NTA JEE Main 27th August 2021 Shift 2 - Mathematics

The set of all values of $$k \gt -1$$, for which the equation $$(3x^2+4x+3)^2 - (k+1)(3x^2+4x+3)(3x^2+4x+2) + k(3x^2+4x+2)^2 = 0$$ has real roots, is:

If $$0 < x < 1$$ and $$y = \frac{1}{2}x^2 + \frac{2}{3}x^3 + \frac{3}{4}x^4 + \ldots$$, then the value of $$e^{1+y}$$ at $$x = \frac{1}{2}$$ is:

Let $$A(a, 0)$$, $$B(b, 2b+1)$$ and $$C(0, b)$$, $$b \neq 0$$, $$|b| \neq 1$$, be points such that the area of triangle $$ABC$$ is 1 sq. unit, then the sum of all possible values of $$a$$ is:

If two tangents drawn from a point $$P$$ to the parabola $$y^2 = 16(x-3)$$ are at right angles, then the locus of point $$P$$ is:

If $$\lim_{x \to \infty} \left(\sqrt{x^2 - x + 1} - ax\right) = b$$, then the ordered pair $$(a, b)$$ is:

The Boolean expression $$(p \wedge q) \Rightarrow ((r \wedge q) \wedge p)$$ is equivalent to:

Two poles $$AB$$ of length $$a$$ metres and $$CD$$ of length $$a + b$$ $$(b \neq a)$$ metres are erected at the same horizontal level with bases at $$B$$ and $$D$$. If $$BD = x$$ and $$\tan \angle ACB = \frac{1}{2}$$, then:

Let $$Z$$ be the set of all integers,
$$A = \{(x,y) \in Z \times Z : (x-2)^2 + y^2 \leq 4\}$$
$$B = \{(x,y) \in Z \times Z : x^2 + y^2 \leq 4\}$$ and
$$C = \{(x,y) \in Z \times Z : (x-2)^2 + (y-2)^2 \leq 4\}$$
If the total number of relations from $$A \cap B$$ to $$A \cap C$$ is $$2^p$$, then the value of $$p$$ is:

Let $$[\lambda]$$ be the greatest integer less than or equal to $$\lambda$$. The set of all values of $$\lambda$$ for which the system of linear equations $$x + y + z = 4$$, $$3x + 2y + 5z = 3$$, $$9x + 4y + (28 + [\lambda])z = [\lambda]$$ has a solution is:

Let $$A = \begin{bmatrix} [x+1] & [x+2] & [x+3] \\ [x] & [x+3] & [x+3] \\ [x] & [x+2] & [x+4] \end{bmatrix}$$, where $$[x]$$ denotes the greatest integer less than or equal to $$x$$. If det$$(A) = 192$$, then the set of values of $$x$$ is in the interval:

If $$y(x) = \cot^{-1}\left(\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\right)$$, $$x \in \left(\frac{\pi}{2}, \pi\right)$$, then $$\frac{dy}{dx}$$ at $$x = \frac{5\pi}{6}$$ is:

A box open from top is made from a rectangular sheet of dimension $$a \times b$$ by cutting squares each of side $$x$$ from each of the four corners and folding up the flaps. If the volume of the box is maximum, then $$x$$ is equal to:

Let $$M$$ and $$m$$ respectively be the maximum and minimum values of the function $$f(x) = \tan^{-1}(\sin x + \cos x)$$ in $$\left[0, \frac{\pi}{2}\right]$$. Then the value of $$\tan(M - m)$$ is equal to:

The value of the integral $$\int_0^1 \frac{\sqrt{x} dx}{(1+x)(1+3x)(3+x)}$$ is:

The area of the region bounded by the parabola $$(y-2)^2 = (x-1)$$, the tangent to it at the point whose ordinate is 3 and the x-axis, is:

If the solution curve of the differential equation $$(2x - 10y^3)dy + ydx = 0$$, passes through the points $$(0, 1)$$ and $$(2, \beta)$$, then $$\beta$$ is a root of the equation?

A differential equation representing the family of parabolas with axis parallel to y-axis and whose length of latus rectum is the distance of the point $$(2, -3)$$ from the line $$3x + 4y = 5$$, is given by:

The equation of the plane passing through the line of intersection of the planes $$\vec{r} \cdot (2\hat{i} + 3\hat{j} - \hat{k}) + 4 = 0$$ and $$\vec{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = 1$$ and parallel to the x-axis, is

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

Each of the persons $$A$$ and $$B$$ independently tosses three fair coins. The probability that both of them get the same number of heads is:

Let $$z_1$$ and $$z_2$$ be two complex numbers such that $$\arg(z_1 - z_2) = \frac{\pi}{4}$$ and $$z_1, z_2$$ satisfy the equation $$|z - 3| = \text{Re}(z)$$. Then the imaginary part $$z_1 + z_2$$ is equal to _________.

Let $$S = \{1, 2, 3, 4, 5, 6, 9\}$$. Then the number of elements in the set $$T = \{A \subseteq S : A \neq \phi$$ and the sum of all the elements of $$A$$ is not a multiple of 3$$\}$$ is _________.

$$3 \times 7^{22} + 2 \times 10^{22} - 44$$ when divided by 18 leaves the remainder _________.

Let $$S$$ be the sum of all solutions (in radians) of the equation $$\sin^4\theta + \cos^4\theta - \sin\theta\cos\theta = 0$$ in $$[0, 4\pi]$$ then $$\frac{8S}{\pi}$$ is equal to _________.

Two circles each of radius 5 units touch each other at the point $$(1, 2)$$. If the equation of their common tangent is $$4x + 3y = 10$$, and $$C_1(\alpha, \beta)$$ and $$C_2(\gamma, \delta)$$, $$C_1 \neq C_2$$ are their centres, then $$|(\alpha + \beta)(\gamma + \delta)|$$ is equal to _________.

Let $$P(a\sec\theta, b\tan\theta)$$ and $$Q(a\sec\phi, b\tan\phi)$$ where $$\theta + \phi = \frac{\pi}{2}$$, be two points on the hyperbola $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$. If the ordinate of the point of intersection of normals at $$P$$ and $$Q$$ is $$-k\left(\frac{a^2+b^2}{2b}\right)$$, then $$k$$ is equal to _________.

An online exam is attempted by 50 candidates out of which 20 are boys. The average marks obtained by boys is 12 with a variance 2. The variance of marks obtained by 30 girls is also 2. The average marks of all 50 candidates is 15. If $$\mu$$ is the average marks of girls and $$\sigma^2$$ is the variance of marks of 50 candidates, then $$\mu + \sigma^2$$ is equal to _________.

$$\int \frac{2e^x+3e^{-x}}{4e^x+7e^{-x}} dx = \frac{1}{14}(ux + v\log_e(4e^x + 7e^{-x})) + C$$, where $$C$$ is a constant of integration, then $$u + v$$ is equal to _________.

Let $$S$$ be the mirror image of the point $$Q(1, 3, 4)$$ with respect to the plane $$2x - y + z + 3 = 0$$ and let $$R(3, 5, \gamma)$$ be a point of this plane. Then the square of the length of the line segment $$SR$$ is _________.

The probability distribution of random variable $$X$$ is given by

Let $$p = P(1 < X < 4 | X < 3)$$. If $$5p = \lambda K$$, then $$\lambda$$ is equal to _________.