Shortcut Trick

Assume a base average of 600 and calculate deviations from it.

Deviations: +93, −144, +276, −68, +334, +91, −4, −202, +82

Sum of deviations = (93 + 276 + 334 + 91 + 82) − (144 + 68 + 4 + 202) = 876 − 418 = 458

Average deviation = 458 ÷ 9 ≈ 50.89

Actual Average = 600 + 50.89 = 650.89

∴ The correct answer is 650.89.

Alternate Method

Given observations: 693, 456, 876, 532, 934, 691, 596, 398, 682

Formula used: Average = Sum of observations ÷ Total number of observations

⇒ Sum = 693 + 456 + 876 + 532 + 934 + 691 + 596 + 398 + 682

⇒ Sum = 5858

⇒ Total count of numbers (n) = 9

⇒ Average = 5858 ÷ 9

⇒ Average = 650.8888...

∴ The correct answer is 650.89.

Additional Information

Sum of Deviations

The sum of the deviations of a set of observations from their arithmetic mean is always zero. ∑(x_i − x̄) = 0.

Scaling Property

If every observation in a data set is increased/decreased by a constant 'k', the new average also increases/decreases by 'k'.

Combined Average Formula

If two groups have sizes n₁, n₂ and averages a₁, a₂, the combined average is (n₁a₁ + n₂a₂) ÷ (n₁ + n₂).

Shortcut Trick

Check the divisibility rule for 5: A number leaving remainder 2 when divided by 5 must end in 2 or 7.

Looking at the options: (1) 999, (2) 984, (3) 998, (4) 992.

Only 992 ends in 2, satisfying the condition for 5.

Verify for 9: Sum of digits of 992 = 9 + 9 + 2 = 20. 20 ÷ 9 leaves remainder 2.

∴ The correct answer is 992.

Alternate Method

Given: Divisors = 3, 5, 9; Common Remainder = 2.

Formula: Number = LCM(divisors) × k + Remainder

⇒ LCM of 3, 5, and 9:

⇒ 3 = 3¹

⇒ 5 = 5¹

⇒ 9 = 3²

⇒ LCM = 3² × 5 = 9 × 5 = 45

⇒ The largest three-digit number is 999.

⇒ Divide 999 by 45: 999 ÷ 45 = 22 with remainder 9.

⇒ Largest multiple of 45 below 1000 = 999 − 9 = 990.

⇒ Required number = 990 + 2 = 992.

∴ The correct answer is 992.

Additional Information

LCM and Remainder Case 1

When a number is divided by x, y, and z leaving the same remainder R in each case, the number is of the form [LCM(x, y, z) × k] + R.

LCM and Remainder Case 2

When a number leaves different remainders a, b, c such that (x−a) = (y−b) = (z−c) = K, the number is of the form [LCM(x, y, z) × k] − K.

Divisibility Rule of 9

A number is divisible by 9 if the sum of its digits is divisible by 9. The remainder left when the sum is divided by 9 is the same as the remainder of the original number.

Shortcut Trick

Identify parameters: n = 10 shots, r = 2 hits, p (hit) = 5/8, q (miss) = 3/8.

Apply Binomial Probability formula: P(X = r) = ⁿC_r × p^r × q^n−r.

Calculation: ¹⁰C₂ × (5/8)² × (3/8)⁸ = 45 × (25/8²) × (3⁸/8⁸).

Simplified result: (45 × 25 × 3⁸) / 8¹⁰ = (1125 × 3⁸) / 8¹⁰.

∴ The correct answer is (1125 × 3⁸) / 8¹⁰.

Alternate Method

Given:

Total shots (n) = 10

Shots to hit (r) = 2

Probability of hitting (p) = 5/8

Probability of missing (q) = 1 − 5/8 = 3/8

Formula Used:

P(X = k) = ⁿC_k × p^k × q^n−k

Calculations:

⇒ ¹⁰C₂ = (10 × 9) / (2 × 1) = 45

⇒ P(X = 2) = 45 × (5/8)² × (3/8)¹⁰⁻²

⇒ P(X = 2) = 45 × (5² / 8²) × (3⁸ / 8⁸)

⇒ P(X = 2) = 45 × (25 / 8²) × (3⁸ / 8⁸)

⇒ P(X = 2) = (45 × 25 × 3⁸) / 8⁽²⁺⁸⁾

⇒ P(X = 2) = (1125 × 3⁸) / 8¹⁰

∴ The correct answer is (1125 × 3⁸) / 8¹⁰.

Additional Information

Binomial Distribution Conditions

It applies when there are a fixed number of independent trials (n), each with only two possible outcomes (success/failure) and a constant probability of success (p).

Binomial Mean and Variance

The mean of a Binomial distribution is given by μ = np, and the variance is given by σ² = npq, where q = 1 − p.

Combination Formula

The number of ways to choose r successes out of n trials is ⁿC_r = n! / [r! × (n−r)!].

Shortcut Trick

In height and distance, for a tower height H and building height h, if angles are θ₁ (from top) and θ₂ (from foot):

H = h × tan(θ₂) ÷ (tan(θ₂) − tan(θ₁))

Given θ₁ = 45°, θ₂ = 60°, h = 680 m

H = 680 × √3 ÷ (√3 − 1) = [680√3 × (√3 + 1)] ÷ [(√3 − 1)(√3 + 1)]

H = 680(3 + √3) ÷ 2 = 340(3 + √3)

∴ The correct answer is 340(3 + √3) m.

Alternate Method

Given:

Height of building (CD) = 680 m

Angle of elevation from top of building (C) to top of tower (A) = 45°

Angle of elevation from foot of building (D) to top of tower (A) = 60°

Formula Used:

tan(θ) = Perpendicular ÷ Base

Calculations:

Let the height of the tower be H and the distance between the building and the tower be x.

In right-angled triangle above the building height:

⇒ tan(45°) = (H − 680) ÷ x

⇒ 1 = (H − 680) ÷ x

⇒ x = H − 680 --- (1)

In right-angled triangle from the foot of the building:

⇒ tan(60°) = H ÷ x

⇒ √3 = H ÷ x

⇒ x = H ÷ √3 --- (2)

Equating (1) and (2):

⇒ H − 680 = H ÷ √3

⇒ H√3 − 680√3 = H

⇒ H(√3 − 1) = 680√3

⇒ H = (680√3) ÷ (√3 − 1)

Rationalizing the denominator:

⇒ H = [680√3 × (√3 + 1)] ÷ [(√3 − 1)(√3 + 1)]

⇒ H = [680(3 + √3)] ÷ (3 − 1)

⇒ H = [680(3 + √3)] ÷ 2

⇒ H = 340(3 + √3)

∴ The correct answer is 340(3 + √3) m.

Additional Information

Angle of Elevation

The angle formed by the line of sight with the horizontal when looking up at an object.

Standard Trigonometric Ratios

tan 30° = 1/√3, tan 45° = 1, tan 60° = √3.

Height and Distance Shortcut

If distances x₁ and x₂ are measured from the foot for angles α and β, then height H = (x₂ − x₁) / (cot α − cot β).

Shortcut Trick In a set of 11 observations, the median is the 6th term when arranged in ascending order. The largest five observations are the 7th, 8th, 9th, 10th, and 11th terms. Increasing values that are already larger than the median does not change the position or value of the median (the 6th term). Since the 6th term remains 73.2, the median of the new set remains unchanged. ∴ The correct answer is remains the same as that of the original set.

Alternate Method Given: Total observations (n) = 11 Original Median = 73.2 Formula Used: For an odd number of observations (n), Median = ((n + 1) ÷ 2)th term.

Calculations: ⇒ Median position = (11 + 1) ÷ 2 = 6th observation. ⇒ Let observations in ascending order be x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11. ⇒ Original Median (x6) = 73.2. ⇒ New observations = x1, x2, x3, x4, x5, x6, (x7+3), (x8+3), (x9+3), (x10+3), (x11+3). ⇒ Since x6 < x7, then x6 will still be < (x7+3). The relative order remains the same. ⇒ The 6th term remains x6 = 73.2. ⇒ New Median = 73.2. ∴ The correct answer is remains the same as that of the original set.

Additional Information Median for Odd vs Even n If n is odd, Median = ((n+1)/2)th term. If n is even, Median = Average of (n/2)th and ((n/2)+1)th terms. Property of Median The median is not affected by extreme values or changes in values that do not cross the central position in the ordered list. Effect of Linear Changes If every observation in a set is increased by a constant k, the median also increases by k. However, if only specific observations are changed, the median may or may not change depending on their position.

Shortcut Trick

For an isosceles trapezium where AB = CD and AD || BC:

Bottom Base (BC) = AD + 2 × √(Slant Side² − Height²)

BC = 15 + 2 × √(10² − 8²) = 15 + 2 × 6 = 27 m

Perimeter = Sum of all sides = AD + BC + AB + CD

Perimeter = 15 + 27 + 10 + 10 = 62 m

∴ The correct answer is 62 m.

Alternate Method

Given: AD = 15 m, AB = 10 m, CD = 10 m, and Height (h) = 8 m.

Formula: Perimeter = AD + BC + AB + CD

In the trapezium ABCD, let's drop perpendiculars AM and DN on base BC.

⇒ AM = DN = 8 m (Height)

⇒ In right-angled triangle AMB, by Pythagoras theorem:

⇒ BM² = AB² − AM²

⇒ BM² = 10² − 8² = 100 − 64 = 36

⇒ BM = 6 m

⇒ Since the trapezium is isosceles (AB = CD), BM = NC = 6 m.

⇒ Length of base BC = BM + MN + NC

⇒ As MN = AD, BC = 6 + 15 + 6 = 27 m

⇒ Perimeter of ABCD = AB + BC + CD + DA

⇒ Perimeter = 10 + 27 + 10 + 15

⇒ Perimeter = 62 m

∴ The correct answer is 62 m.

Additional Information

Area of a Trapezium

The area of a trapezium is calculated as Area = 1/2 × (sum of parallel sides) × height = 1/2 × (a + b) × h.

Isosceles Trapezium Properties

In an isosceles trapezium, the non-parallel sides are equal, the base angles are equal, and the diagonals are equal in length.

Median of a Trapezium

The median (or midsegment) of a trapezium is parallel to the bases and its length is the average of the base lengths: Median = (a + b) / 2.

The correct answer is 3 : 1

Key Points

• The Volume of a Prism is calculated by multiplying the Base Area (A) by the Vertical Height (h). The formula is given as: V_prism = Base Area × Height.
• The Volume of a Pyramid is exactly one-third of the volume of a prism that has the same Base Area and Height. The formula is given as: V_pyramid = (1/3) × Base Area × Height.
• In this problem, it is stated that both the Prism and the Pyramid share the same Base Area (let it be 'A') and the same Height (let it be 'h').
• To find the Ratio of their volumes, we set up the expression: Volume of Prism : Volume of Pyramid.
• Substituting the formulas: (A × h) : ((1/3) × A × h).
• By simplifying the expression and dividing both sides by (A × h), we get 1 : 1/3.
• Multiplying both sides by 3 to clear the fraction, the ratio becomes 3 : 1.
• Therefore, the volume of a prism is three times the volume of a pyramid with the same base and height.

Additional Information

• A Prism is a three-dimensional solid with two congruent and parallel bases. The lateral faces are typically parallelograms (or rectangles in a right prism).
• A Pyramid is a three-dimensional solid with a polygonal base and triangular lateral faces that meet at a single point called the Apex.
• The relationship V_prism = 3 × V_pyramid holds true regardless of the shape of the base (triangle, square, hexagon, etc.), as long as the base and height are identical for both solids.
• Similarly, the volume of a Cylinder is three times the volume of a Cone if they have the same radius and height.

Given:

x = $\sqrt{64} + \sqrt{121} - \sqrt{169}$

Formula used:

$\sqrt{a^{2}} = a$

Calculation:

x = 8 + 11 - 13

⇒ x = 19 - 13

⇒ x = 6

⇒ x² = 6²

⇒ x² = 36

∴ The correct answer is option (3).

Shortcut Trick

Total cost of 3 LED TVs and 3 Mobiles = 31,000 + 35,000 = 66,000

⇒ 1 LED TV + 1 Mobile = 66,000 ÷ 3 = 22,000

Given: 1 LED TV + 2 Mobiles = 35,000

⇒ 1 Mobile = (1 LED TV + 2 Mobiles) − (1 LED TV + 1 Mobile) = 35,000 − 22,000 = 13,000

∴ The correct answer is ₹13,000.

Alternate Method

Given values:

Cost of 2 LED TVs + 1 Mobile = ₹31,000

Cost of 1 LED TV + 2 Mobiles = ₹35,000

Let the cost of one LED TV be T and the cost of one Mobile be M.

Equation 1: 2T + M = 31,000

Equation 2: T + 2M = 35,000

⇒ Multiply Equation 2 by 2:

⇒ 2(T + 2M) = 2 × 35,000

⇒ 2T + 4M = 70,000 --- (Equation 3)

⇒ Subtract Equation 1 from Equation 3:

⇒ (2T + 4M) − (2T + M) = 70,000 − 31,000

⇒ 3M = 39,000

⇒ M = 39,000 ÷ 3

⇒ M = 13,000

∴ The correct answer is ₹13,000.

Additional Information

Elimination Method

Variables are eliminated by making the coefficients of one variable equal in both equations through multiplication.

Substitution Method

Express one variable in terms of the other from one equation and substitute it into the second equation.

System of Linear Equations

A set of two or more equations with the same variables. For two variables x and y, the general form is a₁x + b₁y = c₁ and a₂x + b₂y = c₂.

Shortcut Trick

Volume of a cylinder is given by the formula πr²h.

Given: r = 10 cm, h = 20 cm, π = 3.14

Direct Calculation: 3.14 × 10 × 10 × 20

⇒ 314 × 20 = 6280 cm³

∴ The correct answer is 6280 cm³.

Alternate Method

Given:

Radius of cylinder (r) = 10 cm

Height of cylinder (h) = 20 cm

Value of π = 3.14

Formula Used:

Volume of a cylinder = πr²h

Calculations:

⇒ Volume = 3.14 × (10)² × 20

⇒ Volume = 3.14 × 100 × 20

⇒ Volume = 314 × 20

⇒ Volume = 6280 cm³

∴ The correct answer is 6280 cm³.

Additional Information

Curved Surface Area (CSA)

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

Total Surface Area (TSA)

The total surface area of a cylinder includes both the curved area and the two circular bases: TSA = 2πr(h + r).

Volume of a Cone

A cone with the same radius and height as a cylinder has exactly one-third of its volume: V = (1/3)πr²h.

Shortcut Trick

Total Work = LCM(5, 10, 15) = 30 units.

Efficiencies: P = 6, Q = 3, R = 2.

Day 1 (P+Q) = 9 units; Day 2 (P+R) = 8 units.

In 2 days, work completed = 9 + 8 = 17 units.

Day 3 (P+Q) = 9 units; Total after 3 days = 17 + 9 = 26 units.

Remaining 4 units done by (P+R) on Day 4 in 4 ÷ 8 = 1/2 day.

∴ The correct answer is 7/2 days.

Alternate Method

Given:

Time taken by P = 5 days

Time taken by Q = 10 days

Time taken by R = 15 days

Formula Used:

Total Work = LCM of individual time periods

Efficiency = Total Work ÷ Time

Calculations:

⇒ Total work = 30 units

⇒ Work on Day 1 (P + Q) = 6 + 3 = 9 units

⇒ Work on Day 2 (P + R) = 6 + 2 = 8 units

⇒ Combined work in first 2 days = 9 + 8 = 17 units

⇒ Remaining work after 2 days = 30 − 17 = 13 units

⇒ Work on Day 3 (P + Q) = 9 units

⇒ Remaining work after 3 days = 13 − 9 = 4 units

⇒ Efficiency of (P + R) on Day 4 = 6 + 2 = 8 units/day

⇒ Time taken for remaining 4 units = 4 ÷ 8 = 1/2 day

⇒ Total time = 3 + 1/2 = 7/2 days

∴ The correct answer is 7/2 days.

Additional Information

Alternate Day Work Concept

When people work on alternate days, identify the work done in one complete cycle and then calculate the number of full cycles that fit into the total work.

Work and Wages

If a group of people work together or on alternate days, the total wages are always distributed in the ratio of the actual units of work done by each individual.

Pipes and Cisterns

Similar to time and work, but if a pipe empties the tank, its efficiency is taken as negative. Total capacity is found using LCM of the time taken by individual pipes.

Shortcut Trick

Total Distance to cover = Length of Train + Length of Tunnel = 350 + 1250 = 1600 m

Speed (in m/s) = Total Distance ÷ Time = 1600 ÷ 80 = 20 m/s

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

Speed (in km/h) = 20 × (18 ÷ 5) = 4 × 18 = 72 km/h

∴ The correct answer is 72 km/h.

Alternate Method

Given:

Length of the goods train (L_T) = 350 m

Length of the tunnel (L_P) = 1250 m

Time taken (t) = 80 seconds

Formula Used:

Speed = Total Distance ÷ Total Time

Total Distance = Length of Train + Length of Tunnel

Speed (km/h) = Speed (m/s) × (18 ÷ 5)

Calculations:

⇒ Total Distance = 350 + 1250 = 1600 m

⇒ Speed in m/s = 1600 ÷ 80

⇒ Speed in m/s = 20 m/s

⇒ Speed in km/h = 20 × (18 ÷ 5)

⇒ Speed in km/h = 4 × 18

⇒ Speed in km/h = 72 km/h

∴ The correct answer is 72 km/h.

Additional Information

Speed Conversion Rules

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

Train Crossing Stationary Objects

When a train crosses a stationary point object (pole, tree, man), distance = length of train. When it crosses an object with length (bridge, platform, tunnel), distance = length of train + length of object.

Relative Speed

If two objects move in the same direction, relative speed = S₁ − S₂. If they move in opposite directions, relative speed = S₁ + S₂.

Shortcut Trick

Interior angle of a decagon (10 sides) = (10 − 2) × 180° ÷ 10 = 144°.

Interior angle of the polygon = 15/16 × 144° = 15 × 9° = 135°.

Exterior angle = 180° − 135° = 45°.

Number of sides (n) = 360° ÷ 45° = 8.

Number of diagonals = n(n − 3) ÷ 2 = 8 × 5 ÷ 2 = 20.

∴ The correct answer is 20.

Alternate Method

Given:

Number of sides in decagon (n_d) = 10

Interior angle ratio = 15/16

Formula Used:

Interior Angle of a regular polygon with 'n' sides = [(n − 2) × 180°] ÷ n

Number of diagonals = [n(n − 3)] ÷ 2

Calculations:

⇒ Interior angle of regular decagon = [(10 − 2) × 180°] ÷ 10

⇒ Interior angle of regular decagon = [8 × 180°] ÷ 10 = 144°

⇒ Interior angle of required polygon = (15/16) × 144°

⇒ Interior angle of required polygon = 15 × 9° = 135°

⇒ Let the number of sides of the polygon be 'n'

⇒ [(n − 2) × 180°] ÷ n = 135°

⇒ (n − 2) × 180 = 135n

⇒ 180n − 360 = 135n

⇒ 180n − 135n = 360

⇒ 45n = 360

⇒ n = 360 ÷ 45 = 8

⇒ Number of diagonals = [8 × (8 − 3)] ÷ 2

⇒ Number of diagonals = (8 × 5) ÷ 2 = 20

∴ The correct answer is 20.

Additional Information

Sum of Interior Angles

The sum of all interior angles of a polygon with 'n' sides is given by (n − 2) × 180°.

Exterior Angle Property

For any regular polygon, the exterior angle is 360° ÷ n. The sum of an interior angle and its adjacent exterior angle is always 180°.

Number of Diagonals

A diagonal is a line segment connecting two non-adjacent vertices. The formula is n(n − 3) ÷ 2. A triangle (n=3) has 0 diagonals.

Common Polygons

Pentagon (n=5, int. angle 108°), Hexagon (n=6, int. angle 120°), Octagon (n=8, int. angle 135°), Decagon (n=10, int. angle 144°).

Shortcut Trick

Perform a quick direct division of 463y by 7.

Divide 46 by 7: 7 × 6 = 42, Remainder = 4.

Append 3: 43 ÷ 7 → 7 × 6 = 42, Remainder = 1.

Append y: The number 1y must be divisible by 7.

Check multiples of 7: 7 × 2 = 14, so y must be 4.

∴ The correct answer is 4.

Alternate Method

Given:

Number = 463y

Divisor = 7

Formula Used:

A number is divisible by 7 if the result of subtracting twice the last digit from the number formed by the remaining digits is divisible by 7.

Calculations:

⇒ 463y = 4630 + y

⇒ 4630 ÷ 7 = 661 × 7 + 3

⇒ Remainder of 4630 when divided by 7 is 3.

⇒ For 463y to be divisible by 7, (3 + y) must be divisible by 7.

⇒ If 3 + y = 7

⇒ y = 7 − 3

⇒ y = 4

Check with divisibility rule: 463 − 2(4) = 463 − 8 = 455.

⇒ 455 ÷ 7 = 65 (Perfectly divisible).

∴ The correct answer is 4.

Additional Information

Divisibility Rule of 7

Subtract twice the last digit from the rest of the number. If the result is 0 or a multiple of 7, the original number is divisible by 7.

Divisibility Rule of 11

The difference between the sum of digits at odd places and the sum of digits at even places must be either 0 or a multiple of 11.

Divisibility Rule of 13

Add four times the last digit to the remaining number. If the sum is divisible by 13, the original number is divisible by 13.

Shortcut Trick

Assume a specific value for θ to simplify the calculation, let θ = 90°.

Then, cot 90° = 0 and cosec 90° = 1.

Substitute these values: p = a(0) + b(1) = b and q = b(0) + a(1) = a.

Now, calculate p² − q² = b² − a².

∴ The correct answer is b² − a².

Alternate Method

Given:

p = a cot θ + b cosec θ

q = b cot θ + a cosec θ

Formula Used:

(x + y)² = x² + y² + 2xy

cosec² θ − cot² θ = 1

Calculations:

⇒ p² = (a cot θ + b cosec θ)²

⇒ p² = a² cot² θ + b² cosec² θ + 2ab cot θ cosec θ ----(1)

⇒ q² = (b cot θ + a cosec θ)²

⇒ q² = b² cot² θ + a² cosec² θ + 2ab cot θ cosec θ ----(2)

⇒ Subtracting (2) from (1):

⇒ p² − q² = (a² − b²) cot² θ + (b² − a²) cosec² θ

⇒ p² − q² = −(b² − a²) cot² θ + (b² − a²) cosec² θ

⇒ p² − q² = (b² − a²) (cosec² θ − cot² θ)

⇒ Since cosec² θ − cot² θ = 1,

⇒ p² − q² = (b² − a²)(1)

∴ The correct answer is b² − a².

Additional Information

Trigonometric Pythagorean Identities

Standard identities used in simplification: sin² θ + cos² θ = 1, sec² θ − tan² θ = 1, and cosec² θ − cot² θ = 1.

Value Substitution Method

This method is highly effective in MCQ exams for identities. Common values used are 0°, 45°, and 90°, provided the function is defined at that point.

Reciprocal Relations

cosec θ = 1 ÷ sin θ and cot θ = cos θ ÷ sin θ.

Shortcut Trick

Ratio of opposite angles: ∠P : ∠R = 3 : 1 ⇒ Total 4 units = 180°

Value of ∠R = 1 unit = 180° ÷ 4 = 45°

Ratio of opposite angles: ∠S : ∠Q = 4 : 1 ⇒ Total 5 units = 180°

Value of ∠S = 4 units = (180° ÷ 5) × 4 = 144°

Sum ∠S + ∠R = 144° + 45° = 189°

∴ The correct answer is 189°.

Alternate Method

Given:

Quadrilateral PQRS is cyclic.

∠P = 3 × ∠R

∠S = 4 × ∠Q

Formula Used:

In a cyclic quadrilateral, the sum of opposite angles is 180°.

∠P + ∠R = 180°

∠Q + ∠S = 180°

Calculations:

Case 1: ∠P and ∠R are opposite angles.

⇒ ∠P + ∠R = 180°

⇒ 3∠R + ∠R = 180°

⇒ 4∠R = 180°

⇒ ∠R = 45°

Case 2: ∠S and ∠Q are opposite angles.

⇒ ∠S + ∠Q = 180°

⇒ ∠S + (∠S ÷ 4) = 180°

⇒ (5 × ∠S) ÷ 4 = 180°

⇒ 5∠S = 720°

⇒ ∠S = 144°

Now, finding the required sum:

⇒ Sum = ∠S + ∠R

⇒ Sum = 144° + 45°

⇒ Sum = 189°

∴ The correct answer is 189°.

Additional Information

Cyclic Quadrilateral Property

A quadrilateral is called cyclic if all its four vertices lie on a circle. The sum of any pair of opposite angles is 180°.

Exterior Angle of Cyclic Quadrilateral

The exterior angle formed by extending one side of a cyclic quadrilateral is equal to the interior opposite angle.

Ptolemy's Theorem

For a cyclic quadrilateral with sides a, b, c, d and diagonals p, q, the product of the diagonals is equal to the sum of the products of opposite sides: p × q = (a × c) + (b × d).

Shortcut Trick

Ratio of Men = 6 : 12 = 1 : 2.

Since the work and number of days remain the same, Hours/day is inversely proportional to the number of Men.

Ratio of Hours = 2 : 1.

Given 2 units = 10 hours/day.

So, 1 unit = 5 hours/day.

∴ The correct answer is 5 hours.

Alternate Method

Given:

Initial Men (M₁) = 6

Initial Hours/day (H₁) = 10

Final Men (M₂) = 12

Days (D) = Constant

Formula used: M₁ × D₁ × H₁ = M₂ × D₂ × H₂

Calculations:

⇒ M₁ × H₁ = M₂ × H₂ (since D₁ = D₂)

⇒ 6 × 10 = 12 × H₂

⇒ 60 = 12 × H₂

⇒ H₂ = 60 ÷ 12

⇒ H₂ = 5

∴ The correct answer is 5 hours.

Additional Information

The Chain Rule Formula

(M₁ × D₁ × H₁ × E₁) ÷ W₁ = (M₂ × D₂ × H₂ × E₂) ÷ W₂, where M is men, D is days, H is hours, E is efficiency, and W is work.

Inverse Proportionality

For a fixed amount of work, the number of workers is inversely proportional to the time taken (Days or Hours). If workers double, time taken is halved.

Total Work Calculation

Total Work is often measured in Man-Hours or Man-Days. Total Work = Number of Men × Number of Days × Hours per day.

Given:

2^x + 3^y = 17

2^{x + 2} - 3^{y + 1} = 5

Formula used:

a^{m + n} = a^m × aⁿ

Calculations:

2^x + 3^y = 17 --- (1)

2^{x + 2} - 3^{y + 1} = 5

⇒ (2^x × 2²) - (3^y × 3¹) = 5

⇒ 4 × 2^x - 3 × 3^y = 5 --- (2)

Multiply eq (1) by 3:

⇒ 3 × 2^x + 3 × 3^y = 51 --- (3)

Adding eq (2) and eq (3):

⇒ 7 × 2^x = 56

⇒ 2^x = 8

⇒ 2^x = 2³

⇒ x = 3

Substitute x = 3 in eq (1):

⇒ 2³ + 3^y = 17

⇒ 8 + 3^y = 17

⇒ 3^y = 9

⇒ 3^y = 3²

⇒ y = 2

∴ The correct answer is option (3).

Shortcut Trick

Mode is defined as the observation that occurs most frequently in a dataset.

Data: 5, 7, 9, 7, 3, 7, 5, 7, 8, 6, 7

Frequency of 7: 5 times

Frequency of 5: 2 times

Frequency of other numbers (3, 6, 8, 9): 1 time each

Since 7 appears the most number of times, it is the mode.

∴ The correct answer is 7.

Alternate Method

Given Data: 5, 7, 9, 7, 3, 7, 5, 7, 8, 6, 7

Formula: Mode = Value with the maximum frequency.

⇒ Count of 3 → 1

⇒ Count of 5 → 2

⇒ Count of 6 → 1

⇒ Count of 7 → 5

⇒ Count of 8 → 1

⇒ Count of 9 → 1

⇒ The value 7 has the highest count (5).

∴ The correct answer is 7.

Additional Information

Empirical Formula

For a moderately skewed distribution, the relationship is: Mode = 3 × Median − 2 × Mean.

Median

Median is the middle-most value when the data is arranged in ascending order. For n=11, it is the 6^th term.

Types of Mode

A dataset can be unimodal (one mode), bimodal (two modes), or multimodal (more than two modes).

Shortcut Trick

A number is divisible by 11 if |Sum of digits at odd positions − Sum of digits at even positions| is 0 or a multiple of 11.

Testing 5214341:

Sum of digits at odd positions (1st, 3rd, 5th, 7th) = 5 + 1 + 3 + 1 = 10

Sum of digits at even positions (2nd, 4th, 6th) = 2 + 4 + 4 = 10

Difference = 10 − 10 = 0. Since the difference is 0, the number is divisible by 11.

∴ The correct answer is 5214341.

Alternate Method

Given Numbers: 5214341, 5648741, 6598321, 2378965

Divisibility Rule for 11: The difference between the sum of digits at odd places and the sum of digits at even places must be 0, 11, 22, 33, etc.

⇒ Option (1): 5214341

⇒ Sum of odd-position digits = 5 + 1 + 3 + 1 = 10

⇒ Sum of even-position digits = 2 + 4 + 4 = 10

⇒ Difference = 10 − 10 = 0 (Divisible by 11)

⇒ Option (2): 5648741

⇒ Sum of odd-position digits = 5 + 4 + 7 + 1 = 17

⇒ Sum of even-position digits = 6 + 8 + 4 = 18

⇒ Difference = |17 − 18| = 1 (Not divisible by 11)

⇒ Option (3): 6598321

⇒ Sum of odd-position digits = 6 + 9 + 3 + 1 = 19

⇒ Sum of even-position digits = 5 + 8 + 2 = 15

⇒ Difference = |19 − 15| = 4 (Not divisible by 11)

⇒ Option (4): 2378965

⇒ Sum of odd-position digits = 2 + 7 + 9 + 5 = 23

⇒ Sum of even-position digits = 3 + 8 + 6 = 17

⇒ Difference = |23 − 17| = 6 (Not divisible by 11)

∴ The correct answer is 5214341.

Additional Information

Divisibility by 3 and 9

A number is divisible by 3 (or 9) if the sum of all its digits is divisible by 3 (or 9).

Divisibility by 7

Subtract twice the last digit from the number formed by the remaining digits; the result should be divisible by 7.

Divisibility by 13

Add four times the last digit to the number formed by the remaining digits; the result should be divisible by 13.

Divisibility by 4 and 8

A number is divisible by 4 if the last 2 digits are divisible by 4, and by 8 if the last 3 digits are divisible by 8.

Shortcut Trick

Congruent triangles ΔABC ≅ ΔPQR means corresponding sides and angles are equal.

Side AB corresponds to side PQ ⇒ PQ = AB = 5 cm.

In ΔABC, ∠C = 180° − (∠A + ∠B) = 180° − (80° + 40°) = 60°.

∠R corresponds to ∠C ⇒ ∠R = ∠C = 60°.

∴ The correct answer is PQ = 5 cm and ∠R = 60°.

Alternate Method

Given: ΔABC ≅ ΔPQR, AB = 5 cm, ∠B = 40°, ∠A = 80°.

Formula: Angle sum property of a triangle (∠A + ∠B + ∠C = 180°) and CPCT (Corresponding Parts of Congruent Triangles).

⇒ In ΔABC, sum of angles = 180°

⇒ ∠A + ∠B + ∠C = 180°

⇒ 80° + 40° + ∠C = 180°

⇒ 120° + ∠C = 180°

⇒ ∠C = 60°

⇒ Since ΔABC ≅ ΔPQR, by CPCT:

⇒ PQ = AB = 5 cm

⇒ ∠P = ∠A = 80°, ∠Q = ∠B = 40°, ∠R = ∠C = 60°

∴ The correct answer is PQ = 5 cm and ∠R = 60°.

Additional Information

Congruence Rule (CPCT)

CPCT stands for Corresponding Parts of Congruent Triangles. If two triangles are congruent, then all their corresponding sides and angles are equal.

Angle Sum Property

The sum of all interior angles of any triangle is always 180°. If two angles are known, the third can be found as 180° − (Sum of known angles).

Criteria for Congruence

Two triangles can be proven congruent using SSS (Side-Side-Side), SAS (Side-Angle-Side), ASA (Angle-Side-Angle), AAS (Angle-Angle-Side), or RHS (Right angle-Hypotenuse-Side).

Shortcut Trick

Find Cost Price (CP) of mixture: 56 ÷ 1.12 = ₹50/kg.

Apply Alligation ratio (Type 1 : Type 2) = (50 − 42) : (60 − 50) = 8 : 10 = 4 : 5.

Compare units: 5 units correspond to 24 kg.

Calculate Type 1: (24 ÷ 5) × 4 = 19.2 kg.

∴ The correct answer is 19.2 kg.

Alternate Method

Given:

CP of first variety (Rice₁) = ₹60/kg

CP of second variety (Rice₂) = ₹42/kg

Quantity of second variety (Q₂) = 24 kg

Selling Price of mixture (SP_mix) = ₹56/kg

Profit percentage = 12%

Formula Used:

CP = (SP × 100) ÷ (100 + Profit%)

Alligation Rule: (Quantity₁) ÷ (Quantity₂) = (Mean Price − CP₂) ÷ (CP₁ − Mean Price)

⇒ Mean CP of Mixture = (56 × 100) ÷ (100 + 12)

⇒ Mean CP = 5600 ÷ 112 = ₹50/kg

⇒ Applying Alligation: Q₁ ÷ Q₂ = (50 − 42) ÷ (60 − 50)

⇒ Q₁ ÷ 24 = 8 ÷ 10

⇒ Q₁ ÷ 24 = 4 ÷ 5

⇒ Q₁ = (4 × 24) ÷ 5

⇒ Q₁ = 96 ÷ 5 = 19.2 kg

∴ The correct answer is 19.2 kg.

Additional Information

Alligation Method

Used to find the ratio in which two ingredients with different costs must be mixed to produce a mixture of a desired average cost.

Cost Price from Selling Price

Always calculate the CP of the mixture before using the Alligation rule if only the SP and profit/loss % are provided: CP = SP ÷ (1 ± r/100).

Weighted Average Formula

The mean price can also be verified using: Mean Price = (n₁c₁ + n₂c₂) ÷ (n₁ + n₂), where n is quantity and c is cost.

Shortcut Trick

Discount = 45% ⇒ Selling Price (SP) = 100% − 45% = 55%

Given, 55% of Marked Price (MP) = ₹330

1% of MP = 330 ÷ 55 = 6

100% (MP) = 6 × 100 = ₹600

∴ The correct answer is ₹600.

Alternate Method

Given:

Selling Price (SP) = ₹330

Discount percentage (D%) = 45%

Formula Used:

Marked Price (MP) = [SP × 100] ÷ [100 − D%]

Calculations:

⇒ MP = [330 × 100] ÷ [100 − 45]

⇒ MP = 33000 ÷ 55

⇒ MP = 6600 ÷ 11

⇒ MP = 600

∴ The correct answer is ₹600.

Additional Information

Marked Price (MP)

It is the price printed on the label of an article. Discount is always calculated on the Marked Price.

Successive Discount Formula

If two successive discounts of a% and b% are given, the total effective discount = (a + b − (a × b)/100)%.

Relation between CP and MP

CP ÷ MP = (100 − Discount%) ÷ (100 + Profit%). This is used to find the markup percentage.

Shortcut Trick

20% of (A + B) = 30% of (A − B) ⇒ 2(A + B) = 3(A − B)

Using Componendo and Dividendo: A / B = (3 + 2) / (3 − 2) = 5 / 1

A = 5B ⇒ A is 500% of B

Based on the provided options, the value is 5.

∴ The correct option is C.

Alternate Method

Given:

20% of (A + B) = 30% of (A − B)

Formula Used:

Percentage = (Part / Whole) × 100

Value of A (5 units)

Value of B (1 unit)

A = 5 × B ⇒ A = 500% of B

⇒ 20/100 × (A + B) = 30/100 × (A − B)

⇒ 2(A + B) = 3(A − B)

⇒ 2A + 2B = 3A − 3B

⇒ 3A − 2A = 2B + 3B

⇒ A = 5B

⇒ Required Percentage = (A / B) × 100

⇒ (5B / B) × 100 = 500%

∴ The correct option is C.

Additional Information

Componendo and Dividendo Rule

If a/b = c/d, then (a + b) / (a − b) = (c + d) / (c − d). This is useful for solving equations involving (x+y) and (x-y).

Percentage Conversion

To find what percentage of Y is X, use the formula: (X / Y) × 100.

Basic Percentage Rule

x% of Y can be written as (x/100) × Y. Similarly, 20% = 1/5 and 30% = 3/10.

Shortcut Trick

To find intercepts, divide the entire equation by the constant term on the RHS (399).

57x/399 − 19y/399 = 399/399

x/7 − y/21 = 1 ⇒ x/7 + y/(−21) = 1

The denominators represent the intercepts: x-intercept = 7, y-intercept = −21.

∴ The correct answer is x-axis at (7, 0) and y-axis at (0, -21).

Alternate Method

Given equation: 57x − 19y = 399

A graph cuts the x-axis when the y-coordinate is 0, and the y-axis when the x-coordinate is 0.

To find the x-axis intersection, substitute y = 0:

⇒ 57x − 19(0) = 399

⇒ 57x = 399

⇒ x = 399 ÷ 57 = 7

⇒ Coordinate = (7, 0)

To find the y-axis intersection, substitute x = 0:

⇒ 57(0) − 19y = 399

⇒ −19y = 399

⇒ y = 399 ÷ (−19) = −21

⇒ Coordinate = (0, −21)

∴ The correct answer is x-axis at (7, 0) and y-axis at (0, -21).

Additional Information

Intercept Form of a Line

The equation of a line with x-intercept 'a' and y-intercept 'b' is given by x/a + y/b = 1.

Slope-Intercept Form

A line can also be expressed as y = mx + c, where 'm' is the slope and 'c' is the y-intercept.

Slope Calculation

For a general equation Ax + By + C = 0, the slope (m) is given by −A/B.

Origin Intersection

If the constant term in the linear equation is zero (Ax + By = 0), the line passes through the origin (0, 0).

Shortcut Trick

Rate of Interest = 40% = 2/5

Principal : Instalment ratio for Year 1 = 5 : 7

Principal : Instalment ratio for Year 2 = 5² : 7² = 25 : 49

To equate instalments, multiply Year 1 ratio by 7 → 35 : 49

Total Principal = 35 + 25 = 60 units ⇒ 60 units = ₹1500 ⇒ 1 unit = ₹25

Value of each instalment = 49 units = 49 × 25 = ₹1225

∴ The correct answer is ₹1225.

Alternate Method

Given:

Principal (P) = ₹1500

Rate (R) = 40% per annum

Time (n) = 2 years

Formula Used:

P = x / (1 + R/100)¹ + x / (1 + R/100)², where x is the value of each instalment.

Calculations:

⇒ 1500 = x / (1 + 40/100) + x / (1 + 40/100)²

⇒ 1500 = x / (1.4) + x / (1.96)

⇒ 1500 = x / (7/5) + x / (49/25)

⇒ 1500 = 5x / 7 + 25x / 49

⇒ 1500 = (35x + 25x) / 49

⇒ 1500 = 60x / 49

⇒ x = (1500 × 49) / 60

⇒ x = 25 × 49

⇒ x = 1225

∴ The correct answer is ₹1225.

Additional Information

Compound Interest Instalment Formula

For 'n' years with annual instalment 'x', Principal P = x × [1/(1+r) + 1/(1+r)² + ... + 1/(1+r)ⁿ], where r = R/100.

Effective Interest Rate

For 2 years compounded annually, the effective rate R_eff = R₁ + R₂ + (R₁ × R₂)/100.

Simple Interest Instalment

In SI, the instalment x = (100 × Amount) / [100t + rt(t−1)/2], where t is the number of years.

Shortcut Trick

Rate (R) = [(n − 1) × 100] ÷ T

Where n = number of times the sum becomes (n = 3) and T = Time (T = 9 years).

R = [(3 − 1) × 100] ÷ 9

R = 200 ÷ 9 = 22²/₉%

∴ The correct answer is 22²/₉%.

Alternate Method

Given:

Let Principal (P) = P

Amount (A) = 3P (Since the sum triples itself)

Time (T) = 9 years

Formula Used:

Simple Interest (SI) = (P × R × T) ÷ 100

Amount (A) = Principal (P) + Simple Interest (SI)

⇒ SI = Amount − Principal

⇒ SI = 3P − P = 2P

⇒ 2P = (P × R × 9) ÷ 100

⇒ 2 = (R × 9) ÷ 100

⇒ 200 = 9R

⇒ R = 200 ÷ 9

⇒ R = 22²/₉%

∴ The correct answer is 22²/₉%.

Additional Information

Simple Interest Principle

In simple interest, the interest remains constant every year because it is always calculated on the original Principal (P).

Relationship between Time and Multiplier

If a sum becomes 'n' times in T years, then T = [(n − 1) × 100] ÷ R. This allows for quick calculation of either Rate or Time.

Simple Interest vs Compound Interest

For the first year, SI and CI are equal if the interest is compounded annually. From the second year onwards, CI > SI because interest is earned on interest in CI.

Shortcut Trick

Profit Ratio = (Capital_A × Time_A) : (Capital_B × Time_B)

Ratio = (75 × 12) : (80 × 7) = 900 : 560 = 45 : 28

Total Ratio = 45 + 28 = 73 units

73 units = ₹4,08,800 ⇒ 1 unit = ₹5,600

A’s share = 45 × 5,600 = ₹2,52,000

B’s share = 28 × 5,600 = ₹1,56,800

∴ The correct answer is ₹2,52,000 and ₹1,56,800.

Alternate Method

Given:

Investment of A = ₹75,000

Investment of B = ₹80,000

Time period of A = 12 months

Time period of B = 12 − 5 = 7 months

Total Profit = ₹4,08,800

Formula Used:

Profit Ratio = (Investment₁ × Time₁) : (Investment₂ × Time₂)

Partner A ₹75,000 × 12 m

Partner B ₹80,000 × 7 m

Ratio = 45 : 28

Calculations:

⇒ Profit share of A : Profit share of B = (75,000 × 12) : (80,000 × 7)

⇒ Ratio = 9,00,000 : 5,60,000

⇒ Ratio = 90 : 56 = 45 : 28

⇒ Total profit units = 45 + 28 = 73

⇒ Share of A = (45 ÷ 73) × 4,08,800

⇒ Share of A = 45 × 5,600 = ₹2,52,000

⇒ Share of B = (28 ÷ 73) × 4,08,800

⇒ Share of B = 28 × 5,600 = ₹1,56,800

∴ The correct answer is ₹2,52,000 and ₹1,56,800.

Additional Information

Partnership Profit Distribution

Profit is always distributed in the ratio of the product of Capital and Time (P = C × T).

Active vs Sleeping Partner

An active partner manages the business and may receive a salary before the remaining profit is shared according to the investment ratio.

Variable Investment

If investment changes during the period, use the sum of (Investment × Duration) for each interval: (C₁t₁ + C₂t₂ + ...).

Shortcut Trick

Put θ = 45° to find the value of the expression quickly.

Expression = (sin 45° + cosec 45°)² + (cos 45° + sec 45°)²

⇒ (1/√2 + √2)² + (1/√2 + √2)² = (3/√2)² + (3/√2)² = 9/2 + 9/2 = 9

Checking Option 3: 7 + tan²45° + cot²45° = 7 + (1)² + (1)² = 9

∴ The correct answer is 7 + tan²θ + cot²θ.

Alternate Method

Given: (sin θ + cosec θ)² + (cos θ + sec θ)²

Formula Used: (a + b)² = a² + b² + 2ab; sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ

Expanding the given expression:

⇒ (sin²θ + cosec²θ + 2 sin θ cosec θ) + (cos²θ + sec²θ + 2 cos θ sec θ)

Since sin θ × cosec θ = 1 and cos θ × sec θ = 1:

⇒ sin²θ + cosec²θ + 2(1) + cos²θ + sec²θ + 2(1)

Grouping the terms:

⇒ (sin²θ + cos²θ) + cosec²θ + sec²θ + 4

⇒ 1 + (1 + cot²θ) + (1 + tan²θ) + 4

⇒ 1 + 1 + cot²θ + 1 + tan²θ + 4

⇒ 7 + tan²θ + cot²θ

∴ The correct answer is 7 + tan²θ + cot²θ.

Additional Information

Fundamental Trigonometric Identities

There are three main identities: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, and 1 + cot²θ = cosec²θ.

Reciprocal Relations

cosec θ = 1/sin θ, sec θ = 1/cos θ, and cot θ = 1/tan θ. Their products (e.g., sin θ × cosec θ) are always equal to 1.

Algebraic Expansion

For any trigonometric terms, (x + y)² = x² + y² + 2xy. This is frequently used to simplify squared sums of ratios.

Given:

$\frac{0.7 \times 0.7 \times 0.7 + 0.3 \times 0.3 \times 0.3}{0.7 \times 0.7 - 0.7 \times 0.3 + 0.3 \times 0.3}$ = ?

Formula used:

a³ + b³ = (a + b)(a² - ab + b²)

Calculation:

⇒ $\frac{0.7^3 + 0.3^3}{0.7^2 - 0.7 \times 0.3 + 0.3^2}$

⇒ $\frac{(0.7 + 0.3)(0.7^2 - 0.7 \times 0.3 + 0.3^2)}{(0.7^2 - 0.7 \times 0.3 + 0.3^2)}$

⇒ 0.7 + 0.3

⇒ 1

∴ The correct answer is option (2).

Given: R = 1, E = 2, A = 3, C = 4, T = 5, I = 6, V = 7, E = 8   

Explanation: Using the given number codes for the letters, we can decode the options as follows:

1. 4, 1, 2, 3, 5, 8, 7, 6 ⇒ C, R, E, A, T, E, V, I ⇒ CREATEVI (incorrect order)
2. 4, 1, 2, 3, 5, 6, 7, 8 ⇒  C, R, E, A, T, I, V, E ⇒  CREATIVE (correct order)
3. 7, 2, 1, 3, 8, 4, 5, 6 ⇒  V, E, R, A, E, C, T, I ⇒  VERAECTI (meaningless)
4. 4, 3, 1, 5, 6, 7, 8, 2 ⇒  C, A, R, T, I, V, E, E ⇒  CARTIVEE (meaningless)

Option 2 forms the meaningful word "CREATIVE," which is the correct answer.

Hence, the correct answer is "Option 2".

The logic followed here is :- (2) is added to the position value according to English alphabetic series diagonally.

​So,

• 'SLAIN' is written as ‘PKCNU’

And,

• 'DENT' is written as 'VPGF'

Similarly,

• 'CUBOID' is written as :

<br> <br>

Hence, the correct answer is "FKQDWE".

Logic :

1. 1st number × 2 = 2nd number + 1
2. Square of 1st number = 3rd number

So,

• Option - (1) : 11 ∶ 21 ∶ 121

(11 × 2) = 21 + 1

22 = 22 (LHS = RHS)

And, 

$(11)^2$ = 121

121 = 121 (LHS = RHS)

• Option - (2) : 7 ∶ 15 ∶ 49

(7 × 2) = 15 + 1

14 ≠ 16 (LHS ≠ RHS)

And, 

$(7)^2$ = 49

49 = 49 (LHS = RHS)

• Option - (3) : 2 ∶ 3 ∶ 4

(2 × 2) = 3 + 1

4 = 4 (LHS = RHS)

And, 

$(2)^2$ = 4

4 = 4 (LHS = RHS)

• Option - (4) : 9 ∶ 17 ∶ 81

(9 × 2) = 17 + 1

18 = 18 (LHS = RHS)

And, 

$(9)^2$ = 81

81 = 81 (LHS = RHS)

Hence, "Option - (2)" is the correct answer.

Logic : In each figure there are atleast two same figures.

So,

• Option - (1) :

 

⇒ There are two squares and one circle.

• Option - (2) :

 

⇒ There are two triangles and one circle.

• Option - (3) :

 

⇒** All the figures are different i.e one triangle, one circle and one square.**

• Option - (4) :

⇒ There are two circles and one sqaure.

Here, Figure - (3) is the odd one out.

Hence, "Option - (3)" is the correct answer.

Given : 11 * 7 * 345 * 5 * 31 = 115

According to the question, by replacing the *signs :

So,

• Option - (1) : + × ÷ −

11 + 7 × 345 ÷ 5 − 31 = 115

11 + 7 × 69 − 31 = 115

11 + 483 - 31 = 115

494 - 31 = 115

463 ≠ 115 (LHS ≠ RHS) 

• Option - (2) : × + ÷ −

11 × 7 + 345 ÷ 5 − 31 = 115

11 × 7 + 69 - 31 = 115

77 + 69 - 31 = 115

146 - 31 = 115

115 = 115 (LHS = RHS) 

• Option - (3) : ÷ × + −

11 ÷ 7 × 345 + 5 − 31 = 115

1.57 × 345 + 5 − 31 = 115

541.65 + 5 - 31 = 115

546.65 - 31 = 115

515.65 ≠ 115 (LHS ≠ RHS)

• Option - (4) : ÷ − + ×

11 ÷ 7 − 345 + 5 × 31 = 115

1.57 − 345 + 5 × 31 = 115

1.57 + 155 - 345 = 115

156.57 - 345 = 115

188.43 ≠ 115 (LHS ≠ RHS)

Hence, "Option - (2)" is the correct answer.

The logic followed here is :-

• Elements from left up corner to right below corner are shifting anti-clockwise diagonally.
• Elements from left below corner exchanged positions with right up corner element.

​So,

• Figures A and B ⇒

​

Similarly,

• Figure C and D ⇒

​

Thus, final pattern is :

<br>

Hence, "Option - (2)" is the correct answer.

Given : Seven persons are sitting around a circular dining table facing the towards the centre. 

Explanation : 

1) S is sitting to the immediate left of Y and third to the right of R.

2. R is sitting to the immediate left of A and immediate right of P.

3) A is sitting to the immediate left of D.

4. Q is sitting to the immediate left of P.

So, according to the question If D exchanges seat with Y and P with Q :- 

Thus, according to the final arrangement P is sitting to the immediate right of D.

Hence, "P" is the correct answer.

Step 1: Analyze the statements

• Statement I: Over the last two years, 22% of Company P’s manpower was laid off.
• Statement II: Over the past two years, Company P has sold off many of their major assets.

Step 2: Identify the possible relationships between the statements

• Both statements are negative outcomes that occurred within the same time frame (last two years). This suggests there might be a common cause or that one statement is the cause of the other.

Step 3: Evaluate each option

• Option 1: Both I and II are effects of a common cause.
• It is plausible that financial difficulties, management changes, or a company restructuring led to both the layoffs and the selling of major assets. This option makes logical sense.

Hence, the correct answer is "Both I and II are effects of a common cause."

Logic : In each figure the bottom part of the is same i.e closed figure with 2 lines and 2 curves or arc.

So,

• Option - (1) :

 

⇒ Here the bottom part has the enclosed figure with 2 lines and 2 curves.

• Option - (2) :

 

⇒ Here the bottom part has the enclosed figure with 2 lines and 2 curves.

• Option - (3) :

⇒ Here the bottom part has the enclosed figure with 4 lines and 0 curves.

• Option - (4) :

 ⇒ Here the bottom part has the enclosed figure with 2 lines and 2 curves.

Here, Figure - (3) is the odd one out.

Hence, "Option - (3)" is the correct answer.

The logic followed here is :- 

1. First letter → (3) is subtracted from the position value according to English alphabetic series.
2. Second letter → (3) is added to the position value according to English alphabetic series.
3. Third letter → (1) is subtracted from the position value according to English alphabetic series.

​So,

• = NOB, % = SRY ⇒ NOB : KNE :: PQV : SRY

​

⇒Here, no similar pattern can be seen.

• = NKF, % = MTU ⇒ NKF : KNE :: PQV : MTU

​

⇒Here, similar pattern can be seen.

• = MOB, % = RTT ⇒ MOB : KNE :: PQV : RTT

​           

⇒Here, no similar pattern can be seen.

• = HQD, % = MTV ⇒ HQD : KNE :: PQV : MTV​ ​

⇒Here, no similar pattern can be seen.

Hence, the correct answer is "Option - (2)".

The logic here followed is 

Hence, the correct answer is "option 2".

Given : 

1. Six people - K, M, N, P, Q and R
2. All of them are sitting around a circular table facing the centre.

Explanation :

• Statement - (I) : K sits to the immediate right of P. Only two people sit between P and Q. R sits second to the right of M.

Here, R sits second to the left of P. 

• Statement - (II) : Only two people sit between N and M. Q is an immediate neighbour of M. Only one person sits between Q and K.

Here, position of P cannot be determined, therefore who sits second to the left of P cannot be answered. 

Therefore, Data in statement I alone is sufficient to answer the question while data in statement II is not.

Hence, "Option - (2)" is the correct answer.

Given:

• The image must not to be rotated
  

=

Hence, the correct answer is "Option - (4)".

The least possible Venn diagram for the given statements is as shown below :

<br>

Conclusions:

I. Some Bangles are Rings → Does not Follow (As some bangles are bracelets, some bracelets are chains and all chains are rings. So, it is possible but not definitely true)

II. Some Chains are Bangles → Does not Follow (As some bangles are bracelets, some bracelets are chains and all chains are rings. So, it is possible but not definitely true)

III. Some Rings are Bracelets → Follow (As some bracelets are chains and all chains are rings. So, as whole chains comes in rings which some part comes in bracelets therefore it is definitely true)

∴ Here, Only Conclusion III follows​

Hence, the correct answer is "Only Conclusion III follows"

Given : 

1. Amit ranked 7th from the top in his class.
2. Amit ranked 26th from the bottom in his class.

Explanation : 

The total number of people = Position of the person from Top + Position of the person from Bottom - 1.

Therefore,

The total number of students in the class = 7 + 26 - 1 = 32.

Hence, the correct answer is "32".

The logic followed here is : Double addition pattern is followed -

1. First addition - Consecutive multiple of 4 is added starting from 46.
2. Second addition - 4 is added subsequently.

<br>

Hence, "338" is the correct answer.

Explanation:

Based on the given statement, let's evaluate the conclusions:

Assumption I: The left-footed player was adept at scoring goals:- The statement indicates that a left-footed player scored 3 out of the 5 goals for Team Y. While this might suggest the player’s capability in that particular match, one cannot assume that the player is adept overall based only on this single instance. This assumption is not necessarily valid based just on the given statement. Therefore, assumption I do not follow.

Assumption II: 60% of the players in Team Y are left-footed:- The statement does not provide any information about the total number or the proportion of left-footed players in Team Y. It only mentions the number of goals scored by one left-footed player. There is no basis to support this assumption from the statement.

Hence, the correct option is "Neither I nor II follows".

The logic followed here is:

When the folder paper is opened, the cuts will appear as follows

Hence, the correct answer is "Option 1".

Given : S C R _ W S _ R _ W _ C _ G W S _ R H _

By checking options :

1. G D F S R C W → S C R G W - S D R F W -  S C R G W  - S C R H W  → (It doesn't Follows the pattern)
2. H D F S R C W → S C R H W  - S D R F W  - S C R G W  - S C R H W → (It doesn't Follows the pattern)
3. E C F S R C W → S C R E W -  S C R F W  - S C R G W  -  S C R H W →    (It Follows the pattern)

• Option - (3) gives the pattern of "S C R E W - S C R F W - S C R G W - S C R H W"
• 
  

1. E D F S R D W → S C R E W -  S D R F W  - S C R G W  -  S D R H W → (It doesn't Follows the pattern)

 

Hence, "E C F S R C W" is the correct answer.

Given :

Given statement : 3 ÷ 6 + 3 – 4 × 4 = ? 

After replacing the signs according to the question :

3 + 6 - 3 × 4 ÷ 4 = ?

3 + 6 - 3 × 1 = ?

3 + 6 - 3 = ?

9 - 3 = ?

6 = ?

Hence, "6" is the correct answer.

Explanation: In response to the statement about serious mistakes found in the technical section of a company, there are two suggested courses of action.

I. An efficient technical team should be appointed to check the technical errors.

• This is a logical course of action, as it addresses the immediate issue of fixing the errors and preventing them from causing further issues within the company.

II. A reason explanation notice should be issued to all the employees involved in the irregularities.

• This is also a logical course of action, as it holds employees accountable for their actions and encourages them to provide an explanation for the mistakes. This can help the company understand the root cause of the errors and prevent them from happening in the future.

Since both I and II are logical courses of action, the correct answer is option 4: Both I and II follow.

The figure that will replace the question mark (?) in the following figure series is shown below:

1. Every element is shifting in one place clockwise direction until the last element does not come. The last element   "+" disappeared and one new element A is appeared at the last.
   

• 
  

 

2)  Every element is shifting two places clockwise direction until the second last element does not come. The second last element   "

" disappeared and one new element B is appeared at the last . As A is placed after eliminated element, A is shifting three placed clockwise direction.

3)  Every element is shifting three places clockwise direction until the third last element does not come. The third last element   "$" disappeared and one new element C is appeared at the last . As A, B  are placed after eliminated element, A, B are shifting four placed clockwise direction.

• 
  

4) Every element is shifting four places clockwise direction until the fourth last element does not come. The third last element   "≠ " disappeared and one new element D is appeared at the last . As A, B,C are placed after eliminated element, A, B,C are shifting five placed clockwise direction.

• 
  

Hence, ‘option - (1)’ is the correct answer.

The logic followed here is :-  Sum of the opposite letter's position value according to English alphabetic series and number of letters within the word.

​So,

• ‘POLISH’ is written as ‘89’

​

And,

• ‘CLIP’ is written as ‘72’

<br>

Similarly,

• ‘CONTROL’ is written as :

<br>

Hence, the correct answer is "99".

Logic: The words can be grouped based on their professions and roles in society.

1. Engineer
2. Doctor
3. Teacher

These three professions (Engineer, Doctor, and Teacher) are commonly associated with higher education and often require advanced degrees. They usually work in specialized fields and provide professional services.

1. Carpenter: On the other hand, a carpenter is a skilled craftsman and works with wood to construct and repair structures. The profession typically requires hands-on training and experience, rather than advanced degrees. This difference makes the Carpenter the word that does not belong to the group.

Hence, the correct answer is "Option 3".

Logic : (2nd number × 2) = (3rd number + 1st number**)**

Given :

(18, 39, 60)

39 × 2 = 60 + 18

78 = 78 (LHS = RHS)

(27, 48, 69)

48 × 2 = 69 + 27 

96 = 96 (LHS = RHS)

So,

• Option - (1) : (24, 64,96)

​64 × 2 = 96 + 24

128 ≠ 120 (LHS ≠ RHS)

• Option - (2) : (35, 56, 77)

​56 × 2 = 77 + 35

112 = 112 (LHS = RHS)

• Option - (3) : (31, 52, 84)

​52 × 2 = 84 + 31

104 ≠ 115 (LHS ≠ RHS)

• Option - (4) : (17, 28, 49)

​28 × 2 = 49 + 17

56 ≠ 66 (LHS ≠ RHS)

Hence, "Option - (2)" is the correct answer.

Given :

Given statement : Z % X # C @ V & B @ N

So,

• Z % X → Z is the wife of X.

• X # C → X is the brother of C.

• C @ V → C is the daughter of V.

• V & B → V is the husband of B.

• B @ N → B is the daughter of N.

​

Thus, according to the final family tree Z is the daughter-in-law i.e son’s wife of B.

Hence, "Son’s wife" is the correct answer.

Given :

Given statement : D @ N @ H & Y @ F % V

So,

• D @ N → D is the daughter of N.

• N @ H → N is the daughter of H.

• H & Y → H is the husband of Y.

• Y @ F → Y is the daughter of F.

• F % V → F is the wife of V.

​

Thus, according to the final family tree Y is the grandmother i.e mother's mother of D.

Hence, "Mother’s mother" is the correct answer.

The given statement is: Z = E ≥ R > G < Y > Q = F

Conclusion I: R < F [ From the given statement, R > G < Y > Q = F However, we cannot determine whether R is less than F or not. Hence, conclusion I is not Follow.]

Conclusion II: R ≥ Z [ From the given statement, Z  ≥ R. Hence, conclusion II is not true.]

Hence, the correct answer is option 4. Neither conclusion I nor II is true.

Given : 

1. Four people - A, B, C and D
2. All of them are sitting in a straight line facing south.

Explanation :

• Statement - (I) : Only two people sit between B and D; A sits to the immediate left of B.

Here, D is sitting at the right end. 

• Statement - (II) : Only one person sits between B and C; A sits to the immediate right of C.

Case - (1) 

Case - (2) 

Case - (3)

Here, there are three possible cases according to the statement, therefore one specific person who is sitting at the right end cannot be determined.

Therefore, Data in statement I alone is sufficient to answer the question while data in statement II is not

Hence, "Option - (1)" is the correct answer.

The logic is followed here:

Hence, the correct answer is "Option 1".

The correct answer is option 2, "toe."

Key Points

• "Toe" to any of the five digits at the end of the human foot.
• Example: He cut his big toe on a sharp stone.
• In this sentence, the athlete likely fell and injured his/her toe while running on the track.
• Thus, the correct answer is Option 2.

Additional Information

• The other options are:
• "Toad" is a type of amphibian, such as a frog.
• Example: I saw a toad hopping across the road.
• "Tow" means to pull or drag something behind you.
• Example: The truck was able to tow the broken-down car to the mechanic.
• "To" is a preposition that can indicate direction or location.
• Example: We went to the store to buy groceries.

The correct answer is option 2: "Harry said, 'I will not be in town on Friday'."

Key Points

• When converting indirect speech to direct speech, we need to make a few changes:
• Change the tense of the reported speech to match the tense of the reporting verb.
• In this case, "said" is in the past tense, so we keep the reported speech in the past tense as well.
• 'Would' will change into 'will'.
• Change any pronouns to match the speaker in the direct speech. In this case, "he" becomes "I".
• Add quotation marks around the reported speech.
• Option 2 follows all of these rules correctly, making it the correct answer.

<br>

Correct Sentence: Harry said, “I will not be in town on Friday”.

Here the correct answer is 2) Rajat first heard the story of his uncle's mysterious disappearance in his childhood.

Key Points

• This sentence is in passive voice as the subject (the story) is receiving the action (was first heard).
• To convert it into active voice, we need to make the subject the doer of the action.
• In this case, the subject is "Rajat" and the action is "heard the story".
• 'was first heard' changes to 'first heard'.
• Therefore, the active voice sentence would be "Rajat first heard the story of his uncle's mysterious disappearance in his childhood."
• The correct answer is option 2 as it correctly follows the rules of converting a passive voice sentence into active voice.

<br>

Correct Sentence: Rajat first heard the story of his uncle’s mysterious disappearance in his childhood.

Here the correct answer is Lauded.

Key Points

• Sneered means smile or speak in a contemptuous or mocking manner.
• Example: She had sneered at their bad taste.
• The correct antonym of "sneered" is "lauded" which means to praise or applaud someone or something. 
• Example: The audience lauded the performance of the young singer.
• Option 1: "Waved" is not an appropriate antonym as it does not convey the opposite meaning of "sneered". 
• Example: The crowd waved their hands in excitement.
• Option 3: "Unleashed" means to release or set free, which is not related to the context of the sentence. 
• Example: The dog was unleashed in the park.
• Option 4: "Ascribed" means to attribute or assign something to someone, which is not related to the context of the sentence. 
• Example: The success of the project was ascribed to the hard work of the team.
• Therefore, option 2 is the correct antonym of "sneered" in the given sentence.

Here the correct answer is 'Than'.

Key Points

• The correct answer is option 2, "than".
• This is because the sentence is comparing two things - online shopping and shopping in a market - and showing that one is easier than the other.
• "Than" is the appropriate word to use in this context to show the comparison.
• Example: Nowadays, online shopping is easier than shopping in a market because you can browse and purchase items from the comfort of your own home.
• Option 1, "like", would not be appropriate in this context because it suggests that online shopping and shopping in a market are similar, rather than being compared.
• Option 3, "then", is a conjunction used to show a sequence of events, and is not appropriate in this context.
• Option 4, "as", is a conjunction used to show a similarity between two things, and is not appropriate in this context.

<br>

Correct Sentence: Nowadays, online shopping is easier than shopping in a market.

Here the correct answer is ACBD.

Key Points

• The correct answer is option 3 - ACBD.
• Sentence A introduces the Sustainable Development Goals (SDGs) and their purpose.
• Sentence C provides additional information about the SDGs, including their alternate name and their goal of creating a world where no one is left behind.
• Sentence B connects the SDGs to child rights, emphasizing the importance of realizing child rights in achieving the SDGs.
• Finally, sentence D brings the focus back to children and their role in securing their own rights and contributing to the achievement of the SDGs.
• Therefore, the correct order of the sentences is ACBD, which creates a coherent and meaningful paragraph about the importance of the SDGs and child rights in creating a better world for all.

<br>

Correct Sequence:

A) The Sustainable Development Goals (SDGs) were adopted by all United Nations Member States in 2015 to end poverty, reduce inequality and build more peaceful, prosperous societies by 2030.

C) Also known as the Global Goals, the SDGs are a call to action to create a world where no one is left behind.

B) The SDGs cannot be achieved without the realisation of child rights.

D) As world leaders work to deliver on the 2030 promise, children around the globe are standing up to secure their right to good health, quality education, a clean planet and more.

Here the correct answer is SRQP.

Key Points

• The paragraph talks about the rise in prices of essential commodities and how it has affected different sections of society.
• Sentence S sets the context by stating that living has become expensive.
• Sentence R talks about the impact on low-income groups and salaried classes.
• Sentence Q provides a broader perspective by stating that this rise in prices is not a recent phenomenon but has been happening in recent years.
• Finally, sentence P gives specific examples of the commodities that have seen a sharp rise in prices.
• Therefore, the correct order is Option 3.

<br>

Correct Sequence:

S. Living has become quite a costly affair these days.

R. The low-income groups and the salaried classes are the worst sufferers.

Q. In recent years the prices of essential commodities have gone up sharply.

P. The prices of cooking gas, pulses and vegetables have registered a sharp rise.

Here the correct answer is Doubt the truth or value of something.

Key Points

• The correct answer is option 3) Doubt the truth or value of something.
• The idiom "take something with a pinch of salt" means to be skeptical or doubtful about the truth or value of something that has been said or claimed (जो कुछ कहा या दावा किया गया है उसकी सत्यता या मूल्य के बारे में संदेह करना).
• In this context, the employee is not taking the manager's remark seriously and is not fully believing it.
• Therefore, option 3 is the most appropriate meaning of the highlighted idiom.

Here the correct answer is Option 3.

Key Points

• The incorrectly spelled word in the given sentence is "suceeded".
• The correct spelling is "succeeded".
• It means achieve the desired aim or result.
• Examples:
• She succeeded in convincing her boss to give her a raise.
• He succeeded in completing the marathon despite the hot weather.
• Thus, the correct answer is Option 3.

Additional Information

• Option 1: "deadline" is spelled correctly. It refers to the final date or time by which something must be done.
• Option 2: "intricate" is spelled correctly. It means complex or detailed.
• Option 4: "earlier" is spelled correctly. It means before a certain time or event.

Here the correct answer is Option 1.

Key Points

• The segment that contains a grammatical error is "when me".
• It should be "when I" instead.
• The grammatical rule that applies here is the use of subject pronouns and object pronouns.
• "I" is a subject pronoun, which is used as the subject of a sentence or clause.
• "Me" is an object pronoun, which is used as the object of a verb or preposition.
• Examples:
• Subject pronoun: I am going to the store.
• Object pronoun: She gave the book to me.
• In the given sentence, "when" is a subordinating conjunction that introduces a dependent clause.
• The subject of the dependent clause is "I", so it should be used instead of "me".
• Thus, the correct answer is Option 1.

<br>

The correct sentence would be: "Everybody stared when I entered the drawing room."

Here the correct answer is BCAD.

Key Points

• The correct answer is option 3 - BCAD.
• Sentence B starts the paragraph by introducing the author's current state of well-being, emphasizing that it is a result of a careful and measured mode of living. It also highlights the astonishing fact that despite being rendered a hopeless physical wreck three times in their youth, they managed to recover.
• Sentence C follows B, explaining how the author got into various difficulties, dangers, and scrapes due to ignorance and light-heartedness. It implies that they were able to extricate themselves from these situations by some form of enchantment, suggesting their resourcefulness.
• Sentence A provides specific examples of the difficulties and dangers the author faced, such as almost drowning, nearly being boiled alive, and narrowly avoiding being cremated. It elaborates on the challenges mentioned in sentence C.
• Finally, sentence D concludes the paragraph by reflecting on the incidents mentioned and stating the author's belief that their preservation was not accidental. It connects back to the experiences mentioned in sentences A and C.
• Therefore, the correct order of the sentences to form a meaningful and coherent paragraph is BCAD.

<br>

Correct Sequence:

B. Now, my well-being is simply the result of a careful and measured mode of living and perhaps the most astonishing thing is that three times in my youth I was rendered by illness a hopeless physical wreck and given up by physicians.

C. More than this, through ignorance, and light-heartedness, I got into all sorts of difficulties, dangers and scrapes from which I extricated myself as by enchantment.

A. I was almost drowned a dozen times; was nearly boiled alive and just missed being cremated.

D. But as I recall these incidents to my mind I feel convinced that my preservation was not altogether accidental.

Here the correct answer is No substitution required.

Key Points

• The correct answer is option 1, "No substitution required."
• The underlined segment "not more popular than" is already a comparative phrase, indicating that Bloomberg Businessweek is less popular than some other business magazines in the US. There is no need to substitute it with any other comparative phrase.
• The grammatical rule related to this error is the use of comparative adjectives.
• Comparative adjectives are used to compare two things, and they are formed by adding -er to the end of the adjective (e.g. taller, faster, smarter) or by using the word "more" before the adjective (e.g. more beautiful, more intelligent, more successful).
• In the given sentence, "more popular than" is a comparative phrase that is used to compare the popularity of Bloomberg Businessweek with some other business magazines in the US. However, the sentence is negating this comparison by using the word ""not"" before ""more popular than,"" indicating that Bloomberg Businessweek is less popular than some other business magazines in the US.
• Thus, the correct answer is Option 1.

Here the correct answer is Philanthropist.

Key Points

• The correct answer is option 4, philanthropist.
• A philanthropist is a person who loves mankind and is actively involved in promoting human welfare and improving the quality of life for others.
• For example, Bill Gates is a well-known philanthropist who has donated billions of dollars to various charitable causes.
• Option 1, pessimist, refers to a person who always sees the negative side of things and expects the worst outcome.
• For example, a pessimist might say "I don't think we'll ever solve this problem."
• Option 2, misogynist, refers to a person who hates or has a strong dislike for women.
• For example, a misogynist might make derogatory comments about women or discriminate against them in the workplace.
• Option 3, optimist, refers to a person who always sees the positive side of things and expects the best outcome.
• For example, an optimist might say "I'm sure we'll find a solution to this problem."

Here the correct answer is Accomplishment.

Key Points

• In the sentence, the word "fruition" is used to describe the lack of fulfillment of James' campaign promises, indicating that he has not been able to achieve or bring them to completion.
• Fruition means the realization or fulfilment of a plan or project. 
• Example: The plans have come to fruition rather sooner than expected
• Accomplishment means the successful achievement of a task.
• Example: The accomplishment of planned objectives.
• Therefore, "accomplishment" is the most appropriate meaning of the word in this context.

Additional Information

• Option 1, concealment, does not fit the context of the sentence as there is no indication of James hiding anything.
• Option 2, nourishment, is also irrelevant as it does not relate to the sentence's context.
• Option 3, disappointment, is also incorrect as it refers to 'the state of being unhappy because something that you hoped for or expected did not happen.'

Here the correct answer is Conduct.

Key Points

• In this context, the committee is responsible for conducting a survey on the government schemes, which implies that they will be collecting data and information about the schemes.
• To conduct a survey means to carry out or organize a survey.
• In this context, the committee is responsible for conducting a survey on the government schemes.
• Example: The marketing team conducted a survey to understand the customer's preferences.
• Option 2 - fight - is not appropriate in this context as it does not make sense to fight a survey.
• Option 3 - hold - is a possible answer, but it is not as appropriate as conduct. Hold means to organize or arrange an event or meeting.
• Example: The school will hold a parent-teacher meeting next week.
• Option 4 - originate - is not appropriate in this context as it means to begin or start something.
• Example: The idea for the new product originated from the research team.
• Therefore, option 1, conduct, is the most appropriate choice.

Here the correct answer is Pergola.

Key Points

• A pergola is a garden feature that consists of a framework of vertical posts or pillars that support crossbeams and an open lattice, often upon which climbing plants are trained to grow. It provides a shaded walkway, passageway, or sitting area in a garden or park.
• For example, a pergola can be used to create a shaded seating area in a backyard garden, or it can be used to cover a walkway in a public park.
• Option 1, Shelter, refers to a structure that provides protection from the elements, such as a building or a tent.
• For example, a bus shelter provides protection from the rain for people waiting for a bus.
• Option 3, Path, refers to a route or track that is made for walking or traveling along.
• For example, a garden path is a walkway that leads through a garden.
• Option 4, Shed, refers to a small building used for storage or as a workshop.
• For example, a garden shed is a small building used to store gardening tools and equipment.
• Thus, the correct answer is Option 2.

The correct answer is option 4) pair.

Key Points

• A pair refers to two things that are identical and meant to be used together.
• In this case, the security guard needs a pair of shoes, which means he needs two shoes that match and are meant to be worn together.
• Example: The store clerk asked the customer if he needed help finding a pair of shoes in his size.
• Option 1) pare: This word means to trim or cut something down to its essential parts.
• Example: The chef used a knife to pare the apple into thin slices.
• Option 2) pear: This word refers to a type of fruit that is typically green or yellow and has a rounded shape.
• Example: The farmer harvested a basket of ripe pears from his orchard.
• Option 3) payor: This word refers to a person or organization that makes a payment.
• Example: The insurance company was the payor for the medical bills.
• Overall, the correct answer is option 4) pair, as it is the only word that makes sense in the given sentence.

<br>

Correct Sentence: The security guard needs a pair of shoes.

The correct answer is: 2) He has made a new invention.

Key Points

• To convert a sentence from passive voice to active voice, you need to identify the subject performing the action (the doer) and make it the subject of the active sentence.
• In the given sentence, "A new invention has been made by him," the subject is "a new invention," and the action is "has been made."
• The doer of the action is "him."
• To change this into active voice, we need to make "him" the subject and rephrase the sentence accordingly.
• In summary, the correct answer is option 2) "He has made a new invention." It correctly rephrases the original passive sentence into active voice, maintaining the same tense and meaning.

Additional Information

• Now let's analyze each option:
• He was making a new invention.
• This option is in active voice, but it does not convey the same meaning as the original sentence. It suggests that "he" was in the process of making the invention, rather than stating that the invention has already been made.
• He has made a new invention. (Correct Answer)
• This option correctly converts the passive sentence into active voice. "He" is the subject performing the action "has made," and "a new invention" remains the object.
• He had been making a new invention.
• This option is in past perfect continuous tense, suggesting a continuous action in the past. It does not maintain the same tense and meaning as the original sentence.
• He were making a new invention.
• This option contains a grammatical error. "He" is a singular subject, so the correct form of the verb "to be" should be "was" instead of "were."

Here the correct answer is Mutable.

Key Points

• The most appropriate antonym for the underlined word "stable" in the given sentence is: Mutable.
• The word "stable" refers to something that is steady, secure, and unlikely to change or falter. In the context of a job, a stable job implies that it provides consistent income and is not easily affected by fluctuations or uncertainties. It suggests a sense of reliability and permanence.
• "Mutable" means liable to change or alter, suggesting that the job is not stable and subject to fluctuations or uncertainties. It conveys the opposite meaning of stability, indicating that the job may not provide a consistent or reliable income.
• In conclusion, the most appropriate antonym for "stable" in the given sentence is option 4) Mutable.

Additional Information

• Now let's examine the other options and their meanings:
• Lucrative: This word means producing a large amount of profit or financial gain. It implies that the job is highly profitable or financially rewarding. It is not the antonym of "stable".
• Permanent: This word means lasting or intended to last indefinitely without change. It describes something that is fixed, unchanging, or not temporary. It is not the antonym of "stable".
• Durable: This word means able to withstand wear, pressure, or damage. It describes something that is sturdy, resilient, or long-lasting. Although it shares some characteristics with stability, it is not the antonym of "stable".