Mathematics – Concept Quiz

Sets

Short Questions

1. Define a finite and infinite set with examples.

2. If $A = \{1, 2, 3, 5, 6, 7\}$ and $B = \{3, 4, 5\}$, find $A \cup B$ and $A \cap B$. Is B a subset of A?

Long Question

• Given $U = \{1, 2, 3, 4, 5, 6, 7, 8\}$, $A = \{2, 4, 6\}$, $B = \{4, 5, 6\}$ Find:

  a) $A \cup B$
  b) $A \cap B$
  c) $A - B$
  d) $B'$ (complement of B)

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Triangle

Short Questions

1. State the angle sum property of a triangle.

2. Define an equilateral triangle.

3. In triangle $ABC$, if $\angle A = 50^\circ$, $\angle B = 60^\circ$, find $\angle C$.

Long Question

• Prove that the exterior angle of a triangle is equal to the sum of the two opposite interior angles.

• In triangle $XYZ$, $XY = 6$ cm, $YZ = 8$ cm, $XZ = 10$ cm. Is it a right triangle?

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Trigonometry

Formulas

• $\sin \theta = \dfrac{\text{Opposite}}{\text{Hypotenuse} }$

• $\cos \theta = \dfrac{\text{Adjacent}}{\text{Hypotenuse} }$

• $\tan \theta = \dfrac{\text{Opposite}}{\text{Adjacent} }$

Short Questions

1. Define sine and cosine ratios.

2. If $\sin \theta = \dfrac{3}{5}$, find $\cos \theta$.

Long Question

• In a right triangle, if one angle is $30^\circ$ and hypotenuse is 10 cm, find the opposite and adjacent sides using trigonometric ratios.

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Factorization

Short Questions

1. Factorize: $x^2 + 5x + 6$

2. Factorize: $a^2 - b^2$

3. Expand: $(x + 2)^3$

Long Question

• Factorize: $49a^4 - 154a^2b^2 + 9b^4$

• Factorize: $x^3 + \dfrac{3}{2}x^2 + \dfrac{3}{4}x + \dfrac{1}{8}$

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HCF and LCM

Short Questions

1. Find HCF of 24 and 36.

2. Find LCM of 12 and 15.

3. Verify: $\text{HCF} \cdot \text{LCM} = \text{Product of numbers}$ for 18 and 30.

Long Question

• Two ropes are 60 cm and 84 cm long. Find the greatest length that can be used to cut both without remainder.

• Two bells ring every 45 and 60 seconds. After how many seconds will they ring together?

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Statistics

Short Questions

1. Define mean, median, and mode.

2. Find the mean of: $5, 10, 15, 20, 25$

3. What is the range of: $3, 7, 9, 12, 15$

Long Question

• The marks obtained by 10 students are: $12, 15, 18, 20, 22, 25, 28, 30, 32, 35$ Find:

  a) Mean
  b) Median
  c) Mode
  d) Range

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Probability

Short Questions

1. Define probability.

2. What is the probability of getting a head when tossing a coin?

3. A die is rolled. What is the probability of getting an even number?

Long Question

• A bag contains 3 red, 5 blue, and 2 green balls. One ball is drawn at random.

  Find the probability of:

  a) Getting a red ball

  b) Getting a blue ball

  c) Not getting a green ball

Sets: What is the cardinality of the set A = $\{2, 4, 6, 8, 10\}$?

Options:

• A) 4
• B) 5
• C) 6
• D) 10

Details

Reveal Solution

The set has 5 elements → Answer: B) 5

Triangle: In triangle ABC, if angle A = 60°, angle B = 70°, find angle C.

Options:

• A) 50°
• B) 60°
• C) 70°
• D) 80°

Details

Reveal Solution

Sum of angles = 180° → C = 180° - (60° + 70°) = 50° → Answer: A) 50°

Trigonometry: If sin θ = 3/5, and θ is acute, what is cos θ?

Options:

• A) 4/5
• B) 5/3
• C) 1/2
• D) 2/5

Details

Reveal Solution

Use identity: sin²θ + cos²θ = 1
→ cos²θ = 1 - (9/25) = 16/25 → cos θ = 4/5 → Answer: A) 4/5

Factorization: Factorize: x² + 7x + 12

Options:

• A) (x + 3)(x + 4)
• B) (x + 6)(x + 2)
• C) (x + 1)(x + 12)
• D) (x + 2)(x + 5)

Details

Reveal Solution

Middle term split: 3 + 4 = 7, 3×4 = 12 → Answer: A) (x + 3)(x + 4)

HCF & LCM: Find the LCM of 12 and 15.

Options:

• A) 60
• B) 180
• C) 30
• D) 45

Details

Reveal Solution

Prime factors:
12 = 2² × 3
15 = 3 × 5
LCM = 2² × 3 × 5 = 60 → Answer: A) 60

Statistics: Find the mean of: 5, 10, 15, 20, 25

Options:

• A) 15
• B) 10
• C) 20
• D) 12

Details

Reveal Solution

Mean = (5 + 10 + 15 + 20 + 25)/5 = 75/5 = 15 → Answer: A) 15

Probability: A die is rolled. What is the probability of getting an even number?

Options:

• A) 1/6
• B) 1/2
• C) 2/3
• D) 1/3

Details

Reveal Solution

Even numbers: 2, 4, 6 → 3 outcomes out of 6 → Probability = 3/6 = 1/2 → Answer: B) 1/2

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Conceptual Practice – Grade 9 Math

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Trigonometry

1. If $\sin \theta = \dfrac{3}{5}$ and the hypotenuse is 20 cm, find the base.

2. In a right triangle, $\tan \theta = \dfrac{5}{12}$. Find $\cos \theta$.

3. If $\cos \theta = \dfrac{4}{5}$, find $\tan \theta$.

4. A ladder leans against a wall making an angle of $60^\circ$ with the ground. If the ladder is 10 m long, find the height it reaches on the wall.

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Sets

1. Let $A = \{x: x \text{ is a multiple of 3, } x < 20\}$ and $B = \{x: x \text{ is even, } x < 20\}$. Find $A \cap B$.

2. If $U = \{1,2,3,4,5,6,7,8\}$, $A = \{2,4,6\}$, $B = \{4,5,6\}$, find $A \cup B$ and $A \cap B$.

3. True or False: $A \subseteq A \cup B$ for any sets $A$ and $B$.

4. If $n(A) = 10$, $n(B) = 15$, and $n(A \cap B) = 5$, find $n(A \cup B)$.

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Triangle

1. In triangle $ABC$, if $\angle A = 2x$, $\angle B = 3x$, and $\angle C = 5x$, find the value of $x$.

2. A triangle has sides 5 cm, 12 cm, and 13 cm. Is it a right triangle?

3. The perimeter of a triangle is 36 cm. Two sides are 12 cm and 15 cm. Find the third side.

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Statistics

1. The marks obtained by a student in 5 subjects are: 60, 70, 80, 90, 100. If the highest mark is removed, how does the mean change?

2. Find the median of: 5, 7, 9, 11, 13, 15

3. A data set has a mean of 20. If one value 30 is replaced by 10, what happens to the mean?

4. The mode of a data set is 8. If one more 8 is added, does the mode change?

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Indices

1. Simplify: $2^3 \cdot 2^4$

2. Simplify: $\dfrac{5^6}{5^2}$

3. Evaluate: $(3^2)^3$

4. Simplify: $x^{-2} \cdot x^5$

5. True or False: $a^0 = 0$

6. Write $81$ as a power of $3$.

7. Find $x$: $\dfrac{x^5}{x^2} = x^2$.

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Factorization (Splitting the Middle Term)

1. Factorize: $x^2 + 5x + 6$

2. Factorize: $x^2 + 7x + 12$

3. Factorize: $x^2 - 11x + 24$

4. Factorize: $2x^2 + 7x + 3$

5. Factorize: $6x^2 + 11x + 3$

6. Factorize: $x^2 - 2x - 15$

7. Factorize: $3x^2 - 5x - 2$

8. Factorize: $4x^2 - 4x + 1$