1
JEE Advanced 2023 Paper 1 Online
Numerical
+4
-0
Let $P$ be the plane $\sqrt{3} x+2 y+3 z=16$ and let $S=\left\{\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}: \alpha^2+\beta^2+\gamma^2=1\right.$ and the distance of $(\alpha, \beta, \gamma)$ from the plane $P$ is $\left.\frac{7}{2}\right\}$ Let $\vec{u}, \vec{v}$ and $\vec{w}$ be three distinct vectors in $S$ such that $|\vec{u}-\vec{v}|=|\vec{v}-\vec{w}|=|\vec{w}-\vec{u}|$. Let $V$ be the volume of the parallelepiped determined by vectors $\vec{u}, \vec{v}$ and $\vec{w}$. Then the value of $\frac{80}{\sqrt{3}} V$ is :
2
JEE Advanced 2021 Paper 1 Online
Numerical
+2
-0 Let $$\alpha$$, $$\beta$$ and $$\gamma$$ be real numbers such that the system of linear equations

x + 2y + 3z = $$\alpha$$

4x + 5y + 6z = $$\beta$$

7x + 8y + 9z = $$\gamma$$ $$-$$ 1

is consistent. Let | M | represent the determinant of the matrix

$$M = \left[ {\matrix{ \alpha & 2 & \gamma \cr \beta & 1 & 0 \cr { - 1} & 0 & 1 \cr } } \right]$$

Let P be the plane containing all those ($$\alpha$$, $$\beta$$, $$\gamma$$) for which the above system of linear equations is consistent, and D be the square of the distance of the point (0, 1, 0) from the plane P.

The value of | M | is _________.
3
JEE Advanced 2021 Paper 1 Online
Numerical
+2
-0 Let $$\alpha$$, $$\beta$$ and $$\gamma$$ be real numbers such that the system of linear equations

x + 2y + 3z = $$\alpha$$

4x + 5y + 6z = $$\beta$$

7x + 8y + 9z = $$\gamma$$ $$-$$ 1

is consistent. Let | M | represent the determinant of the matrix

$$M = \left[ {\matrix{ \alpha & 2 & \gamma \cr \beta & 1 & 0 \cr { - 1} & 0 & 1 \cr } } \right]$$

Let P be the plane containing all those ($$\alpha$$, $$\beta$$, $$\gamma$$) for which the above system of linear equations is consistent, and D be the square of the distance of the point (0, 1, 0) from the plane P.

The value of D is _________.
4
JEE Advanced 2021 Paper 1 Online
Numerical
+4
-0 Let $$\overrightarrow u$$, $$\overrightarrow v$$ and $$\overrightarrow w$$ be vectors in three-dimensional space, where $$\overrightarrow u$$ and $$\overrightarrow v$$ are unit vectors which are not perpendicular to each other and $$\overrightarrow u$$ . $$\overrightarrow w$$ = 1, $$\overrightarrow v$$ . $$\overrightarrow w$$ = 1, $$\overrightarrow w$$ . $$\overrightarrow w$$ = 4

If the volume of the paralleopiped, whose adjacent sides are represented by the vectors, $$\overrightarrow u$$, $$\overrightarrow v$$ and $$\overrightarrow w$$, is $$\sqrt 2$$, then the value of $$\left| {3\overrightarrow u + 5\overrightarrow v } \right|$$ is ___________.
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