In a test (+4) marks are given for every correct answer and (-2) marks are given for every incorrect answer. Rohit answered 15 questions correctly and obtained a total score of 36 marks. How many incorrect answers did he give?
Marks from correct answers = 15 × 4 = 60.
Total Score = Marks for Correct + Marks for Incorrect.
36 = 60 + Marks for Incorrect ⇒ Marks for Incorrect = 36 - 60 = -24.
Number of incorrect answers = (-24) ÷ (-2) = 12.
An elevator descends into a mine shaft at the rate of 6 m/min. If the descent starts from 10 m above the ground level, how long will it take to reach -350 m?
Total distance to descend = 10 - (-350) = 360 meters.
Time = Distance ÷ Speed = 360 / 6 = 60 minutes = 1 hour.
Evaluate the value of: [(-100) ÷ 5] ÷ [(-4) × (-1)]
First term = (-100) ÷ 5 = -20.
Second term = (-4) × (-1) = 4.
Result = (-20) ÷ 4 = -5.
Which of the following expressions is NOT distributive for integers?
Division is NOT distributive over addition when the divisor is a sum (e.g., 12 ÷ (2 + 4) = 2, but 12/2 + 12/4 = 6 + 3 = 9).
A freezing process requires room temperature to be lowered from 40°C at the rate of 5°C every hour. What will be the room temperature 11 hours after the process begins?
Initial Temperature = 40°C.
Change in temperature in 11 hours = 11 × (-5°C) = -55°C.
Final Temperature = 40°C + (-55°C) = -15°C.
If a * b = a × b - (a + b), then find the value of (-3) * (-5).
(-3) * (-5) = (-3) × (-5) - [(-3) + (-5)] = 15 - (-8) = 15 + 8 = 23.
The sum of two integers is -16. If one of them is 53, find the other integer.
Let the other integer be x.
x + 53 = -16 ⇒ x = -16 - 53 = -69.
What integer should be multiplied by (-1) to get -85?
Let the integer be x. x × (-1) = -85 ⇒ x = -85 / -1 = 85.
A car travels 16 km using 1 liter of petrol. How much distance will it cover using 2¾ liters of petrol?
Distance = 16 × 2¾ = 16 × (11/4) = 4 × 11 = 44 km.
Solve: 7.75 ÷ 0.25 + 1.25 × 4
7.75 ÷ 0.25 = 31.
1.25 × 4 = 5.
Sum = 31 + 5 = 36.
Ramesh solves 2/7 part of an exercise while Seema solves 4/5 of it. Who solved a lesser part?
Compare fractions by taking LCM of denominators (35):
Ramesh = 2/7 = 10/35.
Seema = 4/5 = 28/35.
Since 10/35 < 28/35, Ramesh solved a lesser part.
If 12.553 is divided by 0.001, the result is:
Dividing by 0.001 (1/1000) is equivalent to multiplying by 1000.
12.553 × 1000 = 12553.
Find the reciprocal of the fraction expression: (3/5 × 10/9) - 1/3
3/5 × 10/9 = 2/3.
Expression = 2/3 - 1/3 = 1/3.
Reciprocal of 1/3 = 3.
The product of two decimals is 42.63. If one of them is 2.1, find the other decimal.
Other decimal = 42.63 ÷ 2.1 = 426.3 ÷ 21 = 20.3.
A ribbon of length 15½ meters is cut into 4 equal pieces. What is the length of each piece?
Length = 15½ ÷ 4 = (31/2) × (1/4) = 31/8 meters.
The mean of 5 observations is 15. If one observation is excluded, their mean becomes 13. What is the excluded observation?
Sum of 5 observations = 5 × 15 = 75.
Sum of 4 observations = 4 × 13 = 52.
Excluded observation = 75 - 52 = 23.
Find the median of the following set of numbers: 13, 16, 12, 14, 19, 12, 14, 13, 14.
Arrange in ascending order: 12, 12, 13, 13, 14, 14, 14, 16, 19.
Number of terms n = 9 (odd). Middle term = (9+1)/2 = 5th term = 14.
The bar graph below shows the number of books sold by a bookstore over 4 consecutive months. What is the ratio of sales in Month 2 to Month 4?
Month 2 sales = 80, Month 4 sales = 40.
Ratio = 80 : 40 = 2 : 1.
A die is thrown once. What is the probability of getting a prime number?
Total outcomes = {1, 2, 3, 4, 5, 6} = 6.
Prime outcomes = {2, 3, 5} = 3.
Probability = 3/6 = 1/2.
What is the mode of the dataset: 2, 6, 5, 3, 0, 3, 4, 3, 2, 4, 5, 2, 4, 4?
Frequencies: '2' appears 3 times, '3' appears 3 times, '4' appears 4 times.
Since 4 has the highest frequency, Mode = 4.
Find the range of the following data: 32, 41, 28, 54, 35, 26, 23, 33, 38, 40.
Maximum value = 54, Minimum value = 23.
Range = Maximum - Minimum = 54 - 23 = 31.
Solve for x: 3(x - 2) + 2(x + 1) = 21
Expand terms: 3x - 6 + 2x + 2 = 21
5x - 4 = 21 ⇒ 5x = 25 ⇒ x = 5.
If 1/5 of a number minus 4 gives 3, then what is the number?
Let the number be n.
(1/5)n - 4 = 3 ⇒ (1/5)n = 7 ⇒ n = 35.
The perimeter of a rectangle is 40 cm. If its length is 4 cm more than its breadth, find the length.
Let breadth = b. Length = b + 4.
Perimeter = 2(l + b) = 2(b + 4 + b) = 40.
2(2b + 4) = 40 ⇒ 2b + 4 = 20 ⇒ 2b = 16 ⇒ b = 8 cm.
Length = 8 + 4 = 12 cm.
Solve for y: (y + 3) / 2 = (2y - 1) / 3
Cross multiply: 3(y + 3) = 2(2y - 1)
3y + 9 = 4y - 2 ⇒ 4y - 3y = 9 + 2 ⇒ y = 11.
Sum of three consecutive odd natural numbers is 87. What is the largest of these three numbers?
Let numbers be x, x+2, x+4.
x + x + 2 + x + 4 = 87 ⇒ 3x + 6 = 87 ⇒ 3x = 81 ⇒ x = 27.
Largest number = x + 4 = 31.
An angle is equal to 8 times its complement. Find the measure of the angle.
Let complement = x. The angle = 8x.
x + 8x = 90° ⇒ 9x = 90° ⇒ x = 10°.
Angle = 8 × 10° = 80°.
In the given diagram, line L1 is parallel to L2. Find the value of x.
The two marked angles are alternate interior angles.
2x + 10 = 110 ⇒ 2x = 100 ⇒ x = 50°.
Two angles forming a linear pair are in the ratio 4 : 5. Find the larger angle.
Linear pair sum = 180°.
4x + 5x = 180° ⇒ 9x = 180° ⇒ x = 20°.
Larger angle = 5 × 20° = 100°.
Find the supplement of an angle which is 2/3 of a right angle.
Angle = 2/3 × 90° = 60°.
Supplement = 180° - 60° = 120°.
A ladder 15 m long reaches a window 12 m high from the ground. Find the distance of the foot of the ladder from the wall.
By Pythagoras Theorem:
d² + 12² = 15² ⇒ d² + 144 = 225 ⇒ d² = 81 ⇒ d = 9 meters.
In a triangle, one angle is 80° and the other two angles are in the ratio 2 : 3. Find the smallest angle of the triangle.
Sum of remaining two angles = 180° - 80° = 100°.
2x + 3x = 100° ⇒ 5x = 100° ⇒ x = 20°.
Smallest angle = 2 × 20° = 40°.
Which of the following CANNOT be the sides of a valid triangle?
By Triangle Inequality Theorem, sum of any two sides must be strictly greater than the third side.
Here 2 + 3 = 5, which is LESS than 6. Thus no triangle can be formed.
In an isosceles triangle ABC with AB = AC, if ∠A = 70°, find ∠B.
Since AB = AC, base angles are equal: ∠B = ∠C.
∠A + ∠B + ∠C = 180° ⇒ 70° + 2∠B = 180° ⇒ 2∠B = 110° ⇒ ∠B = 55°.
A shopkeeper bought an item for $800 and sold it for $960. Find his profit percentage.
Profit = $960 - $800 = $160.
Profit % = (160 / 800) × 100 = 20%.
If $5000 amounts to $6200 in 3 years at Simple Interest, find the rate of interest per annum.
Simple Interest SI = 6200 - 5000 = $1200.
SI = (P × R × T) / 100 ⇒ 1200 = (5000 × R × 3) / 100
1200 = 150 R ⇒ R = 8% per annum.
In a school, 60% of students are boys. If the number of girls is 240, find the total number of students in the school.
Percentage of girls = 100% - 60% = 40%.
Let total students = T.
40% of T = 240 ⇒ (40/100) × T = 240 ⇒ T = 240 × (100/40) = 600.
Convert the ratio 3 : 8 into percentage.
3/8 × 100 = 300 / 8 = 37.5%.
Which of the following rational numbers lies exactly halfway between -1/3 and 1/2?
Midpoint = [(-1/3) + (1/2)] ÷ 2 = [(-2 + 3)/6] ÷ 2 = (1/6) ÷ 2 = 1/12.
Express -36/48 in its standard form.
Divide numerator and denominator by HCF = 12:
-36 ÷ 12 = -3, 48 ÷ 12 = 4 ⇒ Standard form = -3/4.
Simplify: (-7/12) × (-15/14)
(-7/14) = -1/2.
(-15/12) = -5/4.
(-1/2) × (-5/4) = 5/8.
Find the area of a circle whose circumference is 44 cm (Use π = 22/7).
2πr = 44 ⇒ 2 × (22/7) × r = 44 ⇒ r = 7 cm.
Area = πr² = (22/7) × 7 × 7 = 154 cm².
A path 2 m wide runs around the outside of a rectangular garden of length 10 m and breadth 8 m. Find the area of the path.
Inner Area = 10 × 8 = 80 m².
Outer length = 10 + 2 + 2 = 14 m.
Outer breadth = 8 + 2 + 2 = 12 m.
Outer Area = 14 × 12 = 168 m².
Area of path = 168 - 80 = 88 m².
The base of a triangle is 12 cm and its height is 5 cm. Find its area.
Area = ½ × base × height = ½ × 12 × 5 = 30 cm².
Subtract (3x² - 5x + 7) from (5x² - 2x + 1).
(5x² - 2x + 1) - (3x² - 5x + 7)
= 5x² - 2x + 1 - 3x² + 5x - 7
= (5x² - 3x²) + (-2x + 5x) + (1 - 7) = 2x² + 3x - 6.
If a = 2 and b = -3, find the numerical value of: a² + 2ab + b²
a² + 2ab + b² = (a + b)².
(2 + (-3))² = (-1)² = 1.
What should be added to x² + xy + y² to obtain 2x² + 3xy?
Required = (2x² + 3xy) - (x² + xy + y²)
= 2x² - x² + 3xy - xy - y² = x² + 2xy - y².
Simplify and express in exponential form: (2⁵)³ × 2⁻⁴ ÷ 2²
(2⁵)³ = 2¹⁵.
2¹⁵ × 2⁻⁴ = 2¹¹.
2¹¹ ÷ 2² = 2⁹.
Express the number 59860000000 in standard scientific form.
Move the decimal point 10 places to the left: 5.986 × 10¹⁰.
Find the value of x if 3⁺ˣ = 243.
243 = 3 × 3 × 3 × 3 × 3 = 3⁵.
Since bases are equal, x = 5.
What is the order of rotational symmetry for a regular hexagon?
A regular hexagon fits onto itself 6 times during a full 360° turn. Order of rotational symmetry = 6 (Angle of rotation = 60°).
Which of the following English capital letters has BOTH line symmetry and rotational symmetry?
Letter 'H' has 2 lines of symmetry (horizontal & vertical) and rotational symmetry of order 2.
Verify Euler's formula (F + V - E = 2) for a triangular prism. How many faces (F), vertices (V), and edges (E) does it have?
A triangular prism has 5 faces (2 triangular, 3 rectangular), 6 vertices, and 9 edges.
Euler's Check: 5 + 6 - 9 = 2.
Which 3D shape is formed by folding the net shown below consisting of 6 identical squares?
A standard cross-net of 6 squares folds into a 3D Cube.
Find the value of (2⁰ + 3⁰ + 4⁰) × 5⁰.
Any non-zero number to power 0 equals 1.
(1 + 1 + 1) × 1 = 3 × 1 = 3.
The length of a side of a square field is 25 m. Find the cost of turfing the field at the rate of $6 per m².
Area = 25 × 25 = 625 m².
Cost = 625 × $6 = $3750.
What is the degree of the algebraic expression: 4x³y² - 3x²y³ + 7x⁴ - 2?
Degrees of terms: 3+2=5, 2+3=5, 4, 0.
Highest degree among terms = 5.
Solve: 0.3x + 0.4 = 0.28x + 1.16
0.30x - 0.28x = 1.16 - 0.40
0.02x = 0.76 ⇒ x = 0.76 / 0.02 = 38.
Find the average of 4.2, 3.8, 5.6, and 6.4.
Sum = 4.2 + 3.8 + 5.6 + 6.4 = 20.0.
Average = 20.0 / 4 = 5.0.
Product of 11 negative integers and 3 positive integers is:
An odd number of negative integers (11) yields a negative product. Multiplying by positive integers keeps the overall product negative.