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>> Elementary Mathematics
Contents:
  1. Elementary Mathematics
    1. Area and perimeter
    2. Circles
    3. Clocks
    4. Factorisation
    5. Geometry
    6. Height and Distance
    7. Indices and Surd
    8. LCM and HCF
    9. Loci and concurrency
    10. Logarithms
    11. Polynomials
    12. Quadratic Equations
    13. Quadrilateral and parallelogram
    14. Rational expression
    15. Real Analysis
    16. Rectangular and Cartesian products
    17. Set theory
    18. Simple and Decimal fraction
    19. Simplification
    20. Statistics
    21. Straight lines
    22. Triangle
    23. Trigonometric ratio
    24. Trigonometry
    25. Volume and surface area

1). If \( \Large a\ast b = 2a-3b+ab \), then \( \Large 3\ast 5+5\ast 3 \) is equal to:
A). 22 B). 24
C). 26 D). 28
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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
2). If \( \Large p \times q = p+q+\frac{p}{q} \), the va1ue of \( \Large 8 \times 2 \) is:
A). 6 B). 10
C). 14 D). 16
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Correct Answer: 14

\( \Large 8 \times 2 = 8 + 2 + \frac{8}{2} \)

=> 10 + 4 = 14
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;
A). \( \Large \frac{2}{3} \) B). 1
C). \( \Large 1\frac{1}{2} \) D). \( \Large 1\frac{1}{3} \)
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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, \( \Large \frac{3xy}{2 \left(x^{2}-y^{2}\right) }=\frac{3 \times 2y \times y}{2 \left( \left(2y\right)^{2}-y^{2} \right) } \)

= \( \Large \frac{6y^{2}}{2 \times \left(4y^{2}-y^{2}\right) } \)

= \( \Large \frac{6y^{2}}{6y^{2}} = 1 \)
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:
A). \( \Large 1+\frac{1}{x+4} \) B). x+4
C). \( \Large \frac{1}{x} \) D). \( \Large \frac{x+4}{x} \)
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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.

=> \( \Large \left(\frac{x+1}{x}\right) \left(\frac{x+1+1}{x+1}\right) \left(\frac{x+2+1}{x+2}\right) \left(\frac{x+3+1}{x+3}\right) \)

=> \( \Large \frac{1}{x} \times \left(x+4\right) \)

=> \( \Large \frac{x+4}{x} \)
5). If \( \Large x = 7-4\sqrt{3} \), then the value of \( \Large x+\frac{1}{x} \)
A). \( \Large 3\sqrt{3} \) B). \( \Large 8\sqrt{3} \)
C). \( \Large 14+8\sqrt{3} \) D). 14
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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
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:
A). \( \Large \frac{20}{27} \) B). \( \Large \frac{27}{20} \)
C). \( \Large \frac{6}{8} \) D). \( \Large \frac{8}{6} \)
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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, \( \Large \frac{a+b}{b+c} = \frac{20}{27} \)
7). If \( \Large a\ast b=2 \left(a+b\right) \), then \( \Large 5\ast 2 \) is equal to:
A). 3 B). 10
C). 14 D). 20
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Correct Answer: 14

\( \Large a \ast b = 2 \left(a+b\right) \)

\( \Large 5 \ast 2 = 2 \left(5+2\right) = 14 \)
8). If \( \Large \frac{2a+b}{a+4b}=3 \), then find the value of \( \Large \frac{a+b}{a+2b} \)
A). \( \Large \frac{5}{9} \) B). \( \Large \frac{2}{7} \)
C). \( \Large \frac{10}{9} \) D). \( \Large \frac{10}{7} \)
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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, \( \Large \frac{a+b}{a+2b} \)

=> \( \Large \frac{-11b+b}{-11b+2b} \)

= \( \Large \frac{-10b}{-9b} = \frac{10}{9} \)
9). If \( \Large a\ast b=a+b+ab \), then \( \Large 3\ast 4-2\ast 3 \) is equal to:
A). 6 B). 8
C). 10 D). 12
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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, \( \Large 3 \ast 4 - 2 \ast 3 = 19 - 11 = 8 \)
10). If \( \Large x \circledast y = 3x+2y \), then \( \Large 2 \circledast 3+3 \circledast 4 \) is equal to
A). 18 B). 29
C). 32 D). 38
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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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