1
JEE Advanced 2024 Paper 1 Online
MCQ (More than One Correct Answer)
+4
-2
Change Language
Let $S=\{a+b \sqrt{2}: a, b \in \mathbb{Z}\}, T_1=\left\{(-1+\sqrt{2})^n: n \in \mathbb{N}\right\}$, and $T_2=\left\{(1+\sqrt{2})^n: n \in \mathbb{N}\right\}$. Then which of the following statements is (are) TRUE?
A
$\mathbb{Z} \cup T_1 \cup T_2 \subset S$
B
$T_1 \cap\left(0, \frac{1}{2024}\right)=\phi$, where $\phi$ denotes the empty set.
C
$T_2 \cap(2024, \infty) \neq \phi$
D
For any given $a, b \in \mathbb{Z}, \cos (\pi(a+b \sqrt{2}))+i \sin (\pi(a+b \sqrt{2})) \in \mathbb{Z}$ if and only if $b=0$, where $i=\sqrt{-1}$.
2
JEE Advanced 2024 Paper 1 Online
MCQ (More than One Correct Answer)
+4
-2
Change Language

Let $\mathbb{R}^2$ denote $\mathbb{R} \times \mathbb{R}$. Let

$$ S=\left\{(a, b, c): a, b, c \in \mathbb{R} \text { and } a x^2+2 b x y+c y^2>0 \text { for all }(x, y) \in \mathbb{R}^2-\{(0,0)\}\right\} . $$

Then which of the following statements is (are) TRUE?

A
$\left(2, \frac{7}{2}, 6\right) \in S$
B
If $\left(3, b, \frac{1}{12}\right) \in S$, then $|2 b|<1$.
C

For any given $(a, b, c) \in S$, the system of linear equations

$$ \begin{aligned} & a x+b y=1 \\ & b x+c y=-1 \end{aligned} $$

has a unique solution.

D

For any given $(a, b, c) \in S$, the system of linear equations

$$ \begin{aligned} & (a+1) x+b y=0 \\ & b x+(c+1) y=0 \end{aligned} $$

has a unique solution.

3
JEE Advanced 2024 Paper 1 Online
MCQ (More than One Correct Answer)
+4
-2
Change Language

Let $\mathbb{R}^3$ denote the three-dimensional space. Take two points $P=(1,2,3)$ and $Q=(4,2,7)$. Let $\operatorname{dist}(X, Y)$ denote the distance between two points $X$ and $Y$ in $\mathbb{R}^3$. Let

$$ \begin{gathered} S=\left\{X \in \mathbb{R}^3:(\operatorname{dist}(X, P))^2-(\operatorname{dist}(X, Q))^2=50\right\} \text { and } \\ T=\left\{Y \in \mathbb{R}^3:(\operatorname{dist}(Y, Q))^2-(\operatorname{dist}(Y, P))^2=50\right\} . \end{gathered} $$

Then which of the following statements is (are) TRUE?

A
There is a triangle whose area is 1 and all of whose vertices are from $S$.
B
There are two distinct points $L$ and $M$ in $T$ such that each point on the line segment $L M$ is also in $T$.
C
There are infinitely many rectangles of perimeter 48 , two of whose vertices are from $S$ and the other two vertices are from $T$.
D
There is a square of perimeter 48 , two of whose vertices are from $S$ and the other two vertices are from $T$.
4
JEE Advanced 2024 Paper 1 Online
Numerical
+4
-0
Change Language

Let $a=3 \sqrt{2}$ and $b=\frac{1}{5^{1 / 6} \sqrt{6}}$. If $x, y \in \mathbb{R}$ are such that

$$ \begin{aligned} & 3 x+2 y=\log _a(18)^{\frac{5}{4}} \quad \text { and } \\ & 2 x-y=\log _b(\sqrt{1080}), \end{aligned} $$

then $4 x+5 y$ is equal to __________.

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