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

If $$A = {1 \over 2}\left[ {\matrix{ 1 & {\sqrt 3 } \cr { - \sqrt 3 } & 1 \cr } } \right]$$, then :

A
$$\mathrm{A^{30}-A^{25}=2I}$$
B
$$\mathrm{A^{30}+A^{25}-A=I}$$
C
$$\mathrm{A^{30}=A^{25}}$$
D
$$\mathrm{A^{30}+A^{25}+A=I}$$
2
JEE Main 2023 (Online) 1st February Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Two dice are thrown independently. Let $$\mathrm{A}$$ be the event that the number appeared on the $$1^{\text {st }}$$ die is less than the number appeared on the $$2^{\text {nd }}$$ die, $$\mathrm{B}$$ be the event that the number appeared on the $$1^{\text {st }}$$ die is even and that on the second die is odd, and $$\mathrm{C}$$ be the event that the number appeared on the $$1^{\text {st }}$$ die is odd and that on the $$2^{\text {nd }}$$ is even. Then :

A
A and B are mutually exclusive
B
the number of favourable cases of the events A, B and C are 15, 6 and 6 respectively
C
B and C are independent
D
the number of favourable cases of the event $$(\mathrm{A\cup B)\cap C}$$ is 6
3
JEE Main 2023 (Online) 1st February Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let $$\vec{a}=2 \hat{i}-7 \hat{j}+5 \hat{k}, \vec{b}=\hat{i}+\hat{k}$$ and $$\vec{c}=\hat{i}+2 \hat{j}-3 \hat{k}$$ be three given vectors. If $$\overrightarrow{\mathrm{r}}$$ is a vector such that $$\vec{r} \times \vec{a}=\vec{c} \times \vec{a}$$ and $$\vec{r} \cdot \vec{b}=0$$, then $$|\vec{r}|$$ is equal to :

A
$$\frac{11}{7}$$
B
$$\frac{11}{5} \sqrt{2}$$
C
$$\frac{\sqrt{914}}{7}$$
D
$$\frac{11}{7} \sqrt{2}$$
4
JEE Main 2023 (Online) 1st February Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let $$S = \left\{ {x \in R:0 < x < 1\,\mathrm{and}\,2{{\tan }^{ - 1}}\left( {{{1 - x} \over {1 + x}}} \right) = {{\cos }^{ - 1}}\left( {{{1 - {x^2}} \over {1 + {x^2}}}} \right)} \right\}$$.

If $$\mathrm{n(S)}$$ denotes the number of elements in $$\mathrm{S}$$ then :

A
$$\mathrm{n}(\mathrm{S})=0$$
B
$$\mathrm{n}(\mathrm{S})=1$$ and only one element in $$\mathrm{S}$$ is less than $$\frac{1}{2}$$.
C
$$\mathrm{n}(\mathrm{S})=1$$ and the elements in $$\mathrm{S}$$ is more than $$\frac{1}{2}$$.
D
$$\mathrm{n}(\mathrm{S})=1$$ and the element in $$\mathrm{S}$$ is less than $$\frac{1}{2}$$.
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