UP Board Class 10 Mathematics Question Paper 2025 (Code 822 BV) with Answer Key and Solutions PDF is Available to Download

Step 1: Understanding the Concept:

This question is about understanding the relationship between the different methods of calculating the mean of grouped data.

There are three main methods for calculating the mean: the direct method, the assumed mean method, and the step-deviation method.




Step 2: Detailed Explanation:

Direct Method (\(\bar{x} = \frac{\sum f_i x_i}{\sum f_i}\)): This is the basic method of calculating the mean. However, when the values of \(x_i\) and \(f_i\) are large, the calculation becomes complex.

Assumed Mean Method (\(\bar{x} = a + \frac{\sum f_i d_i}{\sum f_i}\)): This is a simplified form of the direct method. In it, we choose an assumed mean ('a') and use deviations (\(d_i = x_i - a\)) to make the calculation easier.

Step-Deviation Method (\(\bar{x} = a + h\left(\frac{\sum f_i u_i}{\sum f_i}\right)\)): This is an even more simplified form of the assumed mean method, used when the class size ('h') is uniform. It further simplifies the calculation by dividing the deviations by the class size (\(u_i = d_i/h\)).

Thus, both the assumed mean and step-deviation methods are ways to simplify the calculations of the basic Direct Method. They are simplified forms of the direct method.




Step 3: Final Answer:

The assumed mean method and the step-deviation method are simplified forms of the direct method.

Therefore, option (B) is correct.
Quick Tip: Think of the three methods as a progression: the direct method is the original, the assumed mean method simplifies it, and the step-deviation method simplifies it even further (when the class size is uniform).

Step 1: Understanding the Concept:

This question requires the calculation of the volume of a right circular cone using the given height and diameter.




Step 2: Key Formula or Approach:

The formula for the volume (V) of a cone is:
\[ V = \frac{1}{3} \pi r^2 h \]
where \(r\) is the radius of the base and \(h\) is the height of the cone.




Step 3: Detailed Explanation:

First, we need to find the radius from the given diameter.

Given:

Height (\(h\)) = 48 cm

Diameter = 28 cm


The radius (\(r\)) is half of the diameter:
\[ r = \frac{Diameter}{2} = \frac{28}{2} = 14 cm \]
Now, substitute the values of \(r\) and \(h\) into the volume formula (using \(\pi = \frac{22}{7}\)):
\[ V = \frac{1}{3} \times \frac{22}{7} \times (14)^2 \times 48 \] \[ V = \frac{1}{3} \times \frac{22}{7} \times 14 \times 14 \times 48 \]
To simplify the calculation, we can cancel the terms:
\[ V = \frac{1}{3} \times 22 \times ( \frac{14}{7} \times 14 ) \times 48 \] \[ V = \frac{1}{3} \times 22 \times (2 \times 14) \times 48 \] \[ V = 22 \times 28 \times (\frac{48}{3}) \] \[ V = 22 \times 28 \times 16 \] \[ V = 616 \times 16 \] \[ V = 9856 cm^3 \]



Step 4: Final Answer:

The volume of the cone is 9856 cm\(^3\).
Quick Tip: Always double-check if the question provides the radius or the diameter. A common mistake is to use the diameter value directly in the formula instead of the radius.

Step 1: Understanding the Concept:

The HCF (Highest Common Factor) of two numbers is the largest number that divides both of them. The prime factorization method involves breaking down each number into a product of its prime factors.




Step 2: Key Formula or Approach:

1. Find the prime factorization of each number.

2. Identify the common prime factors.

3. The HCF is the product of the lowest powers of these common prime factors.




Step 3: Detailed Explanation:

First, find the prime factorization of 96:
\[ 96 = 2 \times 48 = 2 \times 2 \times 24 = 2 \times 2 \times 2 \times 12 = 2 \times 2 \times 2 \times 2 \times 6 = 2 \times 2 \times 2 \times 2 \times 2 \times 3 \] \[ 96 = 2^5 \times 3^1 \]
Next, find the prime factorization of 404:
\[ 404 = 2 \times 202 = 2 \times 2 \times 101 \] \[ 404 = 2^2 \times 101^1 \]
(Note: 101 is a prime number).


Now, identify the common prime factors and their lowest powers:

The only common prime factor is 2.

The lowest power of 2 in both factorizations is \(2^2\).

Therefore, the HCF is the product of these lowest powers:
\[ HCF(96, 404) = 2^2 = 4 \]



Step 4: Final Answer:

The HCF of 96 and 404 is 4.
Quick Tip: To find the HCF, take the lowest power of common factors. To find the LCM (Lowest Common Multiple), take the highest power of all factors.

Step 1: Understanding the Concept:

This question requires proving a trigonometric identity by manipulating one side of the equation (usually the more complex one) to show it is equivalent to the other side.




Step 2: Key Formula or Approach:

We will start with the Left Hand Side (LHS) and use the following fundamental identities:

1. \(\cot A = \frac{\cos A}{\sin A}\)

2. \(\csc A = \frac{1}{\sin A}\)




Step 3: Detailed Explanation:

Starting with the LHS:
\[ LHS = \frac{\cot A - \cos A}{\cot A + \cos A} \]
Substitute \(\cot A = \frac{\cos A}{\sin A}\) into the expression:
\[ LHS = \frac{\frac{\cos A}{\sin A} - \cos A}{\frac{\cos A}{\sin A} + \cos A} \]
Factor out \(\cos A\) from the numerator and the denominator:
\[ LHS = \frac{\cos A \left( \frac{1}{\sin A} - 1 \right)}{\cos A \left( \frac{1}{\sin A} + 1 \right)} \]
Cancel the common factor \(\cos A\):
\[ LHS = \frac{\frac{1}{\sin A} - 1}{\frac{1}{\sin A} + 1} \]
Now, substitute \(\csc A = \frac{1}{\sin A}\):
\[ LHS = \frac{\csc A - 1}{\csc A + 1} \]
This is equal to the Right Hand Side (RHS).
\[ LHS = RHS \]
Hence, proved.




Step 4: Final Answer:

The identity \(\frac{\cot A - \cos A}{\cot A + \cos A} = \frac{\csc A - 1}{\csc A + 1}\) is successfully proven by expressing \(\cot A\) in terms of \(\sin A\) and \(\cos A\) and simplifying.
Quick Tip: When proving trigonometric identities, a good strategy is often to express all terms in the form of \(\sin\) and \(\cos\). Then, use algebraic simplification and other identities to reach the desired expression.

Step 1: Understanding the Concept:

This is a basic probability problem. Probability is the ratio of the number of favorable outcomes to the total number of possible outcomes.




Step 2: Key Formula or Approach:

The formula for probability of an event (E) is:
\[ P(E) = \frac{Number of favorable outcomes}{Total number of possible outcomes} \]



Step 3: Detailed Explanation:

First, we determine the total number of pens.

Number of defective pens = 12

Number of good pens = 132

Total number of pens = Number of defective pens + Number of good pens
\[ Total number of pens = 12 + 132 = 144 \]
This is the total number of possible outcomes.


Next, we determine the number of favorable outcomes.

The event is "the pen taken out is a good one".

Number of good pens = 132

This is the number of favorable outcomes.


Now, calculate the probability:
\[ P(good pen) = \frac{Number of good pens}{Total number of pens} = \frac{132}{144} \]
Simplify the fraction. Both numbers are divisible by 12.
\[ P(good pen) = \frac{132 \div 12}{144 \div 12} = \frac{11}{12} \]



Step 4: Final Answer:

The probability that the pen taken out is a good one is \(\frac{11}{12}\).
Quick Tip: In probability questions, always start by identifying two key numbers: the total number of possible outcomes (the denominator) and the number of outcomes that satisfy the event's condition (the numerator).

Step 1: Understanding the Concept:

This is a word problem that can be solved by setting up a pair of linear equations in two variables representing the numerator and the denominator of the fraction.




Step 2: Key Formula or Approach:

Let the fraction be \(\frac{x}{y}\), where \(x\) is the numerator and \(y\) is the denominator.

We will translate the two given conditions into two separate linear equations.




Step 3: Detailed Explanation:

Condition 1: "A fraction becomes \(\frac{1}{3}\) when 1 is subtracted from the numerator."

This gives us the equation:
\[ \frac{x-1}{y} = \frac{1}{3} \]
Cross-multiplying, we get:
\[ 3(x-1) = y \] \[ 3x - 3 = y \] \[ 3x - y = 3 \] --- (Equation 1)


Condition 2: "it becomes \(\frac{1}{4}\) when 8 is added to its denominator."

This gives us the equation:
\[ \frac{x}{y+8} = \frac{1}{4} \]
Cross-multiplying, we get:
\[ 4x = y+8 \] \[ 4x - y = 8 \] --- (Equation 2)


Now we have a system of two linear equations:

1) \(3x - y = 3\)

2) \(4x - y = 8\)


To solve this, we can subtract Equation 1 from Equation 2:
\[ (4x - y) - (3x - y) = 8 - 3 \] \[ 4x - y - 3x + y = 5 \] \[ x = 5 \]
Now, substitute the value of \(x=5\) into Equation 1 to find \(y\):
\[ 3(5) - y = 3 \] \[ 15 - y = 3 \] \[ y = 15 - 3 = 12 \]
So, the numerator is 5 and the denominator is 12.




Step 4: Final Answer:

The required fraction is \(\frac{5}{12}\).
Quick Tip: Always check your answer by substituting the values back into the original conditions.
Condition 1: \(\frac{5-1}{12} = \frac{4}{12} = \frac{1}{3}\) (Correct).
Condition 2: \(\frac{5}{12+8} = \frac{5}{20} = \frac{1}{4}\) (Correct).

Step 1: Understanding the Concept:

This problem relates distance, speed, and time. We will form a quadratic equation based on the relationship \( Time = \frac{Distance}{Speed} \). The key is that the difference in the time taken at the usual speed and the increased speed is 30 minutes.




Step 2: Key Formula or Approach:

Let the usual speed of the plane be \(x\) km/hr.

Let the distance be \(D = 1500\) km.

Usual time taken (\(T_1\)) = \(\frac{1500}{x}\) hours.

Increased speed = \((x + 250)\) km/hr.

New time taken (\(T_2\)) = \(\frac{1500}{x+250}\) hours.

The plane is late by 30 minutes, which is \(\frac{30}{60} = \frac{1}{2}\) hour.

This means the usual time is longer than the new time by \(\frac{1}{2}\) hour.

So, the equation is: \(T_1 - T_2 = \frac{1}{2}\).




Step 3: Detailed Explanation:

Set up the equation based on the time difference:
\[ \frac{1500}{x} - \frac{1500}{x+250} = \frac{1}{2} \]
Factor out 1500:
\[ 1500 \left( \frac{1}{x} - \frac{1}{x+250} \right) = \frac{1}{2} \] \[ 1500 \left( \frac{(x+250) - x}{x(x+250)} \right) = \frac{1}{2} \] \[ 1500 \left( \frac{250}{x^2 + 250x} \right) = \frac{1}{2} \]
Cross-multiply:
\[ 1500 \times 250 \times 2 = x^2 + 250x \] \[ 750000 = x^2 + 250x \]
Rearrange into a standard quadratic equation:
\[ x^2 + 250x - 750000 = 0 \]
We can solve this by factoring. We need two numbers that multiply to -750000 and have a difference of 250. These numbers are 1000 and -750.
\[ x^2 + 1000x - 750x - 750000 = 0 \] \[ x(x+1000) - 750(x+1000) = 0 \] \[ (x-750)(x+1000) = 0 \]
This gives two possible values for \(x\): \(x = 750\) or \(x = -1000\).

Since speed cannot be negative, we discard \(x = -1000\).




Step 4: Final Answer:

The usual speed of the plane is 750 km/hr.
Quick Tip: In time-speed-distance problems, ensure all units are consistent. If speed is in km/hr, time must be in hours. Convert minutes to hours (e.g., 30 minutes = 0.5 hours) before setting up the equation.

Step 1: Understanding the Concept:

This is another problem on heights and distances. We have a building with a tower on top. We will use the tangent ratio for two different right-angled triangles to find the required height.




Step 2: Key Formula or Approach:

Let BC be the building and CD be the tower on top. Let A be the point on the ground.

Height of the building, BC = 20 m.

Let the height of the tower, CD = \(h\) m.

Let the distance of the point A from the base of the building, AB = \(x\) m.

The angle of elevation of the bottom of the tower is \(\angle CAB = 45^\circ\).

The angle of elevation of the top of the tower is \(\angle DAB = 60^\circ\).

We will use \(\tan \theta = \frac{Perpendicular}{Base}\).




Step 3: Detailed Explanation:

From the right-angled \(\triangle ABC\):
\[ \tan 45^\circ = \frac{BC}{AB} = \frac{20}{x} \]
Since \(\tan 45^\circ = 1\):
\[ 1 = \frac{20}{x} \implies x = 20 m \] --- (Equation 1)


From the right-angled \(\triangle ABD\):

The total height of the perpendicular is BD = BC + CD = \(20 + h\).
\[ \tan 60^\circ = \frac{BD}{AB} = \frac{20+h}{x} \]
Since \(\tan 60^\circ = \sqrt{3}\):
\[ \sqrt{3} = \frac{20+h}{x} \] \[ x\sqrt{3} = 20+h \] --- (Equation 2)


Now, substitute the value of \(x=20\) from Equation 1 into Equation 2:
\[ 20\sqrt{3} = 20+h \]
Solve for \(h\):
\[ h = 20\sqrt{3} - 20 \] \[ h = 20(\sqrt{3} - 1) m \]
We can use the value \(\sqrt{3} \approx 1.732\):
\[ h = 20(1.732 - 1) = 20(0.732) = 14.64 m \]



Step 4: Final Answer:

The height of the tower is 20(\(\sqrt{3}\) - 1) m, which is approximately 14.64 m.
Quick Tip: In problems with two angles of elevation from the same point, you will always get two right-angled triangles sharing a common base or height. Use this common side to equate the two equations you form.