Algebraic Identities

Q.

Verify the following: $(-21)\times \left[\right(-4)+(-6\left)\right]=\left[\right(-21)\times (-4\left)\right]+\left[\right(-21)\times (-6\left)\right]$

Q.

If $1125=3^m\times 5^n;$ find m and n.

Q. Factors of $x^6-y^6$ is:

1. $(x-y)(x+y)$
2. $(x^2+y^2-xy)$
3. $(x^2+y^2+xy)$
4. All of the above

Q.

Using Euclids algorithm, find the HCF of $504\mathrm{and}1188$.

Q. If a square pyramid is cut by a plane parallel to the base, what will the shape of the cross-section of the pyramid be?

1. Rectangle
   
2. Triangle
   
3. Square
   
4. Hexagon
   

Q.

How much is ${(0.1)}^2$?

Q. Which of the following numbers cannot be the repeated digit sum of a perfect square?

1. 9
2. 1
3. 6
4. 4

Q. What is the value of $x?$
$x^x=2^{2048}$

1. 512
2. 1028
3. 128
4. 256

Q. If a prism and pyramid have the same base area and height, the volume of the pyramid is ______ of the volume of the prism.

1. thrice
2. one-third
3. twice
4. one-fourth

Q. Find the value of the square of 51 using a suitable identity.

1. 2601
2. 2701
3. 2801
4. 2901

Q. The value of $\frac{5\times {\left(25\right)}^{n+1}+25\times {\left(5\right)}^{2n-1}}{25\times {\left(5\right)}^{2n}-105{\left(25\right)}^{n-1}}$ is

1. 0
2. 1
3. $6\frac{1}{4}$
4. $5\frac{1}{4}$

Q. Evaluate:
viii) $0.5\times 0.06\times 0.007$

Q. Find the value of $1010\times 990$ using algebraic identities.

Q. Evaluate :
vii) $0.5\times 0.6\times 0.7$

Q.

$Evaluate:997\times 998=$ ___.

Q. A rectangular tank is 120 centimeters long, \(x\) centimeters wide, and 70 centimeters deep. It can hold 840, 000 cubic centimeters of water when full. Find the total surface area of the rectangular prism.

Q. Simplify: $7^2\times 2^2$

Q. Evaluate :
vi) $(0.2{)}^3\times 0.05$

Q. A toy is in the shape of a pyramid mounted on the top of a square prism as shown. What will be the minimum area of the wrapping material needed to cover the toy completely?

1. $316$ sq in
2. $420$ sq in
3. $352$ sq in
4. $416$ sq in

Q. If $A=3x^2–4x+1,B=5x^2+3x–8$ and, $C=4x^2–7x+3$, then find:
(i) (A + B) – C
(ii) B + C – A
(iii) A + B + C

Q.

Using identity $(x-a)(x+a)=$$x^2-a^2$, find $6^2-5^2$.

1. 13

2. 12

3. 10

4. 11

Q. If a cube of edge length 4 ft is cut from the corners a rectangular prism of length 6 ft, width 4 ft, and height 7 ft as shown, then what would be the total surface area of the resulting shape?

1. $150$ sq ft
2. $156$ sq ft
3. $160$ sq ft
4. $162$ sq ft

Q. A rectangular pyramid is placed on the top of a rectangular prism of height 7 ft and width one-third of its length. If both have the same base area and the total height of the structure is19 ft, which has a greater volume?

1. Cannot be determined
2. Pyramid
3. Both have equal area
4. Prism

Q. (4pq+3q)2−(4pq−3q)2 =

1. $44p{q}^2$
2. $48p^{2}q$
3. $48p{q}^2$
4. $44p^{2}q$

Q. When the cube root of 8 is multiplied with the cube root of -8, the result obtained is _____.

1. 4
2. -4
3. -8
4. 8

Q. If $3x+5y=11$ and $xy=2$, find the value of $9x^2+25y^2$.

Q. Factorise $p^2+6p-16$

1. $(p+8)(p-2)$
2. $(p-8)(p-q)$
3. $(p+8)(p-1)$
4. $(p+8)(p-7)$

Q.

The square root of 2.25 is equal to

1. 1.15

2. 1.05

3. 1.55

4. 1.5

Q. A pyramid sits inside a square prism such that its base lies on one of the bases of the prism. If the edge of the base of the square is $40$ inches and the base area of the pyramid is one-fourth of the base area of the prism, then the maximum possible volume of the pyramid is ________.

1. $\frac{400}{3}$ times the edge of the prism
2. $\frac{400}{3}$ times the height of the prism
3. $400$ times the height of the prism
4. $400$ times the base area of the prism

Q. If $a+b+c$ are all non-zero & $a+b+c=0$ prove that,
$\frac{a^2}{bc}+\frac{b^2}{ca}+\frac{c^2}{ab}=3$