Commerce (English Medium) इयत्ता १२ - CBSE Question Bank Solutions

If \[f\left( a + b - x \right) = f\left( x \right)\] , then prove that

Write the equation of a plane which is at a distance of \[5\sqrt{3}\] units from origin and the normal to which is equally inclined to coordinate axes.

Find the distance of the point  \[2 \hat{i} - \hat{j} - 4 \hat{k}\]  from the plane  \[\vec{r} \cdot \left( 3 \hat{i}  - 4 \hat{j}  + 12 \hat{k}  \right) - 9 = 0 .\]

Show that the points \[\hat{i}  - \hat{j}  + 3 \hat{k}  \text{ and }  3 \hat{i}  + 3 \hat{j}  + 3 \hat{k} \] are equidistant from the plane \[\vec{r} \cdot \left( 5 \hat{i}  + 2 \hat{j}  - 7 \hat{k}  \right) + 9 = 0 .\]

Find the distance of the point (3, 3, 3) from the plane \[\vec{r} \cdot \left( 5 \hat{i}  + 2 \hat{j}  - 7k \right) + 9 = 0\]

Find the distance of the point (1, −2, 3) from the plane x − y + z = 5 measured parallel to the line whose direction cosines are proportional to 2, 3, −6.

If the product of the distances of the point (1, 1, 1) from the origin and the plane x − y + z+ λ = 0 be 5, find the value of λ.

Find the distance between the point (7, 2, 4) and the plane determined by the points A(2, 5, −3), B(−2, −3, 5) and C (5, 3, −3). 

Find the distance of the point (1, -2, 4) from plane passing throuhg the point (1, 2, 2) and perpendicular of the planes x - y + 2z = 3 and 2x - 2y + z + 12 = 0 

Find the distance between the parallel planes 2x − y + 3z − 4 = 0 and 6x − 3y + 9z + 13 = 0.

Find the distance between the planes \[\vec{r} \cdot \left( \hat{i}  + 2 \hat{j}  + 3 \hat{k}  \right) + 7 = 0 \text{ and } \vec{r} \cdot \left( 2 \hat{i}  + 4 \hat{j}  + 6 \hat{k}  \right) + 7 = 0 .\]

Evaluate : \[\int(3x - 2) \sqrt{x^2 + x + 1}dx\] .