Class 12 Mathematics - Chapter Vector Algebra NCERT Solutions | Classify the following measures as scala

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 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

Represent graphically a displacement of 40 km, 30° east of north.

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).

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?

Letbe a function defined as. The inverse of f is map g: Range

(A)  

(B) 

(C)  

(D)

 If f: R → R be given by f(x) = , then fof(x) is
(A)  

(B)  x³

(C)  x

(D)  (3 – x³).

Let f: X → Y be an invertible function. Show that the inverse of f ^–1 is f, i.e., (f^–1)^–1 = f.

 Consider f : {1, 2, 3} → {a, b, c} given by f(1) = a, f(2) = b and f(3) = c. Find f ^–1 and show that (f ^–1)^–1 = f. 

 Let f : X → Y be an invertible function. Show that f has unique inverse.

(Hint: suppose g1 and g2 are two inverses of f. Then for all y ∈ Y, fog1(y) = 1Y(y) = fog2(y). Use one-one ness of f). 

Consider f : R₊ → [– 5, ∞) given by f(x) = 9x² + 6x – 5. Show that f is invertible
with .

 Consider f : R₊ → [4, ∞) given by f(x) = x² + 4. Show that f is invertible with the inverse f^–1 of f given by , where R₊ is the set of all non-negative real numbers.