1
JEE Main 2023 (Online) 25th January Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let S$$_1$$ and S$$_2$$ be respectively the sets of all $$a \in \mathbb{R} - \{ 0\} $$ for which the system of linear equations

$$ax + 2ay - 3az = 1$$

$$(2a + 1)x + (2a + 3)y + (a + 1)z = 2$$

$$(3a + 5)x + (a + 5)y + (a + 2)z = 3$$

has unique solution and infinitely many solutions. Then

A
$$\mathrm{n({S_1}) = 2}$$ and S$$_2$$ is an infinite set
B
$$\mathrm{{S_1} = \Phi} $$ and $$\mathrm{{S_2} = \mathbb{R} - \{ 0\}}$$
C
$$\mathrm{{S_1} = \mathbb{R} - \{ 0\}}$$ and $$\mathrm{{S_2} = \Phi} $$
D
S$$_1$$ is an infinite set and n(S$$_2$$) = 2
2
JEE Main 2023 (Online) 25th January Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let $$y = y(x)$$ be the solution curve of the differential equation $${{dy} \over {dx}} = {y \over x}\left( {1 + x{y^2}(1 + {{\log }_e}x)} \right),x > 0,y(1) = 3$$. Then $${{{y^2}(x)} \over 9}$$ is equal to :

A
$${{{x^2}} \over {5 - 2{x^3}(2 + {{\log }_e}{x^3})}}$$
B
$${{{x^2}} \over {3{x^3}(1 + {{\log }_e}{x^2}) - 2}}$$
C
$${{{x^2}} \over {7 - 3{x^3}(2 + {{\log }_e}{x^2})}}$$
D
$${{{x^2}} \over {2{x^3}(2 + {{\log }_e}{x^3}) - 3}}$$
3
JEE Main 2023 (Online) 25th January Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let $$f:(0,1)\to\mathbb{R}$$ be a function defined $$f(x) = {1 \over {1 - {e^{ - x}}}}$$, and $$g(x) = \left( {f( - x) - f(x)} \right)$$. Consider two statements

(I) g is an increasing function in (0, 1)

(II) g is one-one in (0, 1)

Then,

A
Both (I) and (II) are true
B
Neither (I) nor (II) is true
C
Only (II) is true
D
Only (I) is true
4
JEE Main 2023 (Online) 25th January Morning Shift
Numerical
+4
-1
Change Language

The constant term in the expansion of $${\left( {2x + {1 \over {{x^7}}} + 3{x^2}} \right)^5}$$ is ___________.

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