Question 1

1/(1 x 4) + 1/(4 x 7) + 1/(7 x 10) + 1/(10 x 13) + 1/(13 x 16) =?

Accepted Answer

Given expression can be written as = 1/(1 x 4) + 1/(4 x 7) + 1/(7 x 10) + 1/(10 x 13) + 1/(13 x 16) = 1/3{ 3/(1 x 4) + 3/(4 x 7) + 3/(7 x 10) + 3/(10 x 13) + 3/(13 x 16) } = 1/3{ (4 - 1)/(1 x 4) + (7 - 4)/(4 x 7) + (10 - 7 )/(7 x 10) + (13 - 10)/(10 x 13) + (16 - 13/(13 x 16) } = 1/3[(1 - 1/4) + (1/4 - 1/7) + (1/7 - 1/10) + (1/10 - 1/13) + (1/13 - 1/16)] = 1/3[1 - 1/4 + 1/4 - 1/7 + 1/7 - 1/10 + 1/10 - 1/13 + 1/13 - 1/16] = 1/3[1 - 1/16] = 1/3[ (16 - 1)/16] = 1/3[ (15)/16] = 1 x 15/ 3 x 16 = 5/16

Question 2

If x + a/x = 1, then the value of ( x2 + x + a ) / (x3 - x2) is?

Accepted Answer

Given, x + a/x = 1 ⇒ x2 + a = x .................(i) ⇒ x2 - x = -a ................(ii) Now, ( x2 + x + a ) / ( x3 - x2) ( x2 + a + x ) / ( x3 - x2) = ( x + x )/ x(x2 - x) [from Eq. (i)] = 2x/x(-a) [from Eq. (ii) ] = 2/-a = -2/a

Question 3

If √28 - 6√3 = √3a + b, (where a, b are rationals), value of (a + b) is?

Accepted Answer

Given equation is, √28 - 6√3 = √3a + b By expanding equation, √ 1 + 27 - 6√3 = √3a + b ⇒ √(1)2 + (3√3)2 - 2 x 3√3 = ⇒ √3a + b ⇒ √(1 - 3√3)2 = √3a + b ⇒ 1 - 3√3 = √3a + b On comparing the left and right side equation, we get, a = -3, b = 1 ∴ a + b = -3 + 1 = - 2

Question 4

If a = ( √2 -1)1/3, then the value of (a - a-1)3 + 3 ( a - a -1 ) is?

Accepted Answer

Given that, a = (√2 - 1)1/3 By cubing the both side of the given equation. a3 = √2 - 1 Then 1/a3 = 1/√2 - 1 Now multiply and divide the above equation by √2 + 1 ⇒ 1/a3 = ( √2 + 1 ) / ( √2 + 1 ) x 1/(√2 - 1 ) ⇒ 1/a3 = ( √2 + 1 ) x 1 / ( √2 + 1 ) x (√2 - 1 ) [ Use the formula (A + B)(A - B) = A2 - B2 ] ⇒ 1/a3 = ( √2 + 1 ) / ( 2 - 1 ) ⇒ 1/a3 = √2 + 1 According to the question ( a - a-1 )3 + 3 ( a - a-1 ) = (a - 1/a)3 + 3(a - 1/a) Use the formula [ ( A - B)3 = A3 - B3 + 3AB(A - B) ] = a3 - 1/a3 - 3 a x 1/a (a -1/a) + 3(a - 1/a) = a3 - 1/a3 - 3 (a -1/a) + 3(a - 1/a) = a3 - 1/a3 Pu the value of a3 and 1/a3 = √2 -1 - ( √2 + 1 ) = √2 -1 - √2 - 1 = -2

Question 5

If x = ( √ 3 + 1 ) / ( √ 3 - 1 ) and y = ( √ 3 - 1) / ( √ 3 + 1 ) , then the value of x2/y + y2/x is?

Accepted Answer

Given, x = ( √3 + 1) / ( √3 - 1 ) y = ( √3 - 1 ) / ( √3 + 1 ) Then , x + y = { ( √3 + 1) / ( √3 - 1 ) } + { ( √3 - 1 ) / ( √3 + 1 ) } ⇒ x + y = { (√3 + 1)2 + (√3 - 1)2 } / (√3 - 1) (√3 + 1) ⇒ x + y = ( 3 + 1 +2√3 + 3 + 1 - 2√3 ) / ( 3 - 1 ) ⇒ x + y = 8/2 ⇒ x + y = 4 --------------------(1) Now xy = ( √3 - 1 / √3 + 1 ) x ( √3 + 1 / √3 - 1 ) xy = (3 - 1) / (3 - 1) = 2/2 = 1------------------(2) Given in question, ∴ x2/y + y2/x = ( x3 + y3 )/xy ------------------- (3) Apply the algebra formula, A3 + B3 = (A + B)3 - 3AB(A + B) = (x + y)3 - 3xy(x + y)/xy Put the value of x + y from (1) and xy from (2) in Equation (3), we will get = (4)3 - 3 x 1 x (4)/1 = 64 - 3 x 1 x 4 = 64 - 12 = 52

Question 6

If x + a/x = b , then the value of ( x2 + bx + a) / ( bx2 - x3 ) is?

Accepted Answer

Given in question, x + a/x = b ⇒ ( x2 + a )/x = b ⇒ x2 + a = bx .........................(i) ⇒ bx - x2 = a .........................(ii) Now Given in question, ( x2 + bx + a ) / ( bx2 - x3 ) ⇒ ( x2 + a + bx ) / ( bx2 - x3 ) Put the value of x2 + a from equation (i) in above equation, ⇒ ( bx + bx )/(bx2 - x3) = (2bx)/( bx2 - x3 ) = (2bx)/x( bx - x2 ) = 2b/(bx - x2) Put the value of bx - x2 from equation (ii) in above equation, we will get = 2b/a

Question 7

A tailor has 37.5 metres of cloth and he has to make 8 pieces out of a metre of cloth. How many pieces can he make out of half of the cloth he has ?

Accepted Answer

Half of the cloth = 37.5/2 From 1 meter he will make 8 pieces => in 37.5/2 ---- ? 37.5/2 x 8 = 150.

Question 8

In an examination, a student scores 4 marks for every correct answer and loses 1 mark for every wrong answer. If he attempts all 60 questions and secures 130 marks, the no of questions he attempts correctly is :

Accepted Answer

Let the number of correct answers be X. Number of incorrect answers = (60 – X). 4x – (60 – x) = 130 => 5x = 190 => x = 38

Question 9

If 2 tables and 3 chairs cost Rs, 3500 and 3 tables and 2 chairs cost Rs. 4000, then how much does a table cost ?

Accepted Answer

Let the cost of a table and that of a chair be Rs. x and Rs, y respectively. Then, 2x + 3y = 3500 ...(i) and 3x + 2y = 4000 .....(ii) solving (i) and (ii) we get x = 1000, y = 500

Question 10

A fires 5 shots to B's 3 but A kills only once in 3 shots while B kills once in 2 shots. When B has missed 27 times, A has killed :

Accepted Answer

Let the no of shots be x. Then, Shots fired by A =(5/8)x Shots fired by B = (3/8)x Killing shots by A = 1/3 of (5/8)x = (5/24)x Shots missed by B = 1/2 of (3/8)x = (3/16)x (3/16)x = 27 ⇒ x =144 Birds killed by A = (5/24)x = (5/24)*144 = 30

Question 11

If a * b = 2a - 3b + ab, then 3 * 5 + 5 * 3 is equal to :

Accepted Answer

3 * 5 + 5 * 3 = (2 * 3 - 3 * 5 + 3 * 5) + ( 2 * 5 - 3 * 3 + 5 * 3) = (6 + 10 - 9 + 15) = 22.

Question 12

If a + b + c = 13, a2+b2+c2=69 then find the ab + bc + ca :

Accepted Answer

a + b + c 2 = a 2 + b 2 + c 2 + 2 a b + b c + c a 2 a b + b c + c a = a + b + c 2 - a 2 + b 2 + c 2 = 169 - 69 = 100 = a b + b c + c a = 50

Question 13

The difference of 18 and its reciprocal is equal to :

Accepted Answer

Required difference = 19/16 - 16/19 = 105/304..

Question 14

A tank is 2/5 full. If 16 litres of water is added to the tank, it becomes 6/7 full. The capacity of the tank is:

Accepted Answer

Let the capacity of the tank be x litres. Then, (6x/7 - 2x/5) = 16 <=> 30x - 14x = 16*35 <=> x = 35

Question 15

Simplify the following equation. 1943 + 12 + 13 + 11 + 14

Accepted Answer

After solving we get, 62/931. None of the options satisfy.

Question 16

In a regular week, there are 5 working days and for each day, the working hours are 8. A man gets Rs. 2.40 per hour for regular work and Rs. 3.20 per hours for overtime. If he earns Rs. 432 in 4 weeks, then how many hours does he work for ?

Accepted Answer

Suppose the man works overtime for x hours. Now, working hours in 4 weeks = (5 x 8 x 4) = 160. Therefore, 160 * 2.40 + x * 3.20 = 432 => 3.20x = 432 - 384 = 48 => x = 15. Hence, total hours of work = (160 + 15) = 175.

Simplification Questions