10th Standard Matriculation Science Question Bank: Subject-Wise Blueprints and Important Q&A PDF Guide

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Chapter 1 Core Formula Repository

Review these governing mathematical axioms from your data matrix sheets before checking core numerical workouts:

Concept Vector	Governing Algebraic Equation	Operational Criteria
Euclid's Division Lemma	$a = bq + r$	$0 \le r < b$ (Unique integers $q$ and $r$)
HCF & LCM Relationship	$\text{HCF}(a,b) \times \text{LCM}(a,b) = a \times b$	Valid for any two positive integers
Decimal Expansion Rules	$\text{Denominator} = 2^m \times 5^n$	Rational numbers with terminating decimals

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Select Question Type -- Choose Category -- Multiple Choice Questions (1 Mark) Short Answer Questions (2 Marks) Long & Application Problems (5 Marks)

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Multiple Choice Questions (1 Mark)

Q1. The decimal expansion of $7/8$ is:
(a) Terminating | (b) Non-terminating recurring | (c) Non-terminating non-recurring
Step-by-Step Solution: Express the denominator as a product of prime components: $8 = 2^3$. Since the denominator matches the template $2^m \times 5^n$, the decimal expansion terminates precisely.
Correct Answer: (a) Terminating ($7/8 = 0.875$)

Q2. Find the $\text{HCF}(6, 12)$.
(a) 2 | (b) 6 | (c) 12 | (d) 1
Step-by-Step Solution: Prime factors of $6 = 2 \times 3$. Prime factors of $12 = 2^2 \times 3$. The product of the lowest powers of common prime factors is $2^1 \times 3^1 = 6$.
Correct Answer: (b) 6

Q3. Which of the following is an irrational number?
(a) $2/3$ | (b) $\sqrt{4}$ | (c) $\sqrt{2}$ | (d) $0.25$
Step-by-Step Solution: $\sqrt{4} = 2$ and $2/3$ can be neatly written as fractions ($p/q$). But $\sqrt{2}$ creates a non-terminating, non-repeating decimal array, satisfying irrational standards.
Correct Answer: (c) $\sqrt{2}$

Q4. The $\text{LCM}$ of $12$ and $18$ is:
(a) 6 | (b) 12 | (c) 36 | (d) 72
Step-by-Step Solution: Factors: $12 = 2^2 \times 3$, $18 = 2 \times 3^2$. $\text{LCM}$ matches the highest power sequences: $2^2 \times 3^2 = 4 \times 9 = 36$.
Correct Answer: (c) 36

Short Answer Questions (2 Marks)

Q5. Find $\text{HCF}(273, 105)$ using Euclid's Division Lemma.
Solution Sequence: Apply $a = bq + r$ systematically:
$273 = 105 \times 2 + 63$
$105 = 63 \times 1 + 42$
$63 = 42 \times 1 + 21$
$42 = 21 \times 2 + 0$
Since the remainder has reached zero, the final divisor is our highest common factor.
Final Vector Output: $\text{HCF} = 21$

Q6. If $\text{HCF}(a,b) = 12$ and $\text{LCM}(a,b) = 144$, and one number is $36$, find the other number.
Solution Sequence: Apply the formula $\text{HCF} \times \text{LCM} = a \times b$:
$12 \times 144 = 36 \times b$
$1728 = 36b \implies b = \frac{1728}{36} = 48$.
Final Vector Output: $b = 48$

Q7. Determine whether $6/15$ produces a terminating or non-terminating decimal form.
Solution Sequence: First, reduce the rational expression to its simplest terms: $\frac{6}{15} = \frac{2}{5}$. The prime factors of the denominator contain only $5$ ($5^1$). This fits the required $2^m \times 5^n$ layout.
Final Vector Output: Terminating Decimal ($0.4$)

Long & Application Problems (5 Marks)

Q8. Prove that $\sqrt{5}$ is an irrational number.
Proof Steps (Contradiction Method):
1. Assume $\sqrt{5}$ is rational, meaning $\sqrt{5} = \frac{p}{q}$ where $p$ and $q$ are coprime integers ($q \neq 0$).
2. Squaring both sides: $5 = \frac{p^2}{q^2} \implies 5q^2 = p^2$. Therefore, $p^2$ is divisible by $5$, which means $p$ is also divisible by $5$.
3. Substitute $p = 5k$ into our equation: $5q^2 = (5k)^2 \implies 5q^2 = 25k^2 \implies q^2 = 5k^2$.
4. This shows that $q^2$ is divisible by $5$, which means $q$ is divisible by $5$ too.
5. This creates a contradiction because both $p$ and $q$ share a common factor of $5$, meaning they are not coprime. Our initial assumption was wrong.
Conclusion: $\sqrt{5}$ is an irrational number. $\blacksquare$

Q9. Two bells ring at intervals of $12$ minutes and $18$ minutes respectively. If they ring together at 9:00 AM, at what time will they ring together again?
Solution Steps: To find when the bells ring together again, we need to find the Least Common Multiple ($\text{LCM}$) of $12$ and $18$.
Prime factorization: $12 = 2^2 \times 3$ and $18 = 2 \times 3^2$.
$\text{LCM} = 2^2 \times 3^2 = 4 \times 9 = 36 \text{ minutes}$.
Adding $36$ minutes to our starting time of 9:00 AM gives 9:36 AM.
Final Vector Output: 9:36 AM