{S_{4n}}{S_{2n}}$$ is:

If $$b$$ is very small as compared to the value of $$a$$, so that the cube and other higher powers of $$\frac{b}{a}$$ can be neglected in the identity
$$\frac{1}{a-b} + \frac{1}{a-2b} + \frac{1}{a-3b} + \ldots + \frac{1}{a-nb} = \alpha n + \beta n^2 + \gamma n^3$$
then the value of $$\gamma$$ is:

The sum of all values of $$x$$ in $$[0, 2\pi]$$, for which $$\sin x + \sin 2x + \sin 3x + \sin 4x = 0$$, is equal to:

Let a parabola $$P$$ be such that its vertex and focus lie on the positive $$x$$-axis at a distance 2 and 4 units from the origin, respectively. If tangents are drawn from $$O(0, 0)$$ to the parabola $$P$$ which meet $$P$$ at $$S$$ and $$R$$, then the area (in sq. units) of $$\triangle SOR$$ is equal to:

Let an ellipse $$E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$, $$a^2 > b^2$$, passes through $$\left(\sqrt{\frac{3}{2}}, 1\right)$$ and has eccentricity $$\frac{1}{\sqrt{3}}$$. If a circle, centered at focus $$F(\alpha, 0)$$, $$\alpha > 0$$, of $$E$$ and radius $$\frac{2}{\sqrt{3}}$$, intersects $$E$$ at two points $$P$$ and $$Q$$, then $$PQ^2$$ is equal to:

The locus of the centroid of the triangle formed by any point P on the hyperbola $$16x^2 - 9y^2 + 32x + 36y - 164 = 0$$ and its foci is

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

A spherical gas balloon of radius 16 meter subtends an angle 60$$^\circ$$ at the eye of the observer A while the angle of elevation of its center from the eye of A is 75$$^\circ$$. Then the height (in meter) of the top most point of the balloon from the level of the observer's eye is:

The values of $$a$$ and $$b$$, for which the system of equations
$$2x + 3y + 6z = 8$$
$$x + 2y + az = 5$$
$$3x + 5y + 9z = b$$
has no solution, are:

Let $$g : N \to N$$ be defined as
$$g(3n+1) = 3n+2$$
$$g(3n+2) = 3n+3$$
$$g(3n+3) = 3n+1$$, for all $$n \ge 0$$
Then which of the following statements is true?

Let $$f: R \to R$$ be defined as
$$f(x) = \begin{cases} \frac{\lambda x^2 - 5x + 6}{\mu(5x - x^2 - 6)} & x < 2 \\ e^{\frac{\tan(x-2)}{x - [x]}} & x > 2 \\ \mu & x = 2 \end{cases}$$
where $$[x]$$ is the greatest integer less than or equal to $$x$$. If $$f$$ is continuous at $$x = 2$$, then $$\lambda + \mu$$ is equal to:

Let $$f: [0, \infty) \to [0, \infty)$$ be defined as $$f(x) = \int_0^x [y] dy$$ where $$[x]$$ is the greatest integer less than or equal to $$x$$. Which of the following is true?

Let $$f(x) = 3\sin^4 x + 10\sin^3 x + 6\sin^2 x - 3$$, $$x \in \left[-\frac{\pi}{6}, \frac{\pi}{2}\right]$$. Then, $$f$$ is:

The number of real roots of the equation $$e^{6x} - e^{4x} - 2e^{3x} - 12e^{2x} + e^x + 1 = 0$$ is:

The value of the definite integral $$\int_{\pi/24}^{5\pi/24} \frac{dx}{1 + \sqrt[3]{\tan 2x}}$$ is:

The area (in sq. units) of the region, given by the set $$\{x, y \in R \times R | x \ge 0, 2x^2 \le y \le 4 - 2x\}$$ is:

Let $$y = y(x)$$ be the solution of the differential equation $$\frac{dy}{dx} = 1 + xe^{y-x}$$, $$-\sqrt{2} \lt x \lt \sqrt{2}$$, $$y(0) = 0$$, then the minimum value of $$y(x)$$, $$x \in (-\sqrt{2}, \sqrt{2})$$ is equal to:

Let the vectors $$(2 + a + b)\hat{i} + (a + 2b+c)\hat{j} - (b + c)\hat{k}$$, $$(1 + b)\hat{i}+2b\hat{j}-b\hat{k}$$ and $$(2 + b)\hat{i} + 2b\hat{j} + (1 - b)\hat{k}$$, $$ a, b, c \in R$$ be co-planar. Then which of the following is true?

Let the foot of perpendicular from a point $$P(1, 2, -1)$$ to the straight line $$L: \frac{x}{1} = \frac{y}{0} = \frac{z}{-1}$$ be $$N$$. Let a line be drawn from $$P$$ parallel to the plane $$x + y + 2z = 0$$ which meets $$L$$ at point $$Q$$. If $$\alpha$$ is the acute angle between the lines PN and PQ, then $$\cos\alpha$$ is equal to:

Let 9 distinct balls be distributed among 4 boxes $$B_1$$, $$B_2$$, $$B_3$$ and $$B_4$$. If the probability that $$B_3$$ contains exactly 3 balls is $$k\left(\frac{3}{4}\right)^9$$ then $$k$$ lies in the set:

If $$\alpha, \beta$$ are roots of the equation $$x^2 + 5\sqrt{2}x + 10 = 0$$, $$\alpha > \beta$$ and $$P_n = \alpha^n - \beta^n$$ for each positive integer $$n$$, then the value of $$\frac{P_{17}P_{20} + 5\sqrt{2}P_{17}P_{19}}{P_{18}P_{19} + 5\sqrt{2}P_{18}^2}$$ is equal to ___.

There are 5 students in class 10, 6 students in class 11 and 8 students in class 12. If the number of ways, in which 10 students can be selected from them so as to include at least 2 students from each class and at most 5 students from the total 11 students of classes 10 and 11 is 100k, then k is equal to ___.

If the value of $$\left(1 + \frac{2}{3} + \frac{6}{3^2} + \frac{10}{3^3} + \ldots \text{ upto } \infty\right)^{ \log_{(0.25)}\left(\frac{1}{3} + \frac{1}{3^2} + \frac{1}{3^3} + \ldots \text{ upto } \infty\right)}$$ is $$l$$, then $$l^2$$ is equal to ___.

The ratio of the coefficient of the middle term in the expansion of $$(1+x)^{20}$$ and the sum of the coefficients of two middle terms in expansion of $$(1+x)^{19}$$ is ___.

The term independent of $$x$$ in the expansion of $$\left(\frac{x+1}{x^{2/3} - x^{1/3} + 1} - \frac{x-1}{x - x^{1/2}}\right)^{10}$$, where $$x \neq 0, 1$$ is equal to ___.

Consider the following frequency distribution:

If the sum of all frequencies is 584 and median is 45, then $$|\alpha - \beta|$$ is equal to ___.

Let $$M = A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} : a, b, c, d \in \{\pm 3, \pm 2, \pm 1, 0\}$$. Define $$f: M \to Z$$, as $$f(A) = \det A$$, for all $$A \in M$$ where $$Z$$ is set of all integers. Then the number of $$A \in M$$ such that $$f(A) = 15$$ is equal to ___.

Let $$S = \{n \in N, \begin{pmatrix} 0 & i \\ 1 & 0 \end{pmatrix}^n \begin{pmatrix} a & b \\ c & d \end{pmatrix}= \begin{pmatrix} a & b \\ c & d \end{pmatrix}$$ $$\forall a, b, c, d \in R$$, where $$i = \sqrt{-1}\}$$. Then the number of 2-digit numbers in the set $$S$$ is ___.

Let $$y = y(x)$$ be solution of the following differential equation
$$e^y \frac{dy}{dx} - 2e^y \sin x + \sin x \cos^2 x = 0$$, $$y\left(\frac{\pi}{2}\right) = 0$$.
If $$y(0) = \log_e \alpha + \beta e^{-2}$$, then $$4(\alpha + \beta)$$ is equal to ___.

Let $$\vec{p} = 2\hat{i} + 3\hat{j} + \hat{k}$$ and $$\vec{q} = \hat{i} + 2\hat{j} + \hat{k}$$ be two vectors. If a vector $$\vec{r} = \alpha\hat{i} + \beta\hat{j} + \gamma\hat{k}$$ is perpendicular to each of the vectors $$(\vec{p} + \vec{q})$$ and $$(\vec{p} - \vec{q})$$, and $$|\vec{r}| = \sqrt{3}$$, then $$|\alpha| + |\beta| + |\gamma|$$ is equal to ___.