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Concept:

A number is divisible by 10 if its last digit is 0.

Calculation:

We need to check the last digit of each option:

Option 1: 4856789753210 - Last digit is 0

Option 2: 5141781017116791 - Last digit is 1

Option 3: 9171551819071 - Last digit is 1

Option 4: 33517819651819 - Last digit is 9

Conclusion:

Out of the given options, only Option 1 has the last digit as 0.

The correct answer is option 1.

Concept:

To solve this problem, we need to simplify each term and use the properties of exponents.

Calculation:

\(​(\sqrt{11})^6 \times (121)^3 \div (11)^4 = (11)^?\)

\(\sqrt{11} = 11^{1/2}\)]

⇒ \((\sqrt{11})^6 = (11^{1/2})^6 = 11^{(1/2) \times 6} = 11^3\)

\(121 = 11^2\) , so

\((121)^3 = (11^2)^3 = 11^{2 \times 3} = 11^6\)

\((11)^4 = 11^4\)

\(11^3 \times 11^6 \div 11^4\)

⇒ \(11^3 \times 11^6 = 11^{3 + 6} = 11^9\)

⇒ \(11^9 \div 11^4 = 11^{9 - 4} = 11^5 \)

Thus, the value of  ? is 5.

Hence, The correct option is option 2.

Let the age of the younger boy be xx x

.

The older boy is 10 years older than the younger boy.

So, the older boy's age will be x+10x+10 x + 10

The sum of their ages is 40.

According to the problem, the sum of the two boys' ages is 40. This gives us the equation:

x+(x+10)=40x+(x+10)=40

2x+10=402x+10=40

Solve for xx x

:

2x=302x=30

2x = 30

Divide both sides by 2 to solve for xx x

:

x=15x=15

Concept:

To find the number of revolutions the wheel makes, we need to calculate the

circumference of the wheel and then determine how many such

circumferences fit into the total distance travelled.

Formula:

Circumference of the wheel = 2πr

Calculation:

Radius of the wheel = 21 cm

Distance to be travelled = 924 m = 92400 cm

First, we need to find the circumference of the wheel:

Circumference of the wheel = 2 × π × 21 cm

Using π ≈ 22/7:

= 2 × (22/7) × 21

= 2 × 22 × 3

= 132 cm

Next, we need to find out how many such circumferences fit into the total

distance travelled (92400 cm):

Number of revolutions = Total distance / Circumference

Number of revolutions = 92400 / 132

Number of revolutions = 700

∴ The number of revolutions the wheel will make is 700.

Hence, the correct option is option 4.

Concept:

This problem uses the concept of solving systems of linear equations. By

adding all the given equations, we simplify the system and solve for the desired

value.

Calculation:

1. x + 2y + 3z = 14

2. 2x + 3y + z = 11

3. 3x + y + 2z = 11

We need to find x + y + z .

(x + 2y + 3z) + (2x + 3y + z) + (3x + y + 2z) = 14 + 11 + 11

⇒ x + 2y + 3z + 2x + 3y + z + 3x + y + 2z = 36

⇒ (1x + 2x + 3x) + (2y + 3y + 1y) + (3z + 1z + 2z) = 36

⇒ 6x + 6y + 6z = 36

⇒ 6(x + y + z) = 36

⇒ x + y + z = \(\frac{36}{6}\) = 6

The value of x + y + z is 6.

So the correct option is option 1.

Concept:

Make factors of the Given Equation 

Explanation:

x2 - 7x + 12 = 0

x2 - 3x - 4x + 12 = 0

x(x - 3) - 4(x - 3) = 0 

(x - 3)(x - 4) = 0 

x = 3, 4

Hence, α = 3, β = 4 

Hence, The value of α2 + β2  = 3² + 4²  = 9 + 16 = 25

Hence, The Correct Answer is option 4.

Concept:

The square root of a number is a value that, when multiplied by itself, gives the original number. In mathematical terms, if \(n\) is the square root of \(x \), then

\(n \times n= x \)

Calculation:

We need to find the value of √2025.

To calculate the square root of 2025, we can use the prime factorization

method:

Find the prime factors of 2025.

2025 = 5 × 5 × 3 × 3 × 3 × 3

 Pair the prime factors.

2025 = (5 × 5) × (3 × 3) × (3 × 3)

Take one number from each pair and multiply them together.

√2025 = 5 × 3 × 3

√2025 = 45

Therefore, the square root of 2025 is 45.

The correct answer is option 2.

Concept:

A cone is a three-dimensional geometric shape that tapers smoothly from a flat

base (usually circular) to a point called the apex or vertex. It has one flat

surface and one curved surface.

Explanation:

Option 1: A plane figure is a flat, two-dimensional shape like a circle, triangle,

or square. Since a cone is three-dimensional, it cannot be classified as a plane

figure.

Option 2: A curve is a smoothly flowing, continuous line without sharp angles.

Although a cone has a curved surface, it is not solely a curve.

Option 3: Solid Figure: A solid figure is a three-dimensional object. A cone fits

this definition as it has volume and occupies space.

Option 4:  A line is a straight one-dimensional figure having no thickness and

extending infinitely in both directions. A cone is not a line as it is three-dimensional.

The correct option is option 3.

Concept:

This problem uses the concept of the difference of squares. When working with

consecutive numbers, the difference of their squares simplifies to a linear

equation that can easily be solved to find the numbers.

(n+1)2−n2=9(n+1)2−n2=95(n+1)2−n2=95Unknown node type: strong(n+1)2−n2=95Unknown node type: strong(n+1)2−n2=95Unknown node type: strong

Calculation:

Let the two consecutive numbers be \(x\) and \(x\) +1.

Expanding the squares:

\(a^2 - b^2 = (a - b)(a + b)\)

\((x + 1)^2 - x^2\)

⇒ \((x + 1)^2 - x^2 = (x + 1 - x)(x + 1 + x) = 1(2x + 1) = 2x + 1\)

⇒\( 2x + 1 = 95\)

⇒ \(x = \frac{94}{2} = 47 \)

The two consecutive numbers are \(x\)= 47  and \(x\) + 1 = 48.

The two numbers are 47 and  48 .

So the correct option is option 4.

Concept:

This problem uses basic algebra to solve for consecutive integers. When

working with consecutive integers, we can represent them using a variable and

set up an equation to find their values by using the given sum.

Calculation:

 \(x \) (the smallest integer), 

 \(x \) +1 (the middle integer),

 \(x \) + 2  (the largest integer).

The sum of these three integers is given as 54:

\(x + (x + 1) + (x + 2) = 54\)

⇒ \(x + x + 1 + x + 2 = 54\)

This simplifies to:

\(3x + 3 = 54\)

⇒ \(3x = 51\)

⇒ \(x = \frac{51}{3} = 17\)

The first integer  \(x \)= 17,

The second integer is \(x \)+ 1 = 18,

The third (largest) integer is \(x \)+ 2 = 19.

So the correct option is option 4.