1
JEE Main 2023 (Online) 29th January Evening Shift
Numerical
+4
-1
Out of Syllabus
Change Language

Let $$a_1=b_1=1$$ and $${a_n} = {a_{n - 1}} + (n - 1),{b_n} = {b_{n - 1}} + {a_{n - 1}},\forall n \ge 2$$. If $$S = \sum\limits_{n = 1}^{10} {{{{b_n}} \over {{2^n}}}} $$ and $$T = \sum\limits_{n = 1}^8 {{n \over {{2^{n - 1}}}}} $$, then $${2^7}(2S - T)$$ is equal to ____________.

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2
JEE Main 2023 (Online) 29th January Evening Shift
Numerical
+4
-1
Change Language

The total number of 4-digit numbers whose greatest common divisor with 54 is 2, is __________.

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3
JEE Main 2023 (Online) 29th January Evening Shift
Numerical
+4
-1
Change Language

Let $$X=\{11,12,13,....,40,41\}$$ and $$Y=\{61,62,63,....,90,91\}$$ be the two sets of observations. If $$\overline x $$ and $$\overline y $$ are their respective means and $$\sigma^2$$ is the variance of all the observations in $$\mathrm{X\cup Y}$$, then $$\left| {\overline x + \overline y - {\sigma ^2}} \right|$$ is equal to ____________.

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4
JEE Main 2023 (Online) 29th January Evening Shift
Numerical
+4
-1
Out of Syllabus
Change Language

If the equation of the normal to the curve $$y = {{x - a} \over {(x + b)(x - 2)}}$$ at the point (1, $$-$$3) is $$x - 4y = 13$$, then the value of $$a + b$$ is equal to ___________.

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