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1

### JEE Main 2021 (Online) 27th August Morning Shift

Numerical
If the system of linear equations

2x + y $$-$$ z = 3

x $$-$$ y $$-$$ z = $$\alpha$$

3x + 3y + $$\beta$$z = 3

has infinitely many solution, then $$\alpha$$ + $$\beta$$ $$-$$ $$\alpha$$$$\beta$$ is equal to _____________.

## Explanation

2 $$\times$$ (i) $$-$$ (ii) $$-$$ (iii) gives :

$$-$$ (1 + $$\beta$$)z = 3 $$-$$ $$\alpha$$

For infinitely many solution

$$\beta$$ + 1 = 0 = 3 $$-$$ $$\alpha$$ $$\Rightarrow$$ ($$\alpha$$, $$\beta$$) = (3, $$-$$1)

Hence, $$\alpha$$ + $$\beta$$ $$-$$ $$\alpha$$$$\beta$$ = 5
2

### JEE Main 2021 (Online) 26th August Evening Shift

Numerical
Let A be a 3 $$\times$$ 3 real matrix. If det(2Adj(2 Adj(Adj(2A)))) = 241, then the value of det(A2) equal __________.

## Explanation

$$\Rightarrow$$ adj(adj (2A)) = adj(4 adjA) = 16 adj (adj A)

= 16 | A | A

$$\Rightarrow$$ adj (32 | A | A) = (32 | A |)2 adj A

12(32| A |)2 |adj A | = 23 (32 | A |)6 | adj A |

23 . 230 | A |6 . | A |2 = 241

| A |8 = 28 $$\Rightarrow$$ | A | = $$\pm$$2

| A |2 = | A |2 = 4
3

### JEE Main 2021 (Online) 27th July Morning Shift

Numerical
Let $$f(x) = \left| {\matrix{ {{{\sin }^2}x} & { - 2 + {{\cos }^2}x} & {\cos 2x} \cr {2 + {{\sin }^2}x} & {{{\cos }^2}x} & {\cos 2x} \cr {{{\sin }^2}x} & {{{\cos }^2}x} & {1 + \cos 2x} \cr } } \right|,x \in [0,\pi ]$$. Then the maximum value of f(x) is equal to ______________.

## Explanation

$$\left| {\matrix{ { - 2} & { - 2} & 0 \cr 2 & 0 & { - 1} \cr {{{\sin }^2}x} & {{{\cos }^2}x} & {1 + \cos 2x} \cr } } \right|\left( \matrix{ {R_1} \to {R_1} - {R_2} \hfill \cr \& \,{R_2} \to {R_2} - {R_3} \hfill \cr} \right)$$

= $$- 2({\cos ^2}x) + 2(2 + 2\cos 2x + {\sin ^2}x)$$

= $$4 + 4\cos 2x - 2({\cos ^2}x - {\sin ^2}x)$$

$$\therefore$$ $$f(x) = 4 + \underbrace {2\cos 2x}_{\max = 1}$$

$$\Rightarrow$$ $$f{(x)_{\max }} = 4 + 2 = 6$$
4

### JEE Main 2021 (Online) 27th July Morning Shift

Numerical
For real numbers $$\alpha$$ and $$\beta$$, consider the following system of linear equations :

x + y $$-$$ z = 2, x + 2y + $$\alpha$$z = 1, 2x $$-$$ y + z = $$\beta$$. If the system has infinite solutions, then $$\alpha$$ + $$\beta$$ is equal to ______________.

## Explanation

For infinite solutions

$$\Delta$$ = $$\Delta$$1 = $$\Delta$$2 = $$\Delta$$3 = 0

$$\Delta$$ = $$\left| {\matrix{ 1 & 1 & { - 1} \cr 1 & 2 & \alpha \cr 2 & { - 1} & 1 \cr } } \right| = 0$$

$$\Delta = \left| {\matrix{ 3 & 0 & 0 \cr 1 & 2 & \alpha \cr 2 & { - 1} & 1 \cr } } \right| = 0$$

$$\Delta$$ = 3(2 + $$\alpha$$) = 0

$$\Rightarrow$$ $$\alpha$$ = $$-$$2

$${\Delta _2} = \left| {\matrix{ 1 & 2 & { - 1} \cr 1 & 1 & { - 2} \cr 2 & \beta & 1 \cr } } \right| = 0$$

1(1 + 2$$\beta$$) $$-$$2(1 + 4) $$-$$ ($$\beta$$ $$-$$ 2) = 0

$$\beta$$ $$-$$ 7 = 0

$$\beta$$ = 7

$$\therefore$$ $$\alpha$$ + $$\beta$$ = 5

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