Show that the relation R defined in the | Class 12 Mathematics Chapter Relations and Functions, Relations and Functions NCERT Solutions

Show that the function f : R_* → R_* defined by f(x) = 1/x is one-one and onto,where R_* is the set of all non-zero real numbers. Is the result true, if the domain R_* is replaced by N with co-domain being same as R_* ? 

 Let f : {1, 3, 4} → {1, 2, 5} and g : {1, 2, 5} → {1, 3} be given by f = {(1, 2), (3, 5), (4, 1)} and g = {(1, 3), (2, 3), (5, 1)}. Write down gof. 

 Prove that the Greatest Integer Function f : R → R, given by f(x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.

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

(A)  

(B) 

(C)  

(D)

 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. 

The rate of change of the area of a circle with respect to its radius r at r = 6 cm is

(A) 10π (B) 12π (C) 8π (D) 11π

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?

The total revenue in Rupees received from the sale of x units of a product is given by

R (x) = 13x² + 26x + 15

Find the marginal revenue when x = 7.

Maximise Z = 3x + 4y

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

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

In Figure, identify the following vectors.

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

Classify the following as scalar and vector quantities.

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

(iv) velocity (v) work done

The radius of a circle is increasing at the rate of 0.7 cm/s. What is the rate of increase of its circumference?

Maximise Z = 3x + 4y

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