Inequalities – CA Foundation Maths Study Material

This Inequalities – CA Foundation Maths Study Material is designed strictly as per the latest syllabus and exam pattern.

Inequalities – CA Foundation Maths Study Material

Previous Year Exam Questions

Question 1.
Graphs of Inequalities are drawn below: [1 Mark, Nov. 2006]
Inequalities – CA Foundation Maths Study Material 1
L1: 5x + 3y = 30 L2: x + y = 9
L3: y = \(\frac{x}{2}\)
L1: y = \(\frac{x}{2}\)
The common region (shaded part) shown in the diagram refers to the inequalities :
(a) 5x + 3y ≤ 30
x + y ≤ 9
y ≤ \(\frac{1}{2}\)x
y ≤ x/2
x ≥ 0, y ≥ 0

(b) 5x + 3y ≥ 30
x + y ≤ 9
y ≥ x/3
y ≤ x/2
x ≥ 0, y ≥ 0

(c) 5x + 3y > 30
x + y ≥ 9
y ≤ x/3
y ≥ x/2
x ≥ 0, y ≥ 0.

(d) 5x + 3y > 30
x + y < 9 y ≥ 9 y ≤ x/2 x ≥ 0, y ≥ 0 Solution : (b) Tricks : Go by choices Take a point of the common region. Let the testing point is (5 ; 3) of the region. This point satisfies all given inequations of option (b). Question 2. If |x + \(\frac{1}{4}\)| > \(\frac{7}{4}\), then:
(a) x < \(\frac{-3}{2}\) or x > 2
(b) x < -2 or x > \(\frac{3}{2}\)
(c) -2 < x < \(\frac{3}{2}\) (d) None of these [1 Mark, Nov. 2006] Answer: (b) |x + \(\frac{1}{4}\)| > \(\frac{7}{4}\)
Inequalities – CA Foundation Maths Study Material 2

Question 3.
If \(\left|\frac{3 x-4}{4}\right| \leq \frac{5}{12}\), the solution set is:
(a) {x: \(\frac{19}{18}\) ≤ x ≤ \(\frac{29}{18}\)}
(b) {x: \(\frac{7}{9}\) ≤ x ≤ \(\frac{17}{9}\)}
(c) {x: \(\frac{-29}{18}\) ≤ x ≤ \(\frac{-19}{18}\)}
(d) None of these [1 Mark, Feb. 2007]
Answer:
(b) is correct.
Inequalities – CA Foundation Maths Study Material 3
(b) is correct

Inequalities – CA Foundation Maths Study Material

Question 4.
On solving the inequalities 6x + y ≥ 18; x + 4y ≥ 12; 2x + y ≥ 10, we get the following situation: [1 Mark, Feb. 2007]
(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

x 3 2
y 0 6

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

For x + 4y = 12

x 0 4
y 3 2

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

x 0 4
y 4 2

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 5.
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

Conditions A(x) B(y) Total
Assembly 150 man hrs 60 man hrs 30,000 man hrs
Painting 50 man hrs 40 man hrs 13,000 man hrs
Checking & Testing 10 man hrs 20 man hrs 5,000 man hrs

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 6.
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) [1 Mark, Aug. 2007]
(a) y ≥ \(\frac{x}{5}\)
(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 7.
The shaded region represents: [1 Mark, Aug. 2007]
Inequalities – CA Foundation Maths Study Material 4
(a) 3x + 2y ≤ 24, x + 2y ≥ 16, x + y ≤ 10x, x ≥ 0,y ≥ 0
(b) 3x+2y ≤ 24, x + 2y ≤ 16, x + y ≥ 10, x ≥ 0, y ≥ 0
(c) 3x + 2y ≤ 24, x + 2y ≤ 16, x + y ≤ 10, x ≥ 0, y ≥ 0
(d) None of these.
Answer:
(c) Tricks : Go by choices
Take a point (1,1) satisfies all inequations of option (c)
(c) is Correct

Question 8.
The shaded region represents: [1 Mark, Nov. 2007]
Inequalities – CA Foundation Maths Study Material 5
(a) 3x + 5y ≤ 15, 5x + 2y ≥ 10, x, y ≥ 0
(b) 3x + 5y ≤ 15, 5x + 2y ≤ 10, x, y ≥ 0
(c) 3x + 5y ≥ 15, 5x + 2y ≥ 10, x, y, ≥ 0
(d) None of these.
Answer:
(b) Tricks : Go by choices
Let point is (1,1) of the common region Point (1,1) satisfies all ineqns. of option (b)
option (b) is correct.

Inequalities – CA Foundation Maths Study Material

Question 9.
The shaded region represents : [1 Mark, June 2008]
Inequalities – CA Foundation Maths Study Material 6
(a) x + y > 6, 2x – y > 0
(b) x + y < 6, 2x – y > 0
(c) x + y > 6,2x – y < 0 (d) None of these. Answer: (a) Tricks : Go by choices Point (7 ; 1) satisfies all conditions of option (a) (a) is correct Question 10. If a > 0 and b < 0, it followings that: [1 Mark, June 2008] (a) \(\frac{1}{a}>\frac{1}{b}\)
(b) \(\frac{1}{a}<\frac{1}{b}\) (c) \(\frac{1}{a}=\frac{1}{b}\) (d) None of these Answer: (a) a > 0 ⇒ \(\frac{1}{a}\) > 0
b < 0 ⇒ \(\frac{1}{b}\) < 0 ∴ \(\frac{1}{a}>\frac{1}{b}\)
∴ (a) is correct

Question 11.
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 12.
The solution of the inequality \(\frac{(5-2 x)}{3} \leq \frac{x}{6}\) – 5 is: [1 Mark, June 2010]
(a) x ≥ 8
(b) x ≤ 8
(c) x = 8
(d) none of these
Answer:
(a) is correct.
Inequalities – CA Foundation Maths Study Material 7
or 10 – 4x ≤ x – 30
or 10 + 30 ≤ x + 4x
or 5x ≥ 40
or x ≥ 8
option (a) is correct

Question 13.
Solution space of inequalities 2x + y ≤ 10 and x – y ≤ 5: [1 Mark, June 2011]
(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 14.
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. [1 Mark, Dec. 2011 & 12]
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

Inequalities – CA Foundation Maths Study Material

Question 15.
Find the range of real of x satisfying the inequalities 3x – 2 > 7 and 4x – 13 > 15. [1 Mark, June 2012]
(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 16.
The shaded region represents : [1 Mark, Feb. 2008]
Inequalities – CA Foundation Maths Study Material 8
(a) x + y ≤ 5, x ≥ 2, y ≤ 1
(b) x + y ≤ 5, x ≥ 2, y ≥ 1
(c) x + y ≥ 5, x ≥ 2, y ≥ 1
(d) None of these
Answer:
Tricks: Go by choices, option (b)

Question 17.
The union forbids the employer to employ less than 2 experienced person (x) to each fresh person (y),This situation can be expressed as: [1 Mark, June 2013]
(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 = \(\frac{x}{2}\)
\(\frac{x}{2}\) ≥ y (given) ∴ y ≤ \(\frac{x}{2}\)

Question 18.
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 19.
The graph of linear inequalities
7x + 9y < 63; x + y > 1;
0 ≤ x ≤ 6 and 0 ≤ y ≤ 6 has been given below
Inequalities – CA Foundation Maths Study Material 9
(a) BCDB and DEFD
(b) Unbounded
(c) HFGH
(d) ABDFHKA
Answer:
(d) Clearly common region is ABDFHKA.

Question 20.
Which of the following graph represents the in equality x + y ≤ 6 is [1 Mark, Dec. 2014]
Inequalities – CA Foundation Maths Study Material 10
(d) None of these
Answer:
(a) is correct. The graphical representation of x + y ≤ 6 is as follows :
Inequalities – CA Foundation Maths Study Material 11

Question 21.
The graph of linear inequalities
x + y ≥ 5; x + y ≤ 5; 0 ≤ x ≤ 4 and 0 ≤ y ≥ 2 is given below:
Inequalities – CA Foundation Maths Study Material 12
The common region of the inequalities will be: [1 Mark, Dec. 2014]
(a) OABCEO
(b) ECDE
(c) Line Segment DC
(d) Line Segment BC
Answer:
(c)

Question 22.
The common region represented by the inequalities 2x + y ≥ 8, x + y ≥ 12, 3x + 2y ≤ 34 is: [1 Mark, June. 2015]
(a) Unbounded
(b) In feasible i
(c) Feasible and bounded
(d) Feasible and unbounded S
Answer:
(C) is correct.
Inequalities – CA Foundation Maths Study Material 13
Inequalities – CA Foundation Maths Study Material 14
clearly It is Feasible and bounded.

Question 23.
By lines x + y = 6,2x – y = 2, the common region shown is the diagram refers to: [1 Mark, Dec. 2015]
(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).

Inequalities – CA Foundation Maths Study Material

Question 24.
The common region of x + y ≤ 6; x + y ≥ 3, is for shown by shaded region: [1 Mark, June 2016]
Inequalities – CA Foundation Maths Study Material 15
(d) None of these
Answer:
(a) is correct.
Tricks : Go by choices.
Clearly a point of the common region of option (a) satisfy all given constraints x + y ≤ 6 & x + y ≥ 3.

Question 25.
The inequalities x1 + 2x2 < 5; x1 + x2 > 1; x1 > 0; x2 > 0 represents the region. [1 Mark, Dec. 2016]
Inequalities – CA Foundation Maths Study Material 16
Answer:
(a) is correct
Tricks : Go by choices.

Question 26.
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:
Inequalities – CA Foundation Maths Study Material 17
Assuming x kgs of food I is to be mixed with y kgs of food II the situation can be expressed as: [1 Mark, June 2017]
(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 27.
The shaped region represented by the inequalities
4x + 3y ≤ 60, y ≥ 2x, x ≥ 3, x ≥ 0, y ≥ 0
Inequalities – CA Foundation Maths Study Material 18
(d) None
Answer:
Tricks : Go by choices
Option (b) is correct.

Question 28.
In the following diagram, the region represented by the inequalities
x + 2y ≤ 10, x + y ≤ 6. x ≤ 4 & x ≥ 0, y ≥ 0 is: [1 Mark, June 2018]
Inequalities – CA Foundation Maths Study Material 19
(a) OADGO
(b) ADC
(c) ACD
(d) DEG
Answer:
(a)
Tricks : Go by choices

Question 29.
The linear relationship between two variables in an inequality: [1 Mark, May 2018]
(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.

Inequalities – CA Foundation Maths Study Material

Question 30.
On Solving the Inequalities 5x + y ≤ 100, x + y ≤ 60, x ≥ 0, y ≥ 0, we get the following situation: [1 Mark, Nov. 2018]
(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 31.
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: [1 Mark, June 2019]
(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 32.
The solution set of the inequations x + 2 > 0 and 2x – 6 > 0 is
(a) (- 2 , ∞);
(b) ( 3 , ∞)
(c) (- ∞ , – 2 )
(d) (-∞, – 3) [1 Mark, June 2019]
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; ∞)

Question 33.
The common region represented by the following inequalities
L1 = X1 + X2 < 4;
L2 = 2X1 – X2 > 6
Inequalities – CA Foundation Maths Study Material 20
(a) OABC;
(b) Outside of OAB
(c) ΔBCE
(d) ΔABE
Solution:
(d)
Clearly Common region is ΔABE.
(d) is correct.

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