Free practice › Class 11 › Mathematics › Chapter 5
Linear Inequalities — JEE MCQs with answers
Topics in this chapter
- 5.1 Introduction
- 5.2 Inequalities
- 5.3 Algebraic Solutions of Linear Inequalities in One Variable and their Graphical Representation
Free practice questions
Find the solution of the system 5x + 3 ≤ 18, 2x - 1 > -5.
- A. Every real number x
- B. x ≤ 3
- C. -2 < x ≤ 3
- D. -2 ≤ x < 3
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Answer: C — From 5x + 3 ≤ 18 we get x ≤ 3. From 2x - 1 > -5 we get x > -2. The solution of a system is the set of values common to both, i.e. the intersection, so the answer is -2 < x ≤ 3. Hence the correct option is C: -2 < x ≤ 3. Option A (Every real number x) is wrong because it takes the union of the two solution sets instead of the values common to both, and x = -2 lies in it but does not satisfy the given condition. Option B (x ≤ 3) is wrong because it uses only one of the two inequalities and ignores the other, and x = -2 lies in it but does not satisfy the given condition. Option D (-2 ≤ x < 3) is wrong because it carries over the wrong end-point types from the two inequalities, and x = -2 lies in it but does not satisfy the given condition.
NCERT Class 11 Mathematics · Ch 5 · §5.3 ("Also, from inequality (2), we have")
Which sign is conspicuously absent from the four that Section 5.1 lists?
- A. The equality sign, =
- B. The 'less than' sign, <
- C. The 'greater than or equal' sign, ≥
- D. The 'less than or equal' sign, ≤
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Answer: A — The correct option is A. The section lists exactly '<', '>', '≤' and ≥, and the absence of the equality sign is the whole point: the paragraph has just asked whether a statement problem can always be translated in the form of an equation, and it answers by supplying four signs that are not the equality sign. Each of the other three options names a sign that the section does list.
NCERT Class 11 Mathematics · Ch 5 · §5.1 ("get certain statements involving a sign '<' (less than), '>' (greater than), '≤' (less than")
Which group of numbers does NCERT name as the strict inequalities of the list?
- A. (5), (6), (9), (10) and (13)
- B. (5), (6), (9), (10) and (14)
- C. (7), (8), (11), (12) and (13)
- D. (5), (6), (7) and (8)
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Answer: B — NCERT writes that inequalities (5), (6), (9), (10) and (14) are strict inequalities. Hence the correct option is B. Option A is wrong: It replaces (14) by (13); but (13) is ax² + bx + c ≤ 0, which carries '≤' and is named slack. Option C is wrong: That is exactly the group NCERT names as the slack inequalities, not the strict ones. Option D is wrong: Those are the four one-variable linear forms taken together, and (7) and (8) among them are slack, not strict.
NCERT Class 11 Mathematics · Ch 5 · §5.2 ("Inequalities (5), (6), (9), (10) and (14) are strict inequalities while inequalities (7), (8),")
Solve the following inequality for real x: 7x - 4 < 4x + 8
- A. x < 4
- B. x > 4
- C. x < 8/3
- D. x < 12/11
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Answer: A — By Rule 1 add 4 to both sides: 7x < 4x + 12. Again by Rule 1 subtract 4x: 3x < 12. By Rule 2 divide by the positive number 3, so the sign is unchanged: x < 4. Hence the correct option is A. Option B is wrong: the sign has been reversed although the final division was by the positive number 3. Option C is wrong: the -4 on the left has been left where it was, giving 7x < 4x + 8 and then 3x < 8. Option D is wrong: 4x has been ADDED to both sides instead of subtracted, giving 11x < 12.
NCERT Class 11 Mathematics · Ch 5 · §5.3 ("Example 2 Solve 5x - 3 < 3x +1 when")
Section 5.1 announces the chapter's plan; a reader who then works through the whole chapter finds something rather different. Which comparison of the announcement with the chapter is accurate?
- A. The section promises inequalities in one variable only, and the chapter goes further and also solves them in two variables
- B. The section promises inequalities in one and two variables, and the chapter delivers worked solutions for both
- C. The section promises inequalities in two variables only, and the chapter delivers one variable only
- D. The section promises inequalities in one and two variables, but the worked solutions, the figures and the Summary all stay with one variable
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Answer: D — The correct option is D. Section 5.1 says “In this Chapter, we will study linear inequalities in one and two variables”, and Section 5.2 repeats that the book will confine itself “to the study of linear inequalities in one and two variables only”. Yet every solved example and every exercise item in the chapter is in a single variable, the chapter's three figures are all number lines, and no point of the Summary reaches beyond one variable — adding equal numbers to both sides, the reversal of the inequality on multiplying by a negative number, and the representation of x < a and x ≤ a on a number line. The two options that report the promise as one variable alone, or as two variables alone, misstate the announcement itself, which names both. The remaining option reports the announcement correctly but the delivery wrongly: no inequality in two variables is ever solved.
NCERT Class 11 Mathematics · Ch 5 · §5.1 ("In this Chapter, we will study linear inequalities in one and two variables. The")
Changing exactly one symbol in entry (7) turns it into another entry of the same list. Which change turns (7) into (5)?
- A. Replacing '≤' by '>'
- B. Replacing '≤' by '≥'
- C. Replacing '≤' by '<'
- D. Replacing 0 by c
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Answer: C — (7) is ax + b ≤ 0; writing '<' in place of '≤' gives ax + b < 0, which is (5). Hence the correct option is C. Option A is wrong: That gives ax + b > 0, which is entry (6), not entry (5). Option B is wrong: That gives ax + b ≥ 0, which is entry (8), not entry (5). Option D is wrong: That gives ax + b ≤ c, which is not printed anywhere in the list.
NCERT Class 11 Mathematics · Ch 5 · §5.2 ("Some more examples of inequalities are:")
Solve the following inequality for real x: (2x - 1)/3 ≤ (x + 2)/4
- A. x ≤ 2/5
- B. x ≥ 2
- C. x ≤ 11/2
- D. x ≤ 2
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Answer: D — Multiply both sides by the positive number 12 (Rule 2, sign unchanged): 4(2x - 1) ≤ 3(x + 2), i.e. 8x - 4 ≤ 3x + 6. Rule 1 gives 5x ≤ 10, and dividing by the positive number 5 gives x ≤ 2. Hence the correct option is D. Option A is wrong: the -4 has been subtracted from the right side instead of added, giving 5x ≤ 2. Option B is wrong: the sign has been reversed although the final division was by the positive number 5. Option C is wrong: the two multipliers have been swapped, giving 3(2x - 1) ≤ 4(x + 2) instead of 4(2x - 1) ≤ 3(x + 2).
NCERT Class 11 Mathematics · Ch 5 · §5.3 ("Rule 2 Both sides of an inequality can be multiplied (or divided) by the same positive")
Which of the following is printed by NCERT among its examples of literal inequalities?
- A. 7 > 5
- B. y > 2
- C. x + y > 2
- D. y ≥ 2
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Answer: B — NCERT's literal examples are x < 5; y > 2; x ≥ 3, y ≤ 4. The second of them is y > 2. Hence the correct option is B. A is wrong. The mistake is taking the statement with the same relation symbol from the neighbouring list. Check: 7 > 5 stands in the clause ending "are the examples of numerical inequalities". C is wrong. The mistake is building a two-letter analogue by adding a second variable. Check: each of NCERT's four literal examples has a single letter on one side and a number on the other; none adds two letters together. D is wrong. The mistake is assuming the literal examples uniformly admit their bounds. Check: the printed statement is y > 2 with a bare sign, and y = 2 fails it; the only ≥ in that list stands in x ≥ 3.
NCERT Class 11 Mathematics · Ch 5 · §5.2 ("are some examples of")
The chapter contains three numbered figures. What do they have in common, and what does that show about the plan set out in Section 5.1?
- A. All three are number lines, so the two-variable half of the announced plan is never illustrated
- B. All three are graphs in the coordinate plane, matching the two-variable half of the plan
- C. All three show shaded regions bounded by a straight line
- D. One is a number line and two are plane diagrams, so the figures are shared between the two halves of the plan
Show answer
Answer: A — The correct option is A. Section 5.1 announces a study of linear inequalities in one and two variables, but every figure the chapter prints is a number line marking the solution set of an inequality in one variable, two of them sit in the section that develops the algebraic solution of inequalities in one variable and the third accompanies a Miscellaneous Example. Nothing in the chapter is drawn in the coordinate plane, so the second half of the announced plan is never illustrated. Describing the figures as graphs in the plane, or as shaded regions bounded by a straight line, attributes to the chapter diagrams it does not contain — shading in particular is used nowhere. Splitting the three figures between the two cases is equally wrong, since not one of them is a plane diagram.
NCERT Class 11 Mathematics · Ch 5 · §5.1 ("In this Chapter, we will study linear inequalities in one and two variables. The")
Which of the following does the Summary not mention at all?
- A. A dark circle put on the number a
- B. Solutions of the inequality
- C. A circle put on the number a
- D. The modulus (absolute value) of x
Show answer
Answer: D — The Summary's six bullets speak of the symbols that form an inequality, of adding and multiplying on both sides, of solutions of the inequality, and of the plain and the dark circle used on a number line. Modulus or absolute-value notation appears nowhere in this chapter, so it cannot appear in the Summary. Each of the other three is quoted directly from a bullet. Hence the correct option is D.
NCERT Class 11 Mathematics · Ch 5 · §5.3 ("To represent x < a (or x > a) on a number line, put a circle on the number a and")
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