Interest

Interest

Interest is the time value of money. We may say this is the cost of using capital.

Principal:– It is the borrowed amount.

Amount-Sum total of interest and principal.

Rate :– It is the rate percentage payable on the amount borrowed.

Period:-It is the time for which the principal is borrowed.

Simple Interest- Simple interest is payable on principal.

Example question: The compound interest on a certain sum for two years at 20% per annum is Rs 770. If the same sum is lent for 3 years at simple interest for 1% per month, find the interest obtained on the sum. A.Rs 525 B. Rs 550 C. Rs 660 D. Rs 630 E. None of these\frac{Simple Interest}{period.}

\frac{pxrxt}{100.}=where P is the Principal, R is the interest rate, T, is the time Compound Interest \frac{Compound Interest}{interest..}– Compound interest is payable on amount, i.e. on principal and \frac{Simple Interest}{period..} =\frac{pxrxT}{100}, where P is the Principal, R is the interest rate, T, is the time

 

 

 

 

Simplification

Algebric expression contain alphabetic symbol as well as numbers. When an algebraic expression is simplified, an equivalent expression is found and that is simpler than the original.

Let us remember BODMAS rule.

B- Bracket O means Of or Orders, i.e power and square roots.

D means Division

M means Multiplication

A stands for Addition

S stands for Subtraction.

Surds

number of the form\sqrt{n}, i.e irrational numbers.,Ex-\sqrt{2}\sqrt{3} , 6\sqrt{5}-9\sqrt{7} etc

For simplification surds are simplified to same order.

Ex- ,\sqrt{45}-3\sqrt{20},+4\sqrt{5},-6\sqrt{5},+4\sqrt{5} =\sqrt{5}

Some law of indices- and algebra equations.

 a^{m}\times a^{n} = a^{m+n}

 a^{m}\div a^{n} = a^{m-n}

\left ( a^{m} \right )n a^{mn}

 a^{0} =1

Example1.

1\left ( 17^{3.5} \right )\times 17^{2}= 17^{8}

A. 2.29

B. 2.75

C. 4.25

D. 4.5

Answer: Option D

Explanation:

Let \left ( 17^{3.5} \right )\times 17^{x}= 17^{8} .

Then, \left ( 17^{3.5+x} \right ) =  17^{8} .

∴3.5 + x = 8

⇒x = (8 – 3.5)

⇒x = 4.5  

2. If \left ( \frac{a}{b} \right )x-1 =\left ( \frac{b}{a} \right )x-3, then the value of x is:

A.\frac{1}{2}

B. 1

C. 2

D.\frac{7}{2}

Answer: Option C

Explanation:

       

 

 

 

 

  3. Given that  10^{0.48} = x,  10^{0.70}= y and  x^{2}= y^{2}, then the value of z is close to:

A. 1.45

B. 1.88

C. 2.9

D. 3.7

Answer: Option C

Explanation:

 x^{2} = y^{2} ⇔ 10^{0.48} = 10^{(2 x 0.70)}

⇒0(0.48z) = 1.40

⇒z =\frac{140}{48} =\frac{35}{12}= 2.9 (approx.)

4. If  5^{a} = 3125, then the value of  5^{(a - 3)} is:

A. 25

B. 125

C. 625

D. 1625

Answer: Option A

Explanation:

 5^{a }= 3125 ⇒  5^{a } = 5^{5} ⇒a = 5. ∴ 5^{a-3} = 5^{a-3} =  5^{2}= 25. 5. If  3^{(x - y)} = 27 and  3^{(x + y)} = 243, then x is equal to:

A. 0

B. 2

C. 4

D. 6

Answer: Option C

Explanation:

 3^{x - y}= 27 =  3^{3} ⇔ x – y = 3 ….(i)

 3^{x - y} =243 = 3^{5} ⇔  x + y = 5 ….(ii)

On solving (i) and (ii), we get x = 4.

6.  (256)^{0.16} (256)^{0.09} = ?

A. 4

B. 16

C. 64

D. 256.25

Answer: Option A

Explanation:

 (256)^{0.16} ×  (256)^{0.09} =(256) (256)^{0.16} x (256)0.09

= (256) (256)^{(0.16 + 0.09)}

= (256)^{0.25}

= (256)^{(25/100)}

= (256)^{(1/4)}

=\left ( 4^{4} \right )(1/4) 

=\left ( 4^{4} \right )(1.4)

= 4^{1}

= 4

7. The value of  \left [ (10)^{[150}\div (10)^{146} \right ]

A. 1000

B. 10000

C. 100000

D. 106

Answer: Option B

 

 

 

Explanation:

A. 0

B. 1

C.  x^{a - b - c}

D. None of these

Answer: Option B

Explanation:

 

 

 

 

 

Given Exp. =   9.  (25)^{7.5 }× (5)^{2.5 } ÷  (125)^{1.5 } (5)^{*}

A. 8.5

B. 13

C. 16

D. 17.5

E. None of these

Answer: Option B

Explanation:

Let (= (25)^{7.5 }× (5) (5)^{2.5 } ÷  (125)^{1.5 } (5)^{x } .        

 

 

 

 

 

  ∴ x = 13. 10.  (0.04)^{-1.5 }= ?

A. 25

B. 125

C. 250

D. 625

Answer: Option B

Explanation:

 = \left ( \frac{4}{100} \right )-1.5 \left ( \frac{1}{25} \right )-3/2

= 25^{(3/2)}

=\left ( 5^{2} \right )(3/2)

=\left ( 5^{2} \right )2\times(3/2)

= 5^{3} = 125.

 

        

A. 1

B. 2

C. 9

D 3^{n}

Answer: Option

 

 

C Explanation:  A. 1 B. 10 C. 121 D. 1000 Answer: Option D Explanation: We know that  11^{2} = 121. Putting m = 11

 

 

 

 

 

     B.\frac{1}{2}

C. 1

D.  a^{m + n}

Answer: Option C

Explanation:


      

 

 

 

 

13. If m and n are whole numbers such that m n = 121, the value of  (m - 1)^{n + 1} is:

A. 1
B. 10
C. 121
D. 1000
Answer: Option D
Explanation:

and n = 2, we get:

(m - 1) ^{n + 1}(11 - 1) ^{(2 + 1)} 10^{3} = 1000.    

 

 

A.  x^{abc}

B.1

C.  x^{ab + bc + ca}

D.  x^{a + b + c}

Answer:

Option B

Explanation:

Given Exp.

 x^{(b - c)(b + c - a)} .  x^{(c - a)(c + a - b)} . x^{(a - b)(a + b - c)}

= x^{(b - c)(b + c) - a(b - c)} .  x^{(c - a)(c + a) - b(c - a)}

 x^{(a - b)(a + b) - c(a - b)}

 x^{(b2 - c2 + c2 - a2 + a2 - b2)} .  x^{-a(b - c) - b(c - a) - c(a - b)}

=\left ( x^{0}\times x^{0} \right )

= (1 x 1)

= 1.  

15. If x = 3 + 22, then the value of x –\left ( x-\frac{1}{x} \right )

A. 1

B. 2

C. 22

D. 33

Answer: Option B

Explanation:

\left ( x-\frac{1}{x} \right )2 = x+\frac{1}{x}-2

= (3 + 22) +\frac{1}{(3 + 22)}-2

= (3 + 22) +\frac{1}{(3 + 22)} × \frac{(3 - 22)}{(3 - 22)}-2

= (3 + 22) + (3 – 22) – 2 = 4.

\left ( x-\frac{1}{x} \right )=2

 (a+b)^{2}= a^{2}+2ab+ b^{2}

 (a-b)^{2}=a^{2}-2ab+ b^{2}

 a^{2}–   b^{2}=(a+b)(a-b)

 a^{2} +  b^{2} =  (a+b)^{2}– 2ab or  (a-b)^{2}+2ab

(a+b)2 + (a-b)^{2} =2( a^{2}+ b^{2} )

 (a+b)^{2}  (a-b)^{2} =4ab

 (a+b)^{3}= a^{3} +3 a^{2} b+3a b^{2} + b^{3}

(a-b)^{3} = a^{3}-3  a^{2}b+3a b^{2} b^{3}

 a^{3} +  b^{3} (a+b)^{3}-3ab(a+b)also  a^{3} + b^{3}=(a+b)(a^{2}-2ab+ b^{2})

 a^{3} –  b^{3}(a-b)^{3}-3ab(a-b)also  a^{3} – b^{3}=(a-b)(a^{2}+2ab+ b^{2})

 (a+b+c)^{2}= a^{2}+ b^{2}+c^{2}+2ab+2bc+2ca a^{3}+b^{3}+c^{3}-3abc=(a+b+c)( a^{2}+ b^{2}+c^{2}–ab- bc -ca)

Square Root and Cube Root

1. Square Root: If x^{2}= y, we say that the square root of y is x and we write \sqrt{y}= x. Thus,\sqrt{4} = 2,\sqrt{9} = 3,\sqrt{196} = 14.

2. Cube Root:

The cube root of a given number x is the number whose cube is x. i.e multiplied by itself 3 times.

We, denote the cube root of x by 3\sqrt{x} Thus, 8 = 2 x 2 x 2 = 2^{3}, 343 = 7 x 7 x 7 = 7^{3} etc. Note: 1. =\sqrt{xy} =\sqrt{x} × \sqrt{y} \sqrt{\frac{x}{y}} =\frac{\sqrt{x}}{y}  

Examples \sqrt{64}= 8 or – 8 \sqrt{625}= 25 or – 25 In algebra if α is a root and β are two roots of a quadratic equation a x^{2} + bx + c = 0 Then 

         

 

 

 

 

      Cube root of unit Let 3\sqrt[]{1}= x ⇒1 =  x^{3} => (x-1)( x^{2} + x + 1) => x = 1,\frac{-1+\sqrt{1-4}}{2} ,  i.e 1, ,  x has three roots, 1, ,  Say 1, w and w2  1 + w + w2 = 0  And w 3 = 1