NTA JEE Mains 8th April 2024 Shift 2

The sum of all possible values of $$\theta \in [-\pi, 2\pi]$$, for which $$\frac{1 + i\cos\theta}{1 - 2i\cos\theta}$$ is purely imaginary, is equal to :

The number of ways five alphabets can be chosen from the alphabets of the word MATHEMATICS, where the chosen alphabets are not necessarily distinct, is equal to :

In an increasing geometric progression of positive terms, the sum of the second and sixth terms is $$\frac{70}{3}$$ and the product of the third and fifth terms is 49. Then the sum of the $$4^{th}$$, $$6^{th}$$ and $$8^{th}$$ terms is equal to :

If the term independent of $$x$$ in the expansion of $$\left(\sqrt{a}x^2 + \frac{1}{2x^3}\right)^{10}$$ is 105, then $$a^2$$ is equal to :

If the value of $$\frac{3\cos 36° + 5\sin 18°}{5\cos 36° - 3\sin 18°}$$ is $$\frac{a\sqrt{5} - b}{c}$$, where $$a, b, c$$ are natural numbers and $$\gcd(a, c) = 1$$, then $$a + b + c$$ is equal to :

If the image of the point $$(-4, 5)$$ in the line $$x + 2y = 2$$ lies on the circle $$(x + 4)^2 + (y - 3)^2 = r^2$$, then $$r$$ is equal to :

If the line segment joining the points $$(5, 2)$$ and $$(2, a)$$ subtends an angle $$\frac{\pi}{4}$$ at the origin, then the absolute value of the product of all possible values of $$a$$ is :

Let $$A = \{2, 3, 6, 8, 9, 11\}$$ and $$B = \{1, 4, 5, 10, 15\}$$. Let $$R$$ be a relation on $$A \times B$$ defined by $$(a, b)R(c, d)$$ if and only if $$3ad - 7bc$$ is an even integer. Then the relation $$R$$ is :

If $$\alpha \neq a, \beta \neq b, \gamma \neq c$$ and $$\begin{vmatrix} \alpha & b & c \\ a & \beta & c \\ a & b & \gamma \end{vmatrix} = 0$$, then $$\frac{a}{\alpha - a} + \frac{b}{\beta - b} + \frac{\gamma}{\gamma - c}$$ is equal to :

If the system of equations $$x + 4y - z = \lambda$$, $$7x + 9y + \mu z = -3$$, $$5x + y + 2z = -1$$ has infinitely many solutions, then $$(2\mu + 3\lambda)$$ is equal to :