1
JEE Advanced 2022 Paper 2 Online
Numerical
+3
-1
Change Language
Let $\beta$ be a real number. Consider the matrix

$$ A=\left(\begin{array}{ccc} \beta & 0 & 1 \\ 2 & 1 & -2 \\ 3 & 1 & -2 \end{array}\right) $$

If $A^{7}-(\beta-1) A^{6}-\beta A^{5}$ is a singular matrix, then the value of $9 \beta$ is _________.
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2
JEE Advanced 2021 Paper 1 Online
Numerical
+2
-0
Change Language
Let $$\alpha$$, $$\beta$$ and $$\gamma$$ be real numbers such that the system of linear equations

x + 2y + 3z = $$\alpha$$

4x + 5y + 6z = $$\beta$$

7x + 8y + 9z = $$\gamma $$ $$-$$ 1

is consistent. Let | M | represent the determinant of the matrix

$$M = \left[ {\matrix{ \alpha & 2 & \gamma \cr \beta & 1 & 0 \cr { - 1} & 0 & 1 \cr } } \right]$$

Let P be the plane containing all those ($$\alpha$$, $$\beta$$, $$\gamma$$) for which the above system of linear equations is consistent, and D be the square of the distance of the point (0, 1, 0) from the plane P.

The value of | M | is _________.
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3
JEE Advanced 2021 Paper 1 Online
Numerical
+2
-0
Change Language
Let $$\alpha$$, $$\beta$$ and $$\gamma$$ be real numbers such that the system of linear equations

x + 2y + 3z = $$\alpha$$

4x + 5y + 6z = $$\beta$$

7x + 8y + 9z = $$\gamma $$ $$-$$ 1

is consistent. Let | M | represent the determinant of the matrix

$$M = \left[ {\matrix{ \alpha & 2 & \gamma \cr \beta & 1 & 0 \cr { - 1} & 0 & 1 \cr } } \right]$$

Let P be the plane containing all those ($$\alpha$$, $$\beta$$, $$\gamma$$) for which the above system of linear equations is consistent, and D be the square of the distance of the point (0, 1, 0) from the plane P.

The value of D is _________.
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4
JEE Advanced 2020 Paper 2 Offline
Numerical
+3
-1
Change Language
The trace of a square matrix is defined to be the sum of its diagonal entries. If A is a 2 $$ \times $$ 2 matrix such that the trace of A is 3 and the trace of A3 is $$-$$18, then the value of the determinant of A is .............
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