Mathematics and Science PYQs

Concept used:

General quadratic equation is ax² + bx + c 

Then, Sum of the roots = -b/a

Product of the roots = c/a

Formula used:

(a + b)² = a² + b² + 2ab

Calculation:

Let the roots of the equation be α and α + 1,

because as mentioned in the question we have two consecutive integers as the roots.

⇒ α + (α + 1) = b = sum of roots

⇒ 2α + 1 = b = sum of roots

⇒ Product of two roots = α × (α + 1) = c

⇒ Now b² - 4c = (2α + 1)² - 4 (α² + α)

⇒ 4α² + 1 + 4α - 4α² - 4α = 1

Given:

HCF and LCM of two numbers are 13 and 1989 respectively

One of the numbers is 117

Concept used:

First number × Second number = HCF × LCM

Calculation:

Let the second number be m

⇒ 117 × m = 13 × 1989

⇒ m = \(\frac{{13 \times 1989}}{{117}}\)

⇒ m = 221

Given:

Jatin buys an article for 10% less than its value

sells it for 10% more than its value

Formula used:

Profit% = [(S.P – C.P)/C.P] × 100

Calculation:

Let original value be Rs. 10

He bought it at = Rs. 10 × (90/100) = Rs. 90

He sells it at = Rs. 10 × (110/100) = Rs. 110

Gain = Rs. 110 – Rs. 90 = Rs. 20

Gain% = (20/90) × 100%

⇒ 200/9% = \(22\frac{2}{9}\)%

∴ The gain percentage is \(22\frac{2}{9}\)%

Basic quadratic equation is ax² + bx + c  where sum of roots =-b/a and product of roots = c/a

⇒ Here two roots be α and β which are 2x and 3x.

⇒ α × β = c/a  

⇒ α × β = 54

⇒ 6x² = 54

⇒ x = 3 so α = 6 and β = 9

⇒ Now sum of roots = α + β = -b/a 

⇒ p = 15

Given:  

The ratio of the heights of the two right circular cones and their diameters is given.

Formula used:

The volume of cone = \(\frac{1}{3}\pi {r^2}h\)

Calculation:

Here we have two cones with two different radius and height.

Let the radius and the height of the first cone be r₁ and h₁. Similarly the radius and the height of the second cone be r₂ and h₂.

Volume of the first cone be \(\frac{1}{3}\pi {r_1}^2{h_1}\) and the volume of the second cone be \(\frac{1}{3}\pi {r_2}^2{h_2}\)

⇒ r₁ = 2 and r₂ = 2.5, h₁ = 25 and h₂ = 64

⇒ placing the above value in the equation \(\frac{{\frac{1}{3}\pi {r_1}^2{h_1}}}{{\frac{1}{3}\pi {r_2}^2{h_2}}}\)

⇒ \(\frac{{\pi {r_1}^2{h_1}}}{{\pi {r_2}^2{h_2}}}\) = \(\frac{{4 \times 25}}{{6.25 \times 64}}\) = \(\frac{1}{4}\)

∴ Required ratio = 1: 4

Given: 

A = {BELOW}, B = {WOOL}

Concept used:

If a set has “n” elements, then the number of subset of the given set is 2ⁿ

Calculation:

⇒ C = A – B

⇒ C = {B, E} taking out the common letters.

The subsets of C = 2² = 4

∴ The number of subsets of C is 4

Given:

Diagonals BD = 24 and AC = 10.

Concept used: 

Rhombus has all four sides equal and their diagonals bisect each other at 90° 

Let both the diagonals meet at O,

OD = OB = 12 and AO = OC = 5

In Δ OCD use the Pythagoras theorem.

In Δ OCD, OD² + OC² = DC² ,

12² + 5² = DC²

⇒ DC = 13

The perimeter of the Rhombus is 4a where a is the sides of the rhombus.

∴ Perimeter = 4 × 13 = 52 cm.

Formula Used:

 average = (sum of all observation)/ (number of observation)

Solution:

⇒ sum of all observation = 10 × 4.5 + 30 × 3.5 = 150

⇒ number of all observation = 40

⇒ average = 150/40

∴ average = 3.75

so, average = 3.75

Given:

side of cube = 14 cm.

∵ cylinder is cut from cube 

∴ r = 14/2 = 7 cm, h = 14 cm

Formula used:

 If side of a cube is a

∴ total surface area = 6 × a²

If the radius of a cylinder is r and height is h -

∴ total surface area = 2 × π × r × (r + h)

Solution:

total surface area of cylinder = 2 × (22/7) × 7 × (7 + 14)

⇒ total surface area of cylinder = 924 cm²

Formula used:

average = (sum of all observation)/(number of observation)

Concept used:

If we add or subtract anything in the given data, it will happen in the average also.

Solution:

there are 50 number which is subtracted from 53 and their mean (average) is - 3.5

∴ mean of given numbers = 53 - (-3.5) = 56.5

Formula Used:

If the total number of observation given is odd, then the formula to calculate the median is:

Median = {(n+1)/2}thterm

If the total number of observation is even, then the median formula is:

Median  = [(n/2)th term + {(n/2)+1}th]/2

Solution:

5, 9, 14, 15, x + 1, 2x - 13, 28, 30, 32, 34

∵ number of observation is even

∴ median = {(x + 1) + (2x - 13)}/2

⇒ 24 = (3x - 12)/2

⇒ 3x = 60

⇒ x = 20

∴ The value of x is 20

Formula Used:

  c1 × d1 = c2 × d2 

where c = number of cow and d = number of days

Given:

c1 = 44, d1 = 9 days, d2 = 12 days, c2 = x

∴ put these values in given formula:

⇒ 44 × 9 = 12 × x

⇒ x = 33

⇒ number of cows eating in 12 days is 33.

⇒ 11 cows less than the previous one

Mathematics is a branch of science which deals with counting, calculating, and studying numbers, shapes, and structures. 

Key Points

Note that:

• The main branches of mathematics are algebra, number theory, geometry and arithmetic.
• Based on these branches,  other branches have been discovered.
• Before the advent of the modern age, the study of mathematics was very limited.
• But over a period of time, mathematics has been developed as a vast and diverse topic.
• Development in Math continues making large contributions to the field of technology.
• Hence, it is better termed as the "Queen of Science".
• So, if we solve a geometrical figure we used algebra for variables, arithmetic for arithmetical operation, numbers used for calculation, trigonometry used for angles etc.

Hence, we conclude that different branches of Mathematics, viz., algebra, geometry, etc., are dependent on each other.

Mathematics is the study of numbers, shape, quantity, and patterns. The nature of mathematics is logical and it relies on logic and connects learning with learners' day-to-day life.

Key Points

• Pedagogy refers to the set of principles that influence the approaches to the teaching-learning process. It consists of learning environment, teaching-learning arrangements, methods, general educational principles, etc.

Motivating the learners to develop their existing capabilities is the best pedagogical way to teach Mathematics as it will guide them to achieve a goal and will enable them to:

• develop the skill of systematic reasoning.
• develop mathematical and logical thinking.
• initiate mathematical talk in the classroom.
• guess-and-verify the solutions to problems.
• assimilate a sound understanding of mathematics.
• recognize and employ patterns of mathematical thought.

Hence, it is clear that motivating the learners to develop their existing capabilities is the best pedagogical way to teach Mathematics.

Solution:

Let the income of B be 100x.

∴ income of A = 125x

⇒ decrease in income of B = (125x) - 100x = 25x

⇒ percentage decrease in salary of B = {(25x)/(125x)} × 100 = 20%

∴ B's salary is 20% less than that of A.

Given Data:

Sum of two natural number is 8.

The sum of their reciprocals is 8/15.

Concept Used:

(a + b)² = (a – b)² + 4ab

Calculation:

Let the number be a and b.

a + b = 8  …(i)

1/a + 1/b = 8/15

⇒ (b + a)/ab = 8/15

⇒ 8/ab = 8/15

⇒ ab = 15

(a + b)² = (a – b)² + 4ab

⇒ 8² = (a – b)² + 4 × 15

⇒ 64 = (a – b)² + 60

⇒ (a – b)² = 4

⇒ a – b = 2    ….(ii)

Adding (i) and (ii), 2a = 10

⇒ a = 5 and b = 3

∴ The two natural numbers are 5 and 3.

Shortcut:

Directly we can get the value from option 4, as a + b = 8 and ab = 15.

Given Data:

The difference in expenditure between education and travel Rs. 4,000.

Calculation:

The family spends all the money what they earned. So, 100% of the money is utilised.

The difference in expenditure between education and travel Rs. 4,000.

⇒ 25% of total income – 15% of total income = 10% of total income = 4000

⇒ 1% of total income = 400

⇒ 100% of total income = 40,000

∴ Total income of the family = 40,000.

The correct answer is joule.

• The unit of heat in the SI system is Joule.
• A unit of energy equal to the work done when a force of one newton acts through a distance of one meter.

Important Points

• The other common units for heat are-
• BTU (Btu) - British Thermal Unit - also known as a "heat unit" in the United States.
• Calorie: The amount of heat required to raise the temperature of one gram of water 1^oC.
• Erg is a unit of energy.
• It equal to the force of one dyne over a distance of one centimeter.
• 1 erg = 10^-7 joule.
• Watt is a unit (SI) of power.
• It is equal to one joule of work performed per second.
• 1 horsepower = 746 watt

Key Points

• 1 joule = 0.14 calorie = 9.478 x 10^-4 Btu.
• 4.184 joule of heat energy (or one calorie) is required to raise the temperature of a unit weight (1 g) of water from 0^oC to 1oC, or from 32^oF to 33.8^oF.
• 1 dyne = 0.00001 newton.

The correct answer is ​ \(P = \dfrac{\vec{F}}{A}\)

• Pressure is defined as the force divided by the area perpendicular to the force over which the force is applied.
• The SI unit for pressure is the pascal, 1 Pa = 1N/m².

Key Points

• A force has both magnitude and direction, therefore: Force is a vector quantity
• Muscular force is being applied by a vendor when he is pushing the pushcart.
• Opening a door is an example of a Contact force.
• Magnet is an example of a Non-contact force.

Concept:

The correct answer is 3 units.

• Newton's 2^nd law of motion says the rate change of momentum is called force.
• The equation for the change in momentum is F ×  t = m ×  Δv
• If the momentum of a body increases from 10 units to 25 units in 5 seconds.
• Then the change in the momentum is 25 – 10 = 15 units in time 5 seconds.
• So the Force extract from this is 15/5 = 3 Newton.

Key Points

• Momentum is a vector quantity that is the product of the mass (weight) of a particle and its velocity (speed).
• SI unit: kilogram meter per second (kg⋅m/s).
• A spinning object has angular momentum.
• An object travelling with a velocity has linear momentum.
• A change in momentum may result from an acceleration or a force or an impulse.