1
JEE Main 2021 (Online) 1st September Evening Shift
+4
-1
Consider the system of linear equations

$$-$$x + y + 2z = 0

3x $$-$$ ay + 5z = 1

2x $$-$$ 2y $$-$$ az = 7

Let S1 be the set of all a$$\in$$R for which the system is inconsistent and S2 be the set of all a$$\in$$R for which the system has infinitely many solutions. If n(S1) and n(S2) denote the number of elements in S1 and S2 respectively, then
A
n(S1) = 2, n(S2) = 2
B
n(S1) = 1, n(S2) = 0
C
n(S1) = 2, n(S2) = 0
D
n(S1) = 0, n(S2) = 2
2
JEE Main 2021 (Online) 1st September Evening Shift
+4
-1
Let $${J_{n,m}} = \int\limits_0^{{1 \over 2}} {{{{x^n}} \over {{x^m} - 1}}dx}$$, $$\forall$$ n > m and n, m $$\in$$ N. Consider a matrix $$A = {[{a_{ij}}]_{3 \times 3}}$$ where $${a_{ij}} = \left\{ {\matrix{ {{j_{6 + i,3}} - {j_{i + 3,3}},} & {i \le j} \cr {0,} & {i > j} \cr } } \right.$$. Then $$\left| {adj{A^{ - 1}}} \right|$$ is :
A
(15)2 $$\times$$ 242
B
(15)2 $$\times$$ 234
C
(105)2 $$\times$$ 238
D
(105)2 $$\times$$ 236
3
JEE Main 2021 (Online) 31st August Evening Shift
+4
-1
If $$\alpha$$ + $$\beta$$ + $$\gamma$$ = 2$$\pi$$, then the system of equations

x + (cos $$\gamma$$)y + (cos $$\beta$$)z = 0

(cos $$\gamma$$)x + y + (cos $$\alpha$$)z = 0

(cos $$\beta$$)x + (cos $$\alpha$$)y + z = 0

has :
A
no solution
B
infinitely many solution
C
exactly two solutions
D
a unique solution
4
JEE Main 2021 (Online) 31st August Morning Shift
+4
-1
If the following system of linear equations

2x + y + z = 5

x $$-$$ y + z = 3

x + y + az = b

has no solution, then :
A
$$a = - {1 \over 3},b \ne {7 \over 3}$$
B
$$a \ne {1 \over 3},b = {7 \over 3}$$
C
$$a \ne - {1 \over 3},b = {7 \over 3}$$
D
$$a = {1 \over 3},b \ne {7 \over 3}$$
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