AP ICET 2013

Given,  $$a * b = a + b + ab$$  and

$$(1 * 2) * x = -13$$

$$\Rightarrow$$ $$(1 + 2 + 1.2) * x = -13$$

$$\Rightarrow$$ $$5 * x = -13$$

$$\Rightarrow$$ $$5 + x + 5x= -13$$

$$\Rightarrow$$ $$5 + 6x = -13$$

$$\Rightarrow$$ $$6x = -18$$

$$\Rightarrow$$ $$x = -3$$

Hence, the correct answer is Option B

Given,  $$a * b = a + b - \frac{ab}{2}$$

If x * c = x

$$\Rightarrow$$ x + c - $$\frac{\text{xc}}{2}$$ = x

$$\Rightarrow$$ c - $$\frac{\text{xc}}{2}$$ = 0

$$\Rightarrow$$ $$\frac{\text{xc}}{2}$$ = c

$$\Rightarrow$$  x = 2

$$\therefore\ $$The value of x satisfying x * c = x is x = 2

Hence, the correct answer is Option B

Solution

$$\frac{8^3. (27)^4. 6^5}{(36)^2.9^4.(18)^2}$$ =$$\frac{\left(\left(2^3\right)^3\ \left(3^3\right)^4\left(2\cdot3\right)^5\right)}{\left(2^2.3^2\right)\left(3^2\right)^4\left(2\cdot3^2\right)}\ =\frac{\left(2^9.3^{12}.2^5.3^5\right)}{\left(2^4.3^4.3^8.2^2.3^4\right)}=\frac{\left(2^{9+5}.3^{12+5}\right)}{\left(2^6.3^{16}\right)}\ =\ 2^{14-6}.3^{17-16}\ =\ 2^8.3^1$$

On comparing,

$$\alpha\ \ =\ 8,\ \beta\ \ =\ 1.\ And\ \alpha\ \ +\ \beta\ \ =\ 9$$

Hence 9 Answer

Solution :

we need to find the LCM of roots' power i.e. LCM of 3,5,4,2 i.e. 60

$$\sqrt[3]{4}$$ =$$4^{\frac{1}{3}}\ =\ 4^{\left(\frac{20}{60}\right)}$$  =  $$\left(4^{50}\right)^{\frac{1}{60}}$$

$$\sqrt[5]{7}$$ =$$7^{\frac{1}{5}}\ =\ 7^{\left(\frac{12}{60}\right)}$$  = $$\left(7^{12}\right)^{\frac{1}{60}}$$

$$\sqrt[4]{5}$$ =$$5^{\frac{1}{4}}\ =\ 5^{\left(\frac{15}{60}\right)}$$  = $$\left(5^{15}\right)^{\frac{1}{60}}$$

 $$\sqrt[2]{3}$$ =$$3^{\frac{1}{2}}\ =\ 3^{\left(\frac{30}{60}\right)}$$ = $$\left(3^{30}\right)^{\frac{1}{60}}$$

As all the external powers are same, we just need to arrange the internal powers in ascending order to find the lowest one.

On careful observation, and also relying on the fact that 7 has the lowest power.  $$\sqrt[5]{7}$$  is the lowest one.Answer.

Solution:

(a + b) : (b + c) : (c + a) = 2 : 3 : 4

Let the common factor of the ratio be x

So,

a+b = 2x

b+c = 3x

c+a = 4x

Adding all the 3 equations

2(a+b+c) = 9x

A/q 

a + b + c = 9

So, 2.9 = 9x

=> x=2

So a+b = 2.2 =4

c = (a+b+c) - (a+b) = 9 -4 = 5 Answer

Let circle's radius be r and square's side be a

Circle's area = $$\pi r^2$$

Square's Area = $$a^2$$

A both are same ,

$$\pi r^2$$ = $$a^2$$

$$\ \frac{a^2\ }{r^2}$$ = $$\pi\ $$

$$\ \frac{a\ }{r}$$ =  $$\sqrt{\ \pi\ }$$

= $$\sqrt{\ \pi\ }$$ :1 Answer

(Hint:taking the cue from the options we should get the term as a square of 3 added terms.)

Now 

$$\sqrt{6 + 2\sqrt{2} + 2\sqrt{3} + 2\sqrt{6}}$$ = 

$$=\sqrt{\ \ \left(\left(1+2+3\right)+2.1.\sqrt{\ 2}+2.1.\sqrt{\ 3}+2.\sqrt{\ 2}.\sqrt{\ 3}\right)}$$

As $$\sqrt{\ \left(a+b+c\right)^2}=\sqrt{a^2+b^2+c^2+2.a.b+2.b.c+2.c.a\ \ }$$

On comparison, 

$$a=1\ ,\ b=\sqrt{\ 2\ },c\ =\sqrt{\ 3}$$

Hence,

$$\sqrt{6 + 2\sqrt{2} + 2\sqrt{3} + 2\sqrt{6}}$$ = $$1 +\sqrt{2} + \sqrt{3}$$ Answer