Sample QuestionsAlgebraic Identities questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
If $\text{x}+\frac{1}{\text{x}}=3,$ then $\text{x}^6+\frac{1}{\text{x}^6}=$
Answer: D.
View full solution →If $49\text{a}^2-{\text{b}}=\Big(7\text{a}+\frac{1}{2}\Big)\Big(7\text{a}-\frac{1}{2}\Big),$ then the value of $b$ is:
- A
$0$
- ✓
$\frac{1}{4}$
- C
$\frac{1}{\sqrt2}$
- D
$\frac{1}{2}$
Answer: B.
View full solution →$(\mathrm{a}-\mathrm{b})^3+(\mathrm{b}-\mathrm{c})^3+(\mathrm{c}-\mathrm{a})^3=$
Answer: C.
View full solution →If $a + b + c = 0$, then $\frac{\text{a}^2}{\text{bc}}+\frac{\text{b}^2}{\text{ca}}+\frac{\text{c}^2}{\text{ab}}=$
Answer: D.
View full solution →If $\mathrm{a}-\mathrm{b}=-8$ and $\mathrm{ab}=-12$, then $\mathrm{a}^3-\mathrm{b}^3=$
- A
$-244$
- B
$-240$
- ✓
$-224$
- D
$-260$
Answer: C.
View full solution →Statement-1 (A): $a+b+c=6$ and $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{3}{2}$, then $\frac{a}{b}+\frac{a}{c}+\frac{b}{a}+\frac{b}{c}+\frac{c}{a}+\frac{c}{b}=6$
Statement-2 $(R): (a+b+c)^2=a^2+b^2+c^2+2(a b+b c+c a)$
- A
Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
- ✓
Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1.
- C
Statement-1 is True, Statement-2 is False.
- D
Statement-1 is False, Statement-2 is True.
Answer: B.
View full solution →Statement-1 $(A): a^2+b^2+c^2-a b-b c-c a=0$ if and only if $a=b=c$.
Statement-2 (R):$(a+b+c)^2=a^2+b^2+c^2+2 a b+2 b c+2 c a$
- A
Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
- ✓
Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1.
- C
Statement-1 is True, Statement-2 is False.
- D
Statement-1 is False, Statement-2 is True.
Answer: B.
View full solution →Statement-1 $(A): (a-b)^3+(b-c)^3+(c-a)^3=3(a-b)(b-c)(c-a)$
Statement-2 $(R)$; If $a+b+c=0$, then $a^3+b^3+c^3=3 a b c$
- ✓
Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
- B
Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1.
- C
Statement-1 is True, Statement-2 is False.
- D
Statement-1 is False, Statement-2 is True.
Answer: A.
View full solution →Statement-1 (A): $a^3+b^3+3 a b-1=(a+b-1)\left(a^2+b^2+a+b-a b+1\right)$
Statement-2 (R): $ a^3+b^3+c^3-3 a b c=(a+b+c)\left(a^2+b^2+c^2+a b+b c+c a\right)$
- A
Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
- B
Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1.
- ✓
Statement-1 is True, Statement-2 is False.
- D
Statement-1 is False, Statement-2 is True.
Answer: C.
View full solution →Statement-1 (A): $\sqrt{(a+b+c)^2+(a-b+c)^2+2\left(b^2-a^2-c^2-2 a c\right)}=2 b$
Statement-2 (R): $(x+y+z)^2=x^2+y^2+z^2+2(x y+y z+z x)$
- ✓
Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
- B
Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1.
- C
Statement-1 is True, Statement-2 is False.
- D
Statement-1 is False, Statement-2 is True.
Answer: A.
View full solution →If $x^2+y^2-x y=3$ and $y-x=1$, then $\frac{x y}{x^2+y^2}=$ _______________ .
View full solution →If $\frac{a}{b}+\frac{b}{a}=2$, then $\left(\frac{a}{b}\right)^{100}-\left(\frac{b}{a}\right)^{100}=$ _______________ .
View full solution →If $a+b+c=6, \frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{3}{2}$, then $\frac{a}{b}+\frac{a}{c}+\frac{b}{a}+\frac{b}{c}+\frac{c}{a}+\frac{c}{b}=$ _______________ .
View full solution →If $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$ and $a b c=2$, then $a b^2 c^2+a^2 b c^2+a^2 b^2 c=$ _______________
View full solution →If $(a+b+c)\left\{(a-b)^2+(b-c)^2+(c-a)^2\right\}=k\left(a^3+b^3+c^3-3 a b c\right)$, then $k=$ _______________ .
View full solution →If $a+b+c=0$, then write the value of $\frac{a^2}{b c}+\frac{b^2}{c a}+\frac{c^2}{a b}$.
View full solution →If $a^2+\frac{1}{a^2}=102$, find the value of $a-\frac{1}{a}$.
View full solution →If $x-\frac{1}{x}=\frac{1}{2}$, then write the value of $4 x^2+\frac{4}{x^2}$.
View full solution →If $x+\frac{1}{x}=3$, then find the value of $x^6+\frac{1}{x^6}$.
View full solution →If $a+b=7$ and $a b=12$, find the value of $a^2+b^2$.
View full solution →Write the following in the expanded form:
$(x+2 y+4 z)^2$
View full solution →Write the following in the expanded form: $(-3x + y + z)^2$
View full solution →Find the following:
$\left(x^3+1\right)\left(x^6-x^3+1\right)$
View full solution →Find the following products:
$\left(7 p^4+q\right)\left(49 p^8-7 p^4 q+q^2\right)$
View full solution →Evaluate the following using identities:
$(0.98)^2$
View full solution →Evaluate the following:
$(98)^3$
View full solution →Evaluate the following using identities:
$991 \times 1009$
View full solution →Find the following products:
$(4 x-3 y+2 z)\left(16 x^2+9 y^2+4 z^2+12 x y+6 y z-8 z x\right)$
View full solution →Evaluate: $\Big(\frac{1}{2}\Big)^3+\Big(\frac{1}{3}\Big)^3=\Big(\frac{5}{6}\Big)^3$
View full solution →If $9 x^2+25 y^2=181$ and $x y=-6$, find the value of $3 x+5 y$.
View full solution →Simplify the following: $\Big(\text{x}+\frac{2}{\text{x}}\Big)^3+\Big(\text{x}-\frac{2}{\text{x}}\Big)^3$
View full solution →Simplify: $(a + b + c)^2 + (a - b + c)^2 + (a + b - c)^2$
View full solution →Simplify the following expressions: $\big(\text{x}+\text{y}+\text{z}\big)^2+\Big(\text{x}+\frac{\text{y}}{2}+\frac{\text{z}}{3}\Big)^2-\Big(\frac{\text{x}}{2}+\frac{\text{y}}{3}+\frac{\text{z}}{4}\Big)^2$
View full solution →If $\text{x}^4+\frac{1}{\text{x}^4}=119,$ find the valu of $\text{x}^3-\frac{1}{\text{x}^3}.$
View full solution →If $\text{x}+\frac{1}{\text{x}}=3,$ calculate $\text{x}^2+\frac{1}{\text{x}^2},\ \text{x}^3+\frac{1}{\text{x}^3}$ and $\text{x}^4+\frac{1}{\text{x}^4}.$
View full solution →Find the cube of the following binomial expressions:
$\frac{3}{\text{x}}-\frac{2}{\text{x}^2}$
View full solution →If $\text{x}^2+\frac{1}{\text{x}^2}=98,$ find the value of $\text{x}^3+\frac{1}{\text{x}^3}.$
View full solution →Simplify: $(a + b + c)^2 + (a - b + c)^2$
View full solution →Simplify the following: $\Big(\frac{\text{x}}{2}+\frac{\text{y}}{3}\Big)^3-\Big(\frac{\text{x}}{2}-\frac{\text{y}}{3}\Big)^3$
View full solution →If $\text{x}-\frac{1}{\text{x}}=5,$ find the value of $\text{x}^3-\frac{1}{\text{x}^3}.$
View full solution →