Search Previous Year Questions

Shortcut Trick

Given xy = 143 and yz = 195.

Since y is common, find HCF(143, 195).

143 = 11 × 13 and 195 = 15 × 13.

∴ y = 13, x = 11, and z = 15.

Sum = 11 + 13 + 15 = 39.

∴ The correct answer is 39.

Alternate Method

Given: x ≤ y ≤ z are co-prime numbers.

Equations:

⇒ x × y = 143

⇒ y × z = 195

Calculations:

⇒ y = HCF(143, 195)

⇒ 143 = 11 × 13

⇒ 195 = 13 × 15

⇒ Common factor y = 13

⇒ x = 143 ÷ 13 = 11

⇒ z = 195 ÷ 13 = 15

⇒ x (11) ≤ y (13) ≤ z (15) [Condition Satisfied]

⇒ Sum = 11 + 13 + 15 = 39

∴ The correct answer is 39.

Additional Information

Co-prime Numbers

Numbers are co-prime if their Highest Common Factor (HCF) is 1. In this set, HCF(11, 13) = 1, HCF(13, 15) = 1, and HCF(11, 15) = 1.

HCF and Product Property

When two products share a common variable, that variable must be the HCF of the two products if the remaining factors are co-prime.

Prime Factorization

Breaking down a composite number into a product of prime numbers helps in identifying common factors quickly.

Shortcut Trick

A line segment is a part of a line that is bounded by two distinct end points.

Unlike a line (infinite in both directions) or a ray (infinite in one direction), a line segment has a fixed length.

∴ The correct answer is line segment.

Alternate Method

Given: A line joining two end points.

Definition: In geometry, a line segment is a piece or part of a line that is bounded by two distinct end points and contains every point on the line between its endpoints.

⇒ A line extends infinitely in both directions and has no end points.

⇒ A ray has one starting point (end point) and extends infinitely in the other direction.

⇒ A line segment has exactly two end points, which makes it measurable.

⇒ Parallel lines are lines that never meet, and intersecting lines are lines that cross at a point.

∴ The correct answer is line segment.

Additional Information

Ray

A part of a line that starts at a point and goes off to infinity in one direction. It has only one end point.

Parallel Lines

Two lines in the same plane that are at an equal distance from each other and never intersect.

Intersecting Lines

Lines that share exactly one common point, called the point of intersection.

Shortcut Trick

Side of square = √900 = 30 cm.

Perimeter of square = 4 × 30 = 120 cm.

Side of hexagon = 120 ÷ 6 = 20 cm.

Area of hexagon = 6 × (√3/4) × 20² = 600√3 cm².

∴ The correct answer is 600√3 cm².

Alternate Method

Given: Area of square = 900 cm². Perimeter of square = Perimeter of hexagon.

Formula Used: Area of square = a² | Perimeter of square = 4a | Perimeter of hexagon = 6s | Area of regular hexagon = (3√3/2)s².

⇒ Let side of square be a.

⇒ a² = 900 ⇒ a = 30 cm.

⇒ Perimeter of square = 4 × 30 = 120 cm.

⇒ Let side of regular hexagon be s.

⇒ 6s = 120 ⇒ s = 20 cm.

⇒ Area of hexagon = (3√3 ÷ 2) × (20)²

⇒ Area = (3√3 ÷ 2) × 400 = 3√3 × 200 = 600√3 cm².

∴ The correct answer is 600√3 cm².

Additional Information

Regular Hexagon Properties

A regular hexagon is composed of 6 equilateral triangles. Its area is 6 × (√3/4)s².

Square Properties

For a square with side a, the diagonal is a√2 and the area is a² or (diagonal²)/2.

Perimeter Comparison

If the perimeter of different regular polygons is the same, the one with more sides will have a larger area.

The correct answer is Ovary.

Key Points

• A flower is the reproductive part of a plant, and it comprises several parts, including the pistil, ovary, ovule, petal, etc.
• After pollination, the ovules in the ovary get fertilized by the sperm cells, which leads to the formation of seeds.
• The ovary then develops into a fruit that surrounds the seeds and protects them until they are ready for dispersal.
• The fruit can be fleshy or dry, depending on the type of plant and its mode of reproduction.
• Therefore, the ovary is the part of a flower that develops into a fruit after fertilization.

Additional Information

• Pistil is the female reproductive part of a flower that consists of the stigma, style, and ovary.
• ​The stigma receives the pollen, and the style connects it to the ovary.
• Ovule is the structure inside the ovary that contains the female gamete or egg cell.
• It gets fertilized by the male gamete or sperm cell to form a zygote.
• Petal is the colorful and attractive part of a flower that attracts pollinators like bees, butterflies, etc.
• It does not play a direct role in reproduction.

Shortcut Trick

For any point P inside a rectangle ABCD, the British Flag Theorem states: PA² + PC² = PB² + PD².

⇒ 27² + 6² = 21² + PD²

⇒ 729 + 36 = 441 + PD²

⇒ 765 − 441 = PD²

⇒ PD² = 324

⇒ PD = 18 cm

∴ The correct answer is 18 cm.

Alternate Method

Given: Rectangle ABCD, Point P inside. PA = 27 cm, PB = 21 cm, PC = 6 cm.

Formula Used: PA² + PC² = PB² + PD² (British Flag Theorem).

Calculations:

⇒ PA² + PC² = PB² + PD²

⇒ (27)² + (6)² = (21)² + PD²

⇒ 729 + 36 = 441 + PD²

⇒ 765 = 441 + PD²

⇒ PD² = 765 − 441

⇒ PD² = 324

⇒ PD = √324

⇒ PD = 18 cm

∴ The correct answer is 18 cm.

Additional Information

British Flag Theorem

It applies to any point P inside or outside a rectangle, stating that the sums of the squares of distances from P to opposite vertices are equal.

Diagonal Property

In a rectangle, AC = BD. If P is the center (intersection of diagonals), then PA = PB = PC = PD.

Pythagoras Application

The theorem is derived by drawing lines through P parallel to the rectangle sides and using a² + b² = c² on the resulting right triangles.

Given Statements:

1. Some glasses are windows.
2. Some windows are white.
3. All white are brittle.

The least possible Venn diagram for the given statement is:

Conclusions:

I. Some white are glasses - False (Because, No definite relation given between white and glasses, so it can be possible only).

II. Some brittle are glasses - False (Because, No definite relation given between brittle and glasses, so it can be possible only).

III. Some brittle are windows - True (Because, Some windows are white and All white are brittle, so those part of brittle that is white will also windows).

So, Only conclusion III follows.

Hence, the correct answer is "Option 3".

The correct answer is republic.

Key Points

• The Preamble to the Constitution of India starts with the words "We, the people of India".
• This implies that the people of India are the ultimate source of power in the country.
• The Preamble declares India to be a "sovereign, socialist, secular, democratic republic".
• Each of these words has a specific meaning.
• India is a republic because it has an elected head of state.
• The President of India is elected by an Electoral College consisting of the elected members of both houses of Parliament and the Legislative Assemblies of the States.

Additional Information

• Sovereign: India is a free country and has the power to govern itself without any external interference.
• Socialist: The Indian government is committed to creating a society where there is social and economic equality for all.
• Secular: India does not have an official state religion and all religions are treated equally.
• Democratic: The people of India have the right to elect their representatives and participate in the democratic process.

Given Statement: B ≤ T = N > O = Q ≥ M = R

Now,

Conclusions:

(i) N ≥ B → True (B ≤ T = N It gives N ≥ B)

(ii) N < B → False (B ≤ T = N It gives N ≥ B)

(iii) O < R → False ​(O = Q ≥ M = R It gives O ≥ R)

(iv) T < M → False (T = N > O = Q ≥ M It gives T > M)

Here, Only Conclusion i is true.

Hence, the correct answer is "Option 1".

Given numbers: 13 56 46 23 16 34 25 19 78 85 86 24 29 73 94 52 64 63

Now,

The first half is: 13 56 46 23 16 34 25 19 78

The sum of the first half is: 310.

The second half is: 85 86 24 29 73 94 52 64 63

The sum of the first half is: 570.

So, the difference between the sum of the first half and second half is: 310 - 570 = -260.

Hence, the correct answer is "Option 2".

The correct answer is the development of larynx.

Key Points

• Puberty is a phase of physical and sexual development in which the body undergoes various changes due to hormonal changes.
• During puberty, the larynx, which is also known as the voice box, grows larger in both males and females.
• The growth of the larynx causes the vocal cords to lengthen and thicken, resulting in a deeper and more resonant voice.
• The process of voice change during puberty is more prominent in males than in females.

Additional Information

• Cigarette smoking is harmful to the lungs and can cause various respiratory diseases such as bronchitis and lung cancer.
• Lung development is a gradual process that starts before birth and continues throughout childhood and adolescence.
• Muscular development is also a gradual process that occurs throughout childhood and adolescence.
• Puberty usually starts in girls between the ages of 8 and 13, and in boys between the ages of 9 and 14.
• Puberty is characterized by the development of breasts and the onset of menstruation in girls and larger testicles, deeper voice, and more muscular physique in boys.