1). If \( \Large a\ast b = 2a-3b+ab \), then \( \Large 3\ast 5+5\ast 3 \) is equal to:
View Answer Correct Answer: 22 \( \Large 3 \ast 5 + 5 \ast 3 \) => \( \Large 3 \ast 5 = 2 \times 3 - 3 \times 5 + 3 \times 5 \) 6 - 15 + 15 = 6 > \( \Large 5 \ast 3 = 2 \times 5 - 3 \times 3 + 3 \times 5 \) = 10 - 9 + 15 =16 Therefore, \( \Large 3 \ast 5 + 5 \ast 3 \) => 6 + 16 = 22
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2). If \( \Large p \times q = p+q+\frac{p}{q} \), the va1ue of \( \Large 8 \times 2 \) is:
View Answer Correct Answer: 14 \( \Large 8 \times 2 = 8 + 2 + \frac{8}{2} \) => 10 + 4 = 14
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3). Two numbers x and y (x > y) such that their sum is equal to three times their difference. Then value of \( \Large \frac{3xy}{2 \left(x^{2}-y^{2}\right) } \) will be;
View Answer Correct Answer: 1 x > y given :- \( \Large x+y = 3 \left(x-y\right) \) => \( \Large x + y = 3x - 3y \) => \( \Large x - 3x = -3y - y \) => \( \Large -2x = -4y => x = 2y \) Therefore, \( \Large \frac{3xy}{2 \left(x^{2}-y^{2}\right) }=\frac{3 \times 2y \times y}{2 \left( \left(2y\right)^{2}-y^{2} \right) } \) = \( \Large \frac{6y^{2}}{2 \times \left(4y^{2}-y^{2}\right) } \) = \( \Large \frac{6y^{2}}{6y^{2}} = 1 \)
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4). The value of \( \Large \left(1+\frac{1}{x}\right) \left(1+\frac{1}{x+1}\right) \left(1+\frac{1}{x+2}\right) \left(1+\frac{1}{x+3}\right) \) is:
View Answer Correct Answer: \( \Large \frac{x+4}{x} \) \( \Large \left(1+\frac{1}{x}\right) \left(1+\frac{1}{x+1}\right) \left(1+\frac{1}{x+2}\right) \left(1+\frac{1}{x+3}\right) \) Taking L.C.M. of each term. => \( \Large \left(\frac{x+1}{x}\right) \left(\frac{x+1+1}{x+1}\right) \left(\frac{x+2+1}{x+2}\right) \left(\frac{x+3+1}{x+3}\right) \) => \( \Large \frac{1}{x} \times \left(x+4\right) \) => \( \Large \frac{x+4}{x} \)
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5). If \( \Large x = 7-4\sqrt{3} \), then the value of \( \Large x+\frac{1}{x} \)
View Answer Correct Answer: 14 \( \Large x = 7 - 4\sqrt{3} \) \( \Large \frac{1}{x} => \frac{1}{7-4\sqrt{3}} \) By ratonalisation, \( \Large \frac{1}{x}=\frac{1}{7-4\sqrt{3}} \times \frac{7+4\sqrt{3}}{7+4\sqrt{3}} \) = \( \Large \frac{7+4\sqrt{3}}{49-48}= 7 + 4\sqrt{3} \) Therefore, \( \Large x+\frac{1}{x} = 7 - 4\sqrt{3} + 7 + 4\sqrt{3} \) = 14
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6). If \( \Large \frac{a}{b}=\frac{2}{3}\ and\ \frac{b}{c}=\frac{4}{5} \) , then the ratio \( \Large \frac{a+b}{b+c} \) equal to:
View Answer Correct Answer: \( \Large \frac{20}{27} \) \( \Large \frac{a}{b}=\frac{2}{3}\ and\ \frac{b}{c}=\frac{4}{5}\ (given)\ or\ \frac{c}{b}=\frac{5}{4} \) Therefore, \( \Large \frac{a+b}{b+c}=\frac{b \left(\frac{a}{b}+1\right) }{b \left(\frac{c}{b}+1\right) } = \frac{\frac{a}{b}+1}{\frac{c}{b}+1} \) = \( \Large \frac{ \left(\frac{2}{3}+1\right) }{ \left(\frac{5}{4}+1\right) } = \frac{\frac{5}{3}}{\frac{5+4}{4}} \) = \( \Large \frac{5 \times 4}{3 \times 9} = \frac{20}{27} \) Therefore, \( \Large \frac{a+b}{b+c} = \frac{20}{27} \)
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7). If \( \Large a\ast b=2 \left(a+b\right) \), then \( \Large 5\ast 2 \) is equal to:
View Answer Correct Answer: 14 \( \Large a \ast b = 2 \left(a+b\right) \) \( \Large 5 \ast 2 = 2 \left(5+2\right) = 14 \)
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8). If \( \Large \frac{2a+b}{a+4b}=3 \), then find the value of \( \Large \frac{a+b}{a+2b} \)
View Answer Correct Answer: \( \Large \frac{10}{9} \) \( \Large \frac{2a+b}{a+4b}=3 \) \( \Large 2a + b = 3 \left(a+4b\right) \) \( \Large 2a + b = 3a + 12b \) => \( \Large -a = 11b \) a = -11b Therefore, \( \Large \frac{a+b}{a+2b} \) => \( \Large \frac{-11b+b}{-11b+2b} \) = \( \Large \frac{-10b}{-9b} = \frac{10}{9} \)
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9). If \( \Large a\ast b=a+b+ab \), then \( \Large 3\ast 4-2\ast 3 \) is equal to:
View Answer Correct Answer: 8 \( \Large a \ast b = a + b + ab \) \( \Large 3 \ast 4 = 3 + 4 + 3 \times 4 = 19 \) \( \Large 2 \ast 3 = 2 + 3 + 2 \times 3 = 5 + 6 = 11 \) Therefore, \( \Large 3 \ast 4 - 2 \ast 3 = 19 - 11 = 8 \)
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10). If \( \Large x \circledast y = 3x+2y \), then \( \Large 2 \circledast 3+3 \circledast 4 \) is equal to
View Answer Correct Answer: 29 \( \Large x \circledast y = 3x + 2y \) \( \Large \left(2 \circledast 3\right)= 3 \times 2 +2 \times 3 = 6 + 6 = 12 \) \( \Large \left(3 \circledast 4\right) = 3 \times 3 + 2 \times 4 = 9 + 8 = 17 \) Therefore, \( \Large \left(2 \circledast 3\right)+ \left(3 \circledast 4\right) = 12 + 17 = 29 \)
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