NTA JEE Main 22nd July 2021 Shift 1

Let $$n$$ denote the number of solutions of the equation $$z^2 + 3\bar{z} = 0$$, where $$z$$ is a complex number. Then the value of $$\sum_{k=0}^{\infty} \frac{1}{n^k}$$ is equal to

Let $$S_n$$ denote the sum of first $$n$$-terms of an arithmetic progression. If $$S_{10} = 530$$, $$S_5 = 140$$, then $$S_{20} - S_6$$ is equal to:

The number of solutions of $$\sin^7 x + \cos^7 x = 1$$, $$x \in [0, 4\pi]$$ is equal to

Let the circle $$S : 36x^2 + 36y^2 - 108x + 120y + C = 0$$ be such that it neither intersects nor touches the co-ordinate axes. If the point of intersection of the lines, $$x - 2y = 4$$ and $$2x - y = 5$$ lies inside the circle $$S$$, then:

Let $$E_1 : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$, $$a > b$$. Let $$E_2$$ be another ellipse such that it touches the end points of major axis of $$E_1$$ and the foci of $$E_2$$ are the end points of minor axis of $$E_1$$. If $$E_1$$ and $$E_2$$ have same eccentricities, then its value is:

Let a line $$L : 2x + y = k$$, $$k > 0$$ be a tangent to the hyperbola $$x^2 - y^2 = 3$$. If $$L$$ is also a tangent to the parabola $$y^2 = \alpha x$$, then $$\alpha$$ is equal to:

Which of the following Boolean expressions is not a tautology?

Let $$A = [a_{ij}]$$ be a real matrix of order $$3 \times 3$$, such that $$a_{i1} + a_{i2} + a_{i3} = 1$$, for $$i = 1, 2, 3$$. Then, the sum of all entries of the matrix $$A^3$$ is equal to:

The values of $$\lambda$$ and $$\mu$$ such that the system of equations $$x + y + z = 6$$, $$3x + 5y + 5z = 26$$ and $$x + 2y + \lambda z = \mu$$ has no solution, are:

Let $$[x]$$ denote the greatest integer less than or equal to $$x$$. Then, the values of $$x \in R$$ satisfying the equation $$[e^x]^2 + [e^x + 1] - 3 = 0$$ lie in the interval: