Written Instructions:
For each Multiple Choice Question (MCQ), four options are given. One of them is the correct answer. Make your choice (1,2,3 or 4). Write your answers in the brackets provided..
For each Short Answer Question(SAQ) and Long Answer Question(LAQ), write your answers in the blanks provided.
Leave your answers in the simplest form or correct to two decimal places.
| 1) If polynomial x3 + 2x2 - m and 2x3-mx + 5 leave the same remaainder when they are divided by (x-2), find the value of m. Also find the remainder. m = Answer:_______________ |
| 2) If polynomial x3 + 3x2 - m and 3x3-mx + 8 leave the same remaainder when they are divided by (x-3), find the value of m. Also find the remainder. m = Answer:_______________ |
| 3) If polynomial x3 + 3x2 - m and 4x3-mx + 11 leave the same remaainder when they are divided by (x-3), find the value of m. Also find the remainder. m = Answer:_______________ |
| 4) If polynomial x3 + 2x2 - m and 4x3-mx + 7 leave the same remaainder when they are divided by (x-3), find the value of m. Also find the remainder. m = Answer:_______________ |
| 5) If polynomial x3 + 2x2 - m and 3x3-mx + 8 leave the same remaainder when they are divided by (x-2), find the value of m. Also find the remainder. m = Answer:_______________ |
| 6) If polynomial x3 + 2x2 - m and 3x3-mx + 11 leave the same remaainder when they are divided by (x-2), find the value of m. Also find the remainder. m = Answer:_______________ |
| 7) If polynomial x3 + 2x2 - m and 2x3-mx + 5 leave the same remaainder when they are divided by (x-3), find the value of m. Also find the remainder. m = Answer:_______________ |
| 8) If polynomial x3 + 2x2 - m and 2x3-mx + 11 leave the same remaainder when they are divided by (x-2), find the value of m. Also find the remainder. m = Answer:_______________ |
| 9) If polynomial x3 + 2x2 - m and 2x3-mx + 13 leave the same remaainder when they are divided by (x-2), find the value of m. Also find the remainder. m = Answer:_______________ |
| 10) If polynomial x3 + 2x2 - m and 3x3-mx + 7 leave the same remaainder when they are divided by (x-3), find the value of m. Also find the remainder. m = Answer:_______________ |
| 1) If polynomial x3 + 2x2 - m and 2x3-mx + 5 leave the same remaainder when they are divided by (x-2), find the value of m. Also find the remainder. m = Answer: 5 SOLUTION 1 : Let P(x) = x3 + 3x2 - m , 2x3 - mx + 9 When P(x) and q(x) are divided by x - 2, We get same remainder p(2) = q(2) P(2) = (2)3 + 2(2)2 - m = 8 + 8 - m = 16 - m .............→ (1) Let q(x) = 2x3 - mx + 5 q(2) = 2(2)3 - m(2) + 5
= 16 - 2m + 5 = 21 - 2m ...............⇒(2) Given (1) = (2) 16 - m = 21 - 2m ⇒ 2m - m = 21 - 16 ⇒ m = 5 Remainder = P(2) = 16 - m = 16 - 5 = 11
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| 2) If polynomial x3 + 3x2 - m and 3x3-mx + 8 leave the same remaainder when they are divided by (x-3), find the value of m. Also find the remainder. m = Answer: 35 SOLUTION 1 : Let P(x) = x3 + 3x2 - m , 2x3 - mx + 9 When P(x) and q(x) are divided by x - 3, We get same remainder p(3) = q(3) P(3) = (3)3 + 3(3)2 - m = 27 + 27 - m = 54 - m .............→ (1) Let q(x) = 2x3 - mx + 8 q(3) = 3(3)3 - m(3) + 8
= 81 - 3m + 8 = 89 - 3m ...............⇒(2) Given (1) = (2) 54 - m = 89 - 3m ⇒ 3m - m = 89 - 54 ⇒ m = 35 Remainder = P(3) = 54 - m = 54 - 35 = 19
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| 3) If polynomial x3 + 3x2 - m and 4x3-mx + 11 leave the same remaainder when they are divided by (x-3), find the value of m. Also find the remainder. m = Answer: 65 SOLUTION 1 : Let P(x) = x3 + 3x2 - m , 2x3 - mx + 9 When P(x) and q(x) are divided by x - 3, We get same remainder p(3) = q(3) P(3) = (3)3 + 3(3)2 - m = 27 + 27 - m = 54 - m .............→ (1) Let q(x) = 2x3 - mx + 11 q(3) = 4(3)3 - m(3) + 11
= 108 - 3m + 11 = 119 - 3m ...............⇒(2) Given (1) = (2) 54 - m = 119 - 3m ⇒ 3m - m = 119 - 54 ⇒ m = 65 Remainder = P(3) = 54 - m = 54 - 65 = -11
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| 4) If polynomial x3 + 2x2 - m and 4x3-mx + 7 leave the same remaainder when they are divided by (x-3), find the value of m. Also find the remainder. m = Answer: 70 SOLUTION 1 : Let P(x) = x3 + 3x2 - m , 2x3 - mx + 9 When P(x) and q(x) are divided by x - 3, We get same remainder p(3) = q(3) P(3) = (3)3 + 2(3)2 - m = 27 + 18 - m = 45 - m .............→ (1) Let q(x) = 2x3 - mx + 7 q(3) = 4(3)3 - m(3) + 7
= 108 - 3m + 7 = 115 - 3m ...............⇒(2) Given (1) = (2) 45 - m = 115 - 3m ⇒ 3m - m = 115 - 45 ⇒ m = 70 Remainder = P(3) = 45 - m = 45 - 70 = -25
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| 5) If polynomial x3 + 2x2 - m and 3x3-mx + 8 leave the same remaainder when they are divided by (x-2), find the value of m. Also find the remainder. m = Answer: 16 SOLUTION 1 : Let P(x) = x3 + 3x2 - m , 2x3 - mx + 9 When P(x) and q(x) are divided by x - 2, We get same remainder p(2) = q(2) P(2) = (2)3 + 2(2)2 - m = 8 + 8 - m = 16 - m .............→ (1) Let q(x) = 2x3 - mx + 8 q(2) = 3(2)3 - m(2) + 8
= 24 - 2m + 8 = 32 - 2m ...............⇒(2) Given (1) = (2) 16 - m = 32 - 2m ⇒ 2m - m = 32 - 16 ⇒ m = 16 Remainder = P(2) = 16 - m = 16 - 16 = 0
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| 6) If polynomial x3 + 2x2 - m and 3x3-mx + 11 leave the same remaainder when they are divided by (x-2), find the value of m. Also find the remainder. m = Answer: 19 SOLUTION 1 : Let P(x) = x3 + 3x2 - m , 2x3 - mx + 9 When P(x) and q(x) are divided by x - 2, We get same remainder p(2) = q(2) P(2) = (2)3 + 2(2)2 - m = 8 + 8 - m = 16 - m .............→ (1) Let q(x) = 2x3 - mx + 11 q(2) = 3(2)3 - m(2) + 11
= 24 - 2m + 11 = 35 - 2m ...............⇒(2) Given (1) = (2) 16 - m = 35 - 2m ⇒ 2m - m = 35 - 16 ⇒ m = 19 Remainder = P(2) = 16 - m = 16 - 19 = -3
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| 7) If polynomial x3 + 2x2 - m and 2x3-mx + 5 leave the same remaainder when they are divided by (x-3), find the value of m. Also find the remainder. m = Answer: 14 SOLUTION 1 : Let P(x) = x3 + 3x2 - m , 2x3 - mx + 9 When P(x) and q(x) are divided by x - 3, We get same remainder p(3) = q(3) P(3) = (3)3 + 2(3)2 - m = 27 + 18 - m = 45 - m .............→ (1) Let q(x) = 2x3 - mx + 5 q(3) = 2(3)3 - m(3) + 5
= 54 - 3m + 5 = 59 - 3m ...............⇒(2) Given (1) = (2) 45 - m = 59 - 3m ⇒ 3m - m = 59 - 45 ⇒ m = 14 Remainder = P(3) = 45 - m = 45 - 14 = 31
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| 8) If polynomial x3 + 2x2 - m and 2x3-mx + 11 leave the same remaainder when they are divided by (x-2), find the value of m. Also find the remainder. m = Answer: 11 SOLUTION 1 : Let P(x) = x3 + 3x2 - m , 2x3 - mx + 9 When P(x) and q(x) are divided by x - 2, We get same remainder p(2) = q(2) P(2) = (2)3 + 2(2)2 - m = 8 + 8 - m = 16 - m .............→ (1) Let q(x) = 2x3 - mx + 11 q(2) = 2(2)3 - m(2) + 11
= 16 - 2m + 11 = 27 - 2m ...............⇒(2) Given (1) = (2) 16 - m = 27 - 2m ⇒ 2m - m = 27 - 16 ⇒ m = 11 Remainder = P(2) = 16 - m = 16 - 11 = 5
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| 9) If polynomial x3 + 2x2 - m and 2x3-mx + 13 leave the same remaainder when they are divided by (x-2), find the value of m. Also find the remainder. m = Answer: 13 SOLUTION 1 : Let P(x) = x3 + 3x2 - m , 2x3 - mx + 9 When P(x) and q(x) are divided by x - 2, We get same remainder p(2) = q(2) P(2) = (2)3 + 2(2)2 - m = 8 + 8 - m = 16 - m .............→ (1) Let q(x) = 2x3 - mx + 13 q(2) = 2(2)3 - m(2) + 13
= 16 - 2m + 13 = 29 - 2m ...............⇒(2) Given (1) = (2) 16 - m = 29 - 2m ⇒ 2m - m = 29 - 16 ⇒ m = 13 Remainder = P(2) = 16 - m = 16 - 13 = 3
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| 10) If polynomial x3 + 2x2 - m and 3x3-mx + 7 leave the same remaainder when they are divided by (x-3), find the value of m. Also find the remainder. m = Answer: 43 SOLUTION 1 : Let P(x) = x3 + 3x2 - m , 2x3 - mx + 9 When P(x) and q(x) are divided by x - 3, We get same remainder p(3) = q(3) P(3) = (3)3 + 2(3)2 - m = 27 + 18 - m = 45 - m .............→ (1) Let q(x) = 2x3 - mx + 7 q(3) = 3(3)3 - m(3) + 7
= 81 - 3m + 7 = 88 - 3m ...............⇒(2) Given (1) = (2) 45 - m = 88 - 3m ⇒ 3m - m = 88 - 45 ⇒ m = 43 Remainder = P(3) = 45 - m = 45 - 43 = 2
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