Linear Inequalities mcqs

Question 1 
If ∣∣3x−44∣∣≤512, the solution set is:
(a) {x: 1918 ≤ x ≤ 2918}
(b) {x: 79 ≤ x ≤ 179}
(c) {x: −2918 ≤ x ≤ −1918}
(d) None of these [1 Mark, Feb. 2007]
Answer:
(b) is correct

Question 2 On solving the inequalities 6x + y ≥ 18; x + 4y ≥ 12; 2x + y ≥ 10, we get the following situation: 

(а) (0, 18), (12,0), (4, 2) & (7, 6)
(b) (3, 0), (0,3), (4, 2), & (7, 6)
(c) (5, 0), (0, 10), (4, 2) & (7, 6)
(d) (0,18), (12, 0), (4, 2), (0,0) and (7, 6)
Answer:
(a) For 6x + y = 18

Point are (3 ; 0); (2 ; 6)

For x + 4y = 12

Point are (0 ; 3) ; (4 ; 2)
and 2x = y = 10

Points are (3; 4) & (4; 2)
Solving eqn. 6x + y = 18 & 2x + y = 10
Subtracting 4x = 8 x = 2
Putting x = 2 in 2x + y = 10
we get
2 × 2 + y = 10
∴ y = 6
Point is (2; 6)
(a) is correct
Tricks : Go by choices
Point (0; 8) satisfy eqn. 6x + y = 18
Point (12 ; 0) satisfies eqn x + 4y = 12
Point (4 ; 2) satisfies eqns x + 4y = 12 and 2x + y = 10
Point (2 ; 6) satisfies eqns 6x + y = 18 and 2x + y = 10
(a) is Correct

Question 3.
A car manufacturing company manufactures cars of two types A and B. Model A requires 150 man-hours for assembling, 50 man-hours for painting and 10 man¬hours for checking and testing. Model B requires 60 man-hours for assembling, 40 man-hours for painting and 20 man-hours for checking and testing. There are available 30 thousand man-hours for assembling, 13 thousand man-hours for painting and 5 thousand man-hours for checking and testing. Express the above situation using linear inequalities. Let the company manufacture x units of type A model of car and y units of type B model of car.
Then, the inequalities are :
(a) 5x + 2y ≥ 1000; 5x + 4y ≥ 1300, x + 2y ≤ 500; x ≥ 0, y ≥ 0.
(b) 5x + 2y ≤ 1000, 5x + 4y ≤ 1300, x + 2y ≥ 500; x ≥ 0, y ≥ 0.
(c) 5x + 2y ≤ 1,000, 5x + 4y ≤ 1300, x + 2y ≤ 500; x ≥ 0, y ≥ 0.
(d) 5x + 2y = 1000, 5x + 4y ≥ 1300, x + 2y = 500 ; x ≥ 0, y ≥ 0.
Answer:
Models

Ineqns. are
[150x + 60y ≤ 30,000] ÷ 30 ⇒ 5x + 2y ≤ 1,000
[50x+40y ≤ 13000] ÷ 10 ⇒ 5x + 4y ≤ 1300
[10x + 20y ≤ 5000] ÷ 10 ⇒ x + 2y ≤ 500
x ≥ 0 & y ≥ 0
(b) is correct

Question 4.
The rules and regulations demand that the employer should employ not more than 5 experienced hands to 1 fresh one and this fact is represented by : (Taking experienced person as x and fresh person as y) 
(a) y ≥ x5
(b) 5y ≤ x
(c) 5y ≥ x
(d) None.
Answer:
(a) & (c)
1 Fresh with 5 experienced maximum employees.
y Fresh with 5y experienced maximum employees.
From Question
x ≤ 5y ⇒ 5y ≥ x, OR, y ≥ x/5
(a) & (c) are correct.

Question 5.

The Linear relationship between two variables in an inequality :
(a) ax + by ≤ c
(b) ax. by ≤ c
(c) axy + by ≤ c
(d) ax + bxy ≤ c
Answer:
Linear eqn is ax + by = c
∴ (a) option is correct

Question 6
The solution of the inequality (5−2x)3≤x6 – 5 is: 
(a) x ≥ 8
(b) x ≤ 8
(c) x = 8
(d) none of these
Answer:
(a) is correct.

or 10 – 4x ≤ x – 30
or 10 + 30 ≤ x + 4x
or 5x ≥ 40
or x ≥ 8
option (a) is correct

Question 7.
Solution space of inequalities 2x + y ≤ 10 and x – y ≤ 5: 
(i) includes the origin.
(ii) includes the point (4,3) which one is correct? i
(a) Only (i) (b) Only (ii)
(c) both (i) and (ii)
(d) none of the above
Answer:
(a) is correct ;
Tricks : Go by choices
(0, 0) satisfies both ineqns. but (4; 3) does not satisfy 1st
(a) is correct

Question 8.
On the average,experienced person does 5 units work while a fresh one 3 units work daily but the employer have to maintain the output of atleast 30 units of work per day. 
The situation can be expressed as.
(a) 5x + 3y ≤ 30
(b) 5x + 3y ≥ 30
(c) 5x + 3y = 30
(d) None of these
Answer:
(b) Let No. of experienced persons = x and No. of Freshers = y
∴ 5x + 3y ≥ 30

Question 9.
Find the range of real of x satisfying the inequalities 3x – 2 > 7 and 4x – 13 > 15. 
(a) x > 3
(b) x > 7
(c) x < 7
(d) x < 3 Answer: (b) is correct. 3x – 2 > 7 ⇒ 3x > 9 ∴ x > 3 ………..(1)
4x > 15 + 13 ⇒ 4x > 28 ∴ x >7 …………(2)
Clearly From (1) and (2); x > 7 satisfies both
(b) is correct.

Question 10.
The union forbids the employer to employ less than 2 experienced person (x) to each fresh person (y),This situation can be expressed as: 
(a) x ≤ y/2
(b) y ≤ x/2
(c) y ≥ x/2
(d) none
Answer:
(b) is correct
No. of Fresh persons for x Experienced person = x2
x2 ≥ y (given) ∴ y ≤ x2

Question 11.
The solution of the inequality
8x + 6 < 12x + 14 is
(a) (-2, 2)
(b) (-2, 0)
(c) (2, ∞)
(d) (-2, ∞)
Answer:
(d) is correct
8x + 6 < 12x + 14
or – 8 < 4x
or -2 < x x > -2
∴ Soln. is (-2; ∞)

Question 12.
Which of the following graph represents the in equality x + y ≤ 6 is 

(d) None of these
Answer:
(a) is correct. The graphical representation of x + y ≤ 6 is as follows :

Question 13.
By lines x + y = 6,2x – y = 2, the common region shown is the diagram refers to: 
(a) x + y ≥ 6, 2x – y ≤ 2, x ≥ 0, y ≥ 0
(b) x + y ≤ 6, 2x – y ≤ 2, x ≥ 0, y ≥ 0
(c) x + y ≤ 6, 2x – y > 2, x ≥ 0, y ≥ 0
(d) None of these
Answer:
(b) is correct
Tricks: Go by choices
A point (1,1) (let) satisfies all inequations of (b).

Question 14.
A dietitian wishes to mix together two kinds of food so that the vitamin content of the mixture is atleast 9 units of vitamin A, 7 units of vitamin B, 10 units of vitamin C and 12 units of vitaminD. The vitamin content per kg. of each food is shown below:

Assuming x kgs of food I is to be mixed with y kgs of food II the situation can be expressed as: 
(a) 2x + y ≤ 9; x + y ≤ 7; x + 2y ≤ 10; 2x + 3y ≤ 12 ; x ≥ 0, y ≥ 0
(b) 2x + y ≥ 30; x + y ≤ 7; x + 2y ≥ 10; x + 3y ≥ 12; x ≥ 0; y ≥ 0
(c) 2x + y ≥ 9; x + y ≤ 7; x + y ≤ 10; x + 3y ≥ 12; x ≥ 0, y ≥ 0
(d) 2x + y ≥ 9; x + y ≥ 7; x + 2y ≥ 10; 2x + 3y ≥ 12; x ≥ 0; y ≥ 0
Answer:
Atleast → Minimum
So, use > Sign here.
Constraints are:
2x + y ≥ 9;
x + y ≥ 1
x + 2y ≥ 1
2x + 3y ≥ 12
(d) is correct.

Question 15.
The linear relationship between two variables in an inequality: 
(a) ax + by ≤ c
(b) ax.by ≤ c
(c) axy + by ≤ c
(d) ax + bxy ≤ c
Answer:
(a)
Standard form of Linear Eqn. is
ax + by = c.
So; ax + by ≤ c is a Linear Ineqn.

Question 16.
On Solving the Inequalities 5x + y ≤ 100, x + y ≤ 60, x ≥ 0, y ≥ 0, we get the following situation: 
(a) (0, 0), (20, 0), (10, 50) & (0, 60)
(b) (0, 0), (60, 0), (10, 50) & (0, 60)
(c) (0, 0), (20, 0), (0,100) & (10, 50)
(d) None of these
Answer:
(a)
Tricks : Go by choices

Question 17.
An employer recruits experienced (x) and fresh workmen (y) under the condition that he cannot employ more than 11 people, x and y can be related by the inequality: 
(a) x + y ≠ 11 ;
(b) x + y ≤ 11, x ≥ 0, y ≥ 0
(c) x + y ≥ 11, x ≥ 0, y ≥ 0
(d) None of these
Answer:
(b)
Clearly x + y ≤ 11.
and x ; y > 0.

Question 18.
The solution set of the inequations x + 2 > 0 and 2x – 6 > 0 is
(a) (- 2 , ∞);
(b) ( 3 , ∞)
(c) (- ∞ , – 2 )
(d) (-∞, – 3)
Answer:
∵ x + 2 > 0 ⇒ x > -2
and 2x – 6 > 0 ⇒ x > 3
⇒ x = {-1; 0, 1, 2, 3, 4,………..} (1)
and 2x – 6 > 0 ⇒ x > 3
⇒ x = {4 ; 5 ; 6 ; 7 ;………} (2)
From (1) and (2); we get x = {4, 5, 6, ………}satisfies both conditions.
∴ Solution Set = (3; ∞)