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

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.

2
JEE Advanced 2023 Paper 2 Online
MCQ (More than One Correct Answer)
+4
-2
Let $M=\left(a_{i j}\right), i, j \in\{1,2,3\}$, be the $3 \times 3$ matrix such that $a_{i j}=1$ if $j+1$ is divisible by $i$, otherwise $a_{i j}=0$. Then which of the following statements is(are) true?
A
$M$ is invertible
B
There exists a nonzero column matrix $\left(\begin{array}{l}a_1 \\ a_2 \\ a_3\end{array}\right)$ such that $M\left(\begin{array}{l}a_1 \\ a_2 \\ a_3\end{array}\right)=\left(\begin{array}{l}-a_1 \\ -a_2 \\ -a_3\end{array}\right)$
C
The set $\left\{X \in \mathbb{R}^3: M X=\mathbf{0}\right\} \neq\{\mathbf{0}\}$, where $\mathbf{0}=\left(\begin{array}{l}0 \\ 0 \\ 0\end{array}\right)$
D
The matrix $(M-2 I)$ is invertible, where $I$ is the $3 \times 3$ identity matrix
3
JEE Advanced 2021 Paper 1 Online
MCQ (More than One Correct Answer)
+4
-2
For any 3 $$\times$$ 3 matrix M, let | M | denote the determinant of M. Let

$$E = \left[ {\matrix{ 1 & 2 & 3 \cr 2 & 3 & 4 \cr 8 & {13} & {18} \cr } } \right]$$, $$P = \left[ {\matrix{ 1 & 0 & 0 \cr 0 & 0 & 1 \cr 0 & 1 & 0 \cr } } \right]$$ and $$F = \left[ {\matrix{ 1 & 3 & 2 \cr 8 & {18} & {13} \cr 2 & 4 & 3 \cr } } \right]$$

If Q is a nonsingular matrix of order 3 $$\times$$ 3, then which of the following statements is(are) TRUE?
A
F = PEP and $${P^2} = \left[ {\matrix{ 1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & 1 \cr } } \right]$$
B
| EQ + PFQ$$-$$1 | = | EQ | + | PFQ$$-$$1 |
C
| (EF)3 | > | EF |2
D
Sum of the diagonal entries of P$$-$$1EP + F is equal to the sum of diagonal entries of E + P$$-$$1FP
4
JEE Advanced 2021 Paper 1 Online
MCQ (More than One Correct Answer)
+4
-2
For any 3 $$\times$$ 3 matrix M, let |M| denote the determinant of M. Let I be the 3 $$\times$$ 3 identity matrix. Let E and F be two 3 $$\times$$ 3 matrices such that (I $$-$$ EF) is invertible. If G = (I $$-$$ EF)$$-$$1, then which of the following statements is (are) TRUE?
A
| FE | = | I $$-$$ FE| | FGE |
B
(I $$-$$ FE)(I + FGE) = I
C
EFG = GEF
D
(I $$-$$ FE)(I $$-$$ FGE) = I
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