Mathematics and Science PYQs

Formula used:

(a + b)² = a² + b² + 2ab

Calculation : 

\(\rm x = \sqrt{(2019)^2+2019+2020}\)

x = \(\sqrt{(2019)^2+2019+1 +2019}\)

⇒ x = \(\sqrt{(2019)^2+(1)^2 +2\times 2019\times1}\)

⇒ x = \(\sqrt{(2019+1)^2}\)

⇒ x = \(\sqrt{(2020)^2}\) = 2020

Now,

(2x - 1)

⇒ 2 × 2020 - 1

⇒ 4040 - 1= 4039

∴ The correct answer is 4039.

Calculation : 

4/9 = 0.444

5/6 = 0.833

7/12 = 0.583

11/24 = 0.458

8/11 = 0.727

Smallest fractions = 4/9

Largest fractions = 5/6

Sum = 4/9 + 5/6 = [(8 + 15)/18] = 23/18

∴ The correct answer is 23/18.

Calculation : 

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

⇒ (3/2)³/ (1/2)² × (2/3)²

⇒ 27/8 × 4 × 4/9

⇒ 3 × 2 = 6

Reciprocal of x = 1/6

∴ The correct answer is 1/6.

Calculation :

⇒ 3.101 - 2.11 - k = 2.65 - 0.256 

⇒ 0.991 - k = 2.394

⇒ k = -1.403

Now, (1 - k) = 1 - (- 1.403) = 2.403

∴ The correct answer is 2.403.

Calculation :

\(\frac{1.2× 10^3}{2.4× 10^{-4}}\)

⇒ 1/2 × 10³/10^-4

⇒ 0.5 × 10⁷

⇒ 5 × 10⁶

∴ The correct answer is 5 ×106.

Calculation :

\(\rm \frac{(x^2+y^2-z^2)^2-(x^2-y^2-z^2)^2}{(x+z)(x-z)y^2}\)

⇒ x⁴ + y⁴ + z⁴ + 2x²y² - 2y²z² - 2x²z² - [x⁴ + y⁴ + z⁴ - 2x²y² + 2y²z² - 2x²z²]

On canceling out the similar terms with opposite signs we get,

⇒ {4x²y² - 4y²z²}/(x² - z²)y²

⇒  {4x2y2 - 4y2z2}/(x²y² - z²y²)

⇒ 4

∴ The correct answer is 4.

Calculation : 

⇒ (x - 2)(2x2 + x - 15) - (x + 2)(x2 + 11x + 10)

⇒ [2x³ + x² - 15x - 4x² - 2x + 30] - [x³ + 11x² + 10x + 2x² + 22x + 20]

⇒ [2x³ - 3x² -17x + 30] - [x³ + 13x² + 32x + 20]

⇒ x³ - 16x² - 49x + 10

sum of coefficients of x2 and x = -16 - 49 = - 65

∴ The correct answer is - 65.

Calculation : 

\(\rm \frac{y+6}{4}+\rm \frac{y-3}{5}=\rm \frac{5y-4}{8}\)

⇒ 5(y + 6) + 4(y - 3)/20 = 5y - 4/8

⇒ 5y + 30 + 4y -12/20 = 5y - 4/8

⇒ 9y + 18/20 =5y - 4/8

⇒ 72y + 144 = 100y - 80

⇒ 28y = 224

⇒ y = 8

Now by putting value of y in \(\rm \frac{y-5}{2}=\frac{1}{2}+\frac{y-3}{5}\)

⇒ 8 - 5/2 = 1/2 + 8 - 3/5

⇒ 3/2 = 1/2 + 5/5

⇒ 1 = 1

∴ The correct answer is option 3.

Given : 

SP = Rs.980

Loss =  \(6\frac{2}{3}%\)

Gain= 8%

Calculation : 

By applying ratio method,

Loss =   \(6\frac{2}{3}%\) = 20/3% = 2/30

Gain = 8% = 8/100 = 2/25 

Now, 

SP = 28 = 980

⇒ 1 = 35

CP = 30 × 35 = 1050

Now,

CP = 25 = 1050 

⇒ 1 = 42

SP = 27 × 42 = 1134

∴ The correct answer is 1134.

Concept use:

(a + b)² = a² + b² + 2ab

Calculations:

3x + 1/2x = 5 

squaring both sides 

(3x + 1/2x)² = 5²

9x² + 1/4x² + 3 = 25

9x² + 1/4x² = 22

Divide by 3 both sides 

3x² + 1/12x² = 22/3 = \(7\frac{1}{3}\)

Hence, The Correct Option is 1.

Shortcut Trick

In any triangle, the sum of two exterior angles is equal to the interior angle at the third vertex plus 180°.

∠P + 180° = Ext ∠Q + Ext ∠R

∠P + 180° = 104° + 136° = 240°

∠P = 240° − 180° = 60°

Complement of ∠P = 90° − 60° = 30°

∴ The correct answer is 30°.

Alternate Method

Given:

Exterior angle at R, ∠PRT = 136°

Exterior angle at Q, ∠RQX = 104°

Formula Used:

1. Linear Pair: Sum of angles on a straight line is 180°.
2. Angle Sum Property: Sum of interior angles of a triangle is 180°.
3. Complementary Angles: Two angles are complementary if their sum is 90°.

P
Q
R
X
T
136°
104°
P=?

Shortcut Trick

From the first condition: 90° − x = 180° − 4x → 3x = 90° → x = 30°

From the second condition: 180° − y = 4 × (90° − y) → 3y = 180° → y = 60°

Required sum = x + y = 30° + 60° = 90°

∴ The correct answer is 90°.

Alternate Method

Given:

Complement of an angle x = Supplement of angle 4x

Supplement of an angle y = 4 × Complement of y

Formula Used:

Complement of angle θ = 90° − θ

Supplement of angle θ = 180° − θ

y = 60°
x = 30°
Result: x + y = 90°

Calculations:

For finding the angle x:

⇒ 90° − x = 180° − 4x

⇒ 4x − x = 180° − 90°

⇒ 3x = 90°

⇒ x = 30°

For finding the angle y:

⇒ 180° − y = 4 × (90° − y)

⇒ 180° − y = 360° − 4y

⇒ 4y − y = 360° − 180°

⇒ 3y = 180°

⇒ y = 60°

Now, calculate the value of (x + y):

⇒ x + y = 30° + 60° = 90°

∴ The correct answer is 90°.

Additional Information

Complementary Angles

Two angles are complementary if their sum is exactly 90°. If one angle is θ, its complement is (90° − θ).

Supplementary Angles

Two angles are supplementary if their sum is exactly 180°. If one angle is θ, its supplement is (180° − θ).

Straight Angle & Complete Angle

A straight angle measures exactly 180° and forms a straight line, while a complete angle measures exactly 360° around a single central point.

Given : 

∠C = 72° and ∠D = 80°

Bisectors of ∠A and ∠B meet at a point P inside the quadrilateral.

Calculation : 

We know,

∠A  + ∠B + ∠C  + ∠D = 360°

⇒ ∠A  + ∠B + 72 + 80 = 360

⇒ ∠A  + ∠B + 152 = 360

⇒ ∠A  + ∠B = 360 - 152

⇒ ∠A  + ∠B = 208°

Now,

∠APB = 180 - ∠A/2 - ∠B/2

⇒ ∠APB = 180 - 1/2(∠A  + ∠B)

⇒ ∠APB = 180 - 1/2 × 208

⇒ ∠APB = 180 - 104

⇒ ∠APB = 76°

∴ The correct answer is 76°.

Calculation : 

Number of faces = x = 6

Number of edges = y = 10

Number of vertices = z = 6

Now, \(\left(\frac{x+y-z}{2}\right)^2+1\) = (16 - 6/2) + 1 

⇒ (5)² + 1 = 26

∴ The correct answer is 26.

Concept use:

A quadrilateral can be constructed given specific lengths and angles, but the set of conditions must meet certain rules to make it feasible.

If given the lengths of all four sides and one diagonal, a quadrilateral can be constructed if and only if the length of the diagonal is less than the sum of the lengths of the pair of non-adjacent sides.

If given the lengths of two sides and all three angles that are around them, a quadrilateral can be constructed if the sum of angles is less than 360°.

Calculations:

option 1.

The sums of each pair of non-adjacent sides are

AB + CD = 8.4 cm; BC + AD = 8.5 cm; and the given AC = 6 cm.

The given side lengths satisfy the triangle inequality (the sum of any two sides must be greater than the third side),

So we can construct a quadrilateral with these measurements.

Option 2.

The given side lengths also satisfy the triangle inequality. However, the given angles (∠A = 120° and ∠B = 80°) do not add up to 360°, which is necessary for a quadrilateral.

So we can construct a quadrilateral with these measurements.

Option 3.

Again, the side lengths satisfy the triangle inequality. However, the given angles (∠A = 60°, ∠B = 120°, and ∠C = 80°) do not add up to 360°.

So we can construct a quadrilateral with these measurements.

Option 4.

the sum of the three given angles A, B, and C is 360°, which is equal to 360°, not less.

Therefore, a quadrilateral cannot be constructed in this case as it violates the rule about the sum of the three angles being less than 360°.

So, the construction of a quadrilateral is not possible here.

Hence, Option 4 is Correct.

Shortcut Trick

Convert units to cm: Area = 16000 cm², Height = 100 cm, Difference = 80 cm.

Sum of parallel sides = (2 × Area) ÷ Height = (2 × 16000) ÷ 100 = 320 cm.

Longer side = (Sum + Difference) ÷ 2 = (320 + 80) ÷ 2 = 200 cm.

∴ The correct answer is 200 cm.

Alternate Method

Given:

Area of trapezium = 1.6 m² = 1.6 × 10000 = 16000 cm²

Height (distance between sides) = 10 dm = 10 × 10 = 100 cm

Difference between parallel sides = 8 dm = 80 cm

Formula Used:

Area of Trapezium = 1/2 × (Sum of parallel sides) × Height

Calculations:

Let the shorter side be x cm. Then the longer side is (x + 80) cm.

⇒ 16000 = 1/2 × (x + x + 80) × 100

⇒ 16000 = 50 × (2x + 80)

⇒ 320 = 2x + 80

⇒ 2x = 240

⇒ x = 120 cm

⇒ Longer side = x + 80 = 120 + 80 = 200 cm

∴ The correct answer is 200 cm.

Additional Information

Area of a Trapezium

The area is calculated as the product of the average of the parallel sides and the vertical distance between them: Area = (a + b)/2 × h.

Unit Conversions

Always ensure uniform units: 1 m = 10 dm = 100 cm. For area, 1 m² = 100 dm² = 10,000 cm².

Median of a Trapezium

The median (or midsegment) connects the midpoints of non-parallel sides and its length is equal to the average of the parallel sides: M = (a + b)/2.

Shortcut Trick

Given height (h) = 15 cm.

Condition: 4 × (2πrh) = 6 × (2πr²)

8πrh = 12πr² ⇒ 2h = 3r ⇒ r = 2h/3

r = (2 × 15) / 3 = 10 cm.

Volume = πr²h = π × 10² × 15 = 1500π cm³.

∴ The correct answer is 1500π cm³.

Alternate Method

Given: Height (h) = 15 cm. Let radius be 'r'.

Formula Used: Curved Surface Area (CSA) = 2πrh; Area of Bases = 2 × πr²; Volume = πr²h.

⇒ 4 × (2πrh) = 6 × (2πr²)

⇒ 8πrh = 12πr²

⇒ 8 × 15 = 12 × r

⇒ 120 = 12r

⇒ r = 10 cm

⇒ Volume = π × (10)² × 15

⇒ Volume = π × 100 × 15 = 1500π cm³

∴ The correct answer is 1500π cm³.

Additional Information

Total Surface Area of Cylinder

The total surface area includes the curved surface and the two circular bases: TSA = 2πrh + 2πr² = 2πr(h + r).

Volume of a Hollow Cylinder

If R is the outer radius and r is the inner radius, Volume = π(R² − r²)h.

Relationship between Dimensions

When the volume is constant, the height is inversely proportional to the square of the radius: h ∝ 1/r².

Given :

Length of rectangular park = 5 m

Depth of rectangular park = 1.5 m

Capacity of tank  = 30000 liters.

Formula used :

Voulme of cuboid = length × breadth × height

Calculation :

According to question,

Capacity = 30000 Liters = 30 m³

⇒ 30 = 5 × 1.5 × b

⇒ b = 60/15 = 4

∴ The correct answer is 4 m.

Shortcut Trick

Arrange the data in ascending order: 4, 7, 9, 10, 11, 18, 21, 23, 23, 29, 33.

Mode = Most frequent term = 23.

Median = Middle (6^th) term = 18.

Range = Maximum − Minimum = 33 − 4 = 29.

Required Mean = (23 + 18 + 29) ÷ 3 = 70/3.

∴ The correct answer is 70/3.

Alternate Method

Given:

Data set: 9, 11, 4, 7, 10, 23, 21, 29, 23, 33, 18

Formula Used:

Mean = Sum of observations ÷ Total number of observations

Median (for odd n) = ((n + 1) ÷ 2)^th observation

Range = Maximum value − Minimum value

Mode = Observation with the highest frequency

4

7

9

10

11

18

21

23

23

29

33

Mode

Median

Range = 33 − 4 = 29
Required Mean = (23 + 18 + 29) ÷ 3 = 70/3

Calculations:

⇒ Sorted data: 4, 7, 9, 10, 11, 18, 21, 23, 23, 29, 33

⇒ Number of terms (n) = 11

⇒ Median = ((11 + 1) ÷ 2)^th term = 6^th term = 18

⇒ The number 23 appears maximum times (2 times), so Mode = 23

⇒ Minimum value = 4 and Maximum value = 33

⇒ Range = 33 − 4 = 29

⇒ Required Mean = (Mode + Median + Range) ÷ 3

⇒ Required Mean = (23 + 18 + 29) ÷ 3 = 70 ÷ 3 = 70/3

∴ The correct answer is 70/3.

Additional Information

Mean

Mean (Average) is the sum of all observations divided by the total number of observations. Formula: Mean = Sum ÷ n.

Median

For a sorted data set, if the number of terms (n) is odd, Median = ((n+1) ÷ 2)^th term. If n is even, Median is the average of (n ÷ 2)^th and (n ÷ 2 + 1)^th terms.

Mode

Mode is the value that appears most frequently in a data set. A set may have one mode, more than one mode, or no mode at all.

Range

Range is the difference between the highest and lowest values in the data set. It provides a quick measure of data spread.

Concept use:

Probability of an Event = Outcomes in Favour/ Total No of Outcomes

P(A ∪ B) = P(A) + P(B) - P(A∩B)

Calculations:

Number Divisible by 3 = 36, 39, 42, 45, 48, 51, 54, 57, 60, 63 

then P(A) = 10/30 where P(A) is the Probability of divisible by 3 

Number Divisible by 5 = 35, 40, 45, 50, 55, 60

then P(B) = 6/30 where P(B) is the Probability of divisible by 5

Number Divisible by both = 45, 60 

then P(A∩B) = 2/30 where P(A∩B) is the Probability of divisible by 5 and 3 both

P(A ∪ B) = P(A) + P(B) - P(A∩B) = 10/30 + 6/30  - 2/30 = 14/30 = 7/15