1
JEE Advanced 2021 Paper 1 Online
Numerical
+2
-0
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 _________.
2
JEE Advanced 2021 Paper 1 Online
Numerical
+2
-0
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 _________.
3
JEE Advanced 2021 Paper 1 Online
Numerical
+2
-0
Consider the lines L1 and L2 defined by

$${L_1}:x\sqrt 2 + y - 1 = 0$$ and $${L_2}:x\sqrt 2 - y + 1 = 0$$

For a fixed constant $$\lambda$$, let C be the locus of a point P such that the product of the distance of P from L1 and the distance of P from L2 is $$\lambda$$2. The line y = 2x + 1 meets C at two points R and S, where the distance between R and S is $$\sqrt {270}$$. Let the perpendicular bisector of RS meet C at two distinct points R' and S'. Let D be the square of the distance between R' and S'.

The value of $$\lambda$$2 is __________.
4
JEE Advanced 2021 Paper 1 Online
Numerical
+2
-0
Consider the lines L1 and L2 defined by

$${L_1}:x\sqrt 2 + y - 1 = 0$$ and $${L_2}:x\sqrt 2 - y + 1 = 0$$

For a fixed constant $$\lambda$$, let C be the locus of a point P such that the product of the distance of P from L1 and the distance of P from L2 is $$\lambda$$2. The line y = 2x + 1 meets C at two points R and S, where the distance between R and S is $$\sqrt {270}$$. Let the perpendicular bisector of RS meet C at two distinct points R' and S'. Let D be the square of the distance between R' and S'.

The value of D is __________.