Class 12 Mathematics - Chapter Vector Algebra NCERT Solutions | Represent graphically a displacement of

Answer the following as true or false.

 \begin{align}(i) \overrightarrow{a}\;  and\; \overrightarrow{-a}\; are\; collinear.\end{align}

(ii) Two collinear vectors are always equal in magnitude.

(iii) Two vectors having same magnitude are collinear.

(iv) Two collinear vectors having the same magnitude are equal.

Classify the following measures as scalars and vectors.

(i) 10 kg (ii) 2 metres north-west (iii) 40°

(iv) 40 watt (v) 10^–19 coulomb (vi) 20 m/s²

Classify the following as scalar and vector quantities.

(i) time period (ii) distance (iii) force

(iv) velocity (v) work done

In Figure, identify the following vectors.

(i) Coinitial (ii) Equal (iii) Collinear but not equal

Prove that the function f(x) = 5x – 3 is continuous at x = 0, at x = – 3 and at x = 5.

Determine order and degree(if defined) of differential equation \begin{align} \frac{d^4y}{dx^4}\;+\;\sin(y^m)\;=0\end{align}

If a line makes angles 90°, 135°, 45° with x, y and z-axes respectively, find its direction cosines.

Maximise Z = 3x + 4y

Subject to the constraints:x + y ≤ 4, x ≥ 0, y ≥ 0

Find the area of the region bounded by the curve y² = x and the lines x = 1, x = 4 and the x-axis.

Given that E and F are events such that P(E) = 0.6, P(F) = 0.3 and P(E ∩ F) = 0.2, find P (E|F) and P(F|E).

Given that E and F are events such that P(E) = 0.6, P(F) = 0.3 and P(E ∩ F) = 0.2, find P (E|F) and P(F|E).

An edge of a variable cube is increasing at the rate of 3 cm/s. How fast is the volume of the cube increasing when the edge is 10 cm long?

In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.

(i) f : R → R defined by f(x) = 3 – 4x

(ii) f : R → R defined by f(x) = 1 + x² 

Maximise Z = 3x + 4y

Subject to the constraints:x + y ≤ 4, x ≥ 0, y ≥ 0

Determine order and degree(if defined) of differential equation

\begin{align}\left(\frac{d^2y}{dx^2}\right)^2\;+\;cos\left(\frac{dy}{dx}\right)\;=\;0\end{align}

Determine order and degree(if defined) of differential equation yⁿ + (y')² + 2y =0

Determine order and degree(if defined) of differential equation \begin{align} \frac{d^4y}{dx^4}\;+\;\sin(y^m)\;=0\end{align}

State with reason whether following functions have inverse

(i) f : {1, 2, 3, 4} → {10} with

f  = {(1, 10), (2, 10), (3, 10), (4, 10)}

(ii) g : {5, 6, 7, 8} → {1, 2, 3, 4} with

g = {(5, 4), (6, 3), (7, 4), (8, 2)}

(iii) h : {2, 3, 4, 5} → {7, 9, 11, 13} with

h = {(2, 7), (3, 9), (4, 11), (5, 13)}