NTA JEE Main 3rd September 2020 Shift 2 - Mathematics

The set of all real values of $$\lambda$$ for which the quadratic equation $$(\lambda^2 + 1)x^2 - 4\lambda x + 2 = 0$$ always have exactly one root in the interval (0, 1) is:

If $$z_1, z_2$$ are complex numbers such that $$Re(z_1) = |z_1 - 1|$$ and $$Re(z_2) = |z_2 - 1|$$ and $$\arg(z_1 - z_2) = \frac{\pi}{6}$$, then $$Im(z_1 + z_2)$$ is equal to:

If the sum of the series $$20 + 19\frac{3}{5} + 19\frac{1}{5} + 18\frac{4}{5} + \ldots$$ up to $$n^{th}$$ term is 488 and the $$n^{th}$$ term is negative, then:

If the term independent of $$x$$ in the expansion of $$\left(\frac{3}{2}x^2 - \frac{1}{3x}\right)^9$$ is $$k$$, then $$18k$$ is equal to:

If a $$\triangle ABC$$ has vertices $$A(-1, 7)$$, $$B(-7, 1)$$ and $$C(5, -5)$$, then its orthocentre has coordinates:

Let the latus rectum of the parabola $$y^2 = 4x$$ be the common chord to the circles $$C_1$$ and $$C_2$$ each of them having radius $$2\sqrt{5}$$. Then, the distance between the centres of the circles $$C_1$$ and $$C_2$$ is:

Let $$e_1$$ and $$e_2$$ be the eccentricities of the ellipse $$\frac{x^2}{25} + \frac{y^2}{b^2} = 1$$ $$(b < 5)$$ and the hyperbola $$\frac{x^2}{16} - \frac{y^2}{b^2} = 1$$ respectively satisfying $$e_1 e_2 = 1$$. If $$\alpha$$ and $$\beta$$ are the distances between the foci of the ellipse and the foci of the hyperbola respectively, then the ordered pair $$(\alpha, \beta)$$ is equal to:

$$\lim_{x \to a}\frac{(a+2x)^{\frac{1}{3}} - (3x)^{\frac{1}{3}}}{(3a+x)^{\frac{1}{3}} - (4x)^{\frac{1}{3}}}$$ $$(a \neq 0)$$ is equal to:

Let $$p$$, $$q$$, $$r$$ be three statements such that the truth value of $$(p \wedge q) \to (\sim q \vee r)$$ is $$F$$. Then the truth values of $$p$$, $$q$$, $$r$$ are respectively:

Let $$x_i (1 \leq i \leq 10)$$ be ten observations of a random variable X. If $$\sum_{i=1}^{10}(x_i - p) = 3$$ and $$\sum_{i=1}^{10}(x_i - p)^2 = 9$$ where $$0 \neq p \in R$$, then the standard deviation of these observations is:

Let $$R_1$$ and $$R_2$$ be two relations defined as follows:
$$R_1 = \{(a, b) \in R^2 : a^2 + b^2 \in Q\}$$ and $$R_2 = \{(a, b) \in R^2 : a^2 + b^2 \notin Q\}$$, where Q is the set of all rational numbers, then

Let A be a $$3 \times 3$$ matrix such that adj $$A = \begin{bmatrix} 2 & -1 & 1 \\ -1 & 0 & 2 \\ 1 & -2 & -1 \end{bmatrix}$$ and $$B = $$ adj(adjA). If $$|A| = \lambda$$ and $$\left|(B^{-1})^T\right| = \mu$$, then the ordered pair $$(|\lambda|, \mu)$$ is equal to

Suppose $$f(x)$$ is a polynomial of degree four having critical points at -1, 0, 1. If $$T = \{x \in R | f(x) = f(0)\}$$, then the sum of squares of all the elements of $$T$$ is:

If the surface area of a cube is increasing at a rate of 3.6 cm$$^2$$/sec, retaining its shape; then the rate of change of its volume (in cm$$^3$$/sec), when the length of a side of the cube is 10 cm, is:

If $$\int \sin^{-1}\left(\frac{\sqrt{x}}{1+x}\right)dx = A(x)\tan^{-1}(\sqrt{x}) + B(x) + C$$, where C is a constant of integration, then the ordered pair $$(A(x), B(x))$$ can be:

If the value of the integral $$\int_0^{\frac{1}{2}}\frac{x^2}{(1-x^2)^{\frac{3}{2}}}dx$$ is $$\frac{k}{6}$$, then $$k$$ is equal to:

If $$x^3 dy + xy \cdot dx = x^2 dy + 2y dx$$; $$y(2) = e$$ and $$x > 1$$, then $$y(4)$$ is equal to:

Let $$a, b, c \in R$$ be such that $$a^2 + b^2 + c^2 = 1$$. If $$a\cos\theta = b\cos\left(\theta + \frac{2\pi}{3}\right) = c\cos\left(\theta + \frac{4\pi}{3}\right)$$, where $$\theta = \frac{\pi}{9}$$, then the angle between the vectors $$a\hat{i} + b\hat{j} + c\hat{k}$$ and $$b\hat{i} + c\hat{j} + a\hat{k}$$ is:

The plane which bisects the line joining the points (4, -2, 3) and (2, 4, -1) at right angles also passes through the point:

The probability that a randomly chosen 5-digit number is made from exactly two digits is:

The total number of 3-digit numbers whose sum of digits is 10, is ..........

If m arithmetic means (A.Ms) and three geometric means (G.Ms) are inserted between 3 and 243 such that $$4^{th}$$ A.M. is equal to $$2^{nd}$$ G.M., then $$m$$ is equal to:

Let $$S$$ be the set of all integer solutions $$(x, y, z)$$ of the system of equations
$$x - 2y + 5z = 0$$
$$-2x + 4y + z = 0$$
$$-7x + 14y + 9z = 0$$
such that $$15 \leq x^2 + y^2 + z^2 \leq 150$$. Then, the number of elements in the set $$S$$ is equal to ..........

If the tangent to the curve $$y = e^x$$ at a point $$(c, e^c)$$ and the normal to the parabola $$y^2 = 4x$$ at the point (1, 2) intersect at the same point on the $$x$$-axis, then the value of $$c$$ is .....

Let a plane $$P$$ contain two lines $$\vec{r} = \hat{i} + \lambda(\hat{i} + \hat{j})$$, $$\lambda \in R$$ and $$\vec{r} = -\hat{j} + \mu(\hat{j} - \hat{k})$$, $$\mu \in R$$. If $$Q(\alpha, \beta, \gamma)$$ is the foot of the perpendicular drawn from the point M(1, 0, 1) to P, then $$3(\alpha + \beta + \gamma)$$ equals .......