| --- |
| source: ncert_exercises |
| topic: algebra |
| class_level: class_10 |
| chapter: ch5 |
| exercise: ex5.1 |
| difficulty: beginner |
| --- |
| |
| Class 10 | Ch5: Arithmetic Progressions | Exercise 5.1 |
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| AP: Sequence where difference between consecutive terms is constant. |
| Common difference d = a₂-a₁ = a₃-a₂ = ... |
| General form: a, a+d, a+2d, ... |
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| Q1. In which of the following situations, does the list of numbers form AP? |
| (i) Taxi fare after each km when fare is ₹15 for 1st km and ₹8 for each additional km. |
| (ii) Amount of air in a cylinder when a vacuum pump removes 1/4 of air remaining each time. |
| (iii) Cost of digging a well after each metre if it costs ₹150 for first metre and rises ₹50 for each subsequent metre. |
| (iv) Amount of money in the account every year when ₹10000 is deposited at compound interest at 8% p.a. |
| Answer: |
| (i) Fares: 15, 23, 31, 39... d=8 constant. YES, AP. ✅ |
| (ii) Each time removes 1/4, so remaining: V, 3V/4, 9V/16... ratio is constant → geometric, NOT AP. ✅ |
| (iii) Costs: 150, 200, 250... d=50. YES, AP. ✅ |
| (iv) Compound interest: not constant difference → NOT AP. ✅ |
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| Q2. Write first four terms of AP when first term a and common difference d are: |
| (i) a=10, d=10 (ii) a=-2, d=0 (iii) a=4, d=-3 (iv) a=-1, d=1/2 (v) a=-1.25, d=-0.25 |
| Answer: |
| (i) 10,20,30,40. ✅ |
| (ii) -2,-2,-2,-2. ✅ |
| (iii) 4,1,-2,-5. ✅ |
| (iv) -1,-1/2,0,1/2. ✅ |
| (v) -1.25,-1.50,-1.75,-2.00. ✅ |
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| Q3. For the following APs, write first term and common difference: |
| (i) 3,1,-1,-3 (ii) -5,-1,3,7 (iii) 1/3,5/3,9/3,13/3 (iv) 0.6,1.7,2.8,3.9 |
| Answer: |
| (i) a=3, d=-2. ✅ |
| (ii) a=-5, d=4. ✅ |
| (iii) a=1/3, d=4/3. ✅ |
| (iv) a=0.6, d=1.1. ✅ |
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| Q4. Which of the following are APs? If so, find common difference d: |
| (i) 2,4,8,16 (ii) 2,5/2,3,7/2 (iii) -1.2,-3.2,-5.2,-7.2 (iv) -10,-6,-2,2 |
| (v) 3,3+√2,3+2√2,3+3√2 (vi) 0.2,0.22,0.222 (vii) 0,-4,-8,-12 |
| (viii) -1/2,-1/2,-1/2 (ix) 1,3,9,27 (x) a,2a,3a,4a (xi) a,a²,a³,a⁴ |
| Answer: |
| (i) Differences:2,4,8 — not constant. NOT AP. ✅ |
| (ii) d=1/2. YES AP. ✅ |
| (iii) d=-2. YES AP. ✅ |
| (iv) d=4. YES AP. ✅ |
| (v) d=√2. YES AP. ✅ |
| (vi) 0.02,0.002 — not constant. NOT AP. ✅ |
| (vii) d=-4. YES AP. ✅ |
| (viii) d=0. YES AP. ✅ |
| (ix) Differences:2,6,18 — not constant. NOT AP. ✅ |
| (x) d=a. YES AP. ✅ |
| (xi) a(a-1),a²(a-1)— not constant. NOT AP. ✅ |
