1
JEE Main 2023 (Online) 10th April Evening Shift
Numerical
+4
-1
Change Language

Let $$\mathrm{S}$$ be the set of values of $$\lambda$$, for which the system of equations

$$6 \lambda x-3 y+3 z=4 \lambda^{2}$$,

$$2 x+6 \lambda y+4 z=1$$,

$$3 x+2 y+3 \lambda z=\lambda$$ has no solution. Then $$12 \sum_\limits{i \in S}|\lambda|$$ is equal to ___________.

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2
JEE Main 2023 (Online) 10th April Evening Shift
Numerical
+4
-1
Out of Syllabus
Change Language

Suppose $$a_{1}, a_{2}, 2, a_{3}, a_{4}$$ be in an arithmetico-geometric progression. If the common ratio of the corresponding geometric progression is 2 and the sum of all 5 terms of the arithmetico-geometric progression is $$\frac{49}{2}$$, then $$a_{4}$$ is equal to __________.

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3
JEE Main 2023 (Online) 10th April Evening Shift
Numerical
+4
-1
Change Language

If the domain of the function $$f(x)=\sec ^{-1}\left(\frac{2 x}{5 x+3}\right)$$ is $$[\alpha, \beta) \mathrm{U}(\gamma, \delta]$$, then $$|3 \alpha+10(\beta+\gamma)+21 \delta|$$ is equal to _________.

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4
JEE Main 2023 (Online) 10th April Evening Shift
Numerical
+4
-1
Out of Syllabus
Change Language

Let the quadratic curve passing through the point $$(-1,0)$$ and touching the line $$y=x$$ at $$(1,1)$$ be $$y=f(x)$$. Then the $$x$$-intercept of the normal to the curve at the point $$(\alpha, \alpha+1)$$ in the first quadrant is __________.

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