Question : What are the values of $x$ and $y$, respectively, from the following equations? $6x + 7y = 5xy$ $10y – 4x = 4xy$
Option 1: 3 and 4
Option 2: 4 and 5
Option 3: 2 and 4
Option 4: 2 and 5
Correct Answer: 2 and 4
Solution : Here, we have, $6x + 7y = 5xy$ .................( 1) $10y - 4x = 4xy$.................( 2) On dividing eq(1) and eq(2) by xy, we get, $⇒ \frac{6}{y} + \frac{7}{x} = 5$............(3) $⇒ \frac{10}{x} - \frac{4}{y} = 4$..............(4) Now, multiply eq(3) by 2 and eq(4) by 3, we get, $⇒ \frac{12}{y} + \frac{14}{x} = 10$ ...............(5) $⇒ \frac{30}{x} - \frac{12}{y} = 12$..............( 6) Now, eq(5) + eq(6) $⇒ \frac{14}{x} + \frac{30}{x} = 22$ $⇒ \frac{44}{x} = 22$ $⇒x = 2 $ Now, put the value of $x = 2$ in the eq(1), we get, $⇒ 6 × 2 + 7y = 5 × 2y$ $⇒ 12 = 3y$ $⇒ y = 4$ $\therefore$ The values of $x$ and $y$ are 2 and 4, respectively. Hence, the correct answer is 2 and 4.
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Question : If $x^4+x^2 y^2+y^4=21$ and $x^2+xy+y^2=3$, then what is the value of $4xy $?
Option 1: –8
Option 2: 4
Option 3: –4
Option 4: 12
Question : If $\left(5 \sqrt{5} x^3-3 \sqrt{3} y^3\right) \div(\sqrt{5} x-\sqrt{3} y)=\left(A x^2+B y^2+C x y\right)$, then what is the value of $(3 A-B-\sqrt{15} C)$?
Option 1: –3
Option 2: –5
Option 3: 8
Question : If $x+y+z=17, x y z=171$ and $x y+y z+z x=111$, then the value of $\sqrt[3]{\left(x^3+y^3+z^3+x y z\right)}$ is:
Option 1: –64
Option 3: 0
Option 4: –4
Question : Directions: Select the option that arranges the following words in a logical and meaningful way. 1. Mansion 2. Cottage 3. Palace 4. Cabin 5. Villa
Option 1: 2–4–1–5–3
Option 2: 2–1–3–4–5
Option 3: 4–2–5–1–3
Option 4: 3–1–2–4–5
Question : If $x + y + z = 19, x^2 + y^2 + z^2 = 133$ and $xz = y^2, x > z > 0,$ what is the value of $(x - z)$?
Option 1: 5
Option 2: 0
Option 3: –2
Option 4: –5
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