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JEE Main 2023 (Online) 29th January Morning Shift
Numerical
+4
-1
Change Language

Let $$f:\mathbb{R}\to\mathbb{R}$$ be a differentiable function that satisfies the relation $$f(x+y)=f(x)+f(y)-1,\forall x,y\in\mathbb{R}$$. If $$f'(0)=2$$, then $$|f(-2)|$$ is equal to ___________.

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2
JEE Main 2023 (Online) 29th January Morning Shift
Numerical
+4
-1
Change Language

Five digit numbers are formed using the digits 1, 2, 3, 5, 7 with repetitions and are written in descending order with serial numbers. For example, the number 77777 has serial number 1. Then the serial number of 35337 is ____________.

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3
JEE Main 2023 (Online) 29th January Morning Shift
Numerical
+4
-1
Out of Syllabus
Change Language

Let $$\overrightarrow a $$, $$\overrightarrow b $$ and $$\overrightarrow c $$ be three non-zero non-coplanar vectors. Let the position vectors of four points $$A,B,C$$ and $$D$$ be $$\overrightarrow a - \overrightarrow b + \overrightarrow c ,\lambda \overrightarrow a - 3\overrightarrow b + 4\overrightarrow c , - \overrightarrow a + 2\overrightarrow b - 3\overrightarrow c $$ and $$2\overrightarrow a - 4\overrightarrow b + 6\overrightarrow c $$ respectively. If $$\overrightarrow {AB} ,\overrightarrow {AC} $$ and $$\overrightarrow {AD} $$ are coplanar, then $$\lambda$$ is equal to __________.

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4
JEE Main 2023 (Online) 29th January Morning Shift
Numerical
+4
-1
Change Language

Let the co-ordinates of one vertex of $$\Delta ABC$$ be $$A(0,2,\alpha)$$ and the other two vertices lie on the line $${{x + \alpha } \over 5} = {{y - 1} \over 2} = {{z + 4} \over 3}$$. For $$\alpha \in \mathbb{Z}$$, if the area of $$\Delta ABC$$ is 21 sq. units and the line segment $$BC$$ has length $$2\sqrt{21}$$ units, then $$\alpha^2$$ is equal to ___________.

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