Quantitative Aptitude PYQs

Given:

The expression (x³ - 1) (x² – 9x + 14)/ (x² + x + 1)(x² – 8x + 7)

Calculation       

The expression (x³- 1) (x² – 9x + 14)/ (x² + x + 1)(x² – 8x + 7)

(x³- 1³) (x² – 9x + 14)/ (x² + x + 1)(x² – 8x + 7)

(x -1)(x² + x + 1)(x ² – 9x + 14)/ (x² + x + 1) (x² – 8x + 7)

(x -1)(x ² – 9x + 14)/ (x² – 8x + 7)

(x - 1)(x - 2)(x -7)/ (x -1)(x -7)

= (x -2)                        

Hence, (x - 2)

Given:

The expression is 1/(x -2)(x -4) and (2x – 5)/(x² – 5 x + 6)(x -4)

Calculation       

Let y should be added

According to question,

1/(x -2)(x -4) + y = (2x – 5)/(x² – 5 x + 6)(x -4)

1/(x -2)(x -4) +y = (2x – 5)/(x - 2)(x - 3)(x -4)

[1 + y(x - 2) (x - 4)] /(x -2) (x -4) = (2x – 5)/(x - 2)(x - 3)(x -4)

1 + y(x - 2) (x - 4) = (2x – 5)/(x - 3)

y(x - 2) (x - 4) = (2x – 5)/(x - 3) – 1

y(x - 2) (x - 4) = (2x – 5 – x + 3)/ (x - 3)

y(x - 4) = 1/(x - 3)

y = 1/ (x - 3) (x - 4)

y = 1/ (x² -7x + 12)

Hence, 1/ (x² -7x + 12) be added

Given:

The expression x/a + y/b = a + b and x/a² + y/b² = 2

Shortcut Trick

Put x = y = a = b = 1

\(\frac{x}{a} + \frac{y}{b} = a + b\)

⇒ 1/1 + 1/1 = 1 + 1

⇒ 2 = 2

\(\frac{x}{a^2} + \frac{y}{b^2} = 2\)

⇒ 1/1² + 1/12 = 2

⇒ 2 = 2

Now,

\(\frac{x}{a^2} - \frac{y}{b^2}\)

⇒ 1/1² - 1/12 = 0

∴ The correct answer is option (3).

Alternate Method

Calculation       

x/a + y/b = a + b

bx + ay = a²b + b²a …….(1)

and

x/a² + y/b² = 2

b²x + a²y = 2 a² b² …….(2)

In the equation (1) multiply by a and subtract equation (2)

(ab – b²)x = a³b – a²b²

X = a² b(a - b)/ b(a - b)

⇒ X = a²

Putting the value x = a² in equation (1)

Now, a² b + ay = a²b + ab²

⇒Y = b²

Then \(\frac{x}{a^2} - \frac{y}{b^2}\)

Putting the value

We get a²/a² – b² / b² = 1 -1 = 0

Hence, \(\frac{x}{a^2} - \frac{y}{b^2}\) = 0

Given:

If (x - k) is the HCF of x2 + ax + b and x2 + cx + d

Calculation:       

(x - k) is a factor of each expression put x = k in the given expression

x2 + ax + b = 0

Put the value of x = k

(k)2 + ak + b = 0

k2 + ka + b = 0 ……(1)

Also,  x2 + cx + d = 0

Put x = k

(k)2 + ck + d = 0

k2 + ck + d = 0 ……(2)

Now, equation (1) equal to (2)

k2 + ak + b = k2 + ck + d

k(a - c) = d – b

K = (d – b)/(a – c)

∴ The value of k is (d – b)/(a – c).

Given:

1. If x is directly proportional to z and y is directly proportional to z, then (x² - y²) is directly proportional to z².
2. If x is inversely proportional to z and y is inversely proportional to z, then (xy) is inversely proportional to z²

Calculation       

Case I.

According to the question 

x ∝ z and  y ∝ z

⇒ x = zk and y = rz   where k and r are constants 

⇒ x² = z²k² and y² = r²z² 

⇒ x2 - y2 = z2( k2 - r2)

⇒ x2 - y2 ∝ z2

Hence, Case I is correct. 

Case II.

x ∝ 1/z and  y ∝ 1/z

⇒ xz = k and yz = r   where k and r are constants 

⇒ xz. yz = kr

⇒ xyz2 = kr

⇒ xy = kr/z2

⇒ xy ∝ 1/z²

Hence, Case II is correct.

Shortcut TrickLet the income of Rajesh be 100.

 

⇒ Percentage increase in expenditure = [(120 - 95)/95] × 100

⇒ Percentage increase in expenditure = (25/95) × 100 ≈ 26%

∴ The correct answer is Option 4).

Given:

Rajesh saving = 5%

Income increased = 25%

Formula used

Income = expenditure + saving

Calculation

Let income be 100

Given saving = 5% of 100 = 5

Expenditure = 100 – 5 = 95

The income increased by 25%

New income = 25% of 100 + 100 = 125

Saving is the same = 5

Then expenditure = 120

Change in expenditure = 120 – 95 = 25

Therefore, Percentage increase = (25 ÷ 95) × 100 = 26.31% ≈ 26%

∴ Percentage increase in his expenditure is 26%

Shortcut Trick 

LCM of 6, 8 and 4 = 24

⇒ (a + b + c) / d = 20 / 4

⇒ (b + c + d) / a = 21 / 3

⇒ (c + d + a) / b = 18 / 6

⇒ c = 24 - (d + a + b) = 24 - (4 + 3 + 6) = 11

∴ a : b : c : d is 3 : 6 : 11 : 4.

Alternate Method 

Given:

If (a + b + c): d = 5: 1, (b + c + d): a = 7: 1 and, (c + d + a): b = 3: 1

Calculation

(a + b + c): d = 5: 1

Multiplying it by 4 we'll get

(a + b + c): d = 20: 4      ---- (1)

(b + c + d): a = 7: 1

Multiplying it by 3 we'll get

(b + c + d): a = 21 : 3      ---- (2)

(c + d + a): b = 3: 1

Multiplying it by 6 we'll get

(c + d + a): b = 18 : 6      ---- (3)

From (1), (2), and (3)

We can take a = 3, b = 6, d = 4

Putting values in (1)

⇒ a + b + c + d = 20 + 4

⇒ 3 + 6 + c + 4 = 24

⇒ c = 24 - 13

⇒ c = 11

⇒ we'll get c = 11

⇒ a : b : c : d = 3 : 6 : 11 : 4

∴ a : b : c : d is 3 : 6 : 11 : 4.

Given:

The expression 2x/3 + y/ 2 = 4 and x/3 - y/2 = 1

Calculation       

The expression 2x/3 + y/ 2 = 4 and x/3 - y/2 = 1

2x/3 + y/ 2 = 4 …. Equation (1)

And x/3 - y/2 = 1 …. Equation (2)

Adding both 1 and 2

2x/ 3 + x/3 = 4 + 1

⇒ 3x/3 = 5

⇒ X = 5

Hence, the value of x = 5

Given:

The expression \(\frac{3x-6}{(x-6)(x+k)} = \frac{2}{x-6}+\frac{1}{x+k}\)

Calculation:      

The expression \(\frac{3x-6}{(x-6)(x+k)} = \frac{2}{x-6}+\frac{1}{x+k}\)

⇒ \(\frac{3x-6}{(x-6)(x+k)} = \frac{2(x+k)+(x-6)}{(x+k)(x-6)}\)

⇒  3x – 6 = 2(x + k) + 1(x - 6)

⇒ (3x – 6) + 0 = (3x - 6) + 2k

2k = 0

Then k =0

Hence, the value of k = 0

Given:

The expression 1/(a - b)(b - c)  + 1/(b - c)(c - a) + 1/(c - a)(a - b)

Calculation       

The expression 1/(a - b)(b - c)  + 1/ (b - c)(c - a) + 1/ (c - a)(a - b)

= (c – a + a – b + b - c)/(a - b)(b - c) (c - a)

= 0 /(a - b)(b - c) (c - a) = 0

Hence, the value = 0

Given:

The number 413283P759387 is divisible by 13

Concept used:

If adding four times the last digit to the number formed by remaining digits is divisible by 13, then the number is divisible by 13. 

Calculation       

The number 413283P759387 is divisible by 13

By checking options

Option 1,

54 is not divisible by 13, so

4132833759387 is not divisible by 13

Same as, we can try option 2, option 3 and option 4.

Option 2

4132836759387 is not divisible by 13

Option 3

4132837759387 is not divisible by 13

Option 4

 

65 is divisible by 13, so

4132838759387 is divisible by 13 

∴ Value of P is 8.

Given:

The 2^1000000 is divided by 7

Calculation       

The 2^1000000 is written = 2 × 2⁹⁹⁹⁹⁹⁹ = 2 (2³)³³³³³³

= 2 (8)³³³³³³ is divided by 7 = 2 × (1)³³³³³³

= 2 × 1 = 2

Hence, remainders = 2

Given:

The equations x - y = 6 and x/3 +y/2 = 12

Shortcut Trick

Option elimination method

By using option (2)

x = 18, y = 12

x - y = 6

⇒ 18 -12 = 6

⇒ 6 = 6

x/3 +y/2 = 12

⇒ 18/3 + 12/2 = 12

⇒ 12 = 12

Thus, x = 18, y = 12

∴ The correct answer is option (2).

Alternate Method

Calculation       

The equations x - y = 6 and x/3 +y/2 = 12

The equations x – y = 6

Then x = y + 6          

Put the value of x in the equation x/3 +y/2 = 12

Then (y + 6)/3 +y/2 = 12

2y + 12 + 3y = 72

5y = 60

Y = 12

X = 12 + 6 = 18

Hence, the value of x and y is 18 and 12

Given:

The numbers between 1 and 50

Calculation       

The numbers between 1 and 50

All prime number between 1 and 50 are given below

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, and 47

Now twin-prime numbers between 1 and 50 are

(3, 5), (5, 7), (11, 13), (17, 19), (29, 31), and (41, 43)

Hence, numbers of is 6 twin - prime numbers

Given:

There is a remainder of 4 when a number is divided by 7

Shortcut Trick

Let the number be '11'

Square of number = 11² = 121

Now,

Divide 121 by 7 = 121/7

⇒ Remainder = 2

∴ The correct answer is option (4).

Note: Assumed number is (divisor + remainder).

Alternate Method

Formula used:

Dividend = Divisor × Quotient + Remainder

(a + b)² = a² + 2ab + b²

Calculation       

We know that,

Dividend = Divisor × Quotient + Remainder

Let dividend and quotient be a and q

Then a = 7q +4      -----(1)

On square both sides, we get

Then,

a² = 49q² + 56q + 16

= 7[7q² + 8q] + 2

Hence the remainder = 2

Given:

The given expression 27²⁷ - 15²⁷ 

Shortcut Trick

Concept used:

aⁿ - bⁿ is divisible by (a -b) ; if n = odd

Calculations:

2727 - 1527 = an - bn

Here, a = 27 and b = 15

So, (a - b) = 27 - 15 = 12

Thus, 2727 - 1527 is divisible by 12 that means it is also divisible by 6.

So, Remainder = 0

∴ The correct answer is option (1).

Alternate Method

Calculation 

Let,                  

N = Rem of (27²⁷ - 15²⁷)/6

N = Rem of (27²⁷)/6 - Rem of (15²⁷)/6

N = Rem of (24 + 3)²⁷/6 - Rem of (12 + 3)²⁷/6

N = 3 - 3

∴ The remainder when 2727 - 1527 is divided by is 0.

Given:

If a + b + c = 0

Calculation                            

We know that formula, a³ + b³ + c³ – 3abc = (a + b + c) (a² + b² + c² – ab – bc - ca)

Here put a + b + c = 0

Then, a³ + b³ + c³ - 3abc = 0

⇒ a3 + b3 + c3 = 3abc

Option (1) is correct

We know that a² + b² + c² = (a + b + c)² – 2(ab + bc + ca)

Put a + b + c = 0

Then, a² + b² + c² = – 2(ab + bc + ca)

Option (2) is correct.

From above, a3 + b3 + c3 = 3abc 

From the question, c = -(a + b)

⇒ a3 + b3 + c3 = -3ab(a + b)

Option (3) is correct

Hence, 1, 2 and 3 are correct.

Given:

Given, p + 1/p = 4

Calculation                            

Then given p + 1/p = 4

Squaring both side (p + 1/p)² = 4²

p² + 1/p² + 2 = 16

p² + 1/p² = 14

Hence, p² + 1/p²  = 14 

Shortcut Trick⇒ If x + 1/x = k  then x² + 1/x² = k² - 2

⇒ p + 1/p = 4 

​⇒ p2 + 1/p2 = 4² - 2 

​⇒ p2 + 1/p2 = 14

∴ The correct answer is option (2).

Shortcut Trick 

7³² = (7⁴)⁸ = (2401)⁸ = (1)⁸ = 1

∴ The unit digit is 1. 

Alternate Method 

Given:

 The expression 67³²

Calculation

Unit digit of 67³² is 7³²

Now, 7¹ = 7, 7² = 49, 7³ = 343, 7⁴ = 2401, 7⁵ = 16807

Then unit digit is repeated here n = 4

7³² is written as 7²⁸ × 7⁴ = 1 × 1

The unit digit of 7³² = 1 × 1 = 1

∴ The unit digit is 1.

Shortcut Trick 

If y = 1 then

(6x + 4) / (6x - 4) = 8/6

x = 14/3

so, x2 / y2

(14/3)²/1 = 196/9

Alternate Method 

Given:

The expression (6x + 4y) / (6x - 4y) = 8/6

Calculation       

The expression (6x + 4y) / (6x - 4y) = 8/6

Cross multiply,

36x + 24y = 48x – 32y

24y + 32y = 12x

56y = 12x

x/y = 56/12

x/y = 14/3

Then x²/ y² = (14)²/(3)² = 196/9

Therefore, the value of x²/ y² is 196/9.