6 Number Systems
The digital root of 46899 is:
Concept:
1
Digital Root: The digital root of a non-negative integer is the single-digit value obtained by an iterative process of summing digits until a single digit is reached.
Hint:
1
First, calculate $4 + 6 + 8 + 9 + 9$. Take that sum and add its digits together again!
Mistakes to Avoid:
!
Stopping Too Early: If your first sum is 36, you cannot stop there because it is not a single digit. You must add $3 + 6$.
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To find the digital root, add all the digits:
Sum = $4 + 6 + 8 + 9 + 9$
Sum = $36$
Since 36 is a two-digit number, repeat the process:
New Sum = $3 + 6$
New Sum = $9$ [1 M]
Since 9 is a single digit, it is our digital root.
Correct Option: (A)
7 Algebraic Expressions
Which expansion is correct?
Concept:
1
Standard Identity: The square of a binomial difference must always follow the identity $(a - b)^2 = a^2 - 2ab + b^2$.
Hint:
1
Let $a = 15x$ and $b = 6$. Square the first term, square the last term, and calculate the middle term by multiplying the two terms together and doubling the result.
Mistakes to Avoid:
!
The Distributive Illusion: Never assume $(a - b)^2 = a^2 - b^2$ (like Option A). Squaring a binomial always produces three terms, not two!
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We must expand $(15x - 6)^2$.
Using the algebraic identity: $(a - b)^2 = a^2 - 2ab + b^2$
Here, $a = 15x$ and $b = 6$.
Substituting the values:
$= (15x)^2 - 2(15x)(6) + (6)^2$
$= 225x^2 - (30x)(6) + 36$
$= 225x^2 - 180x + 36$ [1 M]
Comparing this, Option (B) matches perfectly.
Correct Option: (B)
8 Proportions
A pump fills 2100 litres of water in 35 minutes. Time required to fill 4800 litres is:
Concept:
1
Unitary Method: In direct proportion, find the value of one single unit (litres per minute) first, then multiply or divide to find the final target.
Hint:
1
Divide 2100 by 35 to find out how many litres are filled in exactly 1 minute. Then divide the target 4800 by this new rate!
Mistakes to Avoid:
!
Flipping the Division: Do not calculate $35 \div 2100$. Keep your units logical: you want Volume per Time (Litres / Minute).
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Applying the Unitary Method:
Water filled in 35 minutes = 2100 litres.
Water filled in 1 minute = $\frac{2100}{35}$ litres.
Water filled in 1 minute = 60 litres. [½ M]
To find the time required for 4800 litres:
Time required = $\frac{4800}{60}$
Time required = 80 minutes. [½ M]
Correct Option: (B)
9 Linear Equations
Two numbers are in the ratio $3 : 5$ and their product is 2160. Then the smaller number is equal to:
Concept:
1
Ratio Variables: Convert the ratio $a : b$ into algebraic terms $ax$ and $bx$. Use the product condition to solve for $x$.
Hint:
1
Let the numbers be $3x$ and $5x$. Multiply them together to equal 2160. Don't forget that $x \times x = x^2$! Solve for $x$, then find $3x$.
Mistakes to Avoid:
!
Ignoring the Square: Writing $3x \times 5x = 15x$ is a fatal error. The variable is squared, so the equation is $15x^2 = 2160$.
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Since the ratio is $3 : 5$,
Let the numbers be $3x$ and $5x$.
The product is given as 2160:
$(3x) \times (5x) = 2160$
$15x^2 = 2160$
$x^2 = \frac{2160}{15}$
$x^2 = 144$ [½ M]
Taking the positive square root: $x = 12$
The question asks for the smaller number:
Smaller number = $3x = 3(12) = 36$. [½ M]
Correct Option: (D)
10 Algebraic Identities
If $52^2 - 48^2$ is equal to:
Concept:
1
Difference of Squares: The identity $a^2 - b^2 = (a - b)(a + b)$ makes calculating large square differences incredibly fast without requiring long multiplication.
Hint:
1
Apply the identity using $a = 52$ and $b = 48$. Subtract them, add them, and multiply the two results.
Mistakes to Avoid:
!
Brute Force Arithmetic: Calculating $52 \times 52$ takes time and increases the risk of multiplication errors. Always use the algebraic shortcut!
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We need to evaluate: $52^2 - 48^2$
Using the standard identity: $a^2 - b^2 = (a - b)(a + b)$
Here, $a = 52$ and $b = 48$.
Substituting the values:
$= (52 - 48)(52 + 48)$ [½ M]
$= (4)(100)$
$= 400$ [½ M]
Correct Option: (D)
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