1
JEE Advanced 2019 Paper 2 Offline
Numerical
+3
-0
Suppose

det$$\left| {\matrix{ {\sum\limits_{k = 0}^n k } & {\sum\limits_{k = 0}^n {{}^n{C_k}{k^2}} } \cr {\sum\limits_{k = 0}^n {{}^n{C_k}.k} } & {\sum\limits_{k = 0}^n {{}^n{C_k}{3^k}} } \cr } } \right| = 0$$

holds for some positive integer n. Then $$\sum\limits_{k = 0}^n {{{{}^n{C_k}} \over {k + 1}}}$$ equals ..............
2
JEE Advanced 2018 Paper 2 Offline
Numerical
+3
-0
Let P be a matrix of order 3 $$\times$$ 3 such that all the entries in P are from the set {$$-$$1, 0, 1}. Then, the maximum possible value of the determinant of P is ............ .
3
JEE Advanced 2017 Paper 1 Offline
Numerical
+3
-0
For a real number $$\alpha$$, if the system

$$\left[ {\matrix{ 1 & \alpha & {{\alpha ^2}} \cr \alpha & 1 & \alpha \cr {{\alpha ^2}} & \alpha & 1 \cr } } \right]\left[ {\matrix{ x \cr y \cr z \cr } } \right] = \left[ {\matrix{ 1 \cr { - 1} \cr 1 \cr } } \right]$$

of linear equations, has infinitely many solutions, then 1 + $$\alpha$$ + $$\alpha$$2 =
4
JEE Advanced 2016 Paper 1 Offline
Numerical
+3
-0

The total number of distinct x $$\in$$ R for which

$$\left| {\matrix{ x & {{x^2}} & {1 + {x^3}} \cr {2x} & {4{x^2}} & {1 + 8{x^3}} \cr {3x} & {9{x^2}} & {1 + 27{x^3}} \cr } } \right| = 10$$ is ______________.

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