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Algebra: Basics Test-1

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Question 1
If a× b = 2a-3b+ab, then 3 × 5 +5 x 3 is equal to:
A
20
B
23
C
24
D
22
Question 1 Explanation: 
We are given by the statement that
$ \displaystyle \begin{array}{l}a\times b=2a-3b+ab\\\Rightarrow 3\times 5=2\times 3-3\times 5+3\times 5=6\\5\times 3=2\times 5-3\times 3+3\times 5\\=10-9+15=16\\\therefore \,\,3\times 5+5\times 3=6+16=22\end{array}$
Question 2
If p ×q = p × q + p/q , the value of 8 ×2 are:
A
13
B
14
C
15
D
16
Question 2 Explanation: 
$ \displaystyle \begin{array}{l}p\,\,\times \,q=p+q+\frac{p}{q}\\\therefore 8\times 2=8+2+\frac{8}{2}\\=10+4=14\end{array}$
Question 3
Two numbers a and b ( a > b) are such that their sum is equal to three times their difference. Then value of $ \displaystyle \frac{3ab}{2\left( {{a}^{2}}-{{b}^{2}} \right)}$ will be:
A
$ \displaystyle 1\frac{1}{2}$
B
$ \displaystyle 1\frac{2}{3}$
C
$ \displaystyle 1\frac{1}{3}$
D
1
Question 3 Explanation: 
From the given statement we can say that
$ \displaystyle \begin{array}{l}\left( a+b \right)=3\left( a-b \right)=3a-3b\\\Rightarrow 3b+b=3a-ra\\\Rightarrow 2a=4b\\\Rightarrow a=2b\\\Rightarrow \frac{a}{b}=\frac{2}{1}\\\Rightarrow \frac{a}{b}=\frac{1}{2}\\\therefore \,a=2,\,b=1\\\frac{3ab}{2\left( {{a}^{2}}-{{b}^{2}} \right)}=\frac{3\times 2\times 1}{2\times \left( 4-1 \right)}=\frac{6}{6}=1\end{array}$
Question 4
The value of <br> $ \displaystyle \left( 1+\frac{1}{p} \right)\left( 1\frac{1}{p+1} \right)\left( 1+\frac{1}{p+2} \right)\left( 1+\frac{1}{p+3} \right)$
is
A
$ \displaystyle 1+\frac{1}{p+4}$
B
$ \displaystyle p+4$
C
$ \displaystyle \frac{p+4}{p}$
D
$ \displaystyle \frac{1}{p}$
Question 4 Explanation: 
We can solve the given fraction as
$ \displaystyle \begin{array}{l}=\left( 1+\frac{1}{p} \right)\left( 1+\frac{1}{p+1} \right)\left( 1+\frac{1}{p+2} \right)\left( 1+\frac{1}{p+3} \right)\\=\frac{p+1}{p}\times \frac{p+2}{p+1}\times \frac{p+3}{p+2}\times \frac{p+4}{p+3}\\=\frac{p+4}{p}\end{array}$
Question 5
If a × b = 2 (a + b), then 5 ×2 is equal to:
A
16
B
12
C
14
D
18
Question 5 Explanation: 
Given statement is
$ \displaystyle \begin{array}{l}a\times b=2\left( a+b \right)\\\therefore \,5\times 2=2\left( 5+2 \right)\,\,\\=2\times 7=14\end{array}$
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