1
JEE Main 2023 (Online) 30th January Morning Shift
Numerical
+4
-1
Change Language

Let $$S=\{1,2,3,4,5,6\}$$. Then the number of one-one functions $$f: \mathrm{S} \rightarrow \mathrm{P}(\mathrm{S})$$, where $$\mathrm{P}(\mathrm{S})$$ denote the power set of $$\mathrm{S}$$, such that $$f(n) \subset f(\mathrm{~m})$$ where $$n < m$$ is ____________.

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2
JEE Main 2023 (Online) 30th January Morning Shift
Numerical
+4
-1
Out of Syllabus
Change Language

If $$\lambda_{1} < \lambda_{2}$$ are two values of $$\lambda$$ such that the angle between the planes $$P_{1}: \vec{r}(3 \hat{i}-5 \hat{j}+\hat{k})=7$$ and $$P_{2}: \vec{r} \cdot(\lambda \hat{i}+\hat{j}-3 \hat{k})=9$$ is $$\sin ^{-1}\left(\frac{2 \sqrt{6}}{5}\right)$$, then the square of the length of perpendicular from the point $$\left(38 \lambda_{1}, 10 \lambda_{2}, 2\right)$$ to the plane $$P_{1}$$ is ______________.

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3
JEE Main 2023 (Online) 30th January Morning Shift
Numerical
+4
-1
Change Language

Let $$f^{1}(x)=\frac{3 x+2}{2 x+3}, x \in \mathbf{R}-\left\{\frac{-3}{2}\right\}$$ For $$\mathrm{n} \geq 2$$, define $$f^{\mathrm{n}}(x)=f^{1} \mathrm{o} f^{\mathrm{n}-1}(x)$$. If $$f^{5}(x)=\frac{\mathrm{a} x+\mathrm{b}}{\mathrm{b} x+\mathrm{a}}, \operatorname{gcd}(\mathrm{a}, \mathrm{b})=1$$, then $$\mathrm{a}+\mathrm{b}$$ is equal to ____________.

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4
JEE Main 2023 (Online) 30th January Morning Shift
Numerical
+4
-1
Change Language

The mean and variance of 7 observations are 8 and 16 respectively. If one observation 14 is omitted and a and b are respectively mean and variance of remaining 6 observation, then $$\mathrm{a+3 b-5}$$ is equal to ___________.

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