RRB NTPC CBT 1 2016 Mathematics Questions with Solutions

Explanation

Using the work formula (M₁ × D₁) / W₁ = (M₂ × D₂) / W₂:

(10 × 30) / 1 = (15 × D₂) / 2

300 = 15 × D₂ / 2

600 = 15 × D₂

D₂ = 600 / 15 = 40 days.

Explanation

Mean = (Sum of observations) / (Number of observations)

16 = (18 + 16 + 22 + 13 + ?) / 5

80 = 69 + ?

? = 80 - 69 = 11

Explanation

Substitute y = 5 into the given expression y(y⁴ - y² - y):

y⁴ = 5⁴ = 625 y² = 5² = 25

y(y⁴ - y² - y) = 5 × (625 - 25 - 5) = 5 × 595 = 2975

Explanation

Let the two positive integers be x and y.

Using the product-sum-difference identity xy = (S² - D²)/4, where S is the sum (34) and D is the difference (8):

xy = (34² - 8²)/4 xy = (1156 - 64)/4 xy = 1092 / 4 = 273

Alternatively, the individual integers are x = (34 + 8)/2 = 21 and y = (34 - 8)/2 = 13. Their product is 21 × 13 = 273.

Explanation

Mother's age = Father's age - 2 = 54 - 2 = 52 years.

Younger sister's age = ½ × Mother's age = ½ × 52 = 26 years.

Elder sister's age = Younger sister's age + 4 = 26 + 4 = 30 years.

Explanation

Total required distance for 7 days = Target average × Total days = 4 × 7 = 28 km

Distance covered in the first 5 days = 3 + 4 + 3.5 + 5 + 4.5 = 20 km

Distance to be covered on the remaining 2 days = Total required distance - Distance covered = 28 - 20 = 8 km

Explanation

Prime factorization of the given numbers: 16 = 2⁴ 24 = 2³ × 3

HCF is the product of the lowest powers of common prime factors: HCF(16, 24) = 2³ = 8

LCM is the product of the highest powers of all prime factors: LCM(16, 24) = 2⁴ × 3 = 48

Explanation

Mean = Sum of observations ÷ Total number of observations

Sum of observations = 2 + 3 + 3 + 4 + 6 + 7 = 25 Total number of observations = 6

Mean = 25 ÷ 6 = 4.166... ≈ 4.17

Explanation

Since equal quantities of tea are sold at two different transaction rates, the net percentage change is the arithmetic mean of the individual percentage changes:

Net percentage = (-20% + 10%) / 2 = -10% / 2 = -5%

A negative value represents a 5% net loss.

Alternatively, using the total values: Total Cost Price = 10 × 400 = Rs. 4000 Selling Price of first 5 kg = 2000 × (1 - 0.20) = Rs. 1600 Selling Price of remaining 5 kg = 2000 × (1 + 0.10) = Rs. 2200 Total Selling Price = 1600 + 2200 = Rs. 3800

Net Loss = 4000 - 3800 = Rs. 200 Net Loss Percentage = (200 / 4000) × 100 = 5% loss

Explanation

By the complementary angle trigonometric identities, the cosecant of an angle's complement equals the secant of that angle: cosec(90° - θ) = sec θ.

Explanation

Using the algebraic identity xy = [(x + y)² - (x - y)²] / 4:

Substituting (x + y) = 19 and (x - y) = 7: xy = (19² - 7²) / 4 xy = (361 - 49) / 4 xy = 312 / 4 = 78

Alternatively, solving the linear system directly by adding the two equations yields 2x = 26, giving x = 13. Subtracting the two equations yields 2y = 12, giving y = 6. Multiplying them gives xy = 13 × 6 = 78.

Explanation

When every quantity in a ratio is increased by the same percentage, the relative proportion between the quantities remains unchanged.

If the initial marks are x, 2x, and 3x, adding a 5% grace mark increases each value by a factor of 1.05. The new marks become 1.05x, 2.10x, and 3.15x. Dividing each term by 1.05x yields the original ratio of 1:2:3.

Explanation

When time is constant, distance is directly proportional to speed (d ∝ s).

Ratio of original speed to new speed = 50 : 60 = 5 : 6

Difference in distance ratio = 6 - 5 = 1 unit Given increase in distance = 20 km 1 unit = 20 km

Actual distance = 5 units = 5 × 20 = 100 km

Explanation

If four numbers a, b, c, and d are in proportion (a : b = c : d), then the fourth proportional d is given by the formula d = (b × c) / a.

Substituting the given values: 4th proportional = (17 × 27) / 9 = 17 × 3 = 51

Explanation

In a rectangle, two adjacent sides and the diagonal form a right-angled triangle. By the Pythagorean theorem, the unknown side is:

other side = √(diagonal² - given side²) = √(13² - 12²) = √(169 - 144) = √25 = 5 m

Area of the rectangle = length × breadth = 12 m × 5 m = 60 sq.m.

Explanation

Actual distance = Map distance × Scale factor

Given: Map distance = 22.25 cm Scale factor = 18.5 km/cm

Actual distance = 22.25 × 18.5 = 411.625 km

Explanation

Using the standard algebraic identity: (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

Substituting the given values into the identity: 6² = 14 + 2(ab + bc + ca) 36 = 14 + 2(ab + bc + ca) 2(ab + bc + ca) = 22 ab + bc + ca = 11

Explanation

Let the capacity of each vessel be LCM(4, 3) = 12 units.

In the first vessel, milk = (1/4) × 12 = 3 units and water = (3/4) × 12 = 9 units. In the second vessel, milk = (2/3) × 12 = 8 units and water = (1/3) × 12 = 4 units.

When mixed together: Total milk = 3 + 8 = 11 units Total water = 9 + 4 = 13 units

The new ratio of milk to water is 11:13.

Explanation

Mother's age = Father's age - 2 = 52 - 2 = 50 years.

Elder sister's age = ½ × Mother's age = ½ × 50 = 25 years.

Younger sister's age = Elder sister's age - 2 = 25 - 2 = 23 years.

Explanation

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

Maturity Value A = P + SI = P × [1 + (R × T) / 100]

Given A = 8,700, R = 15%, and T = 3 years:

8,700 = P × [1 + (15 × 3) / 100] 8,700 = P × 1.45 P = 8,700 / 1.45 = 6,000

Explanation

Number of small notebooks = 5 × 12 = 60 Number of big notebooks = 10 × 12 = 120

Marked price of small notebooks = 60 × 10 = Rs. 600 Marked price of big notebooks = 120 × 15 = Rs. 1800

Total marked price = 600 + 1800 = Rs. 2400

Total discount amount = 5% of 2400 = (5 / 100) × 2400 = Rs. 120

Explanation

Using the algebraic identity (a + b)² - (a - b)² = 4ab:

Let a = 525 and b = 252. The expression simplifies to: [(a + b)² - (a - b)²] / (a × b) = 4ab / ab = 4

Explanation

Using the trigonometric identity tan θ = sin θ / cos θ, the required expression simplifies directly to: cos θ × tan θ = cos θ × (sin θ / cos θ) = sin θ

From the given equation: 2 cos θ = √3 cos θ = √3 / 2

Since cos 30° = √3 / 2, θ = 30°. Substituting θ = 30° into sin θ gives sin 30° = 1/2.

Explanation

The mean of the first n natural numbers is given by the formula (n + 1) / 2.

Substituting n = 11: Mean = (11 + 1) / 2 = 12 / 2 = 6

Explanation

Using the algebraic expansion formula (x - y)³ = x³ - 3x²y + 3xy² - y³:

Substituting x = a and y = 4: (a - 4)³ = a³ - 3(a²)(4) + 3(a)(4²) - 4³ (a - 4)³ = a³ - 12a² + 48a - 64

Explanation

B completes 50% of the work in 20 days, so B's rate of work is ½ ÷ 20 = 1/40 of the total work per day.

The remaining 50% of the work is completed by B and C together in 4 days, so their combined rate of work is ½ ÷ 4 = 1/8 of the total work per day.

C's rate of work = Combined rate - B's rate = 1/8 - 1/40 = 4/40 = 1/10 of the work per day.

Time taken by C alone to complete the whole work = 1 ÷ (1/10) = 10 days.

Explanation

The selling price at a 5% loss corresponds to 95% of the cost price.

95% of Cost Price = Rs. 1235 1% of Cost Price = 1235 / 95 = Rs. 13

To make a profit of 10%, the selling price must be 110% of the cost price:

Required Selling Price = 13 × 110 = Rs. 1430

Explanation

Solving the given linear equation for x:

22x - 40 = 207 + 3x

Rearranging the terms to group x on one side:

22x - 3x = 207 + 40 19x = 247 x = 247 / 19 x = 13

Explanation

For two numbers in the ratio a : b (where a and b are co-prime) with Highest Common Factor (HCF) h, their Least Common Multiple (LCM) is given by the formula:

LCM = HCF × a × b

Substituting the given values: LCM = 8 × 4 × 3 = 96

Explanation

Selling price is directly proportional to (100 + Profit%).

Required Selling Price = Given Selling Price × (100 + New Profit%) / (100 + Old Profit%)

Required Selling Price = 920 × (120 / 115) = 920 × (24 / 23) = 40 × 24 = Rs. 960.

Explanation

Area of a triangle = ½ × base × height

Substituting the given values into the formula: 456 = ½ × base × 24 456 = 12 × base base = 456 / 12 = 38 cm

Explanation

The difference between compound interest and simple interest on a principal amount P for 2 years at an annual interest rate R is given by the formula:

Difference = P × (R / 100)²

Substituting P = 5000 and R = 5:

Difference = 5000 × (5 / 100)² = 5000 × (1 / 20)² = 5000 / 400 = Rs. 12.50

Explanation

Area of a trapezium = ½ × (sum of parallel sides) × height

Substituting the given values into the formula: 18 = ½ × (5 + b) × 3 36 = 3 × (5 + b) 12 = 5 + b b = 7 cm

Explanation

From the given table, for library GHL: Number of Fiction books = 610 Number of Self-help books = 170

The difference between the two categories is calculated as: 610 - 170 = 440

Explanation

Summing the category-wise book counts for MNL gives: 200 + 550 + 700 + 100 + 100 + 150 = 1800 books.

This total of 1800 books is the highest count among the four libraries.

Explanation

Using fundamental trigonometric identities:

cotθ = cosθ / sinθ and cosecθ = 1 / sinθ

Therefore, cotθ / cosecθ = (cosθ / sinθ) / (1 / sinθ) = cosθ

Substituting this back into the expression: (sinθ / cosθ) × (cotθ / cosecθ) = (sinθ / cosθ) × cosθ = sinθ

Explanation

Expanding the bracket by multiplying 5x with each term inside (x + 2): 5x(x + 2) + 4x = 5x² + 10x + 4x

Combining the like terms 10x and 4x: 5x² + (10x + 4x) = 5x² + 14x

Explanation

When a trader cheats by x% while both buying and selling using tampered weights, the total gain percentage is calculated using the successive percentage change formula:

Total Gain % = x + x + (x × x) / 100

Substituting x = 11:

Total Gain % = 11 + 11 + (11 × 11) / 100 = 22 + 121 / 100 = 22 + 1.21 = 23.21%

Explanation

Let the cost price of the article be 100 units.

The first selling price at a 2% profit is 100 + 2 = 102 units. The second selling price at an 18% profit is 100 + 18 = 118 units.

required ratio = First Selling Price : Second Selling Price Ratio = 102 : 118 = 51 : 59

Explanation

To divide 0.592 by 0.8, multiply both the numerator and denominator by 10 to clear the decimal in the divisor:

0.592 ÷ 0.8 = 5.92 ÷ 8

Performing the division gives:

5.92 ÷ 8 = 0.74

Explanation

Rewrite 9999 as (10000 - 1) to simplify the multiplication:

69696 × 9999 = 69696 × (10000 - 1) = 696960000 - 69696 = 696890304

Explanation

The combined time taken by two outlets to empty a container is given by the formula: t = (t₁ × t₂) / (t₁ + t₂)

Substituting t₁ = 9 minutes and t₂ = 16 minutes: t = (9 × 16) / (9 + 16) = 144/25 minutes

Explanation

Standard deviation (σ) is defined as the non-negative square root of variance:

Standard deviation = √(Variance)

Substituting the given variance of 196:

Standard deviation = √196 = 14

Since standard deviation represents a measure of dispersion or distance from the mean, it is always a non-negative quantity.

Explanation

According to the BODMAS rule (Brackets, Orders, Division, Multiplication, Addition, Subtraction), division is performed before subtraction in an arithmetic expression.

First, compute the division step: 4923 ÷ 547 = 9

Next, subtract 10 from the quotient obtained: 9 - 10 = -1

Thus, evaluating the expression 4923 ÷ 547 - 10 gives -1.

Explanation

If each observation in a dataset X is multiplied by a constant k, the variance of the new dataset kX is given by Var(kX) = k² × Var(X).

In the given problem, each number in the second dataset (4, 8, 10, 12, 16, 34) is obtained by multiplying the corresponding number in the first dataset (2, 4, 5, 6, 8, 17) by k = 2.

New Variance = 2² × 23.3333 = 4 × 23.3333 = 93.3333

Explanation

Selling price at a 15% loss represents 85% of the cost price, while the selling price required for a 15% profit represents 115% of the cost price.

Required Selling Price = (Selling Price at loss ÷ 85) × 115 = (22,100 ÷ 85) × 115 = 260 × 115 = ₹29,900

Explanation

Using the exponent rule (aᵐ)ⁿ = aᵐⁿ, rewrite 8 as 2³:

(8)⁽2/3) = (2³)⁽2/3) = 2⁽3 × 2/3) = 2² = 4

Explanation

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

Given Principal (P) = ₹950, Rate (R) = 6%, and Time (T) = 4 years: SI = (950 × 6 × 4) / 100 = ₹228

Total Amount = Principal + Simple Interest Amount = 950 + 228 = ₹1178

Explanation

Sum of ratio parts = 1 + 5 + 8 + 9 = 23 parts

Value of 1 part = 368 / 23 = 16

Multiplying each ratio term by 16 gives: 1st part = 1 × 16 = 16 2nd part = 5 × 16 = 80 3rd part = 8 × 16 = 128 4th part = 9 × 16 = 144

Explanation

The order of rotational symmetry is the number of times a geometric shape fits onto itself during a full 360° rotation about its center point. A rectangle maps onto itself exactly twice in a complete revolution: once after a rotation of 180° and again at 360°. Therefore, the order of rotational symmetry of a rectangle is 2.