1
JEE Advanced 2019 Paper 2 Offline
MCQ (More than One Correct Answer)
+4
-1
Change Language
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
Change Language
$${P_1} = I = \left[ {\matrix{ 1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & 1 \cr } } \right],\,{P_2} = \left[ {\matrix{ 1 & 0 & 0 \cr 0 & 0 & 1 \cr 0 & 1 & 0 \cr } } \right],\,{P_3} = \left[ {\matrix{ 0 & 1 & 0 \cr 1 & 0 & 0 \cr 0 & 0 & 1 \cr } } \right],\,{P_4} = \left[ {\matrix{ 0 & 1 & 0 \cr 0 & 0 & 1 \cr 1 & 0 & 0 \cr } } \right],\,{P_5} = \left[ {\matrix{ 0 & 0 & 1 \cr 1 & 0 & 0 \cr 0 & 1 & 0 \cr } } \right],\,{P_6} = \left[ {\matrix{ 0 & 0 & 1 \cr 0 & 1 & 0 \cr 1 & 0 & 0 \cr } } \right]$$ 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 2 Offline
Numerical
+3
-0
Change Language
Let $$\overrightarrow a = 2\widehat i + \widehat j - \widehat k$$ and $$\overrightarrow b = \widehat i + 2\widehat j + \widehat k$$ be two vectors. Consider a vector c = $$\alpha $$$$\overrightarrow a$$ + $$\beta $$$$\overrightarrow b$$, $$\alpha $$, $$\beta $$ $$ \in $$ R. If the projection of $$\overrightarrow c$$ on the vector ($$\overrightarrow a$$ + $$\overrightarrow b$$) is $$3\sqrt 2 $$, then the
minimum value of ($$\overrightarrow c$$ $$-$$($$\overrightarrow a$$ $$ \times $$ $$\overrightarrow b$$)).$$\overrightarrow c$$ equals ................
Your input ____
4
JEE Advanced 2019 Paper 2 Offline
Numerical
+3
-0
Change Language
Let |X| denote the number of elements in a set X. Let S = {1, 2, 3, 4, 5, 6} be a sample space, where each element is equally likely to occur. If A and B are independent events associated with S, then the number of ordered pairs (A, B) such that 1 $$ \le $$ |B| < |A|, equals .............
Your input ____
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