Class 9 Mathematics Chapter 12 Linear Programming: 30 MCQs with Answers & Solutions

The revised CBSE Class 9 Mathematics curriculum (2024–25) emphasizes conceptual understanding over procedural fluency. MCQs are the fastest way to test this because they require you to: 1. **Identify the problem type**: Distinguish between maximization and minimization problems in seconds. 2. **Recall constraints**: Recognize when inequalities represent viable or non-viable regions (e.g., x ≥ 0, y ≤ 5). 3. **Apply the graphical method**: Mentally plot lines, identify corner points, and evaluate the objective function Z = ax + by without lengthy calculations. 4. **Avoid common traps**: MCQs exploit misconceptions—like confusing the feasible region with its boundary, or misreading optimization direction. In the Term 1 exam (40 marks, 90 min), 8–10 marks come from Linear Programming. Most are 1-mark MCQs and assertion-reason pairs. A student who masters pattern recognition can solve 5–6 MCQs in 10 minutes, leaving time for long-answer questions. Unlike descriptive answers, MCQs leave no room for partial credit—you either know the concept or you don't. This makes them ideal diagnostic tools for your exam readiness.

**Question 1:** A company makes chairs (C) and tables (T). Each chair needs 5 hours, each table needs 8 hours. Maximum available hours = 160. Which inequality represents this constraint? A) 5C + 8T ≤ 160 ✓ B) 5C + 8T ≥ 160 C) 5C + 8T = 160 D) 5C + 8T < 160 **Answer: A** | *Constraint inequalities are ≤ or ≥; here, total hours cannot exceed 160.* **Question 2:** The feasible region in linear programming is: A) The area where all constraints are satisfied simultaneously ✓ B) The area where one constraint is satisfied C) The boundary lines only D) The region outside all constraints **Answer: A** | *Feasibility requires ALL constraints to be true at once.* **Question 3:** For the objective function Z = 3x + 4y (maximization), which point typically gives the maximum value? A) Any point inside the feasible region B) A corner (vertex) of the feasible region ✓ C) The midpoint of the boundary D) The origin (0, 0) **Answer: B** | *The optimal solution always lies at a corner point of the feasible region.* **Question 4:** If a constraint is 2x + y ≤ 10, which point satisfies it? A) (6, 0) B) (2, 5) ✓ C) (0, 15) D) (5, 5) **Answer: B** | *2(2) + 5 = 9 ≤ 10 is true; test others: 2(6) + 0 = 12 > 10 (false).* **Question 5:** The non-negativity constraints in linear programming mean: A) x and y can be any real number B) x ≥ 0 and y ≥ 0 ✓ C) x and y must equal zero D) x + y ≥ 0 **Answer: B** | *Non-negativity ensures we work in the first quadrant (real-world context).* **Question 6:** If the objective function is Z = 5x + 3y (minimization) and corner points are (0,4), (2,1), (4,0), which gives minimum Z? A) (0, 4) → Z = 12 B) (2, 1) → Z = 13 C) (4, 0) → Z = 20 D) Cannot determine without feasible region ✓ **Answer: D** | *You must verify all points lie in the feasible region before comparing Z values.* **Question 7:** A linear inequality like x + 2y ≤ 6 is represented graphically by: A) A line only B) A half-plane below (or on) the line ✓ C) A half-plane above the line D) The point (6, 0) **Answer: B** | *For ≤, shade below the line; for ≥, shade above.* **Question 8:** In the statement "Maximize profit subject to resource constraints," the profit function is: A) A constraint B) The objective function ✓ C) The feasible region D) A non-negativity condition **Answer: B** | *The quantity you want to optimize (maximize/minimize) is the objective function.* **Question 9:** If x represents units produced and x must be at least 10, the inequality is: A) x < 10 B) x ≤ 10 C) x ≥ 10 ✓ D) x = 10 **Answer: C** | *"At least" means ≥; "at most" means ≤.* **Question 10:** The corner points of a feasible region are (0,0), (5,0), (3,2), (0,4). For Z = 2x + y, which point gives the maximum? A) (0, 0) → Z = 0 B) (5, 0) → Z = 10 ✓ C) (3, 2) → Z = 8 D) (0, 4) → Z = 4 **Answer: B** | *Evaluate Z at each corner; (5,0) yields the highest value of 10.*

**Question 11:** A farmer has 100 acres. Wheat needs 2 acres per unit, rice needs 3 acres per unit. Maximum capital = ₹5000; wheat costs ₹50/unit, rice costs ₹75/unit. If W = wheat units, R = rice units, which is a valid constraint? A) 2W + 3R ≤ 100 ✓ B) 50W + 75R ≤ 100 C) 2W + 3R = 100 D) W + R ≤ 5000 **Answer: A** | *Land constraint: 2W + 3R ≤ 100; capital constraint is 50W + 75R ≤ 5000 (different).* **Question 12:** The line 3x + 4y = 12 intersects the x-axis at: A) (4, 0) ✓ B) (0, 3) C) (3, 0) D) (0, 4) **Answer: A** | *At x-axis, y = 0: 3x + 4(0) = 12 → x = 4.* **Question 13:** For constraints x ≥ 0, y ≥ 0, x + y ≤ 5, and 2x + y ≤ 8, the corner points of the feasible region are: A) (0,0), (5,0), (0,5), (3,2) B) (0,0), (4,0), (0,5) ✓ C) (0,0), (5,0), (0,8), (3,2) D) (0,0), (2,3), (4,0) **Answer: B** | *Intersect x + y = 5 with 2x + y = 8: x = 3, y = 2; but (3,2) violates x + y ≤ 5, so corner points are (0,0), (4,0), (0,5).* **Question 14:** A manufacturing problem states: "Produce at least 20 units of Product A." The inequality is: A) A ≤ 20 B) A ≥ 20 ✓ C) A = 20 D) 20A ≥ 1 **Answer: B** | *"At least" always translates to ≥.* **Question 15:** If the objective function Z = 4x + 2y is to be minimized over the feasible region with vertices (0,0), (6,0), (4,3), (0,5), which vertex gives minimum Z? A) (0, 0) → Z = 0 ✓ B) (6, 0) → Z = 24 C) (4, 3) → Z = 22 D) (0, 5) → Z = 10 **Answer: A** | *Z = 4(0) + 2(0) = 0 is the minimum; origin often minimizes when all coefficients are positive.* **Question 16:** The line 2x + 3y = 6 and the line 2x + 3y = 12 are: A) Intersecting B) Parallel ✓ C) The same line D) Perpendicular **Answer: B** | *Both have the form 2x + 3y = c; same coefficients → parallel lines.* **Question 17:** For the feasible region bounded by x ≥ 2, y ≥ 1, x + y ≤ 8, what is the corner point in the second quadrant (x < 0)? A) No corner point in the second quadrant ✓ B) (–1, 2) C) (–2, 3) D) Cannot determine **Answer: A** | *Constraint x ≥ 2 ensures all points have x ≥ 2 > 0, so no second quadrant points exist.* **Question 18:** A profit maximization problem has Z = 10x + 15y. The constraint is 5x + 6y ≤ 300. If (0, 0) and (60, 0) are corner points, which is optimal? A) (0, 0) → Z = 0 B) (60, 0) → Z = 600 ✓ C) Cannot decide without all corner points D) (0, 50) always **Answer: C** | *You must test all corner points of the feasible region, not just two.* **Question 19:** The inequality x + y ≥ 5 is represented by shading: A) Below the line x + y = 5 B) Above the line x + y = 5 ✓ C) Only the line itself D) The region between y = 0 and the line **Answer: B** | *For ≥ inequalities, shade above (or to the right of) the boundary line.* **Question 20:** A company produces two products. Let P₁ = units of product 1, P₂ = units of product 2. If profit per unit is ₹8 and ₹12 respectively, the objective function is: A) 8P₁ + 12P₂ = Z (maximize) ✓ B) 8P₁ + 12P₂ ≤ Z C) P₁ + P₂ = 20 D) 8P₁ + 12P₂ is a constraint **Answer: A** | *Revenue/profit = unit price × quantity; set as Z = 8P₁ + 12P₂ to maximize.*

**Question 21:** **Assertion (A):** The feasible region of a linear programming problem is always a polygon (convex region). **Reason (R):** Linear inequalities define half-planes, and the intersection of half-planes is a convex polygon. A) Both A and R are true, and R is the correct explanation of A ✓ B) Both A and R are true, but R is not the correct explanation C) A is true, but R is false D) Both A and R are false **Answer: A** | *Convexity arises because linear constraints create half-planes; their intersection is convex.* **Question 22:** For the LPP: Maximize Z = 5x + 3y subject to x + 2y ≤ 10, 2x + y ≤ 8, x ≥ 0, y ≥ 0. The corner points are (0,0), (4,0), (2,3), (0,5). Which statement is correct? A) Z is maximum at (2, 3) with Z = 19 B) Z is maximum at (4, 0) with Z = 20 ✓ C) Z is maximum at (0, 5) with Z = 15 D) Multiple optimal solutions exist **Answer: B** | *Test: (0,0)→0, (4,0)→20, (2,3)→19, (0,5)→15; maximum is 20 at (4,0).* **Question 23:** **Assertion (A):** If the objective function is Z = 4x + 2y and two different corner points yield the same maximum Z value, then every point on the line joining them is optimal. **Reason (R):** If two corner points have equal Z values and lie on the objective function line, the entire edge between them lies on the level curve of the objective function. A) Both A and R are true, and R is the correct explanation of A ✓ B) Both A and R are true, but R is not the correct explanation C) A is true, but R is false D) Both A and R are false **Answer: A** | *Multiple optimal solutions occur when the objective function is parallel to a constraint boundary.* **Question 24:** A manufacturing LPP has constraints: 3x + 2y ≤ 120, x + 2y ≤ 80, x ≥ 0, y ≥ 0. To find the feasible region's vertices, which pair of lines would you solve simultaneously? A) 3x + 2y = 120 and x + 2y = 80 ✓ B) 3x + 2y = 120 and x = 0 C) x + 2y = 80 and y = 0 D) All pairs would give vertices **Answer: A** | *Intersecting constraint boundaries yield interior vertices; (3x+2y=120) ∩ (x+2y=80) gives (40,0), but verify via x=40, y=0.* **Question 25:** An LPP aims to minimize Z = 6x + 4y. The feasible region is unbounded, with no maximum value. This means: A) The minimum value of Z is always 0 B) The minimum value might not exist (could be unbounded below) ✓ C) The minimum exists but cannot be calculated D) Z must be maximized instead **Answer: B** | *Unbounded feasible regions can have no optimal solution if the objective function can decrease indefinitely.* **Question 26:** **Assertion (A):** The point (2, 3) lies in the feasible region defined by 2x + 3y ≤ 15, x ≥ 0, y ≥ 0, x + y ≤ 6. **Reason (R):** Substituting x = 2, y = 3: 2(2) + 3(3) = 13 ≤ 15 ✓, 2 ≥ 0 ✓, 3 ≥ 0 ✓, 2 + 3 = 5 ≤ 6 ✓. A) Both A and R are true, and R is the correct explanation of A ✓ B) Both A and R are true, but R is not the correct explanation C) A is true, but R is false D) Both A and R are false **Answer: A** | *All four constraints are satisfied; (2,3) is in the feasible region.* **Question 27:** For the LPP with corner points (0,0), (8,0), (5,5), (0,10), if Z = 3x + ky (maximize), and the maximum occurs at both (8,0) and (5,5), then k equals: A) 3/5 B) 24/5 ✓ C) 5 D) 8 **Answer: B** | *At (8,0): Z = 24; at (5,5): Z = 15 + 5k. For equality: 24 = 15 + 5k → k = 9/5. [Verify: k = 24/5 gives Z(5,5) = 15 + 24 = 39, Z(8,0) = 24]—correction needed—retest: if k = 24/5, Z(8,0) = 24, Z(5,5) = 15 + 5(24/5) = 15 + 24 = 39. Not equal. Correct k: 24 = 15 + 5k → k = 1.8 = 9/5.* **Revised Answer: B = 9/5** | *For both points to be optimal: Z at (8,0) = Z at (5,5); 24 = 15 + 5k → k = 1.8.* **Question 28:** Which of the following statements is FALSE about the graphical method of linear programming? A) The optimal solution always occurs at a corner point of the feasible region. B) If the feasible region is empty (no common area), no solution exists. ✓ (This is TRUE, so the question asks for FALSE) C) The feasible region is the intersection of all constraint half-planes. (TRUE) D) The optimal value of Z can be found by substituting corner point coordinates into the objective function. (TRUE) **Correct FALSE statement:** The optimal solution is always unique. ✓ **Answer:** Multiple optimal solutions can occur (see Question 23). Choose the option stating uniqueness is guaranteed if provided. **Question 29:** In a linear programming problem, if the constraint 2x + y ≤ 8 is changed to 2x + y ≤ 4, the feasible region will: A) Expand B) Shrink ✓ C) Rotate D) Remain unchanged **Answer: B** | *Decreasing the RHS of a ≤ constraint reduces the half-plane, shrinking the feasible region.* **Question 30:** **Assertion (A):** For the LPP: Minimize Z = 5x + 4y subject to 2x + 3y ≥ 12, x ≥ 0, y ≥ 0, the optimal solution lies on the boundary 2x + 3y = 12. **Reason (R):** The constraint is a ≥ inequality, so the feasible region is above the line 2x + 3y = 12, and the minimum Z occurs on the lower boundary of this region. A) Both A and R are true, and R is the correct explanation of A ✓ B) Both A and R are true, but R is not the correct explanation C) A is true, but R is false D) Both A and R are false **Answer: A** | *For minimization with ≥ constraints, the optimal point is on the boundary of the constraint, not in the interior.*

**Trap 1: Confusing ≤ and ≥** Many students reverse inequality signs during problem-solving. Example: "The factory has a maximum of 100 hours" becomes "x + y ≥ 100" (wrong) instead of "x + y ≤ 100" (correct). Always reread the phrase: "at most" = ≤, "at least" = ≥, "no more than" = ≤, "no less than" = ≥. **Trap 2: Mistaking the feasible region for the optimal point** The feasible region is the shaded area where ALL constraints are satisfied. The optimal point is one specific corner of this region. Don't select answers describing the entire region when asked for the maximum/minimum point. **Trap 3: Forgetting non-negativity constraints** Linear programming assumes x ≥ 0 and y ≥ 0 (first quadrant only) unless stated otherwise. A trap answer might show a feasible region in the second or third quadrant, which violates non-negativity. Always check option C and D for this. **Trap 4: Testing only one corner point** The optimal solution MUST be found by evaluating the objective function Z at ALL corner points, not just the largest or closest one. Many wrong answers test Z at only (0,0) or the x-intercept. **Trap 5: Confusing the constraint line with the feasible half-plane** The line 2x + 3y = 12 is just the boundary. Points satisfying 2x + 3y ≤ 12 lie on one side of this line. A trap option might list points on the line itself (which are feasible) but claim they represent the entire feasible region. **Trap 6: Assuming the feasible region is always bounded** Some LPPs have unbounded feasible regions (extending infinitely). If the objective function can increase/decrease indefinitely within this region, no optimal solution exists. Answers claiming "solution always exists" are traps. **Trap 7: Misinterpreting "maximize" vs. "minimize"** If Z = 5x + 3y is to be MINIMIZED, the smallest Z value (not the largest) is optimal. Wrong answers often list the maximum Z value, especially when students misread the problem. **Trap 8: Arithmetic errors in evaluating Z** Z = 3x + 4y at (2, 5) should be 3(2) + 4(5) = 6 + 20 = 26, not 3 + 4 × 2 + 5 = 16 (wrong order of operations). Trap options provide miscalculated Z values. Double-check every substitution. **Trap 9: Ignoring "simultaneous" constraint equations** Finding corner points requires solving pairs of constraint equations simultaneously (e.g., x + y = 5 AND 2x + y = 8 → x = 3, y = 2). Trap answers solve each equation independently or use only one. **Trap 10: Confusing objective function with constraints** The objective function (Z = 5x + 3y) is what you optimize, NOT a constraint. Trap answers treat it as an inequality or vice versa. Always identify which equation is to be maximized/minimized before solving.

**Pre-exam (1–2 weeks before test):** 1. **Master the formula sheet**: Memorize the steps to find corner points: solve constraint pairs simultaneously, verify feasibility (do they satisfy all constraints?), evaluate Z at each valid corner. 2. **Practice 2–3 full LPP problems by hand**: Write out the mathematical formulation, graph (on paper), identify corner points, and compute Z. This builds automaticity. 3. **Drill inequality translation**: In 5 minutes, convert 10 word problems to inequalities (e.g., "budget cannot exceed ₹5000" → 2x + 3y ≤ 5000). Speed here saves exam time. **During the exam (for a 90-minute paper with ~8 Linear Programming marks):** 1. **Allocate 12–15 minutes for all LPP MCQs** (usually 6–8 questions × 1 mark each). 2. **Easy MCQs (Questions 1–10 style): 1 minute per question** - Skim the question, identify the type (constraint formulation, feasibility check, or optimal value). - If it's a definition or translation, answer immediately. - Example: "Maximize profit" → objective function. Spend 20 seconds, mark answer, move on. 3. **Medium MCQs (Questions 11–20 style): 1.5–2 minutes per question** - For corner-point evaluation: quickly test all 4 options in the objective function using a calculator (2 substitutions per option = 6 seconds × 4 = 24 seconds). - Example: Z = 2x + y, corner points (0,0), (4,0), (3,2), (0,5). Compute Z for each: 0, 8, 8, 5. Max = 8. Done in 1 minute. - For constraint verification: test one point per option (30 seconds max). 4. **Hard/Assertion-Reason MCQs (Questions 21–30 style): 2–3 minutes per question** - Read the Assertion (A) carefully (15 seconds). - Read the Reason (R) carefully (15 seconds). - Ask: Is A true? Is R true? Does R explain A? - If both are true and R explains A, mark option (A) and move on (remaining 1 minute for scratch work if needed). - Example: A = "Feasible region is a polygon," R = "Linear constraints are half-planes" → both true, R explains A → answer (A). 5. **Skip-and-return strategy**: - If a Medium MCQ requires graphing feasible regions, skip it on first pass (it may take 3+ minutes). - Answer all Easy MCQs (10 marks in 10 minutes), then return to Medium/Hard. - This ensures you secure easy marks before tackling complex questions. 6. **Mental shortcuts**: - **Optimal point at origin?** If all constraint coefficients are positive and it's minimization, suspect (0,0). Test it first (10 seconds). - **Multiple optimal solutions?** If two different corner points give the same Z, the answer is likely "multiple solutions exist." Verify objective function is parallel to a constraint. - **Unbounded feasible region?** If a constraint uses only one variable (e.g., x ≤ 10, no upper limit on y), the region is unbounded. - **Parallel lines?** If 2x + 3y = 6 and 2x + 3y = 12 appear, they're parallel (same coefficients, different RHS). Use this to predict multiple optimal solutions. 7. **Avoid rework**: - Write down corner points and their Z values on a small table on the side of your answer sheet. This prevents recalculating. - Example: ``` Point | Z = 5x + 3y (0,0) | 0 (4,0) | 20 (2,3) | 19 (0,5) | 15 ``` - Max Z = 20 at (4,0). Done in 2 minutes. 8. **Final check (last 2 minutes)**: - If you have time, revisit any assertion-reason question where you guessed. - Reread the constraint/objective one more time to catch sign errors. **Sample 12-minute timeline for 8 MCQs:** - Easy (5 MCQs): 5 min - Medium (2 MCQs): 4 min - Hard (1 MCQ): 2 min - Buffer: 1 min This ensures you finish Linear Programming questions with time to spare for other sections. Start a 3-day free trial at cbsetutor.ai to practice timed MCQ quizzes with instant feedback and detailed solutions.

**Mathematical Formulation:** - Objective function: Z = ax + by (quantity to maximize or minimize). - Linear constraint: cx + dy ≤ e, cx + dy ≥ e, or cx + dy = e. - Non-negativity: x ≥ 0, y ≥ 0 (unless specified otherwise). **Graphical Method Steps:** 1. Plot all constraint lines (convert ≤/≥ to = for graphing). 2. Shade the half-plane for each ≤ or ≥ constraint. 3. Identify the feasible region (intersection of all shaded areas). 4. Find all corner points (vertices) of the feasible region. 5. Evaluate Z = ax + by at each corner point. 6. **Maximization**: Choose corner with largest Z. **Minimization**: Choose corner with smallest Z. **Special Cases:** - **Empty feasible region**: No solution exists (constraints are inconsistent). - **Unbounded feasible region**: Optimal solution may not exist if Z can increase/decrease indefinitely. - **Degenerate corner point**: More than two constraints meet at one point (rare in Class 9). - **Multiple optimal solutions**: Two or more corners have the same optimal Z; entire edge between them is optimal. Occurs when the objective function is parallel to a constraint. **Translation of Common Phrases:** | Phrase | Inequality | |--------|----------| | At most, no more than, not exceeding | ≤ | | At least, no less than, minimum required | ≥ | | Equal to, exactly | = | | Not equal to | ≠ | **Feasibility Check Example:** Point (3, 2): Does it satisfy 2x + y ≤ 8? → 2(3) + 2 = 8 ✓ (on the boundary, feasible). Point (4, 2): Does it satisfy 2x + y ≤ 8? → 2(4) + 2 = 10 > 8 ✗ (not feasible).