Linear Equations Model Questions Set 1 Section-Wise Topic Notes With Detailed Explanation And Example Questions

Answer: (a)

Let the positive number be x.

According to the question,

x + 10 = ${200}/x$

⇒ $x^2$ + 10x = 200

⇒ $x^2$ + 10x – 200 = 0

⇒ (x – 10) (x + 20) = 0

∴ x = 10, –20

But x ≠ –20, since x is a positive number

The required number is 10.

Answer: (a)

Since a, b are negative numbers.

a < 0 and b < 0

C is a positive real number

⇒ c > 0

(1) a – b < a – c

⇒ – b < – c

⇒ b > c

It is not true as b < c

(2) if a < b ⇒ $a/c < b/c$ this is true.

(3) $1/b < 1/c$

Since c > b ⇒ $1/c < 1/b$ this is not true.

∴ option (a) is correct.

Answer: (b)

Let the two-digit number be 10y + x.

According to question,

x + y = 10 ... (i)

and 10y + x – 18 = 10x + y

⇒9x – 9y = –18

⇒x – y = –2 ... (ii)

On solving equations (i) and (ii), we get

x = 4 and y = 6

∴ Required product = xy = 4 × 6 = 24.

Answer: (a)

Let x be the first digit and y be the second digit of two digit number.

According to question,

x + y = 8 ... (i)

(10x + y) – (10y + x) = 18

⇒9x – 9y = 18

⇒x – y = 2 ... (ii)

Adding (i) and (ii),

x + y = 8

x - y = 2

We get

2x = 10

⇒ x = 5

⇒ y = 8 – 5 = 3

⇒ x = 5 and y = 3

∴ Required difference of digits, x – y

= 5 – 3 = 2.

Answer: (c)

65x – 33y = 97 .... (i)

(given)

33x – 65y = 1 ....(ii)

(given)

From eq (i) + eq (ii)

98x – 98y = 98

98(x – y) = 98 ∴ x – y = 1 ....(iii)

Now

From eq (i) × 1 – eq (iii) × 33

65x - 33y = 97

- 33x + 33y = - 33

32x = 64

∴ $x = 64/32$ = 2

From eq (iii)

x – y = 1

2 – y = 1

∴ y = 2 – 1 = 1

Hence, xy = 2 × 1 = 2

Answer: (c)

$x^2 – y^2$ = 0

(Equation of y = |x|)

$(x – a)^2 + y^2$ = 1

[Equation of a circle with centre (a, 0) and radius 1]

These have single positive solution

⇒ y = |x| to be tangent to the circle

AB = radius of circle

∠AOB = 45°

∠OAB = 90°

[tangent to the circle]

∠ABO = 45°

Length of OB = $√{AB^2 + OA^2} = √{1^2 + 1^2}$

OB = $√2$