1
JEE Main 2023 (Online) 1st February Morning Shift
+4
-1

Let $$S = \left\{ {x:x \in \mathbb{R}\,\mathrm{and}\,{{(\sqrt 3 + \sqrt 2 )}^{{x^2} - 4}} + {{(\sqrt 3 - \sqrt 2 )}^{{x^2} - 4}} = 10} \right\}$$. Then $$n(S)$$ is equal to

A
6
B
4
C
0
D
2
2
JEE Main 2023 (Online) 31st January Evening Shift
+4
-1
The equation $\mathrm{e}^{4 x}+8 \mathrm{e}^{3 x}+13 \mathrm{e}^{2 x}-8 \mathrm{e}^{x}+1=0, x \in \mathbb{R}$ has :
A
two solutions and both are negative
B
two solutions and only one of them is negative
C
four solutions two of which are negative
D
no solution
3
JEE Main 2023 (Online) 31st January Morning Shift
+4
-1

The number of real roots of the equation $$\sqrt{x^{2}-4 x+3}+\sqrt{x^{2}-9}=\sqrt{4 x^{2}-14 x+6}$$, is :

A
0
B
1
C
3
D
2
4
JEE Main 2023 (Online) 29th January Morning Shift
+4
-1

Let $$\lambda \ne 0$$ be a real number. Let $$\alpha,\beta$$ be the roots of the equation $$14{x^2} - 31x + 3\lambda = 0$$ and $$\alpha,\gamma$$ be the roots of the equation $$35{x^2} - 53x + 4\lambda = 0$$. Then $${{3\alpha } \over \beta }$$ and $${{4\alpha } \over \gamma }$$ are the roots of the equation

A
$$7{x^2} - 245x + 250 = 0$$
B
$$49{x^2} - 245x + 250 = 0$$
C
$$49{x^2} + 245x + 250 = 0$$
D
$$7{x^2} + 245x - 250 = 0$$
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