NTA JEE Main 27th August 2021 Shift 2

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: