Grade 9 : Mathematics - 9. Algebraic Identities (Set 3)

Question 1

If $x + \frac{1}{x} = 5$, find the value of $x^2 + \frac{1}{x^2}$.

Accepted Answer

A) 23

Answer

B) 25

Answer

C) 27

Answer

D) 21

Question 2

Evaluate $104 imes 96$ using a suitable algebraic identity.

Accepted Answer

B) 9984

Answer

A) 9976

Answer

C) 9994

Answer

D) 9884

Question 3

Factorize the expression $49a^2 + 70ab + 25b^2$.

Accepted Answer

B) $(7a + 5b)^2$

Answer

A) $(7a - 5b)^2$

Answer

C) $(7a + 5b)(7a - 5b)$

Answer

D) $(49a + 25b)^2$

Question 4

If $a + b + c = 9$ and $ab + bc + ca = 26$, find the value of $a^2 + b^2 + c^2$.

Accepted Answer

C) 29

Answer

A) 31

Answer

B) 55

Answer

D) 43

Question 5

Expand $(2x - 3y + 4z)^2$ using algebraic identities.

Accepted Answer

A) $4x^2 + 9y^2 + 16z^2 - 12xy - 24yz + 16zx$

Answer

B) $4x^2 + 9y^2 + 16z^2 + 12xy - 24yz + 16zx$

Answer

C) $4x^2 - 9y^2 + 16z^2 - 12xy - 24yz + 16zx$

Answer

D) $4x^2 + 9y^2 - 16z^2 - 12xy + 24yz - 16zx$

Question 6

If $x - \frac{1}{x} = 4$, evaluate $x^3 - \frac{1}{x^3}$.

Accepted Answer

C) 76

Answer

A) 52

Answer

B) 68

Answer

D) 80

Question 7

Find the product $(x + 4)(x + 10)$ using identity $(x+a)(x+b) = x^2 + (a+b)x + ab$.

Accepted Answer

D) $x^2 + 14x + 40$

Answer

A) $x^2 + 40x + 14$

Answer

B) $x^2 - 14x + 40$

Answer

C) $x^2 + 14x - 40$

Question 8

Evaluate $99^3$ using algebraic identities.

Accepted Answer

A) 970299

Answer

B) 970301

Answer

C) 997001

Answer

D) 980199

Question 9

If $x + y + z = 0$, evaluate $\frac{x^2}{yz} + \frac{y^2}{zx} + \frac{z^2}{xy}$.

Accepted Answer

C) 3

Answer

A) 0

Answer

B) 1

Answer

D) -3

Question 10

Factorize $27p^3 - \frac{1}{216} - \frac{9}{2}p^2 + \frac{1}{4}p$.

Accepted Answer

A) $(3p - \frac{1}{6})^3$

Answer

B) $(3p + \frac{1}{6})^3$

Answer

C) $(3p - \frac{1}{2})^3$

Answer

D) $(9p - \frac{1}{6})^3$

Question 11

Simplify $(a+b+c)^2 + (a-b-c)^2$.

Accepted Answer

B) $2a^2 + 2b^2 + 2c^2 + 4bc$

Answer

A) $2a^2 + 2b^2 + 2c^2$

Answer

C) $2a^2 + 2b^2 + 2c^2 + 4ab$

Answer

D) $4ab + 4bc + 4ca$

Question 12

Without performing full multiplication, evaluate $(-12)^3 + 7^3 + 5^3$.

Accepted Answer

D) -1260

Answer

A) 1260

Answer

B) -420

Answer

C) 0

Question 13

If $x^2 + \frac{1}{x^2} = 18$, find the value of $x - \frac{1}{x}$ (for $x > 1$).

Accepted Answer

A) 4

Answer

B) 16

Answer

C) $\sqrt{20}$

Answer

D) 2

Question 14

Factorize the cubic polynomial $x^3 + 13x^2 + 32x + 20$.

Accepted Answer

C) $(x+1)(x+2)(x+10)$

Answer

A) $(x-1)(x-2)(x-10)$

Answer

B) $(x+1)(x-2)(x+10)$

Answer

D) $(x+1)(x+4)(x+5)$

Question 15

A square park has side length $(3x + 5)$ meters. An inner square lawn inside it has side length $(3x - 5)$ meters.

What is the area of the path surrounding the lawn?

Accepted Answer

B) $60x m^2$

Answer

A) $30x m^2$

Answer

C) $(18x^2 + 50) m^2$

Answer

D) $120x m^2$

Question 16

Find the coefficient of $x$ in the algebraic expansion of $(x - 3)^3$.

Accepted Answer

A) 27

Answer

B) -27

Answer

C) 9

Answer

D) -9

Question 17

If $x + y = 12$ and $xy = 27$, find the value of $x^3 + y^3$.

Accepted Answer

C) 756

Answer

A) 864

Answer

B) 972

Answer

D) 1296

Question 18

Factorize $8x^3 + y^3 + 27z^3 - 18xyz$.

Accepted Answer

A) $(2x + y + 3z)(4x^2 + y^2 + 9z^2 - 2xy - 3yz - 6zx)$

Answer

B) $(2x + y + 3z)(4x^2 + y^2 + 9z^2 + 2xy + 3yz + 6zx)$

Answer

C) $(2x - y + 3z)(4x^2 + y^2 + 9z^2 - 2xy + 3yz - 6zx)$

Answer

D) $(2x + y - 3z)(4x^2 + y^2 - 9z^2 - 2xy - 3yz - 6zx)$

Question 19

If $a + b + c = 0$, what is the simplified value of $a^3 + b^3 + c^3$?

Accepted Answer

D) $3abc$

Answer

A) $a^2b^2c^2$

Answer

B) 0

Answer

C) $-3abc$

Question 20

Simplify the expression $\frac{0.87 imes 0.87 imes 0.87 + 0.13 imes 0.13 imes 0.13}{0.87 imes 0.87 - 0.87 imes 0.13 + 0.13 imes 0.13}$.

Accepted Answer

A) 1

Answer

B) 0.74

Answer

C) 0.1131

Answer

D) 0

Grade 9 : Mathematics - 9. Algebraic Identities (Set 3)