NTA JEE Mains 1st Feb 2024 Shift 1

Let $$S = x \in R : (\sqrt{3} + \sqrt{2})^x + (\sqrt{3} - \sqrt{2})^x = 10$$. Then the number of elements in $$S$$ is:

Let $$S = z \in \mathbb{C} : z - 1 = 1 \text{ and } (\sqrt{2} - 1)(z + \bar{z}) - i(z - \bar{z}) = 2\sqrt{2}$$. Let $$z_1, z_2 \in S$$ be such that $$z_1 = \max_{z \in S}z$$ and $$z_2 = \min_{z \in S}z$$. Then $$|\sqrt{2}z_1 - z_2^2$$ equals:

If $$n$$ is the number of ways five different employees can sit into four indistinguishable offices where any office may have any number of persons including zero, then $$n$$ is equal to:

Let $$3, a, b, c$$ be in A.P. and $$3, a-1, b+1, c+9$$ be in G.P. Then, the arithmetic mean of $$a$$, $$b$$ and $$c$$ is:

If $$\tan A = \frac{1}{\sqrt{xx^2+x+1}}$$, $$\tan B = \frac{\sqrt{x}}{\sqrt{x^2+x+1}}$$ and $$\tan C=x^{-3}+x^{-2}+x^{-11/2}$$, $$0 < A, B, C < \frac{\pi}{2}$$, then $$A + B$$ is equal to:

Let $$C: x^2 + y^2 = 4$$ and $$C': x^2 + y^2 - 4\lambda x + 9 = 0$$ be two circles. If the set of all values of $$\lambda$$ so that the circles $$C$$ and $$C'$$ intersect at two distinct points, is $$R - a, b $$, then the point $$8a + 12, 16b - 20$$ lies on the curve:

Let $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$, $$a > b$$ be an ellipse, whose eccentricity is $$\frac{1}{\sqrt{2}}$$ and the length of the latus rectum is $$\sqrt{14}$$. Then the square of the eccentricity of $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$ is:

For $$0 < \theta < \pi/2$$, if the eccentricity of the hyperbola $$x^2 - y^2\csc^2\theta = 5$$ is $$\sqrt{7}$$ times eccentricity of the ellipse $$x^2\csc^2\theta + y^2 = 5$$, then the value of $$\theta$$ is:

Let the median and the mean deviation about the median of 7 observations $$170, 125, 230, 190, 210, a, b$$ be $$170$$ and $$\frac{205}{7}$$ respectively. Then the mean deviation about the mean of these 7 observations is:

If $$A = \begin{pmatrix} \sqrt{2} & 1 \\ -1 & \sqrt{2} \end{pmatrix}$$, $$B = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$$, $$C = ABA^T$$ and $$X = A^TC^2A$$, then $$\det X$$ is equal to: