Question 1

If a * b = a + b + a / b, Then the value of 12 * 4 is

Accepted Answer

Since a * b = a + b + a / b ∴ 12 * 4 = 12 + 4 + 12 / 4 = 12 + 4 + 3 = 19

Question 2

(4 x 4 x 4 x 4 x 4 x 4) 5 x (4 x 4 x 4 ) 8 ÷ (4) 3 = (64) ?

Accepted Answer

(4 x 4 x 4 x 4 x 4 x 4)5 x (4 x 4 x 4)8 ÷ (4)3 = (64)? Apply the law of Fractional Exponents and Laws of Exponents if a multiply n times a x a x a x....up to n times, then a x a x a x a ......up to n times = an By simplifying the equation ⇒ (46)5 x (43)8 x 1/(4)3 = (43)? ⇒ (4)30 x (4)24/(4)3 = (4)3 x ? ⇒ 430 + 24 - 3 = 43 x 7 And comparing the exponents both the sides ⇒ 451 = 43 x ? ⇒ 3 x ? = 51 ∴ ? = 51/3 = 17

Question 3

3x + 2y = 12 and xy = 6, them the value of 9x2 + 4y2 is?

Accepted Answer

Given equation are 3x + 2y = 12 ..............(i) xy = 6 .........................(ii) On squaring Eq. (i) on both sides, we will get (3x + 2y)2 = (12)2 ⇒ 9x2 + 4y2 + 12xy = 144 put the value of xy ⇒ 9x2+ 4x2 = 144 - 72 = 72 ⇒ 9x2+ 4x2 = 72

Question 4

If x + 1/x = 3, then x5 + 1/x5 is equal to?

Accepted Answer

x + 1/x = 3 ........................ (i) On squaring both side, we will get (x + 1/x)2 = (3)2 Use the Square algebra formula (a + b)2 = a2 + 2ab + b2 ⇒ x2 + 1/x2 + 2 = 9 ⇒ x2 + 1/x2 = 7 ..................(ii) Again squaring both sides, we will get (x2 + 1/x2)2 = (7)2 Use the Square algebra formula (a + b)2 = a2 + 2ab + b2 ⇒ x4 + 1/4 + 2 = 49 ⇒ x4 + 1/x4 = 47 .....................(iii) On cubing the equation (i) both side, we will get (x + 1/x)3 = (3)3 Use the cube algebra formula (a + b)3 = a3 + 3a2b + 3ab2 + b3 ⇒ x3+ 1/x3 + 3(x + 1/x) = 27 ⇒ x3 + 1/x3 + 9 = 27 [∴ (x + 1/x) = 3] ⇒ x3 + 1/x3 = 18 ...................(iv) On multiplying Eqs. (i) and (iii) , we get ∴ (x4 + 1/x4) (x + 1/x) = 47 x 3 ⇒ x5 + 1/x5 + x3 + 1/x3 = 141 ⇒ x5 + 1/x5 + 18 = 141 [from Eq. (iv)] ⇒ x5 + 1/x5 = 123

Question 5

If x + 1/x = 2, then what is value of x - 1/x ?

Accepted Answer

Given that, x + 1/x = 2 ...................(i) On squaring both sides, we get (x + 1/x)2 = 4 ⇒ x2 + 1/x2 + 2 = 4 ⇒ x2 + 1/x2 = 2 ..................................(ii) Now, we have (x - 1/x)2 = (x2 + 1/x2) - 2x X 1/x now put the value of x2 + 1/x2 (x - 1/x)2 = 2 - 2 = 0 [from Eq. (ii)] ∴ x - 1/x = 0

Question 6

If x + 1/x = 6, then x 4 + 1/x 4 is?

Accepted Answer

x + 1/x = 6 On squaring both sides, we get (x + 1/x)2 = (6)2 ⇒ x2 + 1/x2 + 2 = 36 ⇒ x2 + 1/x2 = 34 On squaring both sides, we get (x2 + 1/x2)2 = (34)2 ⇒ x4 + 1/x4 + 2 = 1156 ⇒ x4 + 1/x4 = 1154

Question 7

If x + y = 18 and xy = 72, what is the value of x2 + y2?

Accepted Answer

Given in question, x + y = 18 By using algebraic formula ∵ x2 + y2 = (x + y)2 - 2xy Put the value of x + y and xy as per given question, ⇒ x2 + y2 = (18)2 - 2 x 72 ⇒ x2 + y2 = 324 - 144 ⇒ x2 + y2 = 180

Question 8

If x + y = 1, then the value of (x3 + y3 + 3xy) is?

Accepted Answer

We know that algebraic formula, (x + y)3 = x3 + y3 + 3xy (x + y) put the value of x + y in given equation. [ given, x + y = 1] 1 = x3 + y3 + 3xy X 1 ⇒ x3 + y3 + 3xy = 1

Question 9

If p + q = 10 and pq = 5, then the numerical value if p/q + q/p will be?

Accepted Answer

p/q + q/p = (p2 + q2)/pq Apply the formula of algebra a 2+ b2 = (a + b)2 - 2ab p/q + q/p = ( (p + q)2 - 2pq ) / pq By substituting the pq and p + q values in given equation. p/q + q/p = ( (p + q)2 - 2pq ) / pq p/q + q/p = ((10)2 - 2 x 5 ) / 5 p/q + q/p = (100 - 10 )/ 5 = 90/5 = 18 p/q + q/p = 90/5 = 18 p/q + q/p = 18

Question 10

[ (m - n) 3 + (n - r) 3 + (r - m) 3 ] / 6[(m - n) (n - r) (r - m)] =?

Accepted Answer

Let, (m - n) = a, (n -r) = b (r - m) = c, Now a + b + c = (m - n) + (n -r) + (r - m) ⇒ a + b + c = m - n + n - r + r - m ⇒ a + b + c = 0............................. (i) As we know the Algebra formula, a3 + b3 + c3 ? 3abc = (a+b+c) X 1/2[(a?b)2+(b?c)2+(a?c)2] Put the value of a + b + c from equation (i). ⇒ a3 + b3 + c3 ? 3abc = 0 X 1/2[(a?b)2+(b?c)2+(a?c)2] ⇒ a3 + b3 + c3 ? 3abc = 0 ∴ a3 + b3 + c3 = 3abc ∴ Given expression in question is [ (m - n)3 + (n - r)3 + (r - m)3 ]/ 6(m - n) (n - r) (r - m) = ( a3 + b3 + c3 ) / 6abc = 3abc/6abc = 1/2

Question 11

(1 - 1/2) (1 - 1/3) (1 -1/4) (1 -1/5) ... (1 - 1/m) =?

Accepted Answer

Given expression = (1 - 1/2) x (1 - 1/3) x (1 - 1/4) x (1 - 1/5) ...(1 - 1/m - 1) x (1 - 1/m) = ((2 - 1)/2) x ((3 - 1)/3) ((4 - 1)/4) ((5 - 1)/5) ...((m-1 - 1)/m - 1) x ((m - 1)/m) = (1/2) x (2/3) x (3/4) x (4/5) x ... x (m -2)/(m-1) x ((m - 1)/m use the multiplication rule of Algebra, = 1 x 1/m = 1/m

Question 12

If a + 1/a = √ 3, then the value of a6 - 1/a6 + 2 will be

Accepted Answer

Given equation is, a + 1/a = √3 .............................(i) On squaring both sides ⇒ a2 + 1/a2 + 2 = 3 ⇒ a2 + 1/a2 = 1 ..............................(ii) Now, multiplying Equation (i) and (ii), we get (a + 1/a) (a2 + 1/a2) = √3 ⇒ a3 + a/a2 + a2/a + 1/a3 = √3 ⇒ a3 + 1/a3 + ( a + 1/a ) = √3 now put the value of a + 1/a in above equation, ⇒ a3 + 1/a3 + √3 = √3 [from Eq. (i)] ⇒ a3 + 1/a3 = 0 ⇒ ( a6 + 1 ) /a3 = 0 ⇒ a6 + 1 = 0 x a3 ⇒ a6 + 1 = 0 ⇒ a6 = -1 ∴ a6 - 1/a6 + 2 put the value of a6 = (-1)6 - 1/(-1)6 + 2 = 1 - 1 + 2 = 2

Question 13

The value of (1 + 1/2) (1 + 1/3) (1 + 1/4) ..... (1 + 1/150) is?

Accepted Answer

(1 + 1/2) (1 + 1/3) (1 + 1/4) .... (1 + 1/150) = 3/2 x 4/3 x 5/4 x ... 151/150 = (1/2) x 151 = 151/2 = 75.5

Question 14

If a + 1/b = 1 and b + 1/c = 1, then c + 1/a =?

Accepted Answer

Given that, a + 1/b = 1 ⇒ a = (1 - 1/b) = (b - 1)/b ⇒ 1/a = b/(b - 1) [reciprocal] ..........................(1) and b + 1/c = 1 ⇒ 1/c = (1 - b) ⇒ c = 1/(1 -b) .........................(2) ∴ c + 1/a ..........................(3) put the value of c from equ. (2) and 1/a from equ. (1) in equ. (3) ⇒ c + 1/a = 1/(1 -b) + b/(b -1) ⇒ c + 1/a = 1/(1 -b) - b/(1 - b) ⇒ c + 1/a = (1 - b)/(1 - b) ⇒ c + 1/a = 1

Question 15

If (a2 + 1/a2) = 17/4, then what is (a3 - 1/a3) equal to?

Accepted Answer

(a2 + 1/a2) = 17/4 subtracts 2 and Add 2 from above equation, we will get ⇒ a2 + 1/a2 - 2 + 2 = 17/4 ⇒ a2 + 1/a2 - 2a x 1/a + 2 = 17/4 Now apply the formula, x2 + x2 - 2xy ⇒ (a - 1/a)2 + 2 = 17/4 ⇒ (a - 1/a)2 = 17/4 - 2 ⇒ (a - 1/a)2 = (17 - 8)/4 ⇒ (a - 1/a)2 = 9/4 ⇒ (a - 1/a) = √9/4 (a -1/a) = 3/2 After that, cubing On both sides, we get (a -1/a)3 = (3/2)3 Apply the formula (x – y)3 = x3 – 3x2y + 3xy2 – c3 ⇒ (x – y)3 = x3 – 3xy(x - y) – y3 ⇒ a3 - 1/a3 - 3 x a x 1/a (a - 1/a) = 27/8 ⇒ a3 - 1/a3 = 27/8 + 3 x (3/2) ⇒ a3 - 1/a3 = 27/8 + 9/2 ⇒ (a3 - 1/a3) = 63/8

Question 16

5/8 x 23/5 ÷ 4/9 =?

Accepted Answer

Simplify 23 to 8 = 5/8 x 8/5÷4/9 By using simple algebraic rule: a ÷ b/c = a x c/b = 5/8 x 8/5 x 9/4 = (5×8×9)/(?8×5×4) = (5×9)/(?5×4) = (9)/(?4) = 9/4

Question 17

(0.49)4 x (0.343)4 ÷ (0.2401)4 x = (70 ÷ 100)? + 3 ?

Accepted Answer

The given question is, (70 ÷ 100)? + 3 = (0.49)4 x (0.343)4 ÷ (0.2401)4 As we know that 70 ÷ 100 = 0.7 0.49 = 0.7 2 0.343 = 0.7 3 0.2401 = 0.7 4 Put these value in given question , we will get ⇒ (0.7)? + 3 = (0.72)4 x ( 0.73 )4 ÷ ( 0.74 )4 ⇒ (0.7) ? + 3 = 0.7 2 x 4 x 0.7 3 x 4 ÷ 0.7 4 x 4 ⇒ (0.7) ? + 3 = 0.7 8 x 0.7 12 ÷ 0.7 16 Apply the law of Fractional Exponents and Laws of Exponents (am)(an) = am+n am÷an=am-n Or am/an=am - n we will get, ⇒ (0.7)? + 3 = (0.7) 8 + 12 - 16 ⇒ (0.7)? + 3 = (0.7) 20 - 16 ⇒ (0.7)? + 3 = (0.7) 4 On comparing the exponents both sides, we get ? + 3 = 4 ∴ ? = 4 - 3 = 1

Question 18

(2.247)3 + (1.730)3 + (1.023)3 - 3 x 2.247 x 1.730 x 1.023 / ( (2.247)2 + (1.730)2 + (1.023)2 - (2.247 x 1.730) - (1.730 x 1.023) - (2.247 x 1.023) )

Accepted Answer

We know that, a3 + b3 + c3 - 3abc = (a + b + c ) ( a2 + b2 + c2 - ab - bc - ca ) ⇒ (a + b + c) = ( a3 + b3 + c3 - 3abc ) / ( a2 + b2 + c2 - ab - bc - ca ) .......................(i) Given that in question, (2.247)3 + (1.730)3 + (1.023)3 - 3 x 2.247 x 1.730 x 1.023 / (2.247)2 + (1.730)2 + (1.023)2 - (2.247 x 1.730) - (1.730 x 1.023) - (2.247 x 1.023) Apply the formula given in Equation (i) and we will get, = ( a + b + c ) = ( 2.247 + 1.730 + 1.023) [ from eq. (i) ] = 5.000 = 5

Question 19

( 8.73 x 8.73 x 8.73 + 4.27 x 4.27 x 4.27 ) / ( 8.73 x 8.73 - 8.73 x 4.27 + 4.27 x 4.27 ) is equal to?

Accepted Answer

Let a = 8.73 and b = 4.27 Then, given expression =( a3 + b3 ) / ( a2 - ab + b2 ) As we Know the Algebra formula, a3 + b3 = (a + b) (a2 - ab + b2) Apply the above algebra formula in given above expression, = (a + b) (a2 - ab + b2) / ( a2 - ab + b2 ) = (a + b) = 8.73 + 4.27 = 13

Question 20

If x = ( √2 + 1 ) / ( √2 - 1 ) and x - y = 4√2, then the value of (x2 + y2) is?

Accepted Answer

Given that, x = ( √2 + 1 ) / ( √2 - 1 ) Expanding given equation by multiplying and dividing with √2 + 1 we will get, ⇒ x = ( ( √2 + 1 ) / ( √2 - 1 ) ) x ( ( √2 + 1 ) / ( √2 + 1 ) ) ⇒ x = ( √2 + 1 ) x ( √2 + 1 ) / ( √2 - 1 ) x ( √2 + 1 ) ⇒ x = ( √2 + 1 ) 2 / ( √2 - 1 ) x ( √2 + 1 ) [ Use the formula (A + B)(A - B) = A2 - B2 ] ⇒ x = ( √2 + 1 ) 2 / ( √22 - 12 ) [ Use the formula (A + B)2 = A2 + 2AB + B2 ] ⇒ x = ( 2 + 1 + 2√2 ) / (2 - 1) ⇒ x = 3 + 2√2 .....................(i) and given in question x - y = 4 √2 ⇒ y = x - 4√2 Put the value of x [ from Equation (i) ] ⇒ y = 3 + 2√2 - 4√2 ⇒ y = 3 - 2√2 Now, x2 + y2 = (3 + 2√2)2 + (3 - 2√2) Use the Algebra formula and solve the equation. [ Use the formula (A + B)2 = A2 + 2AB + B2 ] [ Use the formula (A - B)2 = A2 - 2AB + B2 ] ⇒ x2 + y2 = 9 + 8 + 12√2 + 9 + 8 - 12√2 ⇒ x2 + y2 = 34

Simplification Questions