RRB NTPC CBT 1 2021 Mathematics Questions with Solutions

Explanation

Let N be the total number of children in the class.

According to the first condition, each child receives N / 8 notebooks. Total notebooks = N × (N / 8) = N² / 8

According to the second condition, if the number of children were halved to N / 2, each child would receive 16 notebooks. Total notebooks = (N / 2) × 16 = 8N

Equating both expressions for the total number of notebooks: N² / 8 = 8N N² = 64N N = 64

Substituting N = 64 into the total notebooks formula: Total notebooks = 8 × 64 = 512

Explanation

Let n be the number of years after 2002.

Price of X after n years = 5.20 + 0.40n Price of Y after n years = 6.30 + 0.25n

According to the question, X must cost 0.40 more than Y: (5.20 + 0.40n) − (6.30 + 0.25n) = 0.40 0.15n − 1.10 = 0.40 0.15n = 1.50 n = 10

Adding 10 years to 2002 gives the year 2012.

Explanation

Given the inequality: 1/20 < n/160 < 1/16

Multiplying all parts of the inequality by 160: 160 × (1/20) < n < 160 × (1/16) 8 < n < 10

The only integer n strictly between 8 and 10 is 9.

Explanation

For equal annual installments under compound interest, the principal P and installment amount x are related by P = x / (1 + r/100) + x / (1 + r/100)².

Here, P = ₹2,460 and rate r = 5%, so 1 + r/100 = 21/20.

Substituting the given values into the formula: 2,460 = x / (21/20) + x / (21/20)² 2,460 = x × (20/21) + x × (400/441) 2,460 = x × (420 + 400) / 441 2,460 = x × (820 / 441)

Solving for x: x = (2,460 × 441) / 820 = 3 × 441 = ₹1,323.

Explanation

Rationalizing the denominator of each term:

First term = 3√2 / (√6 + √3) = 3√2(√6 - √3) / (6 - 3) = √2(√6 - √3) = 2√3 - √6

Second term = 4√3 / (√6 + √2) = 4√3(√6 - √2) / (6 - 2) = √3(√6 - √2) = 3√2 - √6

Third term = 6 / (√8 - √12) = 6 / (2√2 - 2√3) = 3 / (√2 - √3) = 3(√2 + √3) / (2 - 3) = -3√2 - 3√3

Substituting these values back into the expression: (2√3 - √6) - (3√2 - √6) - (-3√2 - 3√3) = 2√3 - √6 - 3√2 + √6 + 3√2 + 3√3 = 5√3

Explanation

Total work = 21 women × 40 days = 840 woman-days Total work = 30 men × 21 days = 630 man-days

Days taken by 1 man alone = 630 days Days taken by 1 woman alone = 840 days

Ratio of days taken by a man to a woman = 630 : 840 = 3 : 4

Explanation

Using trigonometry, the height of a point above the ground from distance d with angle of elevation θ is given by h = d × tan(θ).

Let the base of the pole be B and distance AB = 100 m.

Height of the total pole (top S) = BS = AB × tan(60°) = 100 × √3 = 100 × 1.732 = 173.2 m.

Height of the steel portion (top R) = BR = AB × tan(45°) = 100 × 1 = 100 m.

Height of the wooden portion RS = BS - BR = 173.2 - 100 = 73.2 m.

Explanation

To find the greatest possible length that can measure the given lengths exactly, calculate the Highest Common Factor (HCF) of all three lengths in centimeters:

4 m 20 cm = 420 cm 31 m 50 cm = 3150 cm 14 m 70 cm = 1470 cm

Factorizing the numbers: 420 = 210 × 2 3150 = 210 × 15 1470 = 210 × 7

Since 2, 15, and 7 have no common factors other than 1, HCF(420, 3150, 1470) = 210 cm.

Explanation

Let Total Bill = Fixed Cost + (Variable Cost per unit × Units consumed).

Variable cost per unit = Change in bill / Change in units = (2,250 - 1,620) / (690 - 480) = 630 / 210 = ₹3 per unit.

For 500 units, which is 20 units more than 480 units: Bill for 500 units = Bill for 480 units + (20 × ₹3) = 1,620 + 60 = ₹1,680.

Explanation

Since the principal sum and total simple interest remain constant, the product of the rate of interest and time is equal in both cases:

R₁ × T₁ = R₂ × T₂

Substituting the given values: 6 × 7 = R₂ × 5 42 = 5 × R₂ R₂ = 42 / 5 = 8.4%

Explanation

The arithmetic mean of a frequency distribution is given by the formula Mean = Σ(f × x) / Σf, where x represents the given observations and f represents their corresponding frequencies.

Calculating the sum of the products of observations and frequencies: Σ(f × x) = (2 × 5) + (4 × 3) + (8 × 2) + (7 × 6) + (5 × 4) = 10 + 12 + 16 + 42 + 20 = 100

Calculating total frequency: Σf = 5 + 3 + 2 + 6 + 4 = 20

Substituting these values into the formula gives Mean = 100 / 20 = 5.

Explanation

The sum S of an arithmetic series with first term a, last term l, and number of terms n is given by the formula S = (n / 2) × (a + l).

For the series 25 + 26 + ... + 75: Number of terms n = 75 - 25 + 1 = 51 First term a = 25 Last term l = 75

S = (51 / 2) × (25 + 75) = (51 / 2) × 100 = 51 × 50 = 2550

Explanation

Total pieces = Number of pieces per meter × Total length of cloth Total pieces = 6 × 33.5 = 201

Explanation

Simplify the nested square roots step by step, working from the innermost expression outward:

√16 = 4

√(12 + 4) = √16 = 4

√(5 + 4) = √9 = 3

√(6 + 3) = √9 = 3

Explanation

The diagonals of a rhombus intersect perpendicularly at point E, forming a right-angled triangle ΔBEC with m∠BEC = 90°.

In right-angled triangle ΔBEC, the sum of the two acute angles is 90°: m∠ECB + m∠EBC = 90° (5a + 4) + (4a + 5) = 90° 9a + 9 = 90° 9a = 81 a = 9

Substituting a = 9 into the expression for m∠EBC: m∠EBC = 4(9) + 5 = 41°

Explanation

The population after t years with an annual growth rate of r% is given by P = P₀ × (1 + r/100)ᵗ.

Given present population P₀ = 640000, rate r = 5%, and time t = 2 years:

Population after 2 years = 640000 × (1 + 5/100)² = 640000 × (21/20)² = 640000 × 441 / 400 = 1600 × 441 = 705600

Explanation

Given A:B:C = 2:4:6, let A = 2, B = 4, and C = 6.

Calculating each individual ratio: A/B = 2/4 = 1/2 B/C = 4/6 = 2/3 C/A = 6/2 = 3

Therefore, A/B : B/C : C/A = 1/2 : 2/3 : 3.

To express this in whole numbers, multiply each term by 6, the least common multiple of the denominators 2 and 3: (1/2 × 6) : (2/3 × 6) : (3 × 6) = 3 : 4 : 18

Explanation

Total number of triangles that can be formed by choosing 3 points out of 10: Total triangles = 10C3 = (10 × 9 × 8) / (3 × 2 × 1) = 120

The diameter endpoints A and B divide the remaining 8 points into 4 points on each side of AB. Thus, each closed semicircle (one side of AB including the points A and B) contains 6 points in total.

For a triangle to lie completely on one side of AB: Number of triangles on the first side = 6C3 = 20 Number of triangles on the second side = 6C3 = 20

Since the two semicircles share only 2 points (A and B), it is impossible for a 3-vertex triangle to belong to both sides, so there is no overlap.

Total favorable triangles = 20 + 20 = 40

Probability = 40 / 120 = 1/3

Explanation

Let the two observations be x and y.

Arithmetic Mean = (x + y) / 2 = 10, which gives x + y = 20. Geometric Mean = √(xy) = 5, which gives xy = 25.

Using the algebraic identity x² + y² = (x + y)² - 2xy:

x² + y² = 20² - 2(25) = 400 - 50 = 350.

Explanation

Correct Mean = Old Mean + (Correct Value − Incorrect Value) / Total Observations

Substituting the given values into the formula: Correct Mean = 22.5 + (25 − (−25)) / 20 Correct Mean = 22.5 + 50 / 20 Correct Mean = 22.5 + 2.5 = 25

Explanation

For a rate of r% compounded annually for 2 years, the net effective interest rate is given by r₁ + r₂ + (r₁ × r₂)/100.

For Rafique at 15% per annum: Net interest rate = 15 + 15 + (15 × 15)/100 = 32.25%

For Pritam at 10% per annum: Net interest rate = 10 + 10 + (10 × 10)/100 = 21%

The difference in interest rates = 32.25% - 21% = 11.25%

Let the initial investment be P: 11.25% of P = ₹2,250 P = (2250 × 100) / 11.25 = ₹20,000

Explanation

Let the three numbers be x, y, and z.

According to the given conditions: x = 2y ⇒ y = x/2 z = x/3

The sum of the three numbers is given as 253: x + y + z = 253 x + x/2 + x/3 = 253

Taking 6 as the common denominator: (6x + 3x + 2x) / 6 = 253 11x / 6 = 253 x = (253 × 6) / 11 x = 23 × 6 = 138

The second number y is: y = x / 2 = 138 / 2 = 69

Explanation

Let Jesse's efficiency be 5 units per day. Since John is 40% as efficient as Jesse, John's efficiency is 2 units per day.

Total work = John's efficiency × Time taken by John Total work = 2 × 35 = 70 units

Combined efficiency of John and Jesse = 2 + 5 = 7 units per day

Time taken by both working together = Total work ÷ Combined efficiency = 70 ÷ 7 = 10 days

Explanation

Let the two numbers be x and 2x.

The LCM of x and 2x is 2x. Given that the LCM is 24: 2x = 24 x = 12

Thus, the two numbers are 12 and 24.

Product of the two numbers = 12 × 24 = 288.

Explanation

Converting the percentage to a standard fraction: 6.25% = 6.25/100 = 1/16

Using the relation between expenditure and total amount: Total amount = Expenditure ÷ Fraction spent Total amount = 75 ÷ (1/16) = 75 × 16 = 1200

Explanation

Let the distance to the destination be d km. Using the formula Time = Distance ÷ Speed:

Time taken by the first man = d / 3 hours Time taken by the second man = d / 3.75 = 4d / 15 hours

The time difference between them is half an hour (½ hour):

(d / 3) - (4d / 15) = ½

Taking 15 as the common denominator: (5d - 4d) / 15 = ½ d / 15 = ½ d = 15 ÷ 2 = 7.5 km

Explanation

Using the algebraic identity a³ - b³ = (a - b)(a² + ab + b²), let a = 0.96 and b = 0.1.

Since a × b = 0.96 × 0.1 = 0.096, the expression simplifies to:

[(0.96)³ - (0.1)³] / [(0.96)² + 0.096 + (0.1)²] = 0.96 - 0.1 = 0.86

Explanation

Simple Interest (SI) = Amount - Principal = 15,000 - 12,000 = 3,000

Using the formula SI = (P × R × T) / 100:

3,000 = (12,000 × R × 5) / 100 3,000 = 600 × R R = 3,000 / 600 = 5%

Explanation

Using the sum-to-product identity: cos C - cos D = -2 sin((C + D)/2) sin((C - D)/2)

Applying this to cos 64° - cos 4°: cos 64° - cos 4° = -2 sin((64° + 4°)/2) sin((64° - 4°)/2) = -2 sin 34° sin 30°

Since sin 30° = ½: cos 64° - cos 4° = -2 sin 34° × ½ = -sin 34°

Substituting this into the given expression yields: sin 34° + cos 64° - cos 4° = sin 34° - sin 34° = 0

Explanation

Rationalizing each term of the expression separately:

First term: 3√2 / (√6 - √3) = 3√2(√6 + √3) / (6 - 3) = √2(√6 + √3) = 2√3 + √6

Second term: 4√3 / (√6 - √2) = 4√3(√6 + √2) / (6 - 2) = √3(√6 + √2) = 3√2 + √6

Third term: 6 / (√8 - √12) = 6 / (2√2 - 2√3) = 3 / (√2 - √3) = 3(√2 + √3) / (2 - 3) = -3√2 - 3√3

Combining the terms: (2√3 + √6) - (3√2 + √6) - (-3√2 - 3√3) = 2√3 + √6 - 3√2 - √6 + 3√2 + 3√3 = 2√3 + 3√3 = 5√3

Explanation

Length of rectangle (L) = 30 cm Width of rectangle (W) = ⅔ × 30 = 20 cm Perimeter of rectangle (P) = 2 × (L + W) = 2 × (30 + 20) = 100 cm

Since the square has the same perimeter as the rectangle: Perimeter of square = 100 cm Side of square (s) = 100 / 4 = 25 cm Area of square = s² = 25² = 625 cm²

Explanation

Using the trigonometric identity cos A + sin A = √2 cos (45° - A), substitute A = 28°:

cos 28° + sin 28° = √2 cos (45° - 28°) = √2 cos 17°

Given that cos 28° + sin 28° = k, we get:

k = √2 cos 17°

Solving for cos 17°:

cos 17° = k / √2

Explanation

The sum of squares of the first n natural numbers is given by the formula Sₙ = n(n + 1)(2n + 1) / 6.

To find 21² + 22² + 23² + ... + 30², calculate the difference between the sum of the first 30 squares and the sum of the first 20 squares:

Sum up to 30: S₃₀ = (30 × 31 × 61) / 6 = 5 × 31 × 61 = 9455

Sum up to 20: S₂₀ = (20 × 21 × 41) / 6 = 10 × 7 × 41 = 2870

Required sum = 9455 - 2870 = 6585

Explanation

Using the trigonometric identity, if (1 + tan A)(1 + tan B) = 2, then A + B = π/4 + nπ for any integer n.

Here, setting A = α and B = 4α gives: α + 4α = π/4 5α = π/4 α = π/20

Since π/20 lies in the given interval (0, π/16), α = π/20 is the solution.

Explanation

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

Here, Principal (P) = ₹73,000, Rate (R) = 13⅓% = 40/3%, and Time (T) = 10 months = 10/12 years = 5/6 years.

SI = (73,000 × (40/3) × (5/6)) / 100 SI = (73,000 × 200) / (100 × 18) SI = 73,000 / 9 SI ≈ ₹8,111.11

Explanation

Let f(x) = (x - a₁)² + (x - a₂)² + ... + (x - aₙ)².

To find the value of x that minimizes f(x), set the first derivative with respect to x to zero:

f'(x) = 2(x - a₁) + 2(x - a₂) + ... + 2(x - aₙ) = 0 2[nx - (a₁ + a₂ + ... + aₙ)] = 0 nx = a₁ + a₂ + ... + aₙ x = (a₁ + a₂ + ... + aₙ) / n

Since f''(x) = 2n > 0, this critical point corresponds to the global minimum, which is the arithmetic mean of the given numbers.

Explanation

Linear scale ratio is 1 km = 0.05 cm. Area scale ratio is the square of the linear scale: 1 km² = (0.05 cm)² = 0.0025 cm².

Area on map = Ground area × Area scale Area on map = 8000 km² × 0.0025 cm²/km² = 20 cm²

Explanation

Total initial marks of 45 students = 45 × 16.2 = 729

Total marks of remaining 44 students after one student leaves = 44 × 16.5 = 726 Marks of the student who left = 729 - 726 = 3

Total marks of 45 students after new student joins = 45 × 16.4 = 738 Marks of the student who joined = 738 - 726 = 12

Average marks of the two students = (3 + 12) ÷ 2 = 7.5

Explanation

The volume of water displaced in the cylindrical vessel equals the volume of the completely immersed solid sphere.

Volume of displaced water = Volume of sphere π × r² × h = (4/3) × π × R³

Here, radius of cylinder r = 4 cm, radius of sphere R = 3 cm, and h is the rise in water level.

16π × h = (4/3) × π × 3³ 16h = 36 h = 36/16 = 9/4 cm

Explanation

Using the algebraic identity a² + b² - 2ab = (a - b)², where a = 0.2 and b = 0.02:

Numerator = (0.2)² + (0.02)² - 2 × (0.2) × (0.02) = (0.2 - 0.02)² = (0.18)² = 0.0324

Dividing the numerator by the given denominator: 0.0324 ÷ 0.36 = 324 ÷ 3600 = 0.09

Explanation

Distance = (S₁ × S₂ / |S₁ - S₂|) × Δt

Given: S₁ = 40 km/h S₂ = 35 km/h Δt = 15 minutes = 15/60 hours = ¼ hour

Substituting the values into the formula: Distance = (40 × 35 / 5) × (1/4) Distance = (1400 / 5) × (1/4) Distance = 280 × (1/4) = 70 km

Explanation

Let the Marked Price (MP) be ₹100.

Cost Price (CP) = 64% of 100 = ₹64 Selling Price (SP) = MP - Discount = 100 - 12 = ₹88

Gain = SP - CP = 88 - 64 = ₹24

Percentage Gain = (Gain / CP) × 100 = (24 / 64) × 100 = 37.5%

Explanation

The third proportion c to two numbers a and b is given by the formula c = b² / a.

Substituting a = 12 and b = 30 into the formula: c = 30² / 12 = 900 / 12 = 75

Explanation

Let the numerator of the original fraction be x. Since the denominator is 3 more than the numerator, the denominator is x + 3, giving the original fraction x/(x + 3).

When both the numerator and the denominator are increased by 4, the fraction becomes (x + 4)/(x + 7).

Setting this equal to 4/5: (x + 4)/(x + 7) = 4/5

Cross-multiplying: 5(x + 4) = 4(x + 7) 5x + 20 = 4x + 28 x = 8

Substituting x = 8 into x/(x + 3) gives the original fraction as 8/11.

Explanation

Let the costs of the car and the bike be 3x and 2x respectively.

The difference in cost between the car and the bike is given as: 3x - 2x = ₹15,000 x = ₹15,000

Therefore, the cost of the bike is: 2x = 2 × 15,000 = ₹30,000

Explanation

Using the algebraic identity x² - y² = (x + y)(x - y), substitute the given sum x + y = 10 and difference of squares x² - y² = 60: 60 = 10 × (x - y) x - y = 6

Adding the two linear equations x + y = 10 and x - y = 6 yields 2x = 16, giving x = 8. Substituting x = 8 into x + y = 10 yields y = 2.

Explanation

Volume of hollow sphere = (4/3)π(R³ - r³) Volume of cone = (1/3)πr_c²h

For the hollow sphere, external radius R = 4 cm and internal radius r = 2 cm. For the cone, base radius r_c = 4 cm.

Equating the volumes: (4/3) × π × (4³ - 2³) = (1/3) × π × (4²) × h 4 × (64 - 8) = 16h 224 = 16h h = 14 cm

Explanation

Aggregate Percentage = (Total Marks Obtained / Total Maximum Marks) × 100

Marks obtained in the three examinations: First exam = 50% of 100 = 50 Second exam = 60% of 130 = 78 Third exam = 70% of 150 = 105

Total marks obtained = 50 + 78 + 105 = 233 Total maximum marks = 100 + 130 + 150 = 380

Aggregate percentage = (233 / 380) × 100 = 61.315% ≈ 61.32%

Explanation

The given statement translates to the quadratic equation x² - 5x = -6, which simplifies to x² - 5x + 6 = 0.

Factorizing this equation gives: (x - 2)(x - 3) = 0

The roots are x = 2 and x = 3. Between these two positive integers, 2 is the least positive integer.

Explanation

The least number that leaves a constant remainder r when divided by a set of numbers is given by the formula: Required Number = LCM(given numbers) + r

First, calculate the LCM of 6, 10, 15, and 24: 6 = 2 × 3 10 = 2 × 5 15 = 3 × 5 24 = 2³ × 3

LCM(6, 10, 15, 24) = 2³ × 3 × 5 = 120

Substitute the LCM and the remainder r = 5 into the formula: Required Number = 120 + 5 = 125