Class 10 Math Maestros: Conquer The Quiz Challenge!

Explanation
The given equation is
2x + y = 8, 
Plug in  y = -6.
2x-6 = 8
Add 6 on both sides of the equation 2x = 8 + 6 => 14
To get x by itself, divide 2 on both sides of the equation
x = 14/2 = 7

 

This method ensures that we correctly solve for xx by maintaining the balance of the equation through equivalent operations on both sides. This step-by-step approach is crucial for solving linear equations and helps prevent errors, ensuring the accuracy of the solution. Consistent practice of these fundamental techniques strengthens problem-solving skills, making it easier to tackle more complex equations with confidence.

Explanation
To solve the equation 3x - 7 = 11, first add 7 to both sides to get 3x = 18. Then divide both sides by 3 to find x = 6. This process of isolating the variable ensures the equation is balanced and accurate, which is essential for finding the correct value of x. By consistently applying operations equally to both sides of the equation, we prevent errors and maintain the equation's integrity. This approach is fundamental in algebra and helps build a strong foundation for solving a wide range of mathematical problems, from simple linear equations to more complex algebraic expressions.

Explanation
The angle formed by the line of sight with the horizontal level when the point being viewed is above the horizontal plane is known as the angle of elevation. This concept is pivotal in fields such as surveying, navigation, astronomy, and architecture. It helps in determining the angle at which an observer must look upwards to see an object above their horizontal plane. Understanding and applying this concept is crucial in practical scenarios where precise measurements and calculations are required. Mastery of geometric concepts like the angle of elevation enhances problem-solving abilities and enables accurate analysis of real-world situations.

Explanation
The probability of an event represents the likelihood of that event occurring and is expressed as a number between 0 and 1, inclusive. A probability of 0 means the event will not occur, while a probability of 1 indicates certainty that the event will occur. Negative probabilities do not exist in this context as they fall outside the defined range and have no meaningful interpretation within the theory of probability.

Explanation
In the context of a lottery ticket, where the total possible outcomes are calculated by adding the outcomes of not getting a prize (20) to the outcomes of getting a prize (10), the sum equals 30 possible outcomes. To find the probability of winning a prize, we divide the number of favorable outcomes (getting a prize) by the total number of outcomes. This calculation, 10 divided by 30, simplifies to 1/3. Therefore, the probability of securing a prize when purchasing a single ticket in this scenario is 1/3.

Explanation
sin A = sin 30 = 1/2 => A = 30 degree
cos B = cos 45 = 1/ /2 => B = 45 degree
So, A+B => 30 +45 = 75 degree


Therefore, the sum of angles A and B is equal to 75 degrees.  This knowledge is useful in solving trigonometric equations and understanding the relationships between angles. Mastery of trigonometric functions and their corresponding angles is essential for applications in geometry, physics, and engineering, where precise calculations are crucial for problem-solving and designing solutions.

Explanation
Given that the HCF (a, b) is 30 and the LCM (a, b) is 45, we use the relationship HCF × LCM = a × b. Substituting the given values, we have 30 × 45 = 1350. Therefore, a×b=1350a \times b = 1350. Understanding the relationship between HCF and LCM is fundamental in number theory and helps solve problems involving divisibility, simplifying fractions, and finding common multiples. This concept is vital for mathematical problem-solving and practical applications in various fields, including engineering and computer science.

Explanation
tn= a+(n-1)d
Subtract the first term, 4, from 109: 109 - 4 = 105
Divide the result by the common difference, which is 5: 105 / 5 = 21
Add 1 to account for the progression starting at term 1: 21 + 1 = 22
So, the 22nd term of the sequence is 109.

 

Therefore, the 22nd term of the sequence is 109. Understanding arithmetic sequences and their properties is essential for solving problems in algebra and recognizing patterns in data. This knowledge is applicable in various mathematical and real-world contexts, such as finance and science.

Explanation
Start with the Sum Formula: We begin with the formula for the sum of an arithmetic series, which is given as:
Sn=n/2(a1+an)Sn​=2n​(a1​+an​)
Sn​ represents the sum of the series.
n is the number of terms in the series.
a1​ is the first term of the series.
an​ is the nth term of the series.

Plug in the Known Values: In this problem:
Sn​ is 55 because we want the sum to be equal to 55.
a1​ is 1 because the first term in the series is 1.
So, we have:
55=k/2(1+an)55=2k​(1+an​)

Simplify: Since a1=1, we simplify further:
55=k/2(1+k)

Multiply by 2 to Eliminate Fraction: To get rid of the fraction, we multiply both sides by 2:
110=k(1+k)

Quadratic Equation: We now have a quadratic equation in the form k2+k−110=0

Factor or Use Quadratic Formula: To solve this equation, you can either factor it or use the quadratic formula. Factoring it, we get:
(k+11)(k−10)=0

Solve for k: Set each factor equal to zero and solve for k:
k+11=0 → −11
k−10=0 → k=10

Choose the Positive Solution: Since we're looking for a positive value of k that represents the sum of the series, the value of k is 10.
So, the value of k in the equation 1+2+…+k=55 is 10, which means that the sum of the first 10 positive integers is equal to 55.

Explanation
In a parallelogram, opposite angles are equal. This property helps in identifying and solving problems involving parallelograms. Recognizing the equality of opposite angles is essential for understanding geometric shapes and their properties. This knowledge aids in solving various geometric problems and is applicable in fields such as architecture, engineering, and computer graphics. Mastery of geometric concepts like this one enhances problem-solving skills and enables accurate analysis and design of structures and shapes.

Explanation
Explanation: The volume of a cube is given by the formula V = a³. With a side length of 5 cm, the volume is V = 5³ = 125 cm³. Volume calculations are crucial for understanding spatial dimensions and are widely applicable in various fields such as architecture, engineering, and everyday measurements. Knowing how to calculate volume helps in determining the capacity of three-dimensional objects, making it easier to estimate storage space, material usage, and more. This fundamental geometric concept forms the basis for more complex mathematical problems and practical applications.

Explanation
The highest common factor (HCF) of 18 and 24 can be found using prime factorization. 18 = 2 × 3², and 24 = 2³ × 3. The common factors are 2 and 3, and the highest common factor is 6. Finding the HCF is crucial for simplifying fractions and solving problems involving ratios. Understanding how to find the HCF is essential for simplifying fractions, solving problems involving ratios, and determining common divisors in various mathematical contexts. This knowledge is fundamental in number theory and practical applications, such as optimizing resources and solving puzzles.

Explanation
The area of a circle is given by the formula A = πr². Substituting the given radius, 7 cm, into the formula, we get A = π(7)² = 49π cm². Understanding and applying this formula is fundamental in geometry as it relates to circular shapes, allowing us to calculate areas quickly and accurately. Calculating the area of a circle is fundamental in geometry and has practical applications in various fields such as construction, design, and science. Understanding and applying this formula allows us to quickly and accurately determine the space within a circular boundary, helping in efficient planning and resource allocation. This geometric concept is essential for solving problems related to circular shapes and understanding the properties of circles.

Explanation
To find the 10th term of the sequence where the nth term is described by the formula (3n - 2), you can substitute 10 for n in the formula:
Multiply 3 by 10: 3 * 10 = 30
Subtract 2 from the result: 30 - 2 = 28
Therefore, the 10th term of the sequence is 28.

 

Understanding how to work with arithmetic sequences and their general formulas is crucial in algebra and helps in identifying patterns, predicting future terms, and solving related mathematical problems. This knowledge is widely applicable in various contexts, from simple calculations to more advanced studies in mathematics and science.

Explanation
To solve for the value of (sin A + cos A) x cosec A given that cot A = 12/5, we need to first find the values of sin A, cos A, and cosec A.
Step 1: Calculate sin A and cos A From the cotangent identity, cot A = cos A / sin A, and given cot A = 12/5, we can express cos A and sin A using a right triangle where:
The adjacent side (to angle A) = 12 (corresponding to cos A)
The opposite side (to angle A) = 5 (corresponding to sin A)
To find the hypotenuse (h), we use the Pythagorean theorem: Hypotenuse squared = Adjacent squared + Opposite squared h squared = 12 squared + 5 squared h squared = 144 + 25 h squared = 169 h = 13
Thus, the values of sin A and cos A are: sin A = Opposite / Hypotenuse = 5 / 13 cos A = Adjacent / Hypotenuse = 12 / 13
Step 2: Calculate cosec A cosec A = 1 / sin A = 13 / 5
Step 3: Calculate (sin A + cos A) x cosec A sin A + cos A = 5/13 + 12/13 = 17/13 (sin A + cos A) x cosec A = 17/13 x 13/5 = 17 x 13 / 13 x 5 = 17 / 5
Therefore, the value of (sin A + cos A) x cosec A is 17/5.