High School

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28. The (HCF × LCM) for the numbers 50 and 20 is:
(a) 1000
(b) 50
(c) 100
(d) 500

29. HCF of two consecutive even numbers is:
(a) 0
(b) 1
(c) 2
(d) 4

30. The pair of linear equations 2x = 5y + 6 and 15y = 6x - 18 represents two lines which are:
(a) intersecting
(b) parallel
(c) coincident
(d) either intersecting or parallel

31. What is the value of k such that the following pair of equations have infinitely many solutions? x - 2y = 3 and - 3x + ky = - 9
(a) (-6)
(b) -3
(c) 3
(d) 6

32. The sum of the digits of a two-digit number is 9. If 27 is subtracted from the number, its digits are interchanged. Which of these is the product of the digits of the number?
(a) 8
(b) 14
(c) 18
(d) 20

Answer :

Let's address each of the numbered questions separately:

  1. The product of the Highest Common Factor (HCF) and the Least Common Multiple (LCM) of two numbers is equal to the product of the numbers themselves. For the numbers 50 and 20, their product is:

[tex]50 \times 20 = 1000[/tex]

Thus, the correct answer is (a) 1000.

  1. The HCF of two consecutive even numbers is generally 2. Consecutive even numbers are numbers like 2 and 4, 10 and 12, etc., where the difference is always 2, making the smallest factor 2.

Thus, the correct answer is (c) 2.

  1. We need to determine the relationship between the two lines represented by the equations:

  2. [tex]2x = 5y + 6[/tex]

  3. [tex]15y = 6x - 18[/tex]

First, rewrite them into the standard form of lines, i.e., [tex]Ax + By = C[/tex]:

For equation 1, rearrange to get: [tex]2x - 5y = 6[/tex].

For equation 2, rearrange to get: [tex]6x - 15y = 18[/tex].

To check if the lines are parallel, coincident, or intersecting, compare the ratios [tex]\frac{A_1}{A_2}, \frac{B_1}{B_2},[/tex] and [tex]\frac{C_1}{C_2}[/tex]:

  • [tex]\frac{2}{6} = \frac{1}{3}[/tex]

  • [tex]\frac{-5}{-15} = \frac{1}{3}[/tex]

  • [tex]\frac{6}{18} = \frac{1}{3}[/tex]

Since all ratios are equal, the lines are coincident.

Thus, the correct answer is (c) coincident.

  1. For the pair of equations to have infinitely many solutions, they must be equivalent. Consider the equations:

  2. [tex]x - 2y = 3[/tex]

  3. [tex]-3x + ky = -9[/tex]

The second equation is a multiple of the first. Multiply the first equation by 3 to get:

[tex]3(x - 2y) = 9[/tex]

[tex]3x - 6y = 9[/tex]

For the second equation [tex]-3x + ky = -9[/tex] to be equivalent to [tex]3x - 6y = 9[/tex], [tex]k = -6[/tex].

Thus, the correct answer is (a) (-6).

  1. Let's consider the properties of the two-digit number. Let the number be [tex]10a + b[/tex], where [tex]a[/tex] and [tex]b[/tex] are the digits. According to the question:

  • [tex]a + b = 9[/tex]

  • If 27 is subtracted, the digits interchange, so:

    [tex]10a + b - 27 = 10b + a[/tex]

    Simplifying gives: [tex]9a - 9b = 27[/tex] or [tex]a - b = 3[/tex].

Solve the system of equations:

  1. [tex]a + b = 9[/tex]

  2. [tex]a - b = 3[/tex]

Adding the equations gives: [tex]2a = 12[/tex], so [tex]a = 6[/tex].

Substitute back to find [tex]b[/tex]: [tex]6 + b = 9[/tex], so [tex]b = 3[/tex].

Therefore, the number is 63 and the product of the digits is [tex]6 \times 3 = 18[/tex].

Thus, the correct answer is (c) 18.

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