1
JEE Advanced 2019 Paper 2 Offline
MCQ (More than One Correct Answer)
+4
-1 Let x $$\in$$ R and let $$P = \left[ {\matrix{ 1 & 1 & 1 \cr 0 & 2 & 2 \cr 0 & 0 & 3 \cr } } \right]$$, $$Q = \left[ {\matrix{ 2 & x & x \cr 0 & 4 & 0 \cr x & x & 6 \cr } } \right]$$ and R = PQP$$-$$1, which of the following options is/are correct?
A
There exists a real, number x such that PQ = QP
B
For $$x = 0$$, if $$R \left[ {\matrix{ 1 \cr a \cr b \cr } } \right] = 6\left[ {\matrix{ 1 \cr a \cr b \cr } } \right]$$, then a + b =5
C
For x = 1, there exists a unit vector $$\alpha \widehat i + \beta \widehat j + \gamma \widehat k$$ for which $$R\left[ {\matrix{ \alpha \cr \beta \cr \gamma \cr } } \right] = \left[ {\matrix{ 0 \cr 0 \cr 0 \cr } } \right]$$
D
$$\det R = \det \left[ {\matrix{ 2 & x & x \cr 0 & 4 & 0 \cr x & x & 5 \cr } } \right] + 8$$, for all x $$\in$$ R
2
JEE Advanced 2019 Paper 2 Offline
MCQ (More than One Correct Answer)
+4
-1 $${P_1} = I = \left[ {\matrix{ 1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & 1 \cr } } \right],\,{P_2} = \left[ {\matrix{ 1 & 0 & 0 \cr 0 & 0 & 1 \cr 0 & 1 & 0 \cr } } \right],\,{P_3} = \left[ {\matrix{ 0 & 1 & 0 \cr 1 & 0 & 0 \cr 0 & 0 & 1 \cr } } \right],\,{P_4} = \left[ {\matrix{ 0 & 1 & 0 \cr 0 & 0 & 1 \cr 1 & 0 & 0 \cr } } \right],\,{P_5} = \left[ {\matrix{ 0 & 0 & 1 \cr 1 & 0 & 0 \cr 0 & 1 & 0 \cr } } \right],\,{P_6} = \left[ {\matrix{ 0 & 0 & 1 \cr 0 & 1 & 0 \cr 1 & 0 & 0 \cr } } \right]$$ and $$X = \sum\limits_{k = 1}^6 {{P_k}} \left[ {\matrix{ 2 & 1 & 3 \cr 1 & 0 & 2 \cr 3 & 2 & 1 \cr } } \right]P_k^T$$

where $$P_k^T$$ denotes the transpose of the matrix Pk. Then which of the following option is/are correct?
A
X is a symmetric matrix
B
The sum of diagonal entries of X is 18
C
X $$-$$ 30I is an invertible matrix
D
If $$X\left[ {\matrix{ 1 \cr 1 \cr 1 \cr } } \right] = \alpha \left[ {\matrix{ 1 \cr 1 \cr 1 \cr } } \right]$$, then $$\alpha = 30$$
3
JEE Advanced 2019 Paper 1 Offline
MCQ (More than One Correct Answer)
+4
-1 Let $$M = \left[ {\matrix{ 0 & 1 & a \cr 1 & 2 & 3 \cr 3 & b & 1 \cr } } \right]$$ and

adj $$M = \left[ {\matrix{ { - 1} & 1 & { - 1} \cr 8 & { - 6} & 2 \cr { - 5} & 3 & { - 1} \cr } } \right]$$

where a and b are real numbers. Which of the following options is/are correct?
A
B
If $$M\left[ {\matrix{ \alpha \cr \beta \cr \gamma \cr } } \right] = \left[ {\matrix{ 1 \cr 2 \cr 3 \cr } } \right]$$, then $$\alpha - \beta + \gamma = 3$$
C
$${(adj\,M)^{ - 1}} + adj\,{M^{ - 1}} = - M$$
D
a + b = 3
4
JEE Advanced 2018 Paper 2 Offline
MCQ (More than One Correct Answer)
+4
-1
Let S be the set of all column matrices $$\left[ {\matrix{ {{b_1}} \cr {{b_2}} \cr {{b_3}} \cr } } \right]$$ such that $${b_1},{b_2},{b_3} \in R$$ and the system of equations (in real variables)

\eqalign{ & - x + 2y + 5z = {b_1} \cr & 2x - 4y + 3z = {b_2} \cr & x - 2y + 2z = {b_3} \cr}

has at least one solution. Then, which of the following system(s) (in real variables) has (have) at least one solution for each $$\left[ {\matrix{ {{b_1}} \cr {{b_2}} \cr {{b_3}} \cr } } \right]$$$$\in$$S?
A
$$x + 2y + 3z = {b_1}$$, $$\,4y + 5z = {b_2}$$ and $$x + 2y + 6z = {b_3}$$
B
$$x + y + 3z = {b_1}$$, $$5x + 2y + 6z = {b_2}$$ and $$- 2x - y - 3z = {b_3}$$
C
$$- x + 2y - 5z = {b_1}$$, $$\,2x - 4y + 10z = {b_2}$$ and $$x - 2y + 5z = {b_3}$$
D
$$x + 2y + 5z = {b_1}$$, $$2x + 3z = {b_2}$$ and $$x + 4y - 5z = {b_3}$$
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