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 D is _________.
2
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 __________.
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 D is __________.
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

$$E = \left[ {\matrix{ 1 & 2 & 3 \cr 2 & 3 & 4 \cr 8 & {13} & {18} \cr } } \right]$$, $$P = \left[ {\matrix{ 1 & 0 & 0 \cr 0 & 0 & 1 \cr 0 & 1 & 0 \cr } } \right]$$ 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
EXAM MAP
Medical
NEET