SAMACHEER KALVI CLASS 10 ALGEBRA

 SAMACHEER KALVI CLASS 10

CHAPTER 3 ALGEBRA

IMPORTANT CONCEPT BASED QUESTIONS

TOTAL NUMBER OF QUESTIONS: 50

1. Solve the following system of linear equations in three variables

(i) x + y + z = 5
2x – y + z = 9
x – 2y + 3z = 16

2. Discuss the nature of solutions of the following system of equations

(i) x + 2y – z = 6, – 3x – 2y + 5z = -12 , x – 2z = 3

3. There are 12 pieces of five, ten and twenty rupee currencies whose total value is ₹105. But when first 2 sorts are interchanged in their numbers its value will be increased by ₹20. Find the number of currencies in each sort.

4. Vani, her father and her grand father have an average age of 53. One-half of her grand father’s age plus one-third of her father’s age plus one fourth of Vani’s age is 65. If 4 years ago Vani’s grandfather was four times as old as Vani then how old are they all now?

5. Find the GCD of the given polynomials by Division Algorithm

(i) x4 + 3x3 – x – 3, x3 + x2 – 5x + 3

(ii) 3x4 + 6x3 – 12x2 – 24x, 4x4 + 14x3 + 8x2 – 8x

6. Find the LCM of the given polynomials

(i) 4x2y, 8x3y2

(ii) 16m, -12m2n2, 8n2

(iii) p2 – 3p + 2, p2 – 4

7. Find the LCM and GCD for the following and verify that f(x) × g(x) = LCM × GCD

(i) 21x2y, 35xy2

(ii) (x3 – 1)(x + 1),(x3 + 1)

8. Find the LCM of each pair of the following polynomials
(i) a2 + 4a – 12, a2 – 5a + 6 whose GCD is a – 2

9. Reduce each of the following rational expression to its lowest form.

(i) x2−1x2+x

(ii) x2−11x+18x2−4x+4

 

(iii) p2−3p−402p3−24p2+64p

 10. Find the excluded values , if any of the following expressions.

(i) yy2−25

 (ii) x2+6x+8x2+x−2 

 11. Simplify

(i) 2a2+5a+32a2+7a+6÷a2+6a+5−5a2−35a−50

12. If a polynomial p(x) = x2 – 5x – 14 when divided by another polynomial q(x) gets reduced to x−7x+2 find q(x).

13. Subtract 1x2+2 from 2x3+x2+3(x2+2)2

14.

If A = 2x+12x−1, B = 2x−12x+1 find 1A−B – 2BA2−B2

15. Find the square root of the following.

(i) 400x4y12z16100x8y4z4
(ii) 4x2 + 20x + 25

(iii) 9x2 – 24xy + 30xz – 40yz + 25z2 + 16y2

 (iv) (4x2 – 9x + 2)(7x2 – 13x – 2)(28x2 – 3x – 1)

16. Find the square root of the following polynomials by division method

(i) x4 – 12x3 + 42x2 – 36x + 9

(ii) 37x2 – 28x3 + 4x4 + 42x + 9

17. Find the values of a and b if the following polynomials are perfect squares.

(i) 4x4 – 12x3 + 37x2 + bx + a

(ii) x4 – 8x3 + mx2 + nx + 16

18. Determine the quadratic equations, whose sum and product of roots are
(i) -9, 20

(ii) −32, -1

19. Find the sum and product of the roots for each of the following quadratic equations

(i) x2 + 3x – 28 = 0

(ii) 3 + 1a = 10a2

 20. Solve the following quadratic equations by factorization method

(i) 4x2 – 7x – 2 = 0

(ii) 3(p2 – 6) = p(p + 5)

(iii) a(a−7)−−−−−−−√ = 3 2–√

21. Solve the following quadratic equations by completing the square method
(i) 9x2 – 12x + 4 = 0

22. Solve the following quadratic equations by formula method
(i) 2x2 – 5x + 2 = 0

(ii) 36y2 – 12ay + (a2 – b2) = 0

23.If the difference between a number and its reciprocal is 245, find the number.

24. A bus covers a distance of 90 km at a uniform speed. Had the speed been 15 km/hour more it would have taken 30 minutes less for the journey. Find the original speed of the journey.

25.  A girl is twice as old as her sister. Five years hence, the product of their ages – (in years) will be 375. Find their present ages.
 

26.  There is a square field whose side is 10 m. A square flower bed is prepared in its centre leaving a gravel path all round the flower bed. The total cost of laying the flower bed and gravelling the path at ₹3 and ₹4 per square metre respectively is ₹364. Find the width of the gravel path. 

27. A garden measuring 12m by 16m is to have a pedestrian pathway that is meters wide installed all the way around so that it increases the total area to 285 m2. What is the width of the pathway? 

28. Determine the nature of the roots for the following quadratic equations
(i) 15x2 + 11x + 2 = 0

29. Find the value(s) of ‘k’ for which the roots of the following equations are real and equal.
(i) (5k – 6)2 + 2kx + 1 = 0
(ii) kx2 + (6k + 2)x + 16 = 0

30. If the roots of (a – b)x2 + (b – c)x + (c – a) = 0 are real and equal, then prove that b, a, c are in arithmetic progression.

31.  Write each of the following expression in terms of α + β and αβ

(i) α3β+β3α

(ii) (3α – 1) (3β – 1)

32. If α, β are the roots of 7x2 + ax + 2 = 0 and if β – α = 137 Find the values of a.

33. Graph the following quadratic equations and state their nature of solutions.
(i) x2 – 9x + 20 = 0
(ii) x2 – 4x + 4 = 0

34. Draw the graph of y = x2 – 4 and hence solve x2 – x – 12 = 0

35. Draw the graph of y = x2 + 3x – 4 and hence use it to solve x2 + 3x – 4 = 0 

36. If a matrix has 18 elements, what are the possible orders it can have? What if it has 6 elements?

37.  Find x, y and z if
(i) 

(ii) 

38. Find X and Y if  and 

39. Find x, y and z if  [x y – z z + 3] + [y 4 3] = [4 8 16]

40. Find x and y if x 
41.  Solve for x,y :

42.  A has ‘a’ rows and ‘a + 3 ’ columns. B has ‘6’ rows and ‘17 – b’ columns, and if both products AB and BA exist, find a, b? 

43. Given that


verify that A(B + C) = AB + AC

44.  If

then snow that A2 + B2 = I. 

45. 
prove that AAT = I.

46. Verify that A2 = I when
 

47.  

show that A2 – (a + d)A = (bc – ad)I2.
 

 48. 

show that A2 – 5A + 7I2 = 0
 

49. 
verify that (AB)T = BT AT 

50. 

Show that (i) A(BC) = (AB)C
(ii) (A-B)C = AC – BC
(iii) (A-B)T = AT – BT