Mathematics Paper 2 (Algebra - I) - CTET & State TET MCQ

Calculation:

(3x - 2y)³ = (3x)³ - (2y)³ - 3 × (3x)² × (2y) + 3 × 3x × (2y)²

⇒ 27x³ - 8y³ - 54x²y - 4x²y + 36 xy² - 5xy² = (3x - 2y)3 - 4x²y - 5xy²

⇒ 27x3 - 58x2y + 31xy2 - 8y3 = (3x - 2y)3 - 4x2y - 5xy2

Now we put the value of x = - 5 and y = - 7

(3x - 2y)3 - 4x2y - 5xy2 

⇒ {3 × (- 5) - 2 × (- 7)}³ - 4 × (- 5)² × (- 7) - 5 × (- 5) × (- 7)²

⇒ {- 15 + 14}³ + 4 × 25 × 7 + 25 × 49

⇒ (-1)³ + 700 + 1225

⇒ 1925 - 1 = 1924

∴ The correct answer is 1924. 

Formula used:

(a - b)² = a² + b² - 2ab

(a² - b²) = (a + b) × (a - b)

(a + b)2 = a2 + b2 + 2ab

Calculation:

⇒ {(x²)² + 1 - 2 × (x²) × 1}/(x - 1)²

⇒ (x² - 1)²/(x - 1)²

⇒ {(x + 1)²(x - 1)²}/(x - 1)²

⇒ (x + 1)2 = x2 + 2x + 1

∴ The correct option is 3.

Given:

Concept used:

(a + b)³ = a³ + b³ + 3ab (a + b)

(a - b)3 = a³ - b³ - 3ab (a - b)

Calculation:

According to the question,

The expression is =  

We know,

( x + 3)² = x³ + 3³ + 3 × (x) × 3 (x + 3)

⇒ ( x + 3)² = x³ + 27 + 9x(x + 3)

⇒( x + 3)² = x³ + 27 + 9x² + 27x

⇒( x + 3)² = x³ + 9x² + 27x + 27                    ---------(1)

Similarly,

( x - 3)² = x³ - 3³ - 3 × (x) × 3 (x - 3)

⇒ (x - 3)² = x³ - 27 - 9x(x - 3)

⇒( x - 3)² = x³ - 27 - 9x² + 27x

⇒( x - 3)² = x³ - 9x² + 27x - 27                        ---------(2)

Put the value of eq. (1) and eq. (2) in given expression

Formula Used :

x² - y² = (x + y) (x - y)

Calculation:

⇒ (c + d)² - (c - d)²

⇒ (c + d + c - d) (c + d -(c - d))

⇒ 2c × (c + d - c + d)

⇒ 2c × 2d = 4cd

∴ The simplified value is 4cd.

Alternate Method

Formula Used :

(x + y)² = x² + y² + 2xy

(x - y)² = x² + y² - 2xy

Calculation:

⇒ (c + d)² - (c - d)²

⇒ c² + d² + 2cd - (c² + d² - 2cd)

⇒ c² + d² + 2cd - c² - d² + 2cd

⇒ 4cd

∴ The simplified value is 4cd.

Given:

(p - q) = 8,

Formula used:

(a - b)³ = a³ - b³ - 3ab(a - b)

Calculation:

According to the question,

⇒ (p - q) = 8

Take cube on both sides, 

⇒ (p - q)³ = 8³

⇒ p³ - q3 - 3pq(p - q)= 83

⇒ p3 - q3 - 3pq(8) = 512

⇒ p3 - q3 - 24pq = 512

It can be written as:

⇒  q3 - p3 + 24pq = -512

Therefore, "-512" is the required answer.

Given:

The square of the difference between two given natural numbers is 324, while the product of these two given numbers is 144. 

Calculation:

Let the numbers are x and y

(x - y)² = 324

So, x - y = 18, xy = 144

(x + y)2 = (18)² + 4× 144

⇒ 900

⇒ x + y = 30

Then, x is (30 + 18) / 2 = 24 and y = 6

So , x² - y² = 24² - 6²

⇒ 576 - 36 = 540

∴ The correct  option is 2

Given:

​Concept used:

BODMAS rule

​Calculation:

We know from BODMAS the rule that,

Addition is solved first

Then subtraction is solved.

According to the BODMAS rule  -

 = 479

 = 479

  = 479

On solving for x, we get,

x = 4790

Therefore, the correct option is 3 that is 4790.

Calculation:

⇒ 

⇒ 

⇒ 6x - 9y + 3 = 2x + 8y + 16

⇒ 4x - 17y = 13     ------(1)

⇒ 

⇒ 10x - 15y + 5 = 8x - 14y + 4

⇒ (2x - y = -1) × 2

⇒ 4x - 2y = -2        -------(2)

Subtracting equation (1) & (2), we get

⇒ 15y = -15

⇒ y = -1

By putting the value of y in equation (2), we get

⇒ x = -1

⇒ x + y = -1 + (-1) = -2

∴ The correct answer is -2.

Calculation:

∴ The correct option is 1.

Given:

(d + e + f) = 14

(d2 + e2 + f2) = 96

Formula Used:

(a + b + c)² = 2 × (ab + bc + ca) + a2 + b2 + c2

Calculation:

⇒ (d + e + f)2 = 2 × (de + ef + fd) + d2 + e2 + f2

⇒ ( 14)² = 2× (de + ef + fd) + 96

⇒ 2 × (de + ef + fd) = 196 - 96 = 100

⇒ (de + ef + fd) = 100/2 = 50

∴ The value of (de + ef + fd) is 50.