10th Maths Unit 3 Important 5 Mark Questions

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UNIT - 3 (ALGEBRA)

Important 5 Marks Questionss

1. Solve: ¹/_2x + ¹/_4y - ¹/_3z = ¹/₄ ; ¹/_x = ¹/_3y ; ¹/_x - ¹/_5y + ⁴/_z = 2²/₁₅.

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

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

4. Find the GCD of 6x³ - 30x² + 60x - 40 and 3x³ - 12x² + 21x - 18.

5. Find the GCD of x⁴ + 3x³ - x - 3 and x³ + x² - 5x + 3.

6. Find the square root of x⁴ - 12x³ + 42x² - 36x + 9.

7. Find the values of m and n if x⁴ - 8x³ + mx² + nx + 16 is a perfect square.

8. Solve: qx² - (p + q)²x + (p + q)² = 0.

9. A flock of swans contained x² members. As the clouds gathered, 10x went to a lake and one-eighth of the members flew away to a garden. The remaining three pairs played about in the water. How many swans were there in total?

10. Simplify: [^{(b² + 3b - 28)}/_{(b² - 49)}] ÷ [^{(b² + 4b + 4)}/_{(b² - 5b - 14)}].

11. Simplify: ¹/_{(x² - 5x + 6)} + ¹/_{(x² - 3x + 2)} - ¹/_{(x² - 8x + 15)}.

12. 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 bus.

13. A passenger train takes 1 hour more than an express train to travel a distance of 240 km from Chennai to Virudhachalam. The speed of the express train is 20 km/hr more than that of the passenger train. Find the average speed of both the trains.

14. The hypotenuse of a right-angled triangle is 25 cm and its perimeter is 56 cm. Find the length of the smallest side.

15. 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.

16. If the roots of the equation (c² - ab)x² - 2(a² - bc)x + b² - ac = 0 are real and equal, prove that either a = 0 (or) a³ + b³ + c³ = 3abc.

17. Find the value of k for which the roots of the quadratic equation (5k - 6)x² + 2kx + 1 = 0 are real and equal.

18. If α, β are the roots of the equation 3x² + 7x - 2 = 0, find the values of:
(i) ^α/_β + ^β/_α
(ii) ^α²/_β + ^β²/_α

19. The roots of the equation x² + 6x - 4 = 0 are α, β. Find the quadratic equation whose roots are:
(i) α² and β²
(ii) ²/_α and ²/_β
(iii) α²β and β²α

20. If one root of the equation 2y² - ay + 64 = 0 is twice the other then find the values of a.

21. Find x and y if x

-3

+ y

-2

22. Find X and Y if X + Y =

7	0
3	5

and X - Y =

3	0
0	4

23. If A =

1	1
-1	2

, B =

1	-1
2	1
1	3

and C =

1	2
2	-1

, show that (AB)C = A(BC).

24. If A =

1	2	1
2	-1	1

and B =

2	-1
-1	4
0	2

, show that (AB)^T = B^TA^T.

25. Given that A =

1	3
5	-1

, B =

1	-1	2
3	5	2

, C =

1	3	2
-4	1	3

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

26. If A =

cos θ	0
0	cos θ

, B =

sin θ	0
0	sin θ

, then show that A² + B² = I.

27. If A =

a	b
c	d

and I =

1	0
0	1

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

28. If A =

3	1
-1	2

, show that A² - 5A + 7I₂ = 0.

29. Let A =

1	2
1	3

, B =

4	0
1	5

, C =

2	0
1	2

. Show that (A - B)C = AC - BC.

30. Find the value of x, y, z if

x - 3	3x - z
x + y + 7	x + y + z

1	0
1	6

31. Find the values of a and b if ax⁴ + bx³ + 361x² + 220x + 100 is a perfect square.

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