Students and Teacher's Forum

Students and Teachers Forum

1. Find the LCM:

a² – (b + c)², b² – (c + a)²and c² – (a + b)²

Here, 1st expression = a2 – (b + c)2 = (a + b + c) (a – b – c). 2nd expression = b2 – (c + a)2 .....

2. Find the LCM of:

a³ – b³, (a – b)³ and a² – b²

Here, 1st expression, = a3 – b3 =(a – b) (a2 + ab + b2). 2nd expression, = (a – b)3 .....

3. Find the LCM of:

(a – b)²+ 4ab and (a + b)³ – 3ab(a + b)

Here, 1st expression = (a – b)2 + 4ab = a2– 2ab + b2+ 4ab = a2+ 2ab + .....

4. If 4 and 24 are HCF and LCM of 8 and x respectively. Find value of x.

Here, We know, HCF × LCM = 8 × x Or, 4×24 = 8 × x Thus, x = (4×24)/8 = 12 .....

5. Find the L.C.M. of:

x³ – x² + x – 1 and 2x³ + x² +x + 2

Here,

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6. Find the HCF of: 5x+30 and 3x+18

Here, Here, first expression: 5x+30 = 5(x+6) Second expression: 3x+18 = 3(x+6) Thus, the HCF = (x+6) .....

7. Find the H.C.F. of:

x² – y² – 2x + 1 and x² – xy – x

Here,

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Find the H.C.F. of:

x³ + 1, x⁴ + x² + 1 and x³ – 2x² – 2x + 1

Here,

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Find the H.C.F. of:

5a³ – 20a, a³ – 3a² – 10a and a³ – a² – 2a + 8

Here,

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10.

Find the H.C.F. of:

x³y + y⁴, x⁴ + x²y² + y⁴ and 2ax³ – 2ax²y + 2axy²

Here,

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11.

Find the H.C.F. of:

y² – 10y + 24 + 6x – 9x² and 2y⁴ – 18x²y² – 48xy² – 32y²

Here, 1st expression = y2 – 10y + 24 + 6x – 9x2

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12.

Find the H.C.F. of:

4x⁴ + 16x³ – 20x², 3x³ + 14x² – 5x and x⁴ + 125x

Solution:

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13.

Find the H.C.F. of:

x³ – 64y³, x² – 6xy + 8y² and x² – 16y²

Here,

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14. Find the HCF and LCM of:

x³– 1, x4+ x² + 1 and x³ + 2x² + 2x + 1

Here, 1st expression = x3 – 1 = x3 - 13 = (x – 1) (x2 + x + 1). 2nd .....

15. Find the HCF and LCM of:

4x⁴ + 16x³ – 20x², 3x³ + 14x² – 5x and x⁴ + 125x

Here, 1st expression = 4x4 + 16x3 – 20x2 = 4x2(x2+ 4x – 5) .....

16. Find the HCF and LCM of:

16 (a² + 2a – 3), 24(a² + a – 6) and 48(a² – a – 12)

Here, 1st expression = 16 (a2 + 2a – 3) = 16{a2 + (3 – 1)a – .....

17. Find the HCF and LCM of:

x³ – 3x + 2, x² – 4x + 3 and x² – 5x + 6

Here, 1st expression = x3 – 3x + 2 = x3 – x2 + x2 – x – 2x .....

18. Find the HCF and LCM of:

4x⁴ + 16x³ – 20x², 3x³ + 14x² – 5x and x⁴ + 125x

Here, 1st expression = 4x4 + 16x3 – 20x2 = 4x2(x2+ 4x – 5) .....

19. Find the LCM of:

(a + b)² – 4ab and (a – b)³ + 3ab(a – b)

Here, 1st expression = (a + b)2 – 4ab =a2 + 2ab + b2 – 4ab = a2 – .....

20. Find the HCF and LCM of:

x³ – 3x + 2, x² – 4x + 3 and x² – 5x + 6

Here, 1st expression: = x3 – 3x + 2 = x3 – x2 + x2 – x – 2x .....

21. Find the H.C.F and L.C.M of following expression:

4x⁴ + 16x³ – 20x², 3x³ + 14x² – 5x and x4 + 125x

Here, 1st expression = 4x4 + 16x3 – 20x2 = 4x2(x2 + 4x – 5) = 4x2{x2 + .....

22. Find the HCF and LCM of:

x³ – 1, x⁴+ x² + 1 and x³ + 2x² + 2x + 1

Here, 1st expression = x3 – 1 = x3 - 13 .....

23. Find the LCM of:

a⁴ + a² + 1, a³ – 1 and a³ – a² + a

Here, 1st expression = a4+ a2 + 1 = (a4 + 1) + a2 ={(a2)2 + 12} + a .....

24. Find the HCF and LCM of:

16(a² + 2a – 3), 24(a² + a – 6) and 48(a² – a – 12)

Here, 1st expression = 16 (a2 + 2a – 3) = 16{a2 + (3 – 1)a – .....

25. Find the HCF and LCM of:

x³ – 3x + 2, x² – 4x + 3 and x² – 5x + 6

Here, 1st expression: = x3 – 3x + 2 = x3 – x2 + x2 – x – 2x + 2 .....

26. Find HCF and LCM:
2x⁴ +16x and 3x²-6x+12

Here, Expression I: = 2x4 +16x =2x(x3+8) .....

27. Find the H.C.F. of:

x² + 3x + 9 and x⁴ + 9x² + 81

Here,

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28. Find the LCM of:

4x+16 and 5x+20

Here, First expression: = 4x+16 =4(x+4) . Second expression: .....

29. Find the HCF of:

9m² – 4n² – 4nr – r², r² – 4n² – 9m² – 12mn and 9m² + 6mr + r² – 4n²

Here, 1st expression = 9m2 – 4n2 – 4nr – r2 = (3m)2 – [(2n)2 + 2.2n.r +(r)2] .....

30. Find HCF and LCM:
8x²-12x+18 and 8x³+27

Here, Expression I: = 8x2-12x+18 = 2(4x2-6x+9). Expression II: = 8x3+27 .....

31. Find the LCM:

2x³ – 16, x² – 4x + 4 and x² – 3x + 2

Here, 1st expression = 2x3 – 16 = 2(x3 – 8) = 2(x3 – .....

32. Find the LCM:

a⁴ + a² + 1, a³ – 1 and a³ – a² + a

Here, 1st expression = a4+ a2 + 1 = (a4 + 1) + a2 ={(a2)2 + 12} + a2 .....

33. Find the H.C.F and L.C.M of following expression:

x³ – 1, x⁴ + x² + 1 and x³ + 2x² + 2x + 1

Here, 1st expression = x3 – 1 = x3 - 13 = (x – 1) (x2 + x + 1). 2nd .....

34. Find the H.C.F and L.C.M of following expression:

3a⁴ – 8a³ + 4a², a³ – 4a and 4a³ – 10a² + 4a

Here, 1st expression = 3a4 – 8a3 + 4a2 = a2 (3a2 – 8a + 4) = a2 {3a2 .....

35. Find the L.C.M:

2x³ – 16, x² – 4x + 4 and x² – 3x + 2

Here, 1st expression = 2x3 – 16 = 2(x3 – 8) = 2(x3 .....

36. Find the L.C.M:

x³ – 2x² – 3x, x³ – 9x and x⁴ – 27x

Here, 1st expression = x3 – 2x2 – 3x = x(x2 – 2x – 3) .....

37. Find the HCF of:

x⁵y³ and x³y⁵

Here, First expression: = x5y3 = x.x.x.x.x.y.y.y Second expression: .....

38. The L.C.M. of:

x² + 5x + 6 and x² + 3x + 2 is

Here, Eirst Expression : = x2 + 5x + 6 .....

39. Find the H.C.F. of:

m³ – 1, m⁴ + m² + 1 and m³ + 1 + 2m² + 2m

Here,

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40. Find the H.C.F. of:

x² – 2x – 3 and x³ – 9x

Solution:

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41. Find the HCF of:

6x² – x – 1 and 54x⁴ + 2x

Here, First expression = 6x2 – x – 1 = 6x2– 3x+ 2x – 1 .....

42. Find the HCF of:

(1 – x²) (1 – y²) + 4xy & 1 – 2x + y – x²y + x²

Here, 1st expression = (1 – x2) (1 – y2) + 4xy = 1 – y2 – x2 + x2 y2 + 4xy .....

43.

Find the H.C.F. of:

x⁴ + (2b² – a²) x² + b⁴ and x⁴ + 2ax³ + a²x² – b⁴

Here,

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44. If any number x is multiple of 5. Find HCF of 2x and 5.

Here, Here, x is multiple of 5. That means, 2x is also multiple of 5. So, HCF of 2x and 5 = 5 .....

45. If c=a×b. Find the HCF and LCM of c and b.

Here, Here, we can see, c is the multiple of b. Hence, LCM=c and HCF=b. .....

46. Find the HCF of : a³-b³ and a²-b²

Here, Here, first expression: a3-b3 = (a-b)(a2+ab+b2) Second expression: a2-b2=(a-b)(a+b) Thus, the HCF = (a-b) .....

47. Find the LCM and HCF of x, y and z.

Here, The LCM of x,y and z = xyz The HCF of x,y and z .....

48.

Find the H.C.F. of:

9a² – 4b² – 8bc – 4c², 4c² – 4b² – 9a² – 12ab and 9a² + 12ac + 4c² – 4b²

Here,

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49. Find the H.C.F. of:

x² – 9 and x³ – 27

Here, 1st expression = x2 – 9 = x2 – 32 = (x + 3) (x – .....

50. The H.C.F. of:

x² + 2 + 1/x² and x + 1/x is

Here, Eirst expression : =x2 + 2 + 1/x2 = x2 + 2 .x.1/x + (1/x)2 .....