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Shortcut Trick

Check for divisibility by 11 (since π = 22/7 involves a multiple of 11).

Volume = πr²h = (22/7) × 6 × 6 × 10.5

Volume = 11 × 2 × 6 × 6 × 1.5 = 11 × 108 = 1188 cm³.

∴ The correct answer is 1188 cm³.

Alternate Method

Given:

Diameter (d) = 12 cm ⇒ Radius (r) = 12 ÷ 2 = 6 cm

Height (h) = 10.5 cm

Formula Used:

Volume of a Cylinder = πr²h

Calculations:

⇒ Volume = (22 ÷ 7) × 6 × 6 × 10.5

⇒ Volume = (22 ÷ 7) × 36 × (21 ÷ 2)

⇒ Volume = 22 × 36 × (3 ÷ 2)

⇒ Volume = 11 × 36 × 3

⇒ Volume = 11 × 108

⇒ Volume = 1188 cm³

∴ The correct answer is 1188 cm³.

Additional Information

Curved Surface Area (CSA)

The area of the side surface of a cylinder is given by 2πrh.

Total Surface Area (TSA)

The sum of CSA and the area of two circular bases: 2πr(h + r).

Hollow Cylinder Volume

If R is outer radius and r is inner radius, Volume = π(R² − r²)h.

Given:

\(x=\left(-\frac{5}{3}\right) \times\left(\frac{9}{25}\right)+\left(\frac{1}{5}-2\right)\)

Calculation:

⇒\(\left(-\frac{5}{3}\right) \times\left(\frac{9}{25}\right)+\left(\frac{1}{5}-2\right)= x\)

⇒\(x=\left(-\frac{3}{5}\right)+\left(\frac{1-10}{5}\right)\)

⇒\(x=\left(\frac{-3+1-10}{5}\right) :: \)

⇒\(x=\left(\frac{-12}{5}\right):: \)

The value of \(\bf \left(-\frac{5}{3}\right) \times\left(\frac{9}{25}\right)+\left(\frac{1}{5}-2\right)=\left(\frac{-12}{5}\right) : :\).

Shortcut Trick

To find the LCM of 24, 42, and 56, pick the largest number and check its multiples.

Largest number = 56

Multiples of 56: 56 (not div by 24, 42), 112 (not div by 24, 42), 168.

Check 168: 168 ÷ 24 = 7 and 168 ÷ 42 = 4.

∴ The correct answer is 168.

Alternate Method

Given: Numbers are 24, 42, and 56.

Formula Used: Prime Factorization Method for finding LCM.

⇒ Prime factors of 24 = 2 × 2 × 2 × 3 = 2³ × 3¹

⇒ Prime factors of 42 = 2 × 3 × 7 = 2¹ × 3¹ × 7¹

⇒ Prime factors of 56 = 2 × 2 × 2 × 7 = 2³ × 7¹

⇒ LCM = Product of the highest powers of all prime factors involved

⇒ LCM = 2³ × 3¹ × 7¹

⇒ LCM = 8 × 3 × 7

⇒ LCM = 168

∴ The correct answer is 168.

Additional Information

LCM (Least Common Multiple)

The smallest positive integer that is divisible by each of the given numbers without leaving a remainder.

HCF and LCM Relation

For two numbers a and b, the product of the numbers is equal to the product of their HCF and LCM: a × b = HCF(a, b) × LCM(a, b).

Division Method

A method where numbers are divided by common prime factors simultaneously until no further division is possible by a prime other than 1.

Given:

A, B, C, D, E, F and G were seven kings, sitting around a circular table, facing the centre.

Solution :

(1) Only E was sitting between F and C.

 

(2) G was sitting to the immediate left of C. Case 2 is eliminated and only case 1 is possible.

(3) A was third to the left of G.

(4) Only D was sitting between A and B.

Here, king B was sitting to the immediate left of G.

Hence, 'option 2' is the correct answer.

Given: 

P, Q, R, S, T and U are six siblings who passed their class tenth exam in different years viz. 2001, 2003, 2005, 2007, 2010 and 2012 but not necessarily in the same order.

Solution: 

(1) Only Q passed before R.

(2) P passed in one of the years before S and after R. 

(3) U passed in one of the years before P.

(4) T passed in the year immediately after S. 

Here, 'P' passed their class tenth exam in the year 2007.

Hence, 'option 3' is the correct answer.

Shortcut Trick

Substitute a standard angle value, say A = 60°.

LHS = sin² 60° ÷ (1 − cos 60°) = (3/4) ÷ (1 − 1/2) = (3/4) ÷ (1/2) = 1.5.

Now check the options: cos 60° + 1 = 0.5 + 1 = 1.5.

Since the values match, the expression simplifies to cos A + 1.

∴ The correct answer is cos A + 1.

Alternate Method

Given: The trigonometric expression is (sin² A) ÷ (1 − cos A).

Formula Used: Standard trigonometric identity sin² A + cos² A = 1 and algebraic identity a² − b² = (a − b)(a + b).

⇒ From the identity, we know: sin² A = 1 − cos² A

⇒ The given expression becomes: (1 − cos² A) ÷ (1 − cos A)

⇒ Using (a² − b²) formula: [(1 − cos A) × (1 + cos A)] ÷ (1 − cos A)

⇒ Cancelling the common term (1 − cos A) from numerator and denominator:

⇒ Result = 1 + cos A

∴ The correct answer is cos A + 1.

Additional Information

Pythagorean Identities

The three main identities are sin² θ + cos² θ = 1, 1 + tan² θ = sec² θ, and 1 + cot² θ = cosec² θ.

Reciprocal Relations

Always keep in mind that sec A = 1/cos A and cosec A = 1/sin A for quick simplifications.

Factorization in Trigonometry

Expressions like 1 − sin² A or sec² A − 1 can often be factored as a difference of squares to cancel terms in fractions.

The correct option is 'H2S' .

Key Points

• Na₂S+2HCl→2NaCl+H₂S↑
• H₂S
• ​Hydrogen sulfide is slightly denser or heavier than air but fairly soluble in water.
• Hydrogen sulphide is a colourless, poisonous gas that can lead to headaches even if it is inhaled in small quantities.
• It has a distinct rotten egg smell.
• Hydrogen sulfide is a flammable gas.​

The correct answer is Only (i).

Key-points

• According to Dobereiner, when Elements are arranged in Order of Increasing Atomic Mass, groups of three Elements (known as triads) that have Similar Chemical Properties are Obtained.
• The Atomic mass of the Middle Element was nearly equal to the Arithmetic Mean of the Other.
• Examples of triads of elements are as follows-

• This classification was not widely accepted since only a few elements could be arranged into triads.

Additional Information

• Moseley's Modern periodic law
• According to Moseley's Modern periodic law, Elements are Arranged in a Tabular form based on their Atomic Numbers.
• it states that the Chemical and Physical Properties of Elements is a Periodic Function of their Atomic Number.
• Newland 
• Newland arranged the Elements in the Order of Increasing Atomic Masses and found that the Properties of Every 8^th Element are Similar to that of the 1^st Element, Just like musical notes.
• Mendeleev
• Mendeleev arranged the Elements in the Increasing Order of their Atomic Mass.
• Mendeleev's Periodic table is based on the Chemical Properties of the Elements.
• It contains 7 Periods(horizontal rows) and 8 Groups (vertical columns).

The correct answer is \(\frac{u v}{u+v}\).

Key Points

• Mirror Formula: The following formula is known as the mirror formula, which provided a relationship between focal length (f), image distance (v), and objects distant (u) as shown below

 

 

Since the plane mirror is a part of an infinitely large spherical mirror, therefore the radius of curvature and focal length of the plane mirror is Infinity. Spherical mirror formula relating an object distance ‘u’, image distance ‘v’ and focal length of mirror ‘f’ may be applied to a plane mirror when focal length goes to infinity. 

\(\Rightarrow \frac{1}{f} = \frac{1}{v} + \frac{1}{u}\)

so f =  \(\frac{u v}{u+v}\)

​EXPLANATION:

Plane Mirror is a part of the infinitely large spherical mirror:

Additional Information

Sign convention in mirrors

• All distances are measured from the pole of the mirror.
• The distance measured from the pole (p) to the left side of the mirror is taken as negative.
• The distance measured from the pole (p) to the right side of the mirror is taken as positive.
• The heights measured upwards and perpendicular to the principal axis of the mirror is taken as positive.
• The heights measured downwards and perpendicular to the principal axis of the mirror is taken as negative.

The correct answer is reducing agent.

Key-points

• a reducing agent (also known as a reductant, reducer, or electron donor) is a chemical species that "donate" an electron to an electron recipient (called the oxidizing agent, oxidant, oxidizer, or electron acceptor).
• CuII + 2 e- → Cu0 (reduction)
• Mg0 - 2 e- → MgII (oxidation)
• Reduction is the gain of electrons or a reduction in the oxidation state of an atom, ion, or certain atoms in a molecule.
• Mg + CuSO4 = Cu + MgSO4 is a redox reaction where Mg is oxidized and Cu is reduced.
• Mg is a reducing agent (i.e. it lost electrons) and CuSO4 is an oxidizing agent (i.e. it gained electrons).

​

Additional information

• Oxidizing agent
• ​An oxidizing agent (often referred to as an oxidizer or an oxidant) is a chemical species that tends to oxidize other substances, i.e. cause an increase in the oxidation state of the substance by making it lose electrons.
• ​Dissolving agent
• ​A solvent is a dissolving agent and is the liquid portion of a solution.
• The molecules that dissolve in the solvent are called solutes.
• ​Catalyst
• ​A catalyst is a substance that speeds up a chemical reaction or lowers the temperature or pressure needed to start one, without itself being consumed during the reaction.
• Catalysis is the process of adding a catalyst to facilitate a reaction.