NTA JEE Main 15th April 2018 Shift 1

If $$\beta$$ is one of the angles between the normals to the ellipse, $$x^2 + 3y^2 = 9$$ at the points $$(3\cos\theta, \sqrt{3}\sin\theta)$$ and $$(-3\sin\theta, \sqrt{3}\cos\theta)$$; $$\theta \in (0, \frac{\pi}{2})$$; then $$\frac{2\cot\beta}{\sin 2\theta}$$ is equal to:

If the tangents drawn to the hyperbola $$4y^2 = x^2 + 1$$ intersect the co-ordinate axes at the distinct points A and B, then the locus of the mid point of AB is:

If $$(p \wedge \sim q) \wedge (p \wedge r) \rightarrow \sim p \vee q$$ is false, then the truth values of p, q and r are respectively:

The mean of a set of 30 observations is 75. If each observation is multiplied by a nonzero number $$\lambda$$ and then each of them is decreased by 25, their mean remains the same. The $$\lambda$$ is equal to $$\{0\}$$:

An aeroplane flying at a constant speed, parallel to the horizontal ground, $$\sqrt{3}$$ km above it, is observed at an elevation of 60$$^\circ$$ from a point on the ground. If, after five seconds, its elevation from the same point is 30$$^\circ$$, then the speed (in km/hr) of the aeroplane is:

Consider the following two binary relations on the set $$A = \{a, b, c\}$$: $$R_1 = \{(c, a), (b, b), (a, c), (c, c), (b, c), (a, a)\}$$ and $$R_2 = \{(a, b), (b, a), (c, c), (c, a), (a, a), (b, b), (a, c)\}$$. Then:

Let A be a matrix such that $$A \cdot \begin{bmatrix} 1 & 2 \\ 0 & 3 \end{bmatrix}$$ is a scalar matrix and $$|3A| = 108$$. Then $$A^2$$ equals:

If $$f(x) = \begin{vmatrix} \cos x & x & 1 \\ 2\sin x & x^2 & 2x \\ \tan x & x & 1 \end{vmatrix}$$, then $$\lim_{x \to 0} \frac{f'(x)}{x}$$:

Let S be the set of all real values of k for which the system of linear equations
$$x + y + z = 2$$
$$2x + y - z = 3$$
$$3x + 2y + kz = 4$$
has a unique solution. Then S is:

Let $$S = \{(\lambda, \mu) \in R \times R : f(t) = (|\lambda|e^t - \mu) \cdot \sin(2|t|), t \in R$$, is a differentiable function$$\}$$. Then S is a subset of?