1
JEE Main 2023 (Online) 24th January Morning Shift
+4
-1

The equation $${x^2} - 4x + [x] + 3 = x[x]$$, where $$[x]$$ denotes the greatest integer function, has :

A
exactly two solutions in ($$-\infty,\infty$$)
B
no solution
C
a unique solution in ($$-\infty,\infty$$)
D
a unique solution in ($$-\infty,1$$)
2
JEE Main 2022 (Online) 29th July Morning Shift
+4
-1

If $$\frac{1}{(20-a)(40-a)}+\frac{1}{(40-a)(60-a)}+\ldots+\frac{1}{(180-a)(200-a)}=\frac{1}{256}$$, then the maximum value of $$\mathrm{a}$$ is :

A
198
B
202
C
212
D
218
3
JEE Main 2022 (Online) 28th July Evening Shift
+4
-1

$$\text { Let } S=\left\{x \in[-6,3]-\{-2,2\}: \frac{|x+3|-1}{|x|-2} \geq 0\right\} \text { and }$$

$$T=\left\{x \in \mathbb{Z}: x^{2}-7|x|+9 \leq 0\right\} \text {. }$$

Then the number of elements in $$\mathrm{S} \cap \mathrm{T}$$ is :

A
7
B
5
C
4
D
3
4
JEE Main 2022 (Online) 28th July Evening Shift
+4
-1

Let $$\alpha$$, $$\beta$$ be the roots of the equation $$x^{2}-\sqrt{2} x+\sqrt{6}=0$$ and $$\frac{1}{\alpha^{2}}+1, \frac{1}{\beta^{2}}+1$$ be the roots of the equation $$x^{2}+a x+b=0$$. Then the roots of the equation $$x^{2}-(a+b-2) x+(a+b+2)=0$$ are :

A
non-real complex numbers
B
real and both negative
C
real and both positive
D
real and exactly one of them is positive
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