Linear Equations Model Questions Set 1 Practice Questions Answers Test With Solutions & More Shortcuts

LINEAR EQUATIONS PRACTICE TEST [1 - EXERCISES]

Question : 1

The value of k, for which the system of equation 3x – ky – 20 = 0 and 6x – 10y + 40 = 0 has no solution, is

a) 6

b) 5

c) 10

d) 3

Given that system of equations has no solution

⇒ |A| = 0

|A| = – 30 + 6 k

But |A| = 0

⇒ 6k – 30 = 0

⇒ k = 5

∴ Option (b) is correct.

Question : 2

It is given that the equations \$x^2 – y^2\$ = 0 and \$(x – a)^2 + y^2\$ = 1 have single positive solution. For this, the value of 'a' is

a) 2

b) - \$√2\$

c) \$√2\$

d) 1

\$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\$

Question : 3

If 65x – 33y = 97 and 33x – 65y = 1, then what is xy equal to?

a) 3

b) –2

c) 2

d) –3

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

Question : 4

The sum of digits of a two-digit number is 8 and the difference between the number and that formed by reversing the digits is 18. What is the difference between the digits of the number?

a) 2

b) 3

c) 1

d) 4

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.

Question : 5

If 2x + 3y = 87 and 3x – 3y = 48, what is the value of x?

a) 29

b) 11

c) 72

d) 28

e) None of these

 2x + 3y = 87 3x - 3y = 48 5x = 135

∴ x = \${135}/5\$ = 27

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