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Algebraic Expressions & Identities Test - 6

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Algebraic Expressions & Identities Test - 6
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  • Question 1
    1 / -0
    Multiply 4x4 + x3 + y2 by 2x2 + y.
    Solution
    (4x4 + x3 + y2) (2x2 + y)

    = 2x2 (4x4 + x3 + y2) + y(4x4 + x3 + y2)

    = 8x6 + 2x5 + 2x2y2 + 4x4y + x3y + y3
  • Question 2
    1 / -0
    Which of the following is not a standard identity for algebraic expressions? (Where a and b are two variables)
    Solution
    If a and b are any two variables then,
    (a - b)3 = a3 - b3 - 2ab is not a correct identity.
    Therefore, identity in option 4 is not a standard identity.
  • Question 3
    1 / -0
    Simplify .
    Solution
    .

    =

    =


    =
  • Question 4
    1 / -0
    Square of (4y - 2xy) is
    Solution
    (4y - 2xy)2

    We know: (a - b)2 = a2 + b2 - 2ab

    = (4y)2 + (2xy)2 - 2(4y)(2xy)

    = 16y2 + 4x2y2 - 16xy2
  • Question 5
    1 / -0
    If , then find the value of .
    Solution
  • Question 6
    1 / -0
    If 4x + 3y = 30 and = 3, then find the value of 2x + 4y2.
    Solution
    4x + 3y = 30 --- (1)

    = 3
    x = 3y
    Put the value of x in equation (1):
    4(3y) + 3y = 30
    12y + 3y = 30
    y = 2
    x = 6
    Hence, 2x + 4y2 = 2(6) + 4(2)2
    = 12 + 16 = 28
  • Question 7
    1 / -0
    The product of (2xy + 3x2 + 6xy2) and (x2 + y) is
    Solution
    (2xy + 3x2 + 6xy2) (x2 + y)
    = (x2) (2xy + 3x2 + 6xy2) + y(2xy + 3x2 + 6xy2)
    = 2x3y + 3x4 + 6x3y2 + 2xy2 + 3x2y + 6xy3
  • Question 8
    1 / -0
    Find the missing term in the following equation:

    = ?
    Solution


    =

    =

    = 4x2
  • Question 9
    1 / -0
    What should be subtracted from 6x2 + 4y + 8 to get 3x2 + 10y + 4 as the result?
    Solution
    Subtract 3x2 + 10y + 4 from 6x2 + 4y + 8 i.e.

    6x2 + 4y + 8
    (-)3x2 + (-)10y +(-) 4
    -------------------------
    3x2 - 6y + 4
    -------------------------
  • Question 10
    1 / -0
    Simplify:

    .
    Solution


    =

    =

    =
  • Question 11
    1 / -0
    What must be added to 16y2 - 4y + 8 to get 6y2 - 2y - 4 as the result?
    Solution
    Let the term to be added be denoted by 'a'.

    Then,+ a =

    a =- ()

    a =-

    a =
  • Question 12
    1 / -0
    If , then find the value of .
    Solution


    Squaring both sides, we get:





    = 25 - 4

    = 21

  • Question 13
    1 / -0
    Multiply by .
    Solution


    =

    =
  • Question 14
    1 / -0
    Simplify the following expression:

    Solution


    =

    =
  • Question 15
    1 / -0
    Find the value of 'a' if and xy = 2.
    Solution






    Given: xy = 2
    a = 5 × 2
    a = 10
  • Question 16
    1 / -0
    Thomas is making a cuboidal box for his craft class at school. The length and breadth of the box are 7ab and 4a2b2bc2. If the volume of the box is 84a3b4c5, then find the height of the box.
    Solution
    Volume of cuboid = length x breadth x height

    84a3b4c5 = 7ab x (4a2b2bc2) x height



    h = 3bc3
  • Question 17
    1 / -0
    Sahil works as a postman. He gets paid for his work based on a formula: 3l(4l - 5) + 3, where l is the number of letters he delivers. If he delivers 5 letters on Monday, then how much (in Rs.) will he get paid for it?
    Solution
    The formula used to calculate his pay is: 3l(4l - 5) + 3 or (12l2 - 15l + 3)
    Now, he delivers 5 letters.
    Put l = 5 in the formula:
    12(5)2 -15(5) + 3
    = 12(25) - 75 + 3
    = 300 - 75 + 3
    = Rs. 228
  • Question 18
    1 / -0
    Katie makes two rectangular boxes. One box has a length, breadth and height of 5a, 3a2 and 7a4, respectively, while the other box has a length, breadth and height of xy, 2x2y and 2xy2, respectively. Now, she joins them together to make a new rectangular box. Find the volume of the new box.
    Solution
    Volume of new box = Volume of first box + Volume of second box
    Now, volume of first box = 5a x 3a2 x 7a4 = 5 x 3 x 7 x a x a2 x a4 = 105 a7 ... (1)
    Volume of second box = xy x 2x2y x 2xy2 = 2 x 2 x xy x x2y x xy2
    = 4x4y4 ... (2)
    Volume of new box = (1) + (2)
    = 105a7 + 4x4y4
  • Question 19
    1 / -0
    Sumit opens a new account in ABS bank. In the first month, he deposits amount p in his account. In the second month, he deposits amount p2 in his account. In the third month, he deposits thrice the amount deposited by him in the first month. In the fourth month, he deposits an amount equal to the product of amounts deposited by him in the last three months. Find the amount deposited by him in the fourth month.
    Solution
    Amount deposited in the first month = p
    Amount deposited in the second month = p2
    Amount deposited in the third month = 3p
    Amount deposited in the fourth month = (p x p2 x 3p)
    = 3p1 + 2 + 1
    = 3p4
  • Question 20
    1 / -0
    A tortoise and a rabbit were having a race. For every (k(k - j) + j(j - h)) kilometres covered by the tortoise, the rabbit covered three times that distance and an extra h(h - k) kilometres. Find the distance travelled by the rabbit for every (k(k - j) + j(j - h)) kilometres covered by the tortoise.
    Solution
    Required distance = [3 x {(k(k - j) + j(j - h))} + h(h - k)]
    = [3 x {(k2 - kj + j2 - jh)} + h2 - hk)]
    = [3k2 - 3kj + 3j2 - 3jh + h2 - hk]
    = (3k2 + 3j2 + h2 - 3kj - 3jh - hk) kilometres

  • Question 21
    1 / -0
    Directions: Study the following statements and choose the correct option accordingly.

    Statement I: The product of (8p2 + 5q) and (8p2 + 5q) at p = 2 and q = 3 is 541.
    Statement II: The value of is 7.
    Solution
    Statement I:

    (8p2 + 5q)(8p2 + 5q)

    = (8p2 + 5q)2
    = 64p4 + 25q2+ 80p2q
    Given, p = 2 and q = 3.
    So, 64()+ 25()+ 80 ()(3)
    = 1024 + 225 + 960 = 2209

    Statement II:

    =

    =

    = 8.39
  • Question 22
    1 / -0
    Match the following:

    Column - I Column - II
    A. (4x2 - 5xy)(4x2 - 5xy) (i) 16x4 + 25 (xy)2 - 40x3y
    B. (7x - 3x)(4x2 - 9xy) (ii) 16x2 - 18x2y - 8xy + 9xy2
    C. (8x2 - 9xy)(2x2 - 4xy) (iii) 16x3 - 36x2y
    D. (2x - y)(8x - 9xy) (iv) 16x4 - 50x3y + 36(xy)2

    Choose the correct option from the following:

    A B C D
    P (i) (iii) (iv) (ii)
    Q (i) (iv) (iii) (ii)
    R (ii) (iii) (iv) (i)
    S (i) (ii) (iv) (ii)
    Solution
    (A) (4x2- 5xy)(4x2 - 5xy)
    = (4x2 - 5xy)2
    = 16x4 + 25(xy)2 - 40x3y

    (B) (7x - 3x)(4x2 - 9xy)
    = 4x(4x2 - 9xy)
    = 16x3 - 36x2y

    (C) (8x2 - 9xy)(2x2 - 4xy)
    = 16x4 - 32x3y - 18x3y + 36(xy)2
    = 16x4 - 50x3y + 36(xy)2

    (D) (2x - y)(8x - 9xy)
    = 16x2 - 18x2y - 8xy + 9xy2
  • Question 23
    1 / -0
    If p2 + q2 = 26 and pq = 5, then find the values of:

    (i) p + q, (ii) p - q, (iii) p4 + q4

    Choose the correct option from below:

    (i) (ii) (iii)
    A 6 4 626
    B 6 864
    C 4 745
    D 6 4 563
    Solution
    (i) (p + q)2 = p2 + q2 + 2pq
    (p + q)2 = 26 + 2(5) = 26 + 10 = 36
    p + q = = 6

    (ii) (p - q)2 = p2 + q2 - 2pq
    (p - q)2 = 26 - 2(5) = 26 - 10 = 16
    p - q = = 4

    (iii) p2 + q2 = 26
    Squaring both sides, we get:
    (p2)2 + (q2)2 + 2 p2q2 = (26)2
    p4 + q4 + 2(5)2 = (26)2
    p4 + q4 + 50 = 676
    p4 + q4 = 626
  • Question 24
    1 / -0
    Fill in the blanks:
    (i) Monomial is an algebraic expression consisting of ____P____ non-zero term(s) only.
    (ii) A monomialin variable x is an expression of the form CXk, whereC is a ______Q____ and k is a non-negative integer.
    (iii) Any two polynomials can be added, subtracted or multiplied, and the result will be a _____R_____.
    (iv) Terms, _____S_______, and coefficients make up mathematical expressions.

    Choose the correct option from below:

    P Q R S
    A polynomial factors one constant
    B factors constant polynomial one
    C one constant polynomial factors
    D polynomial one factors constant
    Solution
    (i) Monomial is an algebraic expression consisting of one non-zero term(s) only.
    a is a monomial in one variable a, 10ab2 is a monomial in two variables a and b.

    (ii) A monomialin variable x is an expression of the form CXk, whereC is a constant and k is a non-negative integer.
    A Polynomial in x is a sum of finitely many monomials in x. In other words, it is an expression of the form: P(x) = anxn +an−1 +···+a1x + a0.

    (iii) Any two polynomials can be added, subtracted or multiplied, and the result will be a polynomial.When polynomials are added, subtracted, or multiplied, the result is another polynomial. When polynomials are divided, the result is a rational expression. Polynomials can be added under the associative law of addition.


    (iv) Terms, factors, and coefficients make up mathematical expressions.
    A mathematical expression contains numbers, variables, symbols, and operators connected by addition, subtraction, multiplication, and division. These parts are called terms.

    In a mathematical expression, factors are parts of the expression that are connected by multiplication.

    A coefficient is a number that is multiplied by a variable in a mathematical expression.

  • Question 25
    1 / -0
    Simplify the following:

    =
    Solution
    =

    (p2 + 15p - 20)(p + 5) = (p2 + 20p - 3)(p - 8)

    p3 + 15p2 - 20p + 5p2 + 75p - 100 = p3 + 20p2 - 3p - 8p2 - 160p + 24

    p3 + 20p2 + 55p - 100 = p3+ 12p2 - 163p + 24

    8p2 + 218p - 124 = 0
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