1
JEE Main 2023 (Online) 1st February Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let $$P(S)$$ denote the power set of $$S=\{1,2,3, \ldots ., 10\}$$. Define the relations $$R_{1}$$ and $$R_{2}$$ on $$P(S)$$ as $$\mathrm{AR}_{1} \mathrm{~B}$$ if $$\left(\mathrm{A} \cap \mathrm{B}^{\mathrm{c}}\right) \cup\left(\mathrm{B} \cap \mathrm{A}^{\mathrm{c}}\right)=\emptyset$$ and $$\mathrm{AR}_{2} \mathrm{~B}$$ if $$\mathrm{A} \cup \mathrm{B}^{\mathrm{c}}=\mathrm{B} \cup \mathrm{A}^{\mathrm{c}}, \forall \mathrm{A}, \mathrm{B} \in \mathrm{P}(\mathrm{S})$$. Then :

A
only $$R_{2}$$ is an equivalence relation
B
both $$R_{1}$$ and $$R_{2}$$ are not equivalence relations
C
both $$R_{1}$$ and $$R_{2}$$ are equivalence relations
D
only $$R_{1}$$ is an equivalence relation
2
JEE Main 2023 (Online) 1st February Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

The sum of the absolute maximum and minimum values of the function $$f(x)=\left|x^{2}-5 x+6\right|-3 x+2$$ in the interval $$[-1,3]$$ is equal to :

A
13
B
24
C
10
D
12
3
JEE Main 2023 (Online) 1st February Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let $$\vec{a}=5 \hat{i}-\hat{j}-3 \hat{k}$$ and $$\vec{b}=\hat{i}+3 \hat{j}+5 \hat{k}$$ be two vectors. Then which one of the following statements is TRUE ?

A
Projection of $$\vec{a}$$ on $$\vec{b}$$ is $$\frac{-13}{\sqrt{35}}$$ and the direction of the projection vector is opposite to the direction of $$\vec{b}$$.
B
Projection of $$\vec{a}$$ on $$\vec{b}$$ is $$\frac{13}{\sqrt{35}}$$ and the direction of the projection vector is opposite to the direction of $$\vec{b}$$.
C
Projection of $$\vec{a}$$ on $$\vec{b}$$ is $$\frac{13}{\sqrt{35}}$$ and the direction of the projection vector is same as of $$\vec{b}$$.
D
Projection of $$\vec{a}$$ on $$\vec{b}$$ is $$\frac{-13}{\sqrt{35}}$$ and the direction of the projection vector is same as of $$\vec{b}$$.
4
JEE Main 2023 (Online) 1st February Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

For the system of linear equations $$\alpha x+y+z=1,x+\alpha y+z=1,x+y+\alpha z=\beta$$, which one of the following statements is NOT correct?

A
It has infinitely many solutions if $$\alpha=1$$ and $$\beta=1$$
B
It has infinitely many solutions if $$\alpha=2$$ and $$\beta=-1$$
C
$$x+y+z=\frac{3}{4}$$ if $$\alpha=2$$ and $$\beta=1$$
D
It has no solution if $$\alpha=-2$$ and $$\beta=1$$
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