1
JEE Advanced 2022 Paper 1 Online
Numerical
+3
-0
Let $$z$$ be a complex number with a non-zero imaginary part. If

$$\frac{2+3 z+4 z^{2}}{2-3 z+4 z^{2}}$$

is a real number, then the value of $$|z|^{2}$$ is _________.
2
JEE Advanced 2022 Paper 1 Online
Numerical
+3
-0
Let $$\bar{z}$$ denote the complex conjugate of a complex number $$z$$ and let $$i=\sqrt{-1}$$. In the set of complex numbers, the number of distinct roots of the equation

$$\bar{z}-z^{2}=i\left(\bar{z}+z^{2}\right)$$

is _________.
3
JEE Advanced 2022 Paper 1 Online
Numerical
+3
-0
Let $$l_{1}, l_{2}, \ldots, l_{100}$$ be consecutive terms of an arithmetic progression with common difference $$d_{1}$$, and let $$w_{1}, w_{2}, \ldots, w_{100}$$ be consecutive terms of another arithmetic progression with common difference $$d_{2}$$, where $$d_{1} d_{2}=10$$. For each $$i=1,2, \ldots, 100$$, let $R_{i}$ be a rectangle with length $$l_{i}$$, width $$w_{i}$$ and area $A_{i}$. If $$A_{51}-A_{50}=1000$$, then the value of $$A_{100}-A_{90}$$ is __________.
4
JEE Advanced 2022 Paper 1 Online
Numerical
+3
-0
The number of 4-digit integers in the closed interval [2022, 4482] formed by using the digits $$0,2,3,4,6,7$$ is _________.
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