NTA JEE Main 4th September 2020 Shift 1 - Mathematics

Let [t] denote the greatest integer $$\leq$$ t. Then the equation in x, $$[x]^2 + 2[x+2] - 7 = 0$$ has:

Let $$\alpha$$ and $$\beta$$ be the roots of $$x^2 - 3x + p = 0$$ and $$\gamma$$ and $$\delta$$ be the roots of $$x^2 - 6x + q = 0$$. If $$\alpha, \beta, \gamma, \delta$$ form a geometric progression. Then ratio $$(2q + p) : (2q - p)$$ is

Let $$u = \frac{2z+i}{z-ki}$$, $$z = x + iy$$ and $$k \gt 0$$. If the curve represented by Re(u) + Im(u) = 1 intersects the y-axis at points P and Q where PQ = 5 then the value of k is

If $$1 + (1 - 2^2 \cdot 1) + (1 - 4^2 \cdot 3) + (1 - 6^2 \cdot 5) + \ldots + (1 - 20^2 \cdot 19) = \alpha - 220\beta$$, then an ordered pair $$(\alpha, \beta)$$ is equal to:

The value of $$\sum_{r=0}^{20} {}^{50-r}C_6$$ is equal to:

A triangle ABC lying in the first quadrant has two vertices as $$A(1, 2)$$ and $$B(3, 1)$$. If $$\angle BAC = 90°$$, and ar($$\triangle ABC$$) = $$5\sqrt{5}$$ sq. units, then the abscissa of the vertex C is:

Let $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ $$(a > b)$$ be a given ellipse, length of whose latus rectum is 10. If its eccentricity is the maximum value of the function, $$\phi(t) = \frac{5}{12} + t - t^2$$, then $$a^2 + b^2$$ is equal to:

Let $$P(3, 3)$$ be a point on the hyperbola, $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$. If the normal to it at P intersects the $$x$$-axis at (9, 0) and $$e$$ is its eccentricity, then the ordered pair $$(a^2, e^2)$$ is equal to:

Given the following two statements:
$$(S_1)$$ : $$(q \vee p) \to (p \leftrightarrow \sim q)$$ is a tautology
$$(S_2)$$ : $$\sim q \wedge (\sim p \leftrightarrow q)$$ is a fallacy. Then:

The mean and variance of 8 observations are 10 and 13.5, respectively. If 6 of these observations are 5, 7, 10, 12, 14, 15, then the absolute difference of the remaining two observations is:

Two vertical poles AB = 15 m and CD = 10 m are standing apart on a horizontal ground with points A and C on the ground. If P is the point of intersection of BC and AD, then the height of P (in m) above the line AC is:

A survey shows that 63% of the people in a city read newspaper A whereas 76% read news paper B. If $$x$$% of the people read both the newspapers, then a possible value of $$x$$ can be:

If $$A = \begin{bmatrix} \cos\theta & i\sin\theta \\ i\sin\theta & \cos\theta \end{bmatrix}$$, $$(\theta = \frac{\pi}{24})$$ and $$A^5 = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$$, where $$i = \sqrt{-1}$$, then which one of the following is not true?

If $$\left(a + \sqrt{2b}\cos x\right)\left(a - \sqrt{2b}\cos y\right) = a^2 - b^2$$, where $$a > b > 0$$, then $$\frac{dx}{dy}$$ at $$\left(\frac{\pi}{4}, \frac{\pi}{4}\right)$$ is:

Let $$f$$ be a twice differentiable function on $$(1, 6)$$, If $$f(2) = 8$$, $$f'(2) = 5$$, $$f'(x) \geq 1$$ and $$f''(x) \geq 4$$, for all $$x \in (1, 6)$$, then:

The integral $$\int \left(\frac{x}{x\sin x + \cos x}\right)^2 dx$$ is equal to, (where C is a constant of integration):

Let $$f(x) = \int \frac{\sqrt{x}}{(1+x)^2} dx$$ $$(x \geq 0)$$. Then $$f(3) - f(1)$$ is equal to:

Let $$f(x) = |x - 2|$$ and $$g(x) = f(f(x))$$, $$x \in [0, 4]$$. Then $$\int_0^3 (g(x) - f(x)) dx$$ is equal to

Let $$y = y(x)$$ be the solution of the differential equation, $$xy' - y = x^2(x\cos x + \sin x)$$, $$x > 0$$. If $$y(\pi) = \pi$$, then $$y''\left(\frac{\pi}{2}\right) + y\left(\frac{\pi}{2}\right)$$ is equal to:

Let $$x_0$$ be the point of local maxima of $$f(x) = \vec{a} \cdot (\vec{b} \times \vec{c})$$, where $$\vec{a} = x\hat{i} - 2\hat{j} + 3\hat{k}$$, $$\vec{b} = -2\hat{i} + x\hat{j} - \hat{k}$$ and $$\vec{c} = 7\hat{i} - 2\hat{j} + x\hat{k}$$. Then the value of $$\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}$$ at $$x = x_0$$ is:

Let $$(2x^2 + 3x + 4)^{10} = \sum_{r=0}^{20} a_r x^r$$. Then $$\frac{a_7}{a_{13}}$$ is equal to __________

If the system of equations
$$x - 2y + 3z = 9$$
$$2x + y + z = b$$
$$x - 7y + az = 24$$
has infinitely many solutions, then $$a - b$$ is equal to __________

Suppose a differentiable function $$f(x)$$ satisfies the identity $$f(x + y) = f(x) + f(y) + xy^2 + x^2y$$, for all real $$x$$ and $$y$$. If $$\lim_{x \to 0}\frac{f(x)}{x} = 1$$, then $$f'(3)$$ is equal to __________

If the equation of a plane P, passing through the intersection of the planes, $$x + 4y - z + 7 = 0$$ and $$3x + y + 5z = 8$$ is $$ax + by + 6z = 15$$ for some $$a, b \in R$$, then the distance of the point (3, 2, -1) from the plane P is __________

The probability of a man hitting a target is $$\frac{1}{10}$$. The least number of shots required, so that the probability of his hitting the target at least once is greater than $$\frac{1}{4}$$, is ____