## NCERT Solution Class XI Mathematics Linear Inequalities Question 4 (Ex 6.1)

Question 4:

Solve 3x + 8 > 2, when

(i) x is an integer (ii) x is a real number

The given inequality is 3x + 8 > 2.

3x + 8 > 2

⇒ 3x + 8 – 8 > 2 – 8

⇒ 3x > −6

## NCERT Solution Class XI Mathematics Linear Inequalities Question 3 (Ex 6.1)

Question 3:

Solve 5x– 3 < 7, when

(i) x is an integer (ii) x is a real number

The given inequality is 5x– 3 < 7.

−12x > 30

⇒ 5x – 3 + 3 < 7 + 3

⇒ 5x < 10

(i) The integers less than 2 are …, –4, –3, –2, –1, 0, 1.

Thus, when x is an integer, the solutions of the given inequality are …, –4, –3, –2, –1, 0, 1.

Hence, in this case, the solution set is {…, –4, –3, –2, –1, 0, 1}.

(ii) When x is a real number, the solutions of the given inequality are given by x < 2,

that is, all real numbers x which are less than 2.

Thus, the solution set of the given inequality is x ∈ (–∞, 2).

## NCERT Solution Class XI Mathematics Linear Inequalities Question 2 (Ex 6.1)

Question 2:

Solve –12x > 30, when

(i) x is a natural number (ii) x is an integer

The given inequality is –12x > 30.

−12x > 30

(i) There is no natural number less than (−5/2).

Thus, when x is a natural number, there is no solution of the given inequality.

(ii) The integers less than (−5/2) are …, –5, –4, –3.

Thus, when x is an integer, the solutions of the given inequality are …, –5, –4, –3.

Hence, in this case, the solution set is {…, –5, –4, –3}.

## NCERT Solution Class XI Mathematics Linear Inequalities Question 1 (Ex 6.1)

Question 1:

Solve 24x < 100, when (i) x is a natural number (ii) x is an integer

The given inequality is 24x < 100

24x < 100

(i) It is evident that 1, 2, 3, and 4 are the only natural numbers less than 25/6.

Thus, when x is a natural number, the solutions of the given inequality are 1, 2, 3, and 4.

Hence, in this case, the solution set is {1, 2, 3, 4}.

(ii) The integers less than 25/6 are … are …–3, –2, –1, 0, 1, 2, 3, 4.

Thus, when x is an integer, the solutions of the given inequality are …–3, –2, –1, 0, 1, 2, 3, 4.

Hence, in this case, the solution set is {…–3, –2, –1, 0, 1, 2, 3, 4}.

## NCERT Solution Class XI Mathematics Complex Numbers and Quadratic Equations Question 20 (Mis Ex)

Question 20:

∴ m = 4k, where k is some integer.

Therefore, the least positive integer is 1.

Thus, least positive integral value of m is 4(= 4 ×1).

## NCERT Solution Class XI Mathematics Complex Numbers and Quadratic Equations Question 19 (Mis Ex)

Question 19:

If (a + ib) (c + id) (e + if) (g + ih) = A + iB, then show that (a2 + b2) (c2 + d2) (e2 + f2) (g2 + h2) = A2 + B2.

(a + ib) (c + id) (e + if) (g + ih) =A + IB’

∴ |(a + ib) (c + id) (e + if) (g + ih)| = |A + iB|

⇒ |(a + ib) ×|(c + id)| × |(e + if | × |(g + ih)| = |A + iB|   [|z1z2| = |z1||z2|]

On squaring both sides, we obtain

(a2 + b2) (c2 + d2) (e2 + f2) (g2 + h2) = A2 + B2

Hence, proved.

## NCERT Solution Class XI Mathematics Complex Numbers and Quadratic Equations Question 18 (Mis Ex)

Question 18:

Find the number of non-zero integral solutions of the equation |1 –i|x = 2x It is given  that, |β| = 1

|1 – i|x = 2x

Thus, 0 is the only integral solution of the given equation. Therefore, the number of non-zero integral solutions of the given equation is 0.

## NCERT Solution Class XI Mathematics Complex Numbers and Quadratic Equations Question 17 (Mis Ex)

Question 17:

If α and β are different complex numbers with |β| = 1, then find

Let a = a + ib and β = x + iy

It is given  that, |β| = 1

## NCERT Solution Class XI Mathematics Complex Numbers and Quadratic Equations Question 16 (Mis Ex)

Question 16:

(x + iy)3 = u + iv

⇒ x3 + (iy)3 + 3 ∙ x ∙ iy(x + iy) = u + iv

⇒ x3 + i3y3 + 3x2yi + 3xy2i2 = u + iv

⇒ x3 – iy3 + 3x2­yi− 3xy2 = u + iv

⇒ (x3 – 3xy2) + i(3x2y – y3) = u + iv

On equating real and imaginary parts, we obtain

u = x3 – 3xy2, v = 3x2y – y3

Question 15: