Mathematics and Science PYQs

Concept use:

To find the value of 3�−123�−123�−12 3x−12, we can use Euler's formula for polyhedra, which states that for any convex polyhedron, the number of vertices (��� V), edges (�� E), and faces (�� F) are related by the equation:

�−�+�=2�−�+�=2�−�+�=2 V− E + F = 2

Calculations:

Given that �=10�=10�=10 V = 10 and �=15�=15�=15 E = 15, we can substitute these values into Euler's formula to solve for ��� F:

10−15+�=210−15+�=210−15+�=2 10 − 15 + x = 2

�−5=2�−5=2�−5=2 x − 5 = 2

�=2+5�=2+5�=2+5 x = 2 + 5

�=7�=7�=7 x = 7

�=7Unknown node type: span�=7Unknown node type: span�=7Unknown node type: span 3�−123�−123�−12 3x−12, we substitute �=7�=7�=7 x = 7 into the expression:

3�−12=3(7)−123�−12=3(7)−123�−12=3(7)−12 3x − 12 = 3(7) − 12

=21−12=21−12=21−12 = 21 − 12

=9=9=9 = 9

So, Unknown node type: strongUnknown node type: strongUnknown node type: strongUnknown node type: strongUnknown node type: strongUnknown node type: strongUnknown node type: strongUnknown node type: strongUnknown node type: strongUnknown node type: strongUnknown node type: strongUnknown node type: strong 3x−12 equals Unknown node type: strongUnknown node type: strongUnknown node type: strong 9.

p3x + p2(x - y) - p(y + z) - z 

⇒ p3x + p2x - p2y - py - pz - z 

⇒ p²x(p + 1) -py(p + 1) -z(p + 1)

⇒ (p + 1)(p²x - py -z)   

Hence, Option 4 is correct.

Shortcut Trick

For divisibility by 72, the number must be divisible by 8 and 9.

Divisibility by 8: Last 3 digits 9y2 must be divisible by 8 ⇒ y = 1, 5, 9.

Divisibility by 9: Sum of digits (9+4+7+1+x+9+y+2) = 32 + x + y must be divisible by 9.

If y = 1, 32+x+1 = 33+x ⇒ x = 3. If y = 5, 32+x+5 = 37+x ⇒ x = 8. If y = 9, 32+x+9 = 41+x ⇒ x = 4.

Check options:

(1) x=3, y=1 (True), (2) x=8, y=5 (True), (3) x=4, y=9 (True). Option (4) x=9, y=5 is False.

∴ The correct answer is x = 9 and y = 5.

Alternate Method

Given: Number 9471x9y2 is divisible by 72.

Formula Used: Divisibility rules of 8 (last 3 digits) and 9 (sum of digits).

⇒ For divisibility by 8: 9y2 ÷ 8 must leave remainder 0.

⇒ If y = 1, 912 ÷ 8 = 114 (Correct)

⇒ If y = 5, 952 ÷ 8 = 119 (Correct)

⇒ If y = 9, 992 ÷ 8 = 124 (Correct)

⇒ For divisibility by 9: Sum = 9+4+7+1+x+9+y+2 = 32+x+y

⇒ Case 1: y=1, 32+x+1 = 33+x ⇒ x=3 (Option 1 is true)

⇒ Case 2: y=5, 32+x+5 = 37+x ⇒ x=8 (Option 2 is true)

⇒ Case 3: y=9, 32+x+9 = 41+x ⇒ x=4 (Option 3 is true)

⇒ Case 4: If y=5, x must be 8. Thus, x=9 and y=5 is not true.

∴ The correct answer is x = 9 and y = 5.

Additional Information

Divisibility by 72

A number is divisible by 72 if it is simultaneously divisible by its co-prime factors 8 and 9.

Divisibility by 8

A number is divisible by 8 if the last three digits of the number are divisible by 8.

Divisibility by 9

A number is divisible by 9 if the sum of all its digits is a multiple of 9.

Concept use:

Profit = Selling price - Cost price

Profit percentage = (Profit / Cost price) × 100

Explanation:

Cost price of the article = Rs. 300

The selling price after a discount of 28% on the marked price = Rs. 324

Let the marked price be x.

Selling price after 28% discount = 0.72x (since the discount is 28%, the remaining fraction is 0.72)

324 = 0.72x

x = 324/0.72

x = Rs. 450

Selling price after 18% discount = 0.82 × Rs. 450 (since the discount is 18%, the remaining fraction is 0.82)

Selling price after 18% discount = Rs. 369

Profit = Selling price - Cost price

Profit = Rs. 369 - Rs. 300

Profit = Rs. 69

Profit percentage = (Profit / Cost price) × 100

Profit percentage = (69 / 300) × 100

Profit percentage = 23%

Hence, if the shopkeeper sells the article by giving an 18% discount on its marked price, the profit percentage will be 23%.

Schema describes patterns of thinking and behavior that people use to interpret the world.

Key Points

• The concept of 'Schema' was labeled by Jean Piaget.
• A schema is a mental framework that helps individuals organize, process, and store information about their environment.
• These mental structures are essential for understanding the complexities of the world, as they allow us to interpret new experiences through the lens of pre-existing schemas.

The correct answer is Piaget.

Take Option 1:

 ∠P = 55° and QR = 18 cm

if ∠R = 35° , ∠P + ∠R + ∠Q = 180° 

∠Q = 180° - (55° + 35°) = 90° 

Which Contradicts the given statement that ΔPQR can be an obtuse scalene triangle

Option 1 is incorrect.

Take Option 2:

∠R = 25° , ∠P = 55° 

∠Q = 180° - (25° + 55°) = 100° 

But side opposite to the smallest Angle is Largest which means PQ should be greater than 18cm but PQ = 18cm 

Hence, Option 2 is incorrect.

Take Option 4:

∠R = 65° , ∠P = 55° 

∠Q = 180° - (65° + 55°) = 60° which Contradicts the statement that 

ΔPQR can be an obtuse scalene triangle and also PQ > 18cm Contradicts that side opposite to the Largest angle is smallest 

Hence, Option 4 is Incorrect.

Hence, The Correct option is 3.

Key Points

• The student has correctly applied the formula for finding the percentage decrease. However, they made an error in calculating it.

The correct calculation should have been:

• Percentage decrease
• =(28000 − 26500 / 28000) × 100%
• =(1500 / 28000) × 100%
• =5.36%

So, while the student understands the concept of finding a percentage decrease, they made a calculation mistake, possibly by dividing the wrong way, resulting in an incorrect answer.

The correct answer is Students can understand the concept of percentage but are not able to understand the percentage of which quantity is to be found.

Concept use:

Area of the rectangle = length × breadth

Area = 1/2 × (sum of parallel sides) × height

Explanation:

Longer parallel side = 25 cm

Shorter parallel side = 11 cm

Distance between parallel sides (height) = 12 cm

Area = 1/2 × (sum of parallel sides) × height

Area = 1/2 × (25 + 11) × 12

Area of the trapezium = 216 cm²

Now, the area of the rectangle should be equal to this.

The sides of the rectangle are in the ratio 3:2, so if we take the longer side as 3x, the shorter side will be 2x.

Area of the rectangle = length × breadth

216 = 3x × 2x

216 = 6x²

x² = 36

x = 6

Therefore, the longer side of the rectangle = 3x = 18 cm, and the shorter side = 2x = 12 cm.

Perimeter = 2 × (length + breadth)

Perimeter = 2 × (18 + 12)

Perimeter = 60 cm

Hence, The Correct Answer is option (4) 60cm

Concept use:

To solve this problem, we can use the concept of inverse variation, which states that when two variables x and y vary inversely, their product remains constant. Mathematically, this can be represented as:

xy = k

Calculations:

x = 3.5 and y = 2.4 ∴ xy = 8.4 ⇒ 5.6 × y = 8.4 ⇒ y = 8.4/5.6 ⇒ y = 1.5

Hence, The Correct Answer y = 1.5

Calculation:

Let the Two numbers be a and b 

a × b = -40/3 

Let a = -5/2 

-5/2 × b = -40/3 

b = -40/3 × 2/-5 = 16/3 = 5.3333

now, 1/b = 1/5.333 = 0.18761726078

9/50 = 0.18 and 1/5 = 0.20

Hence, The reciprocal of the numbers lies between 9/50 and 1/5 

Hence, The Correct Answer is option 3.

Key Points

• Joyful Learning: Incorporating sports into math classes makes learning enjoyable.
• Inclusive Approach: Besides popular sports, indigenous sports can also teach math effectively.
• Incorrect: Using sports for math teaching is not only possible but beneficial.
• Misconception: Integrating sports doesn't necessarily make teaching time-consuming.

The correct options are (a) and (d), as they accurately reflect the benefits and possibilities of using sports-integrated pedagogy in mathematics classrooms as recommended by the National Education Policy (NEP) 2020.

Shortcut Trick

In a parallelogram, opposite sides are parallel, so alternate interior angles formed by the diagonal are equal.

⇒ ∠CBD = ∠ADB ⇒ 12x = 60° ⇒ x = 5°

⇒ ∠ABD = ∠CDB ⇒ 7y = 28° ⇒ y = 4°

⇒ Value of 2x + 3y = 2(5) + 3(4) = 10 + 12 = 22°

∴ The correct answer is 22°.

Alternate Method

Given:

BD is the diagonal of parallelogram ABCD.

∠CBD = 12x, ∠ABD = 7y, ∠ADB = 60° and ∠CDB = 28°

Concept Used:

Alternate interior angles of a parallelogram are equal.

Calculation:

∠CBD = ∠ADB

⇒ 12x = 60°

⇒ x = 5°

∠ABD = ∠CDB

⇒ 7y = 28°

⇒ y = 4°

Now,

2x + 3y = 2 × 5 + 3 × 4

⇒ 10 + 12 = 22°

∴ The correct answer is 22°.

Additional Information

Properties of a Parallelogram

Opposite sides are equal and parallel. Opposite angles are equal. Consecutive angles are supplementary (sum to 180°).

Alternate Interior Angles

When a transversal intersects two parallel lines, the pairs of angles on opposite sides of the transversal and between the parallel lines are equal.

Diagonal Property

The diagonal of a parallelogram divides it into two congruent triangles, and the diagonals bisect each other.

Assessments of mathematics learning encompass various dimensions, including mathematical reasoning, communication, patterns and procedures, and disposition toward mathematics.

Key Points

Dimension of assessment of mathematics learning:

• Mathematical reasoning: This is a crucial dimension, assessing a student's ability to analyze, justify, and draw conclusions using mathematical concepts.
• Communication: This dimension evaluates a student's ability to explain mathematical ideas clearly, using both written and verbal language.
• Disposition towards mathematics: This dimension assesses a student's attitude, interest, and confidence in learning mathematics.

Hint

• While "patterns" are often linked to mathematical concepts, "procedures" are essentially the steps or methods used to solve problems. These procedures are already covered under "mathematical reasoning" which encompasses applying concepts to solve problems.

Therefore, while patterns are important, having a separate dimension for "patterns and procedures" is redundant. The other options represent distinct aspects of mathematical learning that need to be assessed.

Calculations:

To find out what should be added to ������

    x to get the sum as 1.11.11.11.11.11.1

    1.1, we first need to calculate the value of ������

    x:

�=1.011+10.11−12.101+0.1011�=1.011+10.11−12.101+0.1011�=1.011+10.11−12.101+0.1011�=1.011+10.11−12.101+0.1011�=1.011+10.11−12.101+0.1011�=1.011+10.11−12.101+0.1011

    x = 1.011 + 10.11 − 12.101 + 0.1011

�=1.011+10.11−12.101+0.1011�=1.011+10.11−12.101+0.1011�=1.011+10.11−12.101+0.1011�=1.011+10.11−12.101+0.1011�=1.011+10.11−12.101+0.1011�=1.011+10.11−12.101+0.1011

    x = 1.011 + 10.11 − 12.101 + 0.1011

�=(1.011+0.1011)+10.11−12.101�=(1.011+0.1011)+10.11−12.101�=(1.011+0.1011)+10.11−12.101�=(1.011+0.1011)+10.11−12.101�=(1.011+0.1011)+10.11−12.101�=(1.011+0.1011)+10.11−12.101

    x = (1.011 + 0.1011) + 10.11 − 12.101

�=1.1121+10.11−12.101�=1.1121+10.11−12.101�=1.1121+10.11−12.101�=1.1121+10.11−12.101�=1.1121+10.11−12.101�=1.1121+10.11−12.101

    x = 1.1121 + 10.11 − 12.101

�=11.2221−12.101�=11.2221−12.101�=11.2221−12.101�=11.2221−12.101�=11.2221−12.101�=11.2221−12.101

    x = 11.2221 − 12.101

�=−0.8789�=−0.8789�=−0.8789�=−0.8789�=−0.8789�=−0.8789

    x = −0.8789

Now, to find what should be added to ������

    x to get the sum as 1.11.11.11.11.11.1

    1.1, we subtract ������

    x from 1.11.11.11.11.11.1

    1.1:

Addition needed=1.1−(−0.8789)Addition needed=1.1−(−0.8789)Addition needed=1.1−(−0.8789)Addition needed=1.1−(−0.8789)Addition needed=1.1−(−0.8789)Addition needed=1.1−(−0.8789)

    Addition needed = 1.1 − (−0.8789)

Addition needed=1.1+0.8789Addition needed=1.1+0.8789Addition needed=1.1+0.8789Addition needed=1.1+0.8789Addition needed=1.1+0.8789Addition needed=1.1+0.8789

    Addition needed = 1.1 + 0.8789

Addition needed=1.9789Addition needed=1.9789Addition needed=1.9789Addition needed=1.9789Addition needed=1.9789Addition needed=1.9789

    Addition needed = 1.9789

So, 1.97891.97891.97891.97891.97891.9789

    1.9789 should be added to ������

    x to get the sum as 1.11.11.11.11.11.1

    1.1.

Histograms are graphical representations that display the distribution of continuous data by dividing it into intervals or bins.

Key Points

• By presenting scenarios involving continuous data, such as the heights of participants in an athletics meet and the number of passengers boarding trains over time.
• The teacher is likely aiming to introduce students to histograms as a suitable tool for displaying frequency distributions.
• Through this exercise, students can grasp the idea that histograms are effective for visualizing how data is distributed within specific ranges or intervals, thus enhancing their understanding of this particular type of graph.\

Hence, the correct answer is Teach the use of histograms for data representation.

Key Points

• Ram's ability to accurately speak the definitions of different types of numbers (odd, even, prime, composite) indicates good memorization of information.
• However, the inability to identify numbers accurately when given a set suggests a lack of conceptual understanding. It implies that Ram may have memorized definitions without fully grasping the underlying concepts related to number identification.

The correct answer is Ram has a good memory but lacks conceptual understanding.

Concept use:

Circumference of Circle = 2 × π × r where, r is the radius.

Curved Surface Area of Cylinder = 2πrh

Volume of Cylinder = πr2h where r is the radius and h is the height of Cylinder.

Explanation:

Curved surface area = 4400 cm²

Circumference of the base = 110 cm

π = 22/7

Circumference = 2πr

110 = 2 × (22/7) × r

r = 110 × 7 / (2 × 22) = 17.5 cm

Now, 

Curved surface area = 2πrh

4400 = 2 × (22/7) × 17.5 × h

h = 4400 × 7 / (2 × 22 × 17.5) = 40 cm

Volume = πr²h

Volume = (22/7) × (17.5)² × 40

Volume = 38500 cm³

Volume = 0.0385 m³

Therefore, the volume of the right circular cylinder is 0.0385 m^3.

Hence, The Correct Answer is Option 3) 0.0385

Concept use:

Supplementary angles are those angles that sum up to 180 degrees

Calculations:

Let the angles are ∠A and ∠B 

∠A - ∠B = 20° 

∠A + ∠B = 180° 

Adding both equation we get 2∠A = 200° 

∠A = 100° 

∴ ∠B = 80° 

Now, According to the Question statement p is the smallest angle 

∴ ∠B = 80° 

∴ 3p - 50° = 3 × 80° - 50° = 240° - 50° = 190° 

Hence, The Correct Answer is option 1.

Explanation:

\(\rm \frac{x}{y}=\left(\frac{-1}{3}\right)^{-3}\div \left(\frac{2}{3}\right)^{-4} \)

x/y = (-3)³ / (3/2)⁴ = (-1)(3)³ × 2⁴ / (3) ⁴ = -2⁴/3 = -16/3 

Now \(\rm \left(\frac{x}{y}+\frac{y}{x}\right)^{-1}\) = (-16/3 - 3/16)^-1 = (-(256 + 9)/48)^-1 = (-265/48)^-1 = -48/265

Hence, Option 1 is Correct.

Explanation:

A + B + C - D =  -2x2 + 12x + 11 - 8x + 3x2 + 17 - 4x2 -  x2 + x + 3 

⇒ -2x2 + 3x2 - 4x2 -  x2 + 12x - 8x + x + 11 + 17 + 3 

⇒ -4x² + 5x + 31 = -4x² + 5x + 25

Hence, The Coefficient of x² = -4 Coefficient of x = 5

Hence, The Sum of Coefficient of x² & x = -4 + 5 = 1

Hence, The Correct Answer is option 4.