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Algebra Exercise 2

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  1. Question 1 of 23
    1. Question

    If 2x + 3y = 78 and 3x + 2y = 72, what is the value of x + y?

    Hint

    2x + 3y = 78 …(i)

    3x + 2y = 72 …(ii)

    From equations (i) and (ii),

    x = 12, y = 18

    ∴ x + y = 12 + 18 = 30

  2. Question 2 of 23
    2. Question

    If x + y =18 and xy = 72, what is the value of (x)² + (y)²

    Hint

    x + y = 18

    ⇒ (x + y)² = 18² = 324

    ⇒ x² + y² + 2xy = 324

    ⇒ x² + y² = 324 – 2xy

    ⇒ x² + y² = 324 – 2(72)

    ⇒ x² + y² = 324 – 144 = 180

  3. Question 3 of 23
    3. Question

    If 3x + 5y = 44 and 10x–2y=16, what is the value of x?

    Hint

    3x + 5y = 44 …(i)

    10x – 2y = 16 …(ii)

    By equation (i) × 2 + equation (ii) × 5

    we have,

    6x + 10y + 50x – 10y = 88 + 80

    or 56x = 168

    ∴ x = algebra-26844.png = 3

  4. Question 4 of 23
    4. Question

    If 34x-2 = 729, then the value of X is

    Hint

    729 = 9³ = 3⁶, Now 4X – 2 = 6 ⇒ X = 2.

  5. Question 5 of 23
    5. Question

    If 2x+4 – 2x+2 = 3 then x is equal to

    Hint

    2x + 4 – 2x + 2 = 3

    ⇒ 2x + 2 (2² – 1) = 3

    ⇒ 2x + 2 = 2⁰

    ∴ x + 2 = 0 ⇒ x = –2

  6. Question 6 of 23
    6. Question

    If 4x = 5y = 20z then z is (x, y, z ≠ 0)

    Hint

    4 × 5 = 20z/x × 20z/y = 20z/x + z/y = 20

    i.e. 1 = z/x + z/y

    i.e. z = xy/(x + y)

  7. Question 7 of 23
    7. Question

    If 26754.png, then x is

    Hint

    We have, algebra-26838.png

    ⇒ algebra-26832.png

    ⇒ algebra-26826.png

    ⇒ algebra-26820.png

    ⇒ x = algebra-26813.png

  8. Question 8 of 23
    8. Question

    If xabc = xaxbxc and x are all positive integers, then a² + b² + c² =?

    Hint

    xabc = xaxbxc

    ⇒ xabc = xa + b + c

    ⇒ abc = a + b + c.

    This happens for positive integer value of a, b, c, only when a, b, c take value of 1, 2 & 3.

    Hence, a² + b² + c² = 1² + 2² + 3² = 1 + 4 + 9 = 14.

  9. Question 9 of 23
    9. Question

    The pure form of the surd 25697.pngis

    Hint

    algebra-26353.png

    algebra-26347.png

  10. Question 10 of 23
    10. Question

    A student was asked to solve the fraction 25611.pngand his answer was ¼. By how much was his answer wrong?

    Hint

    algebra-26299.png

    ∴ Required answer

    algebra-26293.png

  11. Question 11 of 23
    11. Question

    A man plants 15376 apple trees in his garden and arranges them so that there are as many rows as there are apples trees in each row. The number of rows is:

    Hint

    algebra-26287.png

    ∴ Number of rows = 124.

  12. Question 12 of 23
    12. Question

    Number of prime factors in 25586.pngis

    Hint

    algebra-26280.png

    = algebra-26272.png

    = algebra-26266.png

    = algebra-26260.png

    = algebra-26254.png

    Hence, the number of prime factors = (2 + 3 + 2) = 7

  13. Question 13 of 23
    13. Question

    The difference between two numbers is 3 and the difference of their squares is 63. Find the greater number.

    Hint

    Let the larger and smaller numbers be x and y respectively.

    Then, x – y = 3 …(i)

    and, x² – y² = 63

    ⇒ (x + y) (x – y) = 63

    algebra-26576.png …(ii)

    From equation (i) and (ii), x = 12

  14. Question 14 of 23
    14. Question

    How much less is ⅘ of 1150 from ⅚ of 1248?

    Hint

    algebra-26570.png

    = 1040 – 920 = 120

  15. Question 15 of 23
    15. Question

    The fourth root of 28 + 1625580.png is:

    Hint

    algebra-26248.png = algebra-26242.png

    = algebra-26236.png = algebra-26230.png

    = algebra-26224.png = algebra-26218.png

    = algebra-26212.png = algebra-26206.png + 1

  16. Question 16 of 23
    16. Question

    Find the value of xa + yb, if ay = 19683, where y is a multiple of a and bx = 1024, where b is a factor of x. a, b, x and y being positive integers:

    Hint

    ay = 19683

    ⇒ 3⁹ = 27³ = (19683)¹ = 19683

    As y is a multiple of a, a = 3, y = 9, bx = 1024

    ⇒ 2¹⁰ = 4⁵ = 32² = (1024)¹ = 1024

    As b is a factor of x, b = 2, x = 10

    Now, xa + yb

    ⇒ 10³ + 9² = 1000 + 81 = 1081.

  17. Question 17 of 23
    17. Question

    What is the difference in the sum of squares and the difference of squares of n and a when25537.pngand the sum the of a and n is 12.

    Hint

    Given, a + n = 12 ⇒ n = 12 – a

    ∴ Given expression becomes

    algebra-26200.png = 212 – a . 312 – a

    By comparing, power of 3, we get a + 2 = 12 – a ⇒ a = 5 and n = 7

    ∴ Required answer = (a² + n²) – (n² – a²) = (49 + 25) – (49 – 25) = 50

  18. Question 18 of 23
    18. Question

    If mxm – nxn = 0, then what is the value of 25530.png+ 25523.pngterms of xn?

    Hint

    mxm = nxn

    ∴ xm = algebra-26193.png

    ∴ Given algebra-26187.png

    =algebra-26180.png

    = algebra-26174.png algebra-26168.png = algebra-26162.png.

  19. Question 19 of 23
    19. Question

    Three of the following four parts are exactly equal. Which parts is not equal to the other three?

    Hint

    Check all options one by one.

    (a) algebra-26151.png

    (b) algebra-26146.png+ (15.2 × 4) = 100

    (c) (15 × 8) – (5 × 4) = 100

    (d) (76 × 1.5) – (5.5 × 2.6) = 99.7

  20. Question 20 of 23
    20. Question

    Three of the following four parts are exactly equal. Which parts is not equal to the other three?

    Hint

    All others are equal to 100. But option (c) gives a value equal to 98.

  21. Question 21 of 23
    21. Question

    Three of the following four parts are exactly equal. Which parts is not equal to the other three?

    Hint

    Others equal 5650.

  22. Question 22 of 23
    22. Question

    Three of the following four parts are exactly equal. Which parts is not equal to the other three?

    Hint

    Except it, others are equal to 550.

  23. Question 23 of 23
    23. Question

    Three of the following four parts are exactly equal. Which parts is not equal to the other three?

    Hint

    Other parts are equal to 300.

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