1
JEE Advanced 2023 Paper 1 Online
+3
-1
Let $\alpha, \beta$ and $\gamma$ be real numbers. Consider the following system of linear equations

\begin{aligned} & x+2 y+z=7 \\\\ & x+\alpha z=11 \\\\ & 2 x-3 y+\beta z=\gamma \end{aligned}

Match each entry in List-I to the correct entries in List-II.

List - I List - II
(P) If $\beta=\frac{1}{2}(7 \alpha-3)$ and $\gamma=28$, then the system has (1) a unique solution
(Q) If $\beta=\frac{1}{2}(7 \alpha-3)$ and $\gamma \neq 28$, then the system has (2) no solution
(R) If $\beta \neq \frac{1}{2}(7 \alpha-3)$ where $\alpha=1$ and $\gamma \neq 28$, then the system has (3) infinitely many solutions
(S) If $\beta \neq \frac{1}{2}(7 \alpha-3)$ where $\alpha=1$ and $\gamma=28$, then the system has (4) $x=11, y=-2$ and $z=0$ as a solution
(5) $x=-15, y=4$ and $z=0$ as a solution

The correct option is:
A
$(P) \rightarrow(3) ~~ (Q) \rightarrow(2) ~~(R) \rightarrow(1)~~ (S) \rightarrow(4)$
B
$(P) \rightarrow(3) ~~(Q) \rightarrow(2) ~~(R) \rightarrow(5)~~ (S) \rightarrow(4)$
C
$(P) \rightarrow(2)~~ (Q) \rightarrow(1) ~~ (R) \rightarrow(4) ~~ (S) \rightarrow(5)$
D
$(P) \rightarrow(2) ~~ (Q) \rightarrow(1) ~~ (R) \rightarrow(1) ~~ (S) \rightarrow(3)$
2
JEE Advanced 2022 Paper 2 Online
+3
-1
If $M=\left(\begin{array}{rr}\frac{5}{2} & \frac{3}{2} \\ -\frac{3}{2} & -\frac{1}{2}\end{array}\right)$, then which of the

following matrices is equal to $M^{2022} ?$
A
$\left(\begin{array}{rr}3034 & 3033 \\ -3033 & -3032\end{array}\right)$
B
$\left(\begin{array}{ll}3034 & -3033 \\ 3033 & -3032\end{array}\right)$
C
$\left(\begin{array}{rr}3033 & 3032 \\ -3032 & -3031\end{array}\right)$
D
$\left(\begin{array}{rr}3032 & 3031 \\ -3031 & -3030\end{array}\right)$
3
JEE Advanced 2022 Paper 1 Online
+3
-1

Let $$p, q, r$$ be nonzero real numbers that are, respectively, the $$10^{\text {th }}, 100^{\text {th }}$$ and $$1000^{\text {th }}$$ terms of a harmonic progression. Consider the system of linear equations

$$\begin{gathered} x+y+z=1 \\ 10 x+100 y+1000 z=0 \\ q r x+p r y+p q z=0 \end{gathered}$$\$

List-I List-II
(I) If $$\frac{q}{r}=10$$, then the system of linear equations has (P) $$x=0, \quad y=\frac{10}{9}, z=-\frac{1}{9}$$ as a solution
(II) If $$\frac{p}{r} \neq 100$$, then the system of linear equations has (Q) $$x=\frac{10}{9}, y=-\frac{1}{9}, z=0$$ as a solution
(III) If $$\frac{p}{q} \neq 10$$, then the system of linear equations has (R) infinitely many solutions
(IV) If $$\frac{p}{q}=10$$, then the system of linear equations has (S) no solution
(T) at least one solution

The correct option is:

A
(I) $$\rightarrow$$ (T); (II) $$\rightarrow$$ (R); (III) $$\rightarrow$$ (S); (IV) $$\rightarrow$$ (T)
B
(I) $$\rightarrow$$ (Q); (II) $$\rightarrow$$ (S); (III) $$\rightarrow$$ (S); (IV) $$\rightarrow$$ (R)
C
(I) $$\rightarrow(\mathrm{Q})$$; (II) $$\rightarrow$$ (R); (III) $$\rightarrow(\mathrm{P})$$; (IV) $$\rightarrow$$ (R)
D
(I) $$\rightarrow$$ (T); (II) $$\rightarrow$$ (S); (III) $$\rightarrow$$ (P); (IV) $$\rightarrow$$ (T)
4
JEE Advanced 2019 Paper 1 Offline
+3
-1
Let $$M = \left[ {\matrix{ {{{\sin }^4}\theta } \cr {1 + {{\cos }^2}\theta } \cr } \matrix{ { - 1 - {{\sin }^2}\theta } \cr {{{\cos }^4}\theta } \cr } } \right] = \alpha I + \beta {M^{ - 1}}$$,

where $$\alpha$$ = $$\alpha$$($$\theta$$) and $$\beta$$ = $$\beta$$($$\theta$$) are real numbers, and I is the 2 $$\times$$ 2 identity matrix. If $$\alpha$$* is the minimum of the set {$$\alpha$$($$\theta$$) : $$\theta$$ $$\in$$ [0, 2$$\pi$$)} and {$$\beta$$($$\theta$$) : $$\theta$$ $$\in$$ [0, 2$$\pi$$)}, then the value of $$\alpha$$* + $$\beta$$* is
A
$$- {{17} \over {16}}$$
B
$$- {{31} \over {16}}$$
C
$$- {{37} \over {16}}$$
D
$$- {{29} \over {16}}$$
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