Science (English Medium) Class 12 - CBSE Question Bank Solutions for Mathematics

For the function f (x) = x + \[\frac{1}{x}\] ∈ [1, 3], the value of c for the Lagrange's mean value theorem is 

If from Lagrange's mean value theorem, we have \[f' \left( x_1 \right) = \frac{f' \left( b \right) - f \left( a \right)}{b - a}, \text { then }\]

Rolle's theorem is applicable in case of ϕ (x) = a^sin x, a > a in

The value of c in Rolle's theorem when
f (x) = 2x³ − 5x² − 4x + 3, x ∈ [1/3, 3] is

When the tangent to the curve y = x log x is parallel to the chord joining the points (1, 0) and (e, e), the value of x is ______.

The value of c in Rolle's theorem for the function \[f\left( x \right) = \frac{x\left( x + 1 \right)}{e^x}\] defined on [−1, 0] is

The value of c in Lagrange's mean value theorem for the function f (x) = x (x − 2) when x ∈ [1, 2] is

The value of c in Rolle's theorem for the function f (x) = x³ − 3x in the interval [0,\[\sqrt{3}\]] is 

If f (x) = e^x sin x in [0, π], then c in Rolle's theorem is

Find the points on the curve x² + y² − 2x − 3 = 0 at which the tangents are parallel to the x-axis ?

Find the angle between the line \[\vec{r} = \left( 2 \hat{i}+ 3 \hat {j}  + 9 \hat{k}  \right) + \lambda\left( 2 \hat{i} + 3 \hat{j}  + 4 \hat{k}  \right)\]  and the plane  \[\vec{r} \cdot \left( \hat{i}  + \hat{j}  + \hat{k}  \right) = 5 .\]

Find the angle between the line \[\frac{x - 1}{1} = \frac{y - 2}{- 1} = \frac{z + 1}{1}\]  and the plane 2x + y − z = 4.

The line  \[\vec{r} = \hat{i} + \lambda\left( 2 \hat{i} - m \hat{j}  - 3 \hat{k}  \right)\]  is parallel to the plane  \[\vec{r} \cdot \left( m \hat{i}  + 3 \hat{j}  + \hat{k}  \right) = 4 .\] Find m. 

Show that the line whose vector equation is \[\vec{r} = 2 \hat{i}  + 5 \hat{j} + 7 \hat{k}+ \lambda\left( \hat{i}  + 3 \hat{j}  + 4 \hat{k}  \right)\] is parallel to the plane whose vector  \[\vec{r} \cdot \left( \hat{i} + \hat{j}  - \hat{k}  \right) = 7 .\]  Also, find the distance between them.

Find the angle between the line \[\frac{x - 2}{3} = \frac{y + 1}{- 1} = \frac{z - 3}{2}\] and the plane

3x + 4y + z + 5 = 0.

State when the line \[\vec{r} = \vec{a} + \lambda \vec{b}\]  is parallel to the plane  \[\vec{r} \cdot \vec{n} = d .\]Show that the line  \[\vec{r} = \hat{i}  + \hat{j}  + \lambda\left( 3 \hat{i}  - \hat{j}  + 2 \hat{k}  \right)\]  is parallel to the plane  \[\vec{r} \cdot \left( 2 \hat{j} + \hat{k} \right) = 3 .\]   Also, find the distance between the line and the plane.

Show that the plane whose vector equation is \[\vec{r} \cdot \left( \hat{i}  + 2 \hat{j}  - \hat{k}  \right) = 1\] and the line whose vector equation is  \[\vec{r} = \left( - \hat{i}  + \hat{j} + \hat{k}  \right) + \lambda\left( 2 \hat{i}  + \hat{j}  + 4 \hat{k}  \right)\]   are parallel. Also, find the distance between them. 

Find the angle between the line

Write the angle between the line \[\frac{x - 1}{2} = \frac{y - 2}{1} = \frac{z + 3}{- 2}\]  and the plane x + y + 4 = 0.