RRC Group D Previous Year Paper 2 (Held On: 18 Sep 2018 Shift 1)

Using BODMAS we get,

⇒ (136 ÷ 17) + (17 × 13) – (103 – 85) × (62 + 145) ÷ 23,

⇒ 8 + 221 – 18 × 207 ÷ 23

⇒ 8 + 221 – 18 × 9

⇒ 8 + 221 – 162

⇒ 67

Given data

⇒ Rated value of the electric lamp (P) = 120 watts

⇒ Amount of time the lamp is being used = 8 hours/day

⇒ t = 8 h

We know that, Power(P) = E/t

Where ‘P’ is power in watts and ‘E’ is energy consumed in watt-hours and ‘t’ is time.

Substituting the values in the equation above,

120 = E/8

E = 960 Watt-Hours

E = 0.96 kWh or units. (1 Kilo Watt Hour = 1 unit)

Hence, the energy consumed by the lamp in 1 day (E) = 0.96 units

The correct answer is 1815.

Key Points

• The Battle of Waterloo, which took place in Belgium on June 18, 1815, was marked as the final defeat of Napoleon Bonaparte.
• He conquered much of Europe in the early 19th century.
• Napoleon rose through the ranks of the French army during the French Revolution and seized control of the French government in 1799.
• He became emperor in 1804.
• The French Emperor Napoleon Bonaparte escaped from exile in March 1815 and returned to power.
• He decided to go on the offensive, hoping to win a quick victory that would tear apart the coalition of European armies formed against him.
• He expanded his empire across western and central Europe. The Battle of Waterloo, in which Napoleon's forces were defeated by the British and Prussians, marked the end of his reign and of France's domination in Europe.
• The Battle of Waterloo, in which Napoleon’s forces were defeated by the British and Prussians, marked the end of his reign and of France’s domination in Europe.

Important Points 

The battle of Waterloo is still considered to be important due to the following reasons:

• ​It paved a way for the UK to become a global power.
• The Prussians’ contribution to the defeat of the French Army at Waterloo entered the mythology of the Prussian state, creating a sense of nationalism
• which then played a key role in the formation of the new German Empire after 1870
• Historian John Sadler states that "Many who died that day in Waterloo were buried in shallow graves but their bodies were later disinterred and their skeletons were taken.

Cotton, wool and rayon are fibres but Cardboard is not a fibre.

Hence, “Cardboard” is the correct answer.

As the Resistors are connected in series the equivalent resistance will be -

R_eqv = R₁ + R₂

= 10 Ω + 20 Ω

= 30 Ω

According to Ohm’s law -

→ V = IR

→ I = V/R

→ I = 6/30 = 1/5 [V = 6 V given]

→ I = 0.2 A

Given expression: {(36 + 6) + 6} × 12

After changing the symbols:

{(36 ÷ 6) ÷ 6} + 12

= {6 ÷ 6} + 12

= 1 + 12 = 13

Hence, “13” is the correct answer.

Shortcut Trick

Convert speed to m/s: 20 × (5/18) = 50/9 m/s.

Time = Distance ÷ Speed.

Time = 450 ÷ (50/9) = 9 × 9 = 81 seconds.

∴ The correct answer is 81 sec.

Alternate Method

Given:

Distance = 450 m

Speed = 20 km/hr

Formula Used:

Time = Distance ÷ Speed

1 km/hr = 5/18 m/s

Calculations:

⇒ Speed in m/s = 20 × (5/18) m/s

⇒ Speed = 100/18 = 50/9 m/s

⇒ Time = 450 ÷ (50/9)

⇒ Time = 450 × (9/50)

⇒ Time = 9 × 9

⇒ Time = 81 seconds

∴ The correct answer is 81 sec.

Additional Information

Unit Conversion

To convert km/hr to m/s, multiply by 5/18. To convert m/s to km/hr, multiply by 18/5.

Basic Speed Formula

Speed = Distance ÷ Time; Distance = Speed × Time; Time = Distance ÷ Speed.

Average Speed

Average Speed = Total Distance ÷ Total Time. For two equal distances with speeds x and y, Average Speed = 2xy/(x+y).

Shortcut Trick

Loss of 5% means SP₁ is 95% of CP.

95% → Rs. 25.65

1% → 25.65 ÷ 95 = 0.27

New SP₂ as percentage of CP = 31.05 ÷ 0.27 = 115%

Since 115% > 100%, Gain% = 115% − 100% = 15%.

∴ The correct answer is Gain 15%.

Alternate Method

Given: First Selling Price (SP₁) = Rs. 25.65, Loss = 5%, Second Selling Price (SP₂) = Rs. 31.05.

Formulas: CP = SP × 100 ÷ (100 − Loss%), Gain% = (SP − CP) ÷ CP × 100

⇒ Cost Price (CP) = 25.65 × 100 ÷ (100 − 5) = 25.65 × 100 ÷ 95

⇒ CP = 2565 ÷ 95 = Rs. 27

⇒ Profit = New SP − CP = 31.05 − 27 = Rs. 4.05

⇒ Gain% = (4.05 ÷ 27) × 100 = 0.15 × 100 = 15%

∴ The correct answer is Gain 15%.

Additional Information

Profit and Loss on CP

Unless otherwise stated, Profit percentage and Loss percentage are always calculated on the Cost Price (CP).

Unitary Method for SP

If CP is assumed as 100 units, then SP at x% profit is (100 + x) units and SP at y% loss is (100 − y) units.

The statement given us no information to support the fact that younger children learn swimming faster. Hence assumption I is not implicit.

It is given in the statement school has made it compulsory for the children to learn swimming, this implies that the school is serious about the overall development of children. Hence, assumption II is also implicit.

• Pravasi Bharatiya Divas is held traditionally on 9th January every year to commemorate the return of Mahatma Gandhi as a Pravasi from South Africa to India.
• It also marks the contribution of the overseas Indian community in the development of India.
• The 15^th Pravasi Bharatiya Divas 2019 was celebrated with the theme "Role of Indian Diaspora in building a New India.”

When the numbers are arranged in ascending order we get,

0.658, 0.8, 5.74, 6.03, 7.2

∴ The required decimal number is 5.74

Given,

Similarly,

Hence, “IFO” is the correct answer.

The minimum possible Venn diagram of the given statements,

Conclusions:

I. Some metals are pins → True

II. Some pins are clips → False (its possible but not definite)

Hence, only conclusion I follows.

We can count the rectangles as follows,

Hence, there are “24” rectangles in the given figure.

Given:

 {(.98)3 + (0.02)3 + 3 × 0.98 × 0.02 – 1}

Concept used:

(a + b)3 = a3 + b3 + 3ab(a + b)

Calculation:

{(.98)³ + (0.02)³ + 3 × 0.98 × 0.02 – 1

⇒ [(.98)³ + (0.02)³ + 3 × 0.98 × 0.02(0.98 + 0.02)} – 1]

Now, first term is of the form (a + b)3 where a = .98 and b = 0.02

⇒ (0.98 + 0.02)³ – 1 = 1 – 1 = 0

∴ The value is 0.

• Union Defence Minister of India Nirmala Sitharaman gave away Republic Day Parade 2018 best tableau award to Maharashtra.
• Tripura bagged the best state tableau in 2019 Republic Day parade With the theme ‘Empowering Rural Economy the Gandhian Way’.
• The second position was bagged by Jammu and Kashmir while the state of Punjab was placed in the third spot.
• India celebrated the 69^th republic day this year and Cyril Ramaphosa, President of South Africa was the chief guest for this year republic day parade.

• Xylem and Phloem are Complex Permanent Tissue while the other three are simple permanent tissues.

Shortcut Trick

Directly substitute the standard trigonometric values.

sin 60° = √3/2, cos 30° = √3/2, cot 45° = 1, sec 60° = 2.

(√3/2)² + (√3/2)² + 1² + 2² = 3/4 + 3/4 + 1 + 4 = 6/4 + 5 = 1.5 + 5 = 6.5.

6.5 = 13/2.

∴ The correct answer is 13/2.

Alternate Method

Given: sin² 60° + cos² 30° + cot² 45° + sec² 60°

Formula Used: Standard trigonometric ratios at specific angles.

⇒ sin 60° = √3/2

⇒ cos 30° = √3/2

⇒ cot 45° = 1

⇒ sec 60° = 2

Calculations:

⇒ (√3/2)² + (√3/2)² + (1)² + (2)²

⇒ (3/4) + (3/4) + 1 + 4

⇒ 6/4 + 5

⇒ 3/2 + 5

⇒ (3 + 10) / 2

⇒ 13/2

∴ The correct answer is 13/2.

Additional Information

Trigonometric Identities

sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ.

Complementary Angles

sin(90°−θ) = cosθ and cot(90°−θ) = tanθ. This explains why sin 60° = cos 30°.

Reciprocal Relations

secθ = 1/cosθ; cosecθ = 1/sinθ; cotθ = 1/tanθ.

Shortcut Trick

In Data Sufficiency, a statement is sufficient if it yields a unique value for the required variable.

Equation 1 is a quadratic equation in x, which simplifies to (x − 9)² = 0, giving a unique value x = 9.

Equation 2 involves independent variables (p, q, r) and provides no information regarding x.

∴ The correct answer is Only 1 is sufficient.

Alternate Method

Given Equations:

1. x² − 18x + 81 = 0
2. p + q + r = 0

Analysis of Equation 1:

⇒ x² − 9x − 9x + 81 = 0

⇒ x(x − 9) − 9(x − 9) = 0

⇒ (x − 9)(x − 9) = 0

⇒ x = 9, 9

Since Equation 1 provides a definite value for x, it is sufficient.

Analysis of Equation 2:

⇒ p + q + r = 0

This equation contains three variables p, q, and r. There is no relationship or data provided to link these variables to x.

Thus, Equation 2 is insufficient to determine the value of x.

∴ The correct answer is Only 1 is sufficient.

Additional Information

Quadratic Identity

The expression a² − 2ab + b² is a perfect square which equals (a − b)². In Equation 1, x² − 2(x)(9) + 9² fits this form perfectly.

Nature of Roots

For ax² + bx + c = 0, if the discriminant D = b² − 4ac is equal to 0, the equation has two equal real roots, providing a unique solution for the variable.

Data Sufficiency Rule

A statement is considered sufficient if it leads to one and only one specific answer. If multiple values or no values can be determined, the statement is insufficient.

• The pH level is a measure of the number of Hydrogen ions in a solution.
• The lower the solution or compound on the pH scale, the higher will be the concentration of hydrogen ions.
• Therefore the solution having a pH of 2.4 will have the highest concentration of hydrogen ions.
• The pH scale ranges from 0 to 14.
• A pH value of 7 is considered neutral, example - water and pH value > 7 is Basic > pH value < 7 is Acidic

Shortcut Trick

Speed = Difference in Distance ÷ Difference in Time

Distance difference = 205 − 110 = 95 m

Time difference = 18.25 − 13.5 = 4.75 seconds

Speed in m/s = 95 ÷ 4.75 = 20 m/s

Speed in km/h = 20 × (18/5) = 72 km/h

∴ The correct answer is 72 km/h.

Alternate Method

Given: Platform₁ = 110 m, Time₁ = 13.5 s, Platform₂ = 205 m, Time₂ = 18.25 s.

Formula Used: Speed = (Train Length + Platform Length) ÷ Time

⇒ Let the length of the train be L.

⇒ Speed = (L + 110) / 13.5 = (L + 205) / 18.25

⇒ 18.25(L + 110) = 13.5(L + 205)

⇒ 18.25L + 2007.5 = 13.5L + 2767.5

⇒ 4.75L = 760

⇒ L = 160 m

⇒ Speed = (160 + 110) / 13.5 = 270 / 13.5 = 20 m/s

⇒ Speed in km/h = 20 × (18/5) = 72 km/h

∴ The correct answer is 72 km/h.

Additional Information

Unit Conversion

To convert m/s to km/h, multiply by 18/5. To convert km/h to m/s, multiply by 5/18.

Train Crossing Stationary Object

When a train crosses a platform or bridge, the total distance covered is the sum of the train length and the object length.

Relative Speed

For two objects moving in the same direction, relative speed = S₁ − S₂. For opposite directions, relative speed = S₁ + S₂.

Shortcut Trick

Net filling rate = (1/Time A&B) − (1/Time C)

Net filling rate = (1/4) − (1/8) = (2 − 1)/8 = 1/8

Time taken = 1 ÷ (1/8) = 8 hours

∴ The correct answer is 8 hr.

Alternate Method

Given: Filling time for A & B = 4 hours, Emptying time for C = 8 hours.

Formula Used: Total Work = LCM of time taken; Efficiency = Total Work ÷ Time.

⇒ Net efficiency of A + B + C = (+2) + (−1) = +1 unit/hr

⇒ Total time required = Total Work ÷ Net Efficiency

⇒ Total time = 8 ÷ 1 = 8 hours

∴ The correct answer is 8 hr.

Additional Information

Inlet and Outlet Pipes

An inlet pipe fills the tank (positive work), while an outlet pipe empties it (negative work).

Net Work Formula

If an inlet fills in x hours and an outlet empties in y hours, net part filled in 1 hour = (1/x) − (1/y).

Pipes and Cisterns Logic

This topic follows the same principles as Time and Work, where efficiency is inversely proportional to time.

Hence, Aadhya is in North-West direction with respect to her initial position.

• Marsilea is a Pteridophyte, and unlike algae (like Chara and Spirogyra) or bryophytes (like Riccia), Pteridophytes have a plant body differentiated into true roots, stems, and leaves. However, in the context of typical biology questions regarding non-differentiated bodies, algae are the primary examples. If the question asks for the plant body NOT differentiated, and we must remove ambiguity, we should change the options to exclude multiple thallophytes. Given the constraint to only correct the ambiguity: Marsilea is the only one here that IS differentiated. If the question asks for NOT differentiated, and we have Riccia (Bryophyte - thalloid), Chara (Algae - thalloid), Spirogyra (Algae - thalloid), the question is flawed. Let's adjust the question to ask for one that IS differentiated to make Marsilea the unique answer.

From the given statement, we understand the following things: the word ‘reminded’ tells us that Rina ‘wants them to remember about their study schedule’ and because ‘the final exams are getting near’, she wants them to ‘give serious thought to their exams’. From this we can safely state that both the assumptions are implied.

• The Pradhan Mantri Jan Dhan Yojana was launched on 28th August 2014 by Prime Minister Narendra Modi.
• It also entered into the Guinness Book of World Record for having most bank accounts opened in one week.
• The scheme sought to financially empower the poor by providing them access to the formal banking system.

• The President of India Ram Nath Kovind released Rs 1,000 and Rs 10 commemorative coins on the occasion of Nabakalebar festival.
• The government had announced the release of the coins marking the festival which is a symbolic recreation of sibling deities -- Lord Jagannath, Balabhadra and Devi Subhadra.
• The commemorative coins were released at a function of Rashtriya Sanskrit Sansthan at Puri in Odisha

⇒ 4 + 4.44 + 44.4 + 4.04 + 444 – 20 = 480.88

The correct answer is 17th.

• The Halogens are the elements belonging to Group 17 in the periodic table.
• A periodic table is an arrangement of elements based on their atomic numbers and chemical properties.
• It is divided into 18 groups and 7 periods running vertically and horizontally respectively.
• The periodic table accommodates a discrete combination of metal, non-metals, and metalloids.
• The elements in the periodic table are arranged horizontally in ascending order of their atomic numbers known as periods.
• Vertically the elements having identical chemical properties are clubbed together known as groups.
• The 18 groups are also known as columns and the 7 periods are also known as the rows.

Explanation:

Periods:

• Elements are arranged in increasing the atomic number of elements in a period.
• One extra electron gets added to the outermost shell as we move along the periods from left to right.
• The electron gets added to the same shell or orbit and thus the electrons present for bonding increase by one unit.
• Thus, the shell number remains the same but the number of electrons present for bonding increases along a period.

Groups:

• Elements having the same number of outer electrons are put in the same group of the periodic table.
• When we move down a group, new shells get added.
• The outermost shell has electrons present for bonding.
• Though the number of shells increases as we go down in a group, the number of electrons in the outermost shell remains the same.
• For example, the Halogens F, Cl, Br, I, At all belong to group 17 and have 7 electrons in the outermost shell.
• Similarly, Group 16 elements have 6 electrons in the outermost shell, group 15 has 5 electrons in the outermost shell, and so on.

Hence, group 17 is called halogens.

• The government appointed senior IAS officer S Selvakumar as Chairman and Managing Director (CMD) of Security Printing and Minting Corporation of India Ltd (SPMCIL).
• Mr. Selvakumar has replaced Mr. Anurag Agarwal.
• The SPMCIL is responsible for producing coins, banknotes, postage stamps, non-judicial stamps, and other official documents.

Shortcut Trick

Convert mixed fraction to improper: 2(3/4) = 11/4 litres.

Distance for 1 litre = Total Distance ÷ Total Petrol

⇒ 44 ÷ (11/4) = 44 × (4/11) = 4 × 4 = 16 km.

∴ The correct answer is 16 km.

Alternate Method

Given:

Total Distance = 44 km

Total Petrol used = 2(3/4) litres = 11/4 litres

Calculations:

⇒ Distance covered in 11/4 litres = 44 km

⇒ Distance covered in 1 litre = 44 ÷ (11/4)

⇒ Distance covered in 1 litre = 44 × (4/11)

⇒ Distance covered in 1 litre = 4 × 4

⇒ Distance covered in 1 litre = 16 km

∴ The correct answer is 16 km.

Additional Information

Unitary Method

It is a technique where we find the value of a single unit first to calculate the value of the required number of units.

Mileage/Fuel Efficiency

The distance a vehicle travels per unit of fuel consumed. Formula: Mileage = Total Distance ÷ Total Fuel.

Mixed Fraction Conversion

To convert a mixed fraction a(b/c) to improper, use (a×c + b)/c.

Two lines are intersecting each other in figure A, B and C, but in figure D Both the lines are parallel to each other.

Hence, “D” is the odd one.

Explanation:

• Testosterone is the hormone that develops male characteristics in the human body. When XY chromosomes meet, they form a male body. It is a hormone secreted by the male reproductive organ Testes.
• Testosterone plays a key role in the development of male reproductive tissues such as the testes and prostate, as well as promoting secondary sexual characteristics such as increased muscle and bone mass, and the growth of body hair.

Additional Information

• Insulin is a hormone secreted by the β cells of islets of Langerhans. Insulin is released into the bloodstream to lower down blood sugar levels and prevent hyperglycemia.
• Thyroxine: It is an endocrine secretion of the thyroid gland. The thyroid gland requires 120 microgram Iodine per day for the production of thyroxine It regulates the basal metabolic rate of the body.

• The first Khelo India School Games was held in New Delhi where Haryana topped the overall medal tally with 102 medals- 38 gold and 26 silver.
• Maharashtra won 110 medals (36 gold) and finished second while Delhi with 94 medals came third.
• Sports Minister, Col Rajyavardhan Rathore presented the trophy to the top state Haryana.
• The first Khelo India School Games were launched by Prime Minister Narendra Modi at Indira Gandhi Indoor Stadium in New Delhi.
• It was introduced to revive the sports culture in India at the grass-root level.

• Velu Thampi Dalava was the ruler of the Indian kingdom of Travancore between 1802 and 1809.
• He is best known to rebel against the British East India Company's supremacy in India.

Here, analogy pair is Glass that is transparent and wood will be opaque.

Hence, “opaque” is the correct answer.

CONCEPT:

• Kinetic energy (K.E): The energy possessed by a body by the virtue of its motion is called kinetic energy.

The expression for kinetic energy is given by:

$KE = \frac{1}{2}m{v^2}$

Where m = mass of the body and v = velocity of the body

• Momentum (p): The product of mass and velocity is called momentum.

Momentum (p) = mass (m) × velocity (v)

The relationship between the kinetic energy and Linear momentum is given by:

As we know,

 $KE = \frac{1}{2}m{v^2}$

Divide numerator and denominator by m, we get

$KE = \frac{1}{2}\frac{{{m^2}{v^2}}}{m} = \frac{1}{2}\frac{{;{{\left( {mv} \right)}^2}}}{m} = \frac{1}{2}\frac{{{p^2}}}{m};$ [p = mv]

$\therefore KE = \frac{1}{2}\frac{{{p^2}}}{m};$

$p = \sqrt {2mKE}$

EXPLANATION:

Given that:

K.E1 = K.E2= K.E (let say)

The relation between the momentum and the kinetic energy is given by:

$P = \sqrt {2m;K.E}$

But as K.E is same

∴ $P \propto \sqrt m$

Or, $\frac{{{P_1}}}{{{P_2}}} = \sqrt {\frac{{{m_1}}}{{{m_2}}}}$

Here, if m₁ > m₂ then p₁ > p₂

• If two unequal masses possess the same kinetic energy, then the heavier mass has greater momentum. So option 3 is correct.

Shortcut Trick

Use the MDH/W formula: (M₁ × D₁) / W₁ = (M₂ × D₂) / W₂

Substitute values: (15 × 5) / 750 = (25 × 6) / W₂

75 / 750 = 150 / W₂ ⇒ 1 / 10 = 150 / W₂

W₂ = 150 × 10 = ₹1500

∴ The correct answer is ₹1500.

Alternate Method

Given:

Case 1: Boys (M₁) = 15, Days (D₁) = 5, Earnings (W₁) = ₹750

Case 2: Boys (M₂) = 25, Days (D₂) = 6, Earnings (W₂) = ?

Formula Used:

Work (Earnings) is directly proportional to the number of men and the number of days: W ∝ M × D

Calculations:

⇒ Earnings per boy per day = 750 / (15 × 5)

⇒ Earnings per boy per day = 750 / 75 = ₹10

⇒ Earnings for 25 boys in 1 day = 25 × 10 = ₹250

⇒ Earnings for 25 boys in 6 days = 250 × 6 = ₹1500

∴ The correct answer is ₹1500.

Additional Information

Chain Rule in Work and Wages

The amount of work done or wages earned is directly proportional to the number of persons and the time duration: (M₁D₁H₁)/W₁ = (M₂D₂H₂)/W₂.

Direct vs Inverse Proportion

Men and Days are inversely proportional for a fixed work (M₁D₁ = M₂D₂), while Men and Wages are directly proportional (M₁/W₁ = M₂/W₂).

• The number of water molecules that combine chemically in definite molecular proportion, with the concerned salt in the crystalline state is known as the water of crystallization.
• Baking soda does NOT possess water of crystallization while other three do so.

The correct answer is Upwards positive.

Key Points

• The velocity of a vertically thrown ball, with time, will be “up” in the positive direction.
• Gravity pulls the ball down, and since that is the only force acting on the ball, neglecting air resistance, the acceleration is always 9.8 m/s² down or -9.8 m/s².
• Since, u=0

           v(t) = u + at, so v(t) = - 9.8t.

CONCEPT:

• Velocity: The rate of change of position i.e. rate of displacement with time is called velocity.

$v = \frac{{{s_2} - {s_1}}}{{{t_2} - {t_1}}} = \frac{{{\rm{\Delta }}s}}{{{\rm{\Delta }}t}}$

Where s2 = displacement of the object at t2 and s1 = displacement of the object at t1

• It is a vector quantity having the symbol $\vec v$.

EXPLANATION:

• When the ball goes upwards the direction of the velocity of the ball decreases but in the cartesian plane when the ball goes upward its speed is positive and when the ball comes downward its velocity is negative.

Additional Information

• The velocity and time are linearly proportional to each other which is the same as the equation of a straight line.
• Therefore the correct velocity-time graph is

The statement talks about the definition and process of sustainability and its concurrent practices. It also talks about the human capital is the underlying basis for all sustainable approaches. Hence assumption I follows.

Assumption II seems not to connect with the statement, as, it is moving to investors and invest. So, Assumption II not follows.

Hence, the correct answer is "Only assumption I is implicit".

• Table is made of wood and cloth is different.

Therefore, the best possible Venn diagram representation will be,

Hence, “option 3” is the correct answer.

Mistake Points

• Some tables are made of plastic too. However, an option indicating this is not given.
• We must choose the best possible option from the given ones.

Hence, North become “north-west”.

Shortcut Trick

Milk required for 1 kg paneer = 1000 / 200 = 5 Litres

Total milk for 20 kg paneer = 20 × 5 = 100 Litres

Total cost per day = 100 × 40 = Rs. 4,000

∴ The correct answer is Rs. 4,000.

Alternate Method

Given:

Total paneer produced daily = 20 kg

Paneer produced from 1 litre of milk = 200 gms

Cost of 1 litre of milk = Rs. 40

Calculations:

⇒ Total weight of paneer in grams = 20 × 1000 = 20,000 gms

⇒ Total milk required = Total paneer ÷ Paneer per litre

⇒ Total milk required = 20,000 ÷ 200 = 100 Litres

⇒ Money spent on milk = Total milk × Cost per litre

⇒ Total money spent = 100 × 40 = Rs. 4,000

∴ The correct answer is Rs. 4,000.

Additional Information

Unit Conversion

Always ensure units are consistent before calculation. 1 kg = 1000 grams.

Unitary Method

Value of a required quantity = (Value of known unit quantity × required quantity) ÷ known quantity.

Total Cost Calculation

Total Expenditure = Quantity × Rate per unit.

Shortcut Trick

From the equation x² − x − 1 = 0, the sum of roots (α + β) = 1 and product of roots (αβ) = −1.

The sum of the new roots (α/β + β/α) = (α² + β²) / αβ = [(α + β)² − 2αβ] / αβ.

New Sum = [1² − 2(−1)] / (−1) = 3 / −1 = −3.

The product of the new roots (α/β × β/α) = 1.

The required equation is x² − (Sum)x + (Product) = 0 ⇒ x² − (−3)x + 1 = 0 ⇒ x² + 3x + 1 = 0.

∴ The correct answer is x² + 3x + 1 = 0.

Alternate Method

Given:

Quadratic equation: x² − x − 1 = 0 with roots α and β.

Formula Used:

For a quadratic equation ax² + bx + c = 0, sum of roots = −b/a and product of roots = c/a.

The equation with roots r₁ and r₂ is x² − (r₁ + r₂)x + (r₁r₂) = 0.

Calculations:

⇒ α + β = −(−1)/1 = 1

⇒ αβ = −1/1 = −1

⇒ New roots are r₁ = α/β and r₂ = β/α.

⇒ Sum (S) = α/β + β/α = (α² + β²) / αβ

⇒ α² + β² = (α + β)² − 2αβ = (1)² − 2(−1) = 1 + 2 = 3

⇒ S = 3 / (−1) = −3

⇒ Product (P) = (α/β) × (β/α) = 1

⇒ Required equation: x² − (S)x + P = 0

⇒ x² − (−3)x + 1 = 0

⇒ x² + 3x + 1 = 0

∴ The correct answer is x² + 3x + 1 = 0.

Additional Information

Relation between Roots and Coefficients

For any quadratic equation ax² + bx + c = 0, the sum of roots is −b/a and the product of roots is c/a.

Symmetric Functions of Roots

Functions of roots that remain unchanged when roots are interchanged, such as α² + β² = (α + β)² − 2αβ and α³ + β³ = (α + β)³ − 3αβ(α + β).

Reciprocal Roots

If the roots of ax² + bx + c = 0 are reciprocals of each other, then the product of roots c/a = 1, which implies c = a.

Transformation of Equations

To find an equation with roots 1/α and 1/β, replace x with 1/x in the original equation ax² + bx + c = 0 to get cx² + bx + a = 0.

Shortcut Trick

Use the Value Substitution Method: Assume a = 2, b = 2, and c = 2.

According to the condition a + b + c = 2s, we have 2 + 2 + 2 = 2s ⇒ s = 3.

Substitute these values into the given expression: (3 − 2)² + (3 − 2)² + (3 − 2)² + 3²

⇒ 1² + 1² + 1² + 9 = 1 + 1 + 1 + 9 = 12.

Check Option (3): a² + b² + c² = 2² + 2² + 2² = 4 + 4 + 4 = 12.

∴ The correct answer is (a² + b² + c²).

Alternate Method

Given: a + b + c = 2s

We need to find the value of the algebraic expression: (s − a)² + (s − b)² + (s − c)² + s²

Using the identity (x − y)² = x² − 2xy + y², expand each term:

⇒ (s² − 2as + a²) + (s² − 2bs + b²) + (s² − 2cs + c²) + s²

Now, group the terms by combining like powers:

⇒ (s² + s² + s² + s²) − (2as + 2bs + 2cs) + (a² + b² + c²)

⇒ 4s² − 2s(a + b + c) + (a² + b² + c²)

Substitute the given condition a + b + c = 2s into the equation:

⇒ 4s² − 2s(2s) + (a² + b² + c²)

⇒ 4s² − 4s² + a² + b² + c²

⇒ a² + b² + c²

∴ The correct answer is (a² + b² + c²).

Additional Information

Trinomial Square Identity

The expansion of the square of a sum of three terms is (a + b + c)² = a² + b² + c² + 2(ab + bc + ca).

Semi-perimeter Property

In geometry, specifically in Hero's formula for the area of a triangle, s represents the semi-perimeter where 2s = a + b + c.

Difference of Squares Expansion

The square of a binomial difference is always expanded as (x − y)² = x² − 2xy + y².

In figure A, B and C arrow is moving in a clockwise direction but in figure, D arrow is moving in an anticlockwise direction.

Hence, “figure D” is odd one out.

The minimum possible Venn diagram of the given statements,

Conclusions:

1. All brakes are paddles → False (it is possible but not definite)

2. Some brakes are paddles → True.

3. No brake is glass → False (it is possible but not definite)

Hence, only conclusion 2 follows.

• Hydra can reproduce by regeneration and budding.
• In budding, initially buds appear as a small bulge. They further grow, and within 2-4 days they become fully-formed hydra. Then it detaches from the parent body.
• In regeneration, regrowth of damaged or lost part takes place.
• Hydra belongs to Phylum Coelenterata.
• Yeast is a fungi and it reproduces asexually through budding.
• Plasmodium is a protozoan and it has sexual cycle in mosquito and asexual cycle in humans. In asexual cycle it undergoes multiple fission.
• Planaria reproduces asexually through regeneration. In fact it possess the highest power of regeneration.

Here, axes are shifting 45 degrees in the Anticlockwise direction, hence, northeast would become north direction.

Similarly, the northeast will be there in place of the East.

Hence, East will become “North-East”

• Karewas are lacustrine deposits (deposits in a lake) in the valley of Kashmir and in Bhadarwah Valley of the Jammu division of Jammu and Kashmir.
• These are the flat-topped mounds that border the Kashmir valley on all sides. They are characterized by fossils of mammals and at places by peat. The various crops grown in this area are saffron, almond, apples, walnut, peaches, pears, cherry and plum.
• Ling nut is grown in tropical and subtropical areas of West Bengal, Bihar and Jharkhand.

Shortcut Trick

For 987x54 to be divisible by 6, it must be divisible by both 2 and 3.

Since the last digit is 4, it is divisible by 2.

Sum of digits = 9 + 8 + 7 + x + 5 + 4 = 33 + x.

For (33 + x) to be divisible by 3, x must be 3 (from options, 33 + 3 = 36 is divisible by 3).

∴ The correct answer is 3.

Alternate Method

Given number is 987x54.

Check divisibility by 2:

⇒ Last digit is 4, which is even, so the number is divisible by 2.

Check divisibility by 3:

⇒ Sum of digits = 9 + 8 + 7 + x + 5 + 4

⇒ Sum of digits = 33 + x

Let us test the given options for x:

⇒ If x = 5, sum = 33 + 5 = 38 (not divisible by 3)

⇒ If x = 3, sum = 33 + 3 = 36 (divisible by 3)

⇒ If x = 2, sum = 33 + 2 = 35 (not divisible by 3)

⇒ If x = 1, sum = 33 + 1 = 34 (not divisible by 3)

⇒ Therefore, the value of x must be 3.

∴ The correct answer is 3.

Additional Information

Divisibility Rule of 3

A number is completely divisible by 3 if the sum of its individual digits is a multiple of 3.

Divisibility Rule of 6

A number is divisible by 6 if it is simultaneously divisible by both 2 (even unit digit) and 3 (sum of digits is divisible by 3).

Divisibility Rule of 9

A number is completely divisible by 9 if the sum of all its digits is divisible by 9.

• Electric bulbs are usually filled with chemically-inactive gases like Argon and Nitrogen.
• When a bulb is filled with an inert gas, it slows down the evaporation of the tungsten filament compared to operating it in a vacuum.
• This allows greater temperatures and therefore greater efficacy with less reduction in filament life.

Shortcut Trick

Apply Successive Percentage formula: x + y + (x × y)/100

Here, x = 8 and y = 12

Net increase = 8 + 12 + (8 × 12)/100

Net increase = 20 + 0.96 = 20.96%

∴ The correct answer is 20.96%.

Alternate Method

Given: Increase in length (L) = 8%, Increase in breadth (B) = 12%.

Formula Used: Area of Rectangle = Length × Breadth

⇒ Let initial Length = 100 and initial Breadth = 100

⇒ Initial Area = 100 × 100 = 10,000

⇒ New Length = 100 + 8% of 100 = 108

⇒ New Breadth = 100 + 12% of 100 = 112

⇒ New Area = 108 × 112 = 12,096

⇒ Increase in Area = 12,096 − 10,000 = 2,096

⇒ Percentage Increase = (2,096 / 10,000) × 100 = 20.96%

∴ The correct answer is 20.96%.

Additional Information

Successive Percentage Change

Used when a value is changed and then the resultant value is changed again. Net % = x + y + xy/100.

Area and Volume Changes

For 2D shapes (Area), use successive change twice. For 3D shapes (Volume), apply the formula thrice.

Successive Discount

Equivalent discount for two discounts d₁ and d₂ is (d₁ + d₂ − d₁d₂/100)%.

The correct answer is Moseley.

Key Points

• Henry Moseley classified elements based on Atomic numbers.
• In 1913 A.D. the English scientist Henry Moseley demonstrated, with the help of the experiments done using an X-ray tube, that the atomic number (Z) of an element corresponds to the positive charge on the nucleus or the number of protons in the nucleus of the atom of that element.
• This revealed that ‘atomic number’ is a more fundamental property of an element than its atomic mass.

• Tamil Nadu's Anukreethy Vas was named Miss India 2018.
• The 55^th Femina Miss India 2018 was held Mumbai.
• Miss World 2017 Manushi Chhillar, crowned Anukreethy, who will go on to represent India at Miss World.
• Haryana's Meenakshi Chaudhary stood second and Andhra Pradesh's Shreya Rao Kamavarapu ranked third. Malaika Arora and Irfan Pathan were among the judges.

• Yuzvendra Chahal is a former chess player and an Indian Cricketer who represented India in both One day international and Twenty-20 Internationals
• He played for Mumbai Indians, Royal Challengers Bangalore and now playing for Rajasthan Royals (2022) in the IPL and for Haryana in domestic cricket.

• Aamir Khan won the best actor award at 2017 Filmfare awards for playing the character of Mahavir Singh Phogat in the movie Dangal.
• The 1st Filmfare Awards were held on March 21, 1954, which honoured the best in Hindi cinema in 1953.
• That year they were known as Clare Awards, after Clare Mendonca, a film critic of The Times of India who had died in the same year. The award for best actor was won by Dilip Kumar for his movie Daag.

Nail, hair and toe are parts of the body and thus form a group whereas acne is a skin disease.

Hence, acne(C) does not belong to the group.

Here, the pattern followed here is,

Hence, “34” is the correct answer.

The minimum possible Venn diagram of the given statements,

Conclusions:

All plastic are covers → False (it is possible but not definite)

All toxic are covers → False (it is possible but not definite)

Hence, “No conclusions follow”.

Shortcut Trick

Range of n numbers when means of subsets are given = (Difference in Means) × (Number of elements in subset).

Difference in mean = 11 − 9 = 2.

Number of elements in each subset = 3.

Range = 2 × 3 = 6.

∴ The correct answer is 6.

Alternate Method

Given:

Mean of the three smallest numbers = 9

Mean of the three largest ones = 11

Formula Used: Range = Largest value − Smallest value

Let the four numbers in ascending order be a, b, c, and d.

⇒ Sum of three smallest numbers (a + b + c) = 9 × 3 = 27 → (Eq. 1)

⇒ Sum of three largest ones (b + c + d) = 11 × 3 = 33 → (Eq. 2)

Subtracting (Eq. 1) from (Eq. 2):

⇒ (b + c + d) − (a + b + c) = 33 − 27

⇒ d − a = 6

∴ The correct answer is 6.

• Centre of gravity is an imaginary point in a body of matter where the entire weight of an object acts.
• This is sometimes useful in designing static structures (e.g., buildings and bridges) or tp predict the behaviour of a moving body when it is acted on by gravity.

Given:

\(22-\left[ {23-\left{ {24-\left( {27 - \overline {25 - 30} } \right)} \right}} \right]\)

Concept Used:

Calculation:

⇒ \(22-\left[ {23-\left{ {24-\left( {27 - ({ - 5} } \right)} \right}} \right]\)

⇒ 22 – [23 – {24 – (27 + 5)}]

⇒ 22 – [23 – {24 – 32}]

⇒ 22 – [23 – (-8)]

⇒ 22 – [23 + 8]

⇒ 22 – 31

∴ -9

Given expression: 80 + 20 ÷ 5 - 12 × 92

After changing the symbols: 80 ÷ 20 - 5 × 12 + 92

= 4 - 5 × 12 + 92

= 4 - 60 + 92

= 96 - 60

= 36

Hence, 36 is the correct answer.

• The mass of the moon is 1/100 times the mass of Earth and the radius of the moon is ¼ times the radius of the Earth.
• As a result, the gravitational attraction on the moon is about one-sixth when compared to that of the Earth.
• The weight of an object depends on the value of g, i.e. acceleration due to gravity.
• Hence the weight of an object on the moon is 1/6^th of the weight on the Earth.

Hence, “21” is the correct answer.

Here, by combining A, C and D we can get the question figure, as following,

Hence, “C, D and A” is the correct answer.

• Calcium Hydroxide is Ca(OH)₂ which is also known as Slaked lime.
• It is an inorganic compound.
• It is a white powder or colorless crystal which is obtained when Quicklime (calcium oxide) is mixed with water.
• It is used in the Cement Industry.

• The Gospel of Yudas is the book about the politics of faithfulness and the price of idealism written by K. R. Meera.
• Aarachar and Nethromeelanam are also other famous novels written by K.R Meera.

Shortcut Trick

Apply the Alligation Method on the mean scores of Boys and Girls.

Ratio of Boys to Girls = (15 − 14.28) : (14.28 − 13) = 0.72 : 1.28.

Simplified ratio = 72 : 128 ⇒ 9 : 16.

Since the number of boys is 9 (9 units = 9), the number of girls must be 16 (16 units = 16).

Total students = 9 + 16 = 25.

∴ The correct answer is 25 students.

Alternate Method

Given:

Number of boys (n_B) = 9

Mean score of boys (M_B) = 13

Mean score of girls (M_G) = 15

Overall average (M_avg) = 14.28

Formula Used:

Total Average = [(n_B × M_B) + (n_G × M_G)] ÷ (n_B + n_G)

Calculations:

Let the number of girls be x.

⇒ 14.28 = [(9 × 13) + (x × 15)] ÷ (9 + x)

⇒ 14.28 × (9 + x) = 117 + 15x

⇒ 128.52 + 14.28x = 117 + 15x

⇒ 15x − 14.28x = 128.52 − 117

⇒ 0.72x = 11.52

⇒ x = 11.52 ÷ 0.72 = 16

⇒ Total number of students = 9 + 16 = 25

∴ The correct answer is 25 students.

Additional Information

Weighted Average Concept

It is used when groups have different sizes. The formula is A_w = (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).

Rule of Alligation

Ratio of quantities = (Higher value − Mean) : (Mean − Lower value). This provides the inverse ratio of the weights of the two components.

Deviation Method

The sum of deviations of all observations from their mean is always zero. This can be used to check if the calculated mean or student count is correct.

From the given information we can draw following family tree,

Clearly, Mamata’s brother is maternal uncle of Surya’s brother.

Hence, “Maternal uncle” is the correct answer.

• Launchpad Accelerator India is a mentorship program launched in 2018 that would aim to support Indian startups that deploy artificial intelligence and machine learning to build solutions for local needs.
• The three-month accelerator will provide a cohort of 8-10 Indian startups with the mentorship of Google.
• The start-ups will also get up to $1,00,000 of Google Cloud credits.

• It is known that light ray that is travelling from a Rarer medium to a Denser medium will bend towards the normal at the point of incidence and conversely, a ray of light travelling from a Denser medium to the Rarer medium bends away from the normal.
• Glass is denser than Water, therefore, a ray of light travelling from Water to Glass will bend towards the normal and slow down.
• Glass has a higher refractive index than water. The refractive index of window glass is 1.52, and that of pure water is 1.33.

Here, if the question figure is rotated by 90° clockwise we get the figure in option D.

Hence, “D” is the correct answer.

→ Given data -

• Mass, m = 40 kg

Height, h = 5m

Time, t = 4s

• Work done, W = mgh

W = 40 × 10 × 5 [g = 10 m/s², given]

W = 2000 joules

→ Power, P = work done /time

P = 2000/4

P = 500 joule/sec

P = 500 W

Shortcut Trick

Convert ratio 1/2 : 2/3 : 3 : 4 to integers by multiplying by LCM(2, 3) = 6.

New ratio ⇒ 3 : 4 : 18 : 24. Total units = 3 + 4 + 18 + 24 = 49 units.

49 units = ₹686 → 1 unit = 686 ÷ 49 = ₹14.

First part (3 units) = 3 × 14 = ₹42.

∴ The correct answer is ₹42.

Alternate Method

Given: Total amount = ₹686, Ratio = 1/2 : 2/3 : 3 : 4.

Step 1: Simplify the ratio

⇒ (1/2 × 6) : (2/3 × 6) : (3 × 6) : (4 × 6)

⇒ 3 : 4 : 18 : 24

Step 2: Sum of ratios

⇒ 3 + 4 + 18 + 24 = 49

Step 3: Calculate the first part

⇒ First part = (3 ÷ 49) × 686

⇒ First part = 3 × 14

⇒ First part = 42

∴ The correct answer is ₹42.

Additional Information

Ratio Simplification

To simplify a fractional ratio a/b : c/d, multiply all terms by the Least Common Multiple (LCM) of the denominators b and d.

Proportional Division

If a total amount T is divided in the ratio a : b : c, the first share is calculated as [a / (a + b + c)] × T.

Compound Ratio

The compound ratio of two ratios (a : b) and (c : d) is found by multiplying corresponding terms: (a × c) : (b × d).

• National Voters Day is celebrated on January 25 every year to encourage more young voters to take part in the political process.
• It started from January 25^th, 2011 to mark the Commission's foundation day.
• The Election Commission of India (ECI) celebrated 8^th National Voters Day across the nation on 25^th January 2018 for enhanced participation of citizens in the electoral process.

• The Volga is the longest river in Europe.
• It is the national river of Russia flowing at a distance of 2294 miles.
• It originates in the Valdai Hills northwest of Moscow and discharges into the Caspian Sea. Out of the 20 largest cities of Russia, 11 are located in the river’s drainage basin.
• It has two main tributaries named Kama and Oka.
• The Danube is Europe's second longest river, after the Volga and flows through 10 countries.

Shortcut Trick

For quarterly compounding, the Quarterly Rate = 12% ÷ 4 = 3%.

The Time Periods (n) for 6 months = 2 quarters.

Effective Percentage Rate for 2 periods = 3 + 3 + (3 × 3)/100 = 6.09%.

Compound Interest = 50000 × 6.09% = 50000 × 0.0609 = 3045.

∴ The correct answer is Rs. 3045.

Alternate Method

Given:

Principal (P) = Rs. 50000

Rate (R) = 12% per annum

Time (T) = 6 months

Compounding: Quarterly

Formula Used:

Amount (A) = P(1 + r/100)ⁿ

Compound Interest (CI) = A − P

Calculations:

⇒ Quarterly Rate (r) = 12% ÷ 4 = 3%

⇒ Number of quarters (n) = 6 months ÷ 3 months = 2

⇒ A = 50000 × (1 + 3/100)²

⇒ A = 50000 × (103/100)²

⇒ A = 50000 × (1.03)²

⇒ A = 50000 × 1.0609

⇒ A = 53045

⇒ CI = 53045 − 50000

⇒ CI = 3045

∴ The correct answer is Rs. 3045.

Additional Information

Quarterly Compounding Rule

When interest is compounded quarterly, the annual rate is divided by 4 and the time in years is multiplied by 4.

Effective Rate of Interest

For two successive interest periods with rate r%, the effective rate is given by (r + r + r²/100)%.

Half-Yearly Compounding

When interest is compounded half-yearly, the annual rate is divided by 2 and the time in years is multiplied by 2.

• The Women and Child Development Minister Maneka Gandhi launched a social media campaign to mark Daughters’ Day on August 11, 2016.
• In this campaign, the entire week was celebrated as Daughters’ Week as part of the government’s ‘Beti Bachao, Beti Padhao’ programme to check female foeticide, improve sex ratio and educate girls across India.

• The induced current is highest when the direction of motion of the coil is at 90° or perpendicular with respect to the magnetic field, as the flux cutting is maximum in this case.
• While it is minimum when the direction of motion of the coil is at 180° or parallel with respect to the magnetic field, as here the flux cutting by the coil will be minimum.

⇒ Area cultivated in West Bengal = 22%

⇒ Area cultivated in Tamil Nadu = 16%

⇒ Required % = [(22 – 16)%/16%] × 100 = 37.5%

⇒ Mobile phone manufactured by DEF = 5% of 1240000 = 62000

• The breakdown of pyruvate using O2 takes place in the mitochondria matrix.
• When glucose gets converted to pyruvate by glycolysis, only a small fraction of the total free energy potentially available from the glucose is released.
• In mitochondria, when the metabolism of sugars is completed: the pyruvate is imported into the mitochondria and oxidized by O₂ to CO₂ and H₂O.
• This allows 15 times more ATP to be made as compared to that produced by glycolysis alone.

The correct answer is India.

Key Points

• India is the only country to win the ICC Under-19 Cricket World Cup four times.
• India won the ICC Under-19 Cricket World Cup in the year 2000, 2008, 2012 and 2018.
• The 2018 Under-19 Cricket World Cup was held in New Zealand.
• India beat Australia by 8 wickets in the finals to win their fourth Under-19 World Cup, the most by any side.
• Rahul Dravid is the coach of the Indian Under-19 cricket team.
• In 2022, India beat England by four wickets to win their fifth Under-19 Cricket World Cup

Additional Information

Shortcut Trick

A number ending in 25 is a perfect square only if the digits before 25 are a product of two consecutive integers (n × (n+1)).

110 = 10 × 11 (Perfect Square)

90 = 9 × 10 (Perfect Square)

30 = 5 × 6 (Perfect Square)

60 is not a product of consecutive integers (7 × 8 = 56, 8 × 9 = 72).

∴ The correct answer is 6025.

Alternate Method

Given: Four numbers ending in 25: 11025, 6025, 9025, 3025.

Property: To find the square root of a number ending in 25, take the square root of 25 (which is 5) and find the factor n such that n × (n+1) equals the remaining part.

⇒ For 11025: 110 = 10 × 11 ∴ √11025 = 105 (Rational)

⇒ For 9025: 90 = 9 × 10 ∴ √9025 = 95 (Rational)

⇒ For 3025: 30 = 5 × 6 ∴ √3025 = 55 (Rational)

⇒ For 6025: 60 cannot be expressed as n(n+1). Nearest products are 7 × 8 = 56 and 8 × 9 = 72.

⇒ Since 6025 is not a perfect square, its square root is irrational.

∴ The correct answer is 6025.

Additional Information

Perfect Square Unit Digits

A perfect square always ends in 0, 1, 4, 5, 6, or 9. It can never end in 2, 3, 7, or 8.

Irrational Square Roots

The square root of any natural number that is not a perfect square is always an irrational number.

Rule for 5

If a number ends in 5, its square must always end in 25. Additionally, the preceding digits must form a product of two consecutive integers.

• Mamang Dai was the winner of the Sahitya Akademi Award 2017 from the above.
• She won the award for her English novel The Black Hill.
• Ramesh Kuntal Megh also won 2017 Sahitya Akademi award for the work Vishw Mithak Sarit Sagar, a literary criticism in Hindi.
• Sahitya Akademi announces its annual Sahitya Akademi Awards in 24 languages. The award is in the form of a casket containing an engraved copper-plaque, a shawl and a cheque of Rs. 1 lakh.
• Anees Salim won the 2018 Sahitya Akademi award in the English language for his novel ‘The Blind Lady's Descendants’.

Shortcut Trick

The cyclicity of both 7 and 3 is 4.

For 7⁹⁵, remainder of 95 ÷ 4 is 3 → 7³ ends in 3.

For 3⁵⁸, remainder of 58 ÷ 4 is 2 → 3² ends in 9.

Unit digit ⇒ 3 − 9 → (borrow 10) → 13 − 9 = 4.

∴ The correct answer is 4.

Alternate Method

Given expression: 7⁹⁵ − 3⁵⁸

To find the unit digit, we use the power cycle of the base numbers.

⇒ Cyclicity of 7: 7¹=7, 7²=9, 7³=3, 7⁴=1 (repeats every 4 powers).

⇒ Divide the power 95 by 4: 95 = (4 × 23) + 3. Remainder = 3.

⇒ Unit digit of 7⁹⁵ = Unit digit of 7³ = 3.

⇒ Cyclicity of 3: 3¹=3, 3²=9, 3³=7, 3⁴=1 (repeats every 4 powers).

⇒ Divide the power 58 by 4: 58 = (4 × 14) + 2. Remainder = 2.

⇒ Unit digit of 3⁵⁸ = Unit digit of 3² = 9.

⇒ Required unit digit = (Unit digit of 7⁹⁵) − (Unit digit of 3⁵⁸)

⇒ Unit digit = 3 − 9.

⇒ Since 3 is less than 9, we take a carry from the tens place: 13 − 9 = 4.

∴ The correct answer is 4.

Additional Information

Cyclicity of Digits

The unit digit of any number raised to a power repeats in a specific cycle. Digits 2, 3, 7, and 8 have a cyclicity of 4.

Special Cases (Cyclicity 1 and 2)

Digits 0, 1, 5, and 6 have a cyclicity of 1 (always end in themselves). Digits 4 and 9 have a cyclicity of 2.

Negative Unit Digit Rule

If the calculation of unit digits results in a negative value (e.g., a − b where a < b), simply add 10 to the result to find the correct unit digit.

Shortcut Trick

Since the side and one diagonal are both 6 cm, the rhombus consists of two equilateral triangles joined base-to-base.

Area of one equilateral triangle = (√3/4) × (side)²

Total Area = 2 × (√3/4) × 6²

Total Area = 2 × (√3/4) × 36 = 18√3 cm².

∴ The correct answer is 18√3 cm².

Alternate Method

Given: Side (a) = 6 cm, Diagonal 1 (d₁) = 6 cm.

Formula Used: a² = (d₁/2)² + (d₂/2)² and Area = (1/2) × d₁ × d₂

⇒ 6² = (6/2)² + (d₂/2)²

⇒ 36 = 3² + (d₂/2)²

⇒ 36 − 9 = (d₂/2)²

⇒ 27 = (d₂/2)²

⇒ d₂/2 = √27 = 3√3

⇒ d₂ = 6√3 cm

⇒ Area = (1/2) × 6 × 6√3

⇒ Area = 3 × 6√3 = 18√3 cm²

∴ The correct answer is 18√3 cm².

Additional Information

Properties of Rhombus

All sides are equal, and diagonals bisect each other at 90°.

Rhombus Perimeter

Perimeter of a rhombus with side 'a' is given by 4 × a.

Relation between Side and Diagonals

4a² = d₁² + d₂², where a is the side and d₁, d₂ are diagonals.

Shortcut Trick

Identify the Highest Common Factor (HCF) of 75 and 125.

HCF(75, 125) = 25.

Divide both terms by 25: 75 ÷ 25 = 3 and 125 ÷ 25 = 5.

The simplest ratio is 3 : 5.

∴ The correct answer is 3 : 5.

Alternate Method

Given: Ratio = 75 : 125

Formula Used: Simplest form = (Term₁ ÷ HCF) : (Term₂ ÷ HCF)

⇒ Express as a fraction: 75 / 125

⇒ Divide by 5: 15 / 25

⇒ Divide by 5 again: 3 / 5

⇒ Simplest form = 3 : 5

∴ The correct answer is 3 : 5.

Additional Information

Simplest Form of a Ratio

A ratio a : b is in its simplest form if the Highest Common Factor (HCF) of a and b is 1.

Equivalent Ratios

Ratios obtained by multiplying or dividing both terms of a given ratio by the same non-zero number are called equivalent ratios.

HCF Calculation

To find the HCF of 75 (3 × 5²) and 125 (5³), we take the lowest power of common factors: 5² = 25.