NTA JEE Main 15th April 2018 Shift 2 - Mathematics

If $$|z - 3 + 2i| \leq 4$$ then the difference between the greatest value and the least value of $$|z|$$ is:

The number of four letter words that can be formed using the letters of the word BARRACK is:

Let $$A_n = \left(\frac{3}{4}\right) - \left(\frac{3}{4}\right)^2 + \left(\frac{3}{4}\right)^3 - \ldots + (-1)^{n-1}\left(\frac{3}{4}\right)^n$$ and $$B_n = 1 - A_n$$. Then, the least odd natural number p, so that $$B_n > A_n$$, for all $$n \geq p$$ is:

If a, b, c are in A.P. and $$a^2, b^2, c^2$$ are in G.P. such that $$a < b < c$$ and $$a + b + c = \frac{3}{4}$$, then the value of a is:

The coefficient of $$x^{10}$$ in the expansion of $$(1+x)^2(1+x^2)^3(1+x^3)^4$$ is equal to:

The number of solutions of $$\sin 3x = \cos 2x$$, in the interval $$\left(\frac{\pi}{2}, \pi\right)$$ is:

Consider the following two statements.
Statement p: The value of $$\sin 120^\circ$$ can be divided by taking $$\theta = 240^\circ$$ in the equation $$2\sin\frac{\theta}{2} = \sqrt{1 + \sin\theta} - \sqrt{1 - \sin\theta}$$.
Statement q: The angles A, B, C and D of any quadrilateral ABCD satisfy the equation $$\cos\left(\frac{1}{2}(A+C)\right) + \cos\left(\frac{1}{2}(B+D)\right) = 0$$.
Then the truth values of p and q are respectively:

The foot of the perpendicular drawn from the origin, on the line, $$3x + y = \lambda(\lambda \neq 0)$$ is P. If the line meets x-axis at A and y-axis at B, then the ratio BP : PA is:

The sides of a rhombus ABCD are parallel to the lines, $$x - y + 2 = 0$$ and $$7x - y + 3 = 0$$. If the diagonals of the rhombus intersect at P(1, 2) and the vertex A (different from the origin) is on the y axis, then the ordinate of A is:

The tangent to the circle $$C_1: x^2 + y^2 - 2x - 1 = 0$$ at the point (2, 1) cuts off a chord of length 4 from a circle $$C_2$$ whose centre is (3, -2). The radius of $$C_2$$ is:

Tangents drawn from the point (-8, 0) to the parabola $$y^2 = 8x$$ touch the parabola at P and Q. If F is the focus of the parabola, then the area of the triangle PFQ (in sq. units) is equal to:

A normal to the hyperbola, $$4x^2 - 9y^2 = 36$$ meets the co-ordinate axes x and y at A and B, respectively. If the parallelogram OABP (O being the origin) is formed, then the locus of P is:

$$\lim_{x \to 0} \frac{x\tan 2x - 2x\tan x}{(1 - \cos 2x)^2}$$ equals:

If the mean of the data: 7, 8, 9, 7, 8, 7, $$\lambda$$, 8 is 8, then the variance of this data is:

A tower T$$_1$$ of height 60 m is located exactly opposite to a tower T$$_2$$ of height 80 m on a straight road. From the top of T$$_1$$, if the angle of depression of the foot of T$$_2$$ is twice the angle of elevation of the top of T$$_2$$, then the width (in m) of the road between the feet of the towers T$$_1$$ and T$$_2$$ is:

Suppose A is any $$3 \times 3$$ non-singular matrix and $$(A - 3I)(A - 5I) = O$$, where $$I = I_3$$ and $$O = O_3$$. If $$\alpha A + \beta A^{-1} = 4I$$, then $$\alpha + \beta$$ is equal to:

If the system of linear equations
$$x + ay + z = 3$$
$$x + 2y + 2z = 6$$
$$x + 5y + 3z = b$$
has no solution, then:

Let $$f : A \to B$$ be a function defined as $$f(x) = \frac{x-1}{x-2}$$, where $$A = R - \{2\}$$ and $$B = R - \{1\}$$. Then f is:

Let $$f(x) = \begin{cases} (x-1)^{\frac{1}{2-x}}, & x > 1, x \neq 2 \\ k, & x = 2 \end{cases}$$. The value of k for which f is continuous at $$x = 2$$ is:

If $$f(x) = \sin^{-1}\left(\frac{2 \times 3^x}{1 + 9^x}\right)$$, then $$f'\left(-\frac{1}{2}\right)$$ equals:

If $$f(x)$$ is a quadratic expression such that $$f(1) + f(2) = 0$$, and -1 is a root of $$f(x) = 0$$, then the other root of $$f(x) = 0$$ is:

Let $$f(x)$$ be a polynomial of degree 4 having extreme values at $$x = 1$$ and $$x = 2$$. If $$\lim_{x \to 0}\left(\frac{f(x)}{x^2} + 1\right) = 3$$, then $$f(-1)$$ is equal to:

$$\int \frac{2x+5}{\sqrt{7 - 6x - x^2}} dx = A\sqrt{7 - 6x - x^2} + B\sin^{-1}\left(\frac{x+3}{4}\right) + C$$
(where C is a constant of integration), then the ordered pair (A, B) is equal to:

The value of integral $$\int_{\frac{\pi}{4}}^{\frac{3\pi}{4}} \frac{x}{1 + \sin x} dx$$ is:

If $$I_1 = \int_0^1 e^{-x} \cos^2 x \, dx$$; $$I_2 = \int_0^1 e^{-x^2} \cos^2 x \, dx$$ and $$I_3 = \int_0^1 e^{-x^2} dx$$; then:

The curve satisfying the differential equation, $$(x^2 - y^2)dx + 2xydy = 0$$ and passing through the point (1, 1) is:

If the position vectors of the vertices A, B and C of a $$\triangle$$ABC are respectively $$4\hat{i} + 7\hat{j} + 8\hat{k}$$, $$2\hat{i} + 3\hat{j} + 4\hat{k}$$ and $$2\hat{i} + 5\hat{j} + 7\hat{k}$$, then the position vector of the point, where the bisector of $$\angle A$$ meets BC is:

An angle between the lines whose direction cosines are given by the equations, $$l + 3m + 5n = 0$$ and $$5lm - 2mn + 6nl = 0$$, is:

A plane bisects the line segment joining the points (1, 2, 3) and (-3, 4, 5) at right angles. Then this plane also passes through the point:

A player X has a biased coin whose probability of showing heads is p and a player Y has a fair coin. They start playing a game with their own coins and play alternately. The player who throws a head first is a winner. If X starts the game, and the probability of winning the game by both the players is equal, then the value of 'p' is: