Mathematics

Linear Equations in Two Variables

Question:

if x = 1 2 3 , find the value of x 3 2x2 7x + 5

Answer:

We have,

x = 1 2 3 = 1 2 3 × 2 + 3 2 + 3 = 2 + 3 22 ( 3 )2 = 2 + 3 4 3 = 2 + 3

=> x 2 = 3

=> (x 2) 2 = ( 3 )2

=> x 2 4 x+4 = 3 => x 2 - 4x + 1 = 0

x 3 2 x 27x +5 = x (x 2 - 4x + 1 )+ 2(x 2 - 4x + 1 )+ 3 = x × 0 + 2 × 0 + 3 = 3
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Linear Equations in Two Variables

Q 1.

Rationalise the denominator of :
2 3

Q 2.

if a = 2 5 2 + 5 and b = 2 + 5 2 5 , find a2 b2

Q 3.

If both a and b are rational numbers, find the values of a and b in each of the following equalities :

(i) 3 1 3 + 1 = a + b 3 (ii) 3 + 7 3 7 = a + b 7 (iii) 5 + 2 3 7 + 4 3 = a + b 3

(iv) 5 + 3 7 4 3 = 47 a + 3 b (v) 5 + 3 5 3 = a + b 15 (vi) 2 + 3 3 2 2 3 = a b 6

Q 4.

Simplify:
(i) 5 + 2 2 [NCERT] (ii) 11 - 5 2

Q 5.

Find the value to three places of decimals, of each of the following: It is given that 2 = 1.414 , 3 = 1.732, 10 = 3.162 and 5 = 2.236 (approx.)

(i) 1 2 (ii) 1 3 (iii) 1 10

Q 6.

Simplify:
(i) 5 + 5 5 - 5 (ii) 3 + 2 2 3 - 2 2

Q 7.

Simply the following expression:
(i) 3 + 3 2 + 2 [NCERT] (ii) 5 + 7 2 + 5

Q 8.

Rationalise the denominator of : 1 3 + 2

Q 9.

Prove that :

1 1 + 2 + 1 2 + 3 + 1 3 + 4 + 1 4 + 5 + 1 5 + 6 + 1 6 + 7 + 1 7 + 8 + 1 8 + 9 = 2

Q 10.

if x = 1 2 3 , find the value of x 3 2x2 7x + 5

Q 11.

Rationalise the denominator in each of the following:

(i) 2 7 [NCERT] (ii) 2 3 3 (iii) 2 7 11

Q 12.

If 2 = 1.414 , find the value of 3 ÷ 6 upto three places of decimals.

Q 13.

if x = 3 + 2 3 2 and y = 3 2 3 + 2 , find x2 + y2

Q 14.

Simplify each of the following :

(i) 3 5 3 + 2 5 + 3 (ii) 4 + 5 4 5 + 4 5 4 + 5 (iii) 5 2 5 + 2 5 + 2 5 2

Q 15.

Prove that :

= 1 3 8 1 8 7 + 1 7 6 1 6 5 + 1 5 2 = 5

Q 16.

Simplify each of the following by rationalising the denominator:

(i) 1 5 + 2 (ii) 5 + 6 5 - 6 (iii) 7 + 3 5 7 - 3 5 (iv) 2 3 5 2 3 + 3 3

Q 17.

If x = 2 + 3 , find the value of x2 + 1 x2

Q 18.

Given that 5 = 2.236 approximately, find to three places of decimals the value of 2 5

Q 19.

Simplify each of the following by rationalising the denominator:

(i) 1 5 + 2 (ii) 5 + 6 5 - 6 (iii) 7 + 3 5 7 - 3 5 (iv) 2 3 5 2 3 + 3 3

Q 20.

Rationalise the denominator of : 1 7 + 3 2

Q 21.

Simply each of the following: (i) 5 16 × 5 2 (ii) 4 243 4 3

Q 22.

Rationalise the denominator of : 5 3 - 5

Q 23.

If x = 3 - 2 2 , find the value of x2 + 1 x2

Q 24.

If x = 1 2 , find the value of x 1 x 3

Q 25.

Evaluate 15 10 + 20 + 40 - 5 - 80 , is being given that 5 = 2.236 and 10 = 3.162

Q 26.

Find the value to three places of decimals, of each of the following:
It is given that 2 = 1.414 , 3 = 1.732, 10 = 3.162 and 5 = 2.236 (approx.).
(i) 2 + 1 5 (ii) 2 - 3 3 (iii) 10 - 5 2