The problem states that the polynomial \( px^{3} + x^{2} - 2x - q \) is exactly divisible by \((x - 1)\) and \((x + 1)\). According to the Factor Theorem, if a polynomial \(P(x)\) is exactly divisible by \((x - a)\), then \(P(a) = 0\). We will apply this theorem for both given factors to form a system of two linear equations with two variables, \(p\) and \(q\), and then solve for \(p\) and \(q\).
Let the given polynomial be \( P(x) = px^{3} + x^{2} - 2x - q \).
Since \(P(x)\) is exactly divisible by \((x - 1)\), we have \(P(1) = 0\).
\[ P(1) = p(1)^{3} + (1)^{2} - 2(1) - q = 0 \]\[ p + 1 - 2 - q = 0 \]\[ p - q - 1 = 0 \]\[ p - q = 1 \quad \text{(Equation 1)} \]Since \(P(x)\) is exactly divisible by \((x + 1)\), we have \(P(-1) = 0\).
\[ P(-1) = p(-1)^{3} + (-1)^{2} - 2(-1) - q = 0 \]\[ -p + 1 + 2 - q = 0 \]\[ -p - q + 3 = 0 \]\[ -p - q = -3 \]\[ p + q = 3 \quad \text{(Equation 2)} \]We have two linear equations:
Equation 1: \( p - q = 1 \)
Equation 2: \( p + q = 3 \)
Add Equation 1 and Equation 2:
\[ (p - q) + (p + q) = 1 + 3 \]\[ 2p = 4 \]\[ p = \frac{4}{2} \]\[ p = 2 \]Using Equation 1:
\[ p - q = 1 \]\[ 2 - q = 1 \]\[ -q = 1 - 2 \]\[ -q = -1 \]\[ q = 1 \]Thus, \( p = 2 \) and \( q = 1 \).
A) p = 2 and q = 1. This matches our calculated values for \(p\) and \(q\).
The question asks for the full form of the abbreviation SAARC. This is a direct knowledge-based question about international organizations. Knowing the full forms of important abbreviations is crucial for competitive exams.
Correct Option: D) South Asian Association for Regional Cooperation is the correct full form of SAARC. It is an intergovernmental organization and geopolitical union of states in South Asia.
The question asks to identify the last satellite launched by ISRO in the year 2020. This requires knowledge of ISRO's launch history for that specific year.
ISRO's last launch in 2020 was the communication satellite CMS-01 (formerly known as GSAT-12R). It was launched on December 17, 2020, by PSLV-C50 from the Satish Dhawan Space Centre SHAR, Sriharikota.
A) CMS-01 — CMS-01 was indeed the last satellite launched by ISRO in 2020. It was a communication satellite launched on December 17, 2020.
The question asks for the establishment year of the Export-Import (EXIM) Bank of India. This is a factual question requiring knowledge of the history of financial institutions in India.
B) 1982 is the correct year when the Export-Import (EXIM) Bank of India was established and began its operations. It was set up as a principal financial institution for coordinating the working of institutions engaged in financing export and import of goods and services with a view to promoting the country's international trade.
Simplify the expression \(\sqrt{81}+\sqrt{0.81}+\sqrt{0.0081}\)\(\sqrt{10000}\).
To simplify the given expression, we need to evaluate each square root term separately and then perform the addition and multiplication operations in the correct order. We will convert decimal numbers into fractions to easily find their square roots.
We know that:
\[ (9 + 0.9 + 0.09) \times 100 \]
\[ 9 + 0.9 + 0.09 = 9.99 \]
\[ 9.99 \times 100 = 999 \]
C) 999 is the correct answer. The step-by-step calculation shows that the simplified value of the expression is 999.
The question asks to identify which of the given options is NOT a volcano. This requires knowledge of famous mountains and their geological classification.
Correct Option: A) Mont Blanc
Mont Blanc is the highest mountain in the Alps and Western Europe, located between France and Italy. It is part of a crystalline massif and was formed primarily through tectonic plate collision and uplift, not volcanic activity. It is not a volcano.
This problem involves the relationship between the Highest Common Factor (HCF) and Least Common Multiple (LCM) of two numbers. The fundamental property states that the product of two numbers is equal to the product of their HCF and LCM. We are given the HCF and LCM, and information to find the first number. Using this, we can determine the second number.
Identify the given values:
Find the first number:
We are told that if the first number is divided by 2, the quotient is 44. Let the first number be \(N_1\).
\[ \frac{N_1}{2} = 44 \]Multiply both sides by 2 to find \(N_1\):
\[ N_1 = 44 \times 2 = 88 \]So, the first number is 88.
Apply the HCF-LCM property:
For any two positive integers, the product of the numbers is equal to the product of their HCF and LCM.
\[ N_1 \times N_2 = \text{HCF} \times \text{LCM} \]Substitute the known values:
\[ 88 \times N_2 = 44 \times 264 \]Solve for the second number (\(N_2\)):
Divide both sides by 88:
\[ N_2 = \frac{44 \times 264}{88} \]Simplify the expression. We can see that \(88 = 2 \times 44\):
\[ N_2 = \frac{44 \times 264}{2 \times 44} \]Cancel out 44 from the numerator and denominator:
\[ N_2 = \frac{264}{2} \] \[ N_2 = 132 \]Therefore, the second number is 132.
B) 132 is the correct answer, as calculated using the HCF-LCM property and the given information about the first number.
The question asks to identify the Indian-born Scottish inventor who pioneered the development of the Automated Teller Machine (ATM). This requires knowledge of the history of the ATM and its key inventors.
Correct Option: C) John Shepherd-Barron is credited with inventing the first cash dispenser, which was installed in 1967 at a Barclays bank in Enfield, North London. He was born in Shillong, Meghalaya, India (then British India), to Scottish parents, making him an Indian-born Scottish inventor.
Find the HCF of \( \frac{54}{9} \), 3 \( \frac{9}{17} \) and \( \frac{36}{51} \).
\( \frac{5}{17} \)
\( \frac{6}{17} \)
\( \frac{7}{17} \)
\( \frac{8}{17} \)
To find the HCF (Highest Common Factor) of fractions, we use the formula: HCF of fractions = (HCF of numerators) / (LCM of denominators). First, we need to convert all given numbers into improper fractions in their simplest form.
Convert the given numbers into improper fractions:
The first number is \( \frac{54}{9} \). This simplifies to:
\[ \frac{54}{9} = 6 \]We can write this as \( \frac{6}{1} \).
The second number is 3 \( \frac{9}{17} \). Convert the mixed fraction to an improper fraction:
\[ 3 \frac{9}{17} = \frac{(3 \times 17) + 9}{17} = \frac{51 + 9}{17} = \frac{60}{17} \]The third number is \( \frac{36}{51} \). We can simplify this fraction by dividing the numerator and denominator by their HCF, which is 3:
\[ \frac{36}{51} = \frac{36 \div 3}{51 \div 3} = \frac{12}{17} \]List the simplified fractions:
The fractions are \( \frac{6}{1} \), \( \frac{60}{17} \), and \( \frac{12}{17} \).
Find the HCF of the numerators:
The numerators are 6, 60, and 12.
Factors of 6: 1, 2, 3, 6
Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
Factors of 12: 1, 2, 3, 4, 6, 12
The highest common factor among 6, 60, and 12 is 6.
So, HCF (6, 60, 12) = 6.
Find the LCM of the denominators:
The denominators are 1, 17, and 17.
The least common multiple of 1, 17, and 17 is 17.
So, LCM (1, 17, 17) = 17.
Calculate the HCF of the fractions:
Using the formula HCF of fractions = (HCF of numerators) / (LCM of denominators):
\[ \text{HCF} = \frac{\text{HCF}(6, 60, 12)}{\text{LCM}(1, 17, 17)} = \frac{6}{17} \]B) \( \frac{6}{17} \) is the HCF of the given numbers, as calculated by finding the HCF of the numerators and the LCM of the denominators.
The problem states that a boy was supposed to multiply 495 by 36. Due to an error in reading one digit, he obtained 16740. We need to find which digit was read erroneously. We can solve this by first calculating the correct product. Then, we can find the number the boy actually multiplied by 36 to get 16740. By comparing this incorrect number with the original 495, we can identify the erroneously read digit.
The correct multiplication should have been \(495 \times 36\).
\[ 495 \times 36 = 17820 \]The boy obtained 16740 as his answer after multiplying by 36. Let the erroneously read number be \(X\). So, we have:
\[ X \times 36 = 16740 \]To find \(X\), we divide 16740 by 36:
\[ X = \frac{16740}{36} = 465 \]Original number = 495
Erroneously read number = 465
Comparing 495 and 465, we can see that the digit in the tens place has changed from 9 to 6. This means the digit '9' was read erroneously as '6'.
Correct Option: A) 9