Mathematics and Science PYQs

Calculation:

According to option 1, every non-zero rational number has a reciprocal,  If the rational number is 0, it does not have a reciprocal since dividing by zero is undefined. (True)

According to option 2, There are only two non-zero rational numbers that are equal to their reciprocals. These two numbers are 1 and -1. (True)

According to option 3, The multiplicative inverse of \(\frac{8}{9}\) is 9/8 so option 3 is (False)

According to option 4, 3/8 = 0.375 and 1/4 = 0.25 and 1/2 = 0.5, so 3/8 lies between the rational numbers \(\frac{1}{4}\) and \(\frac{1}{2}\) (True)

∴ The correct answer is option 3

Calculation:

According to option 1, 

\(\frac{2}{5}\times \left(\frac{-3}{7}\right)-\frac{1}{14}-\frac{3}{7}\times \frac{3}{5} \)

⇒ - 6/35 - 1/14 - 9/35

⇒ -15/35 - 1/14

⇒ (- 30 - 5)/70

⇒ - 35/70

⇒ - 1/2 

So, option 1 is True

According to option 2,

• 11/8 = - 1.375, - 8/5 = - 1.6, and - 9/5 = - 1.8 all lying between rational numbers -2 and -1. so option 2 is True

According to option 3,

• 3/5 = - 0.6 and - 17/30 = - 0.566, so \(\frac{-3}{5}<\frac{-17}{30}\), option 3 is True

According to option 4,

The difference between rational numbers \(\frac{5}{7}\) and \(\frac{3}{8}\) is,

5/7 - 3/8 = 19/56 which is ≠ \(\frac{17}{56}\) , so option 4 is False

∴ The correct answer is option 4

Calculation:

From the given options, 

According to options 4,

The successor of a two digit number 99 is 100 which is a three digit number, so option 4 is False

∴ The correct answer is option 4

Concept use:

A number is divisible by 3 if the sum of its digits is divisible by 3. 

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

A number is divisible by 6 if it is divisible by both 2 and 3.

Calculations:

option 1.

2 + 1 + 0 + 4 + 3 + 6 + 0 + 5 + 8 + 7 = 36 is divisble by 9 

Hence, Correct 

option 2. 

1 + 5 + 1 + 2 + 8 + 7 + 5 + 1  = 30 is not divisible by 9.  

Hence, Correct.

option 3.

All possible values of digit z in the number 31z5 so that this number is a multiple of 3 are 3, 6 and 8. 

take z = 8 

3185 is not divisible by 3  because sum of the digits of number is 17 not divisible by 3

Hence, Incorrect

option 4.

For the number 21y5 to be a multiple of 9, the digit y must be 1. 

take y = 1

2 + 1 + 1 + 5 = 9 is divieble by 9 

Hence, Correct

Calculation:

A film show lasted for \(3\frac{1}{9}\) hours where \(1\frac{2}{3}\) hours of advertisements included,

So the actual duration of the film

\(3\frac{1}{9}\) - \(1\frac{2}{3}\) 

⇒ 28/9 - 5/3

⇒ (28 - 15)/9

⇒ 13/9

⇒ \(1\frac{4}{9}\) hours

∴ The correct answer is  \(1\frac{4}{9}\) hours

Concept used:

(a + b)² = a² + 2ab + b²

Calculation:

(xy + yz)2 -2xy2z

⇒ x²y² + 2xy²z + y²z² - 2xy2z

⇒ x2y2 + y2z2

∴ The correct answer is x2y2 + y2z2

Concept used:

(a - b)² = a² - 2ab + b²

Calculation:

\(\rm \left(\frac{4}{3}a-\frac{3}{4}b\right)^2+2ab\)

⇒ \(\frac{16}{9}\)a² - 2ab + \(\frac{9}{16}\)b² + 2ab

⇒ \(\rm \frac{16}{9}a^2+\frac{9}{16}b^2\) 

∴ The correct answer is option 1

Calculation:

In the school, there are 7 periods of 50 minutes duration, total duration = (7 × 50) = 350 min

Now, 3 periods were included, then total periods = (7 + 3) = 10

If the total duration remains unchanged, then the new duration of each periods = 350/10 = 35 min

∴ The correct answer is 35 min

Concept used:

15% = 3/20

Calculation:

When the shopkeeper sells at 15% profit, then CP : SP = 20 : 23

When the shopkeeper sells at 15% loss, then CP : SP = 20 : 17

According to the question, we have to do equal SP in both the ratio,

 

So, all over loss% = (800 - 782)/800 × 100 = 18/800 × 100 = 2.25%

∴ The correct answer is 2.25% loss

Calculation:

Let's assume that the age of R = k, then age of Q = 3k and the age of P = 3k + 5,

Now, according to the question,

(3k + 5) + 3k + k = 47

⇒ 7k + 5 = 47

⇒ 7k = 42

⇒ k = 42/7 = 6

Then the age of P = (3 × 6 + 5) = 23 years

∴ The correct answer is 23 years

Concept used:

To form a triangle, the sum of any two sides of a triangle must be greater than the other side.

Calculation:

The sum of the two sides of the triangle = (5 + 9) = 14 cm

So, the third side must be any length greater than 4 cm and less than 14 cm

∴ The correct answer is option 2

Concept used:

Let n be the number of sides in the regular polygon. Then, the measure of each exterior angle (θ) is = 360/n

Calculation:

To find the maximum exterior angle, we need to consider the minimum value of n.

A polygon with the minimum number of sides is a triangle (n = 3).

So, the maximum exterior angle possible for a regular polygon is 360/3 = 120° 

∴ The correct answer is 120°

Explanations:

 

A pyramid is a three-dimensional geometric shape that is formed by connecting a polygon base to a point, called the apex, that is not in the same plane as the base. The base can be any polygon, but is often a rectangle or square.

 

Now let's take a triangle

If we draw lines from all three vertices of the triangle to a point not on the plane of the triangle, we get a tetrahedron, which is a kind of pyramid where the base and the faces are all triangles.

Essentially, the number of sides in the polygon determines the number of triangular faces the pyramid will have. The lines you draw between the polygon vertices and the external point form the edges of the triangular faces.

 

So, when all the vertices of a polygon are joined to a point which is not in the plane of the polygon, the resulting shape obtained is a pyramid,

where the base is the original polygon and the vertices become the corners of the triangular faces of the pyramid.

Hence, The Correct Answer is pyramid.

Calculation:

A Quadrilateral cannot be constructed uniquely if only its two sides and three angles are known, so option 3 is not correct

To uniquely construct a quadrilateral, we need to know either:

• All four side lengths and one angle (SSSSA criterion).
• Two adjacent sides and their included angle (SAS criterion).
• Three consecutive sides (SSS criterion).
• Two opposite sides and the angle between them (SAA criterion).

If we know any of the above combinations, you can uniquely construct the quadrilateral.

∴ The correct answer is option 3

Concept used:

To form a triangle, the sum of any two sides of a triangle must be greater than the other side.

Calculation:

From the given options, 

According to option 3, in the ΔXYZ, (3 + 4) = 7 > 5, again (4 + 5) = 9 > 3 again, (5 + 3) = 8 > 4

So, only the ΔXYZ is possible

∴ The correct answer is option 3

Concept used:

• Diagonals of a rhombus bisect each other at right-angle
• Perimeter of rhombus = 4 × side

Calculation:

According to the given picture, ABCD is a rhombus

Then in the right-angle ΔOBC, OB = 5, OC = 12, the BC = 13 [ as 5, 12, 13 are Pythagorean triplet ]

The perimeter of the rhombus = 4 × 13 = 52 cm

∴ The correct answer is

Concept used:

• The volume of a cylinder = π × r² × h, where, r = radius, h = height.
• 1000 litre = 1 m³

Calculation:

According to the question, 

The volume of water in the tank = π × 1502 × 70 = 4950000 cm³

To convert the volume to liters, we have to convert it into m³

4950000 cm³ = 4950000/1000000 m³ = 4.950 m³

4.950 m³ = 4950 L

∴ The correct answer is 4950 L

Shortcut Trick

Square Area = (Diagonal)² ÷ 2 = (6√2)² ÷ 2 = 36 m².

Area of Trapezium = Average of parallel sides × Height.

Height = Area ÷ [(Sum of parallel sides) ÷ 2]

Height = 36 ÷ [(12 + 6) ÷ 2] = 36 ÷ 9 = 4 m.

∴ The correct answer is 4 m.

Alternate Method

Given:

Parallel sides of trapezium: a = 12 m, b = 6 m.

Diagonal of square (d) = 6√2 m.

Formula Used:

Area of Trapezium = (1/2) × (a + b) × h

Area of Square = d² / 2

Calculations:

⇒ Area of Square = (6√2)² ÷ 2

⇒ Area of Square = (36 × 2) ÷ 2 = 36 m²

⇒ Area of Trapezium = Area of Square = 36 m²

⇒ (1/2) × (12 + 6) × h = 36

⇒ (1/2) × 18 × h = 36

⇒ 9 × h = 36

⇒ h = 36 ÷ 9 = 4 m

∴ The correct answer is 4 m.

Additional Information

Area of Square from Diagonal

Area = d² ÷ 2. If the side is 's', then diagonal d = s√2.

Properties of Trapezium

It has one pair of parallel sides. The area is the product of the average of these parallel sides and the perpendicular distance (height) between them.

Median of a Trapezium

The line segment joining the midpoints of non-parallel sides is called the median. Length of median = (Sum of parallel sides) ÷ 2.

Calculation:

The mean of the given data set is 

(135 + 150 + 139 + 128 + 151 + 132 + 146 + 149 + 143 + 141)/10 = 1414/10 = 141.4

So, the number of girls having height more than the mean height is 5

∴ The correct answer is 5

Concept used:

Probability of an event P(A) = Number of favorable events/ Total number of events

Calculation:

If a die is thrown, total number of results = 1, 2, 3, 4, 5, 6

The numbers divisible by three is 3, 6

So, required probability = 2/6 = 1/3

∴ The correct answer is 1/3