Arts (English Medium) कक्षा १२ - CBSE Question Bank Solutions for Mathematics

Find the direction cosines of the following vectors:
\[3 \hat{i} - 4 \hat{k}\]

Find the angles at which the following vectors are inclined to each of the coordinate axes:
\[\hat{i} - \hat{j} + \hat{k}\]

Find the angles at which the following vectors are inclined to each of the coordinate axes:
\[\hat{j} - \hat{k}\]

Find the angles at which the following vectors are inclined to each of the coordinate axes:
\[4 \hat{i} + 8 \hat{j} + \hat{k}\]

Show that the vector \[\hat{i} + \hat{j} + \hat{k}\] is equally inclined with the axes OX, OY and OZ.

Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ are \[\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} .\]

If a unit vector \[\vec{a}\] makes an angle \[\frac{\pi}{3}\] with \[\hat{i} , \frac{\pi}{4}\] with \[\hat{j}\]  and an acute angle θ with \[\hat{k}\], then find θ and hence, the components of \[\vec{a}\].

Write the direction cosines of the vector \[\overrightarrow{r} = 6 \hat{i} - 2 \hat{j} + 3 \hat{k} .\]

A unit vector \[\overrightarrow{r}\] makes angles \[\frac{\pi}{3}\] and \[\frac{\pi}{2}\] with \[\hat{j}\text{ and }\hat{k}\]  respectively and an acute angle θ with \[\hat{i}\]. Find θ.

What is the cosine of the angle which the vector \[\sqrt{2} \hat{i} + \hat{j} + \hat{k}\] makes with y-axis?

Write the direction cosines of the vector \[\hat{i} + 2 \hat{j} + 3 \hat{k}\].

Write the direction cosines of the vectors \[- 2 \hat{i} + \hat{j} - 5 \hat{k}\].

The vector cos α cos β \[\hat{i}\] + cos α sin β \[\hat{j}\] + sin α \[\hat{k}\] is a

Show that the function f given by f(x) = tan^–1 (sin x + cos x) is decreasing for all \[x \in \left( \frac{\pi}{4}, \frac{\pi}{2} \right) .\]

If A and B are square matrices of order 3 such that |A| = –1, |B| = 3, then find the value of |2AB|.

The radius r of a right circular cylinder is increasing uniformly at the rate of 0·3 cm/s and its height h is decreasing at the rate of 0·4 cm/s. When r = 3·5 cm and h = 7 cm, find the rate of change of the curved surface area of the cylinder. \[\left[ \text{ Use } \pi = \frac{22}{7} \right]\]

Find the general solution of the differential equation \[x \cos \left( \frac{y}{x} \right)\frac{dy}{dx} = y \cos\left( \frac{y}{x} \right) + x .\]

Using integration, find the area of the region {(x, y) : x² + y² ≤ 1 ≤ x + y}.

Let A = ℝ − {3}, B = ℝ − {1}. Let f : A → B be defined by \[f\left( x \right) = \frac{x - 2}{x - 3}, \forall x \in A\] Show that f is bijective. Also, find
(i) x, if f⁻¹(x) = 4
(ii) f⁻¹(7)