To identify a pair of co-prime numbers, we need to find the Greatest Common Divisor (GCD) of each pair. Two numbers are co-prime (or relatively prime) if their GCD is 1. We will find the prime factors of each number in the given pairs and then determine their GCD.
A) 81, 16 — The prime factors of 81 are \(3^4\) and the prime factors of 16 are \(2^4\). They share no common prime factors, so their Greatest Common Divisor (GCD) is 1. Thus, 81 and 16 are co-prime numbers.
This problem involves calculating equal annual instalments for a loan repaid with compound interest on a reducing balance. We can use the formula for present value of an annuity or work step-by-step by calculating the outstanding balance after each instalment.
The present value of a loan repaid in equal instalments can be given by:
\[ P = \frac{X}{(1+r)^1} + \frac{X}{(1+r)^2} + \dots + \frac{X}{(1+r)^n} \]For this problem, with 2 instalments:
\[ P = \frac{X}{(1+r)} + \frac{X}{(1+r)^2} \]Substitute the given values:
\[ 11550 = \frac{X}{(1+0.10)} + \frac{X}{(1+0.10)^2} \] \[ 11550 = \frac{X}{1.10} + \frac{X}{(1.10)^2} \] \[ 11550 = \frac{X}{1.10} + \frac{X}{1.21} \]To combine the terms on the right side, find a common denominator:
\[ 11550 = X \left( \frac{1.10}{1.21} + \frac{1}{1.21} \right) \] \[ 11550 = X \left( \frac{1.10 + 1}{1.21} \right) \] \[ 11550 = X \left( \frac{2.10}{1.21} \right) \]Now, solve for X:
\[ X = 11550 \times \frac{1.21}{2.10} \] \[ X = 11550 \times \frac{121}{210} \] \[ X = 55 \times 121 \] \[ X = 6655 \]Let the loan amount be \(P = 11550\).
End of Year 1:
End of Year 2:
Correct Option: C) 6,655
This question tests knowledge of Mendeleev's periodic table and his predictions for undiscovered elements. Mendeleev predicted the existence of several elements, naming them using the Sanskrit prefix 'Eka-' meaning 'one' or 'next to' the element above it in the same group. We need to identify the element that corresponds to Eka-aluminium and then determine the formula of its oxide.
A) Ga₂O₃ is the oxide of Gallium (Eka-aluminium). This matches our derivation.
The problem involves a quadrilateral and the angle bisectors of two adjacent angles. We will use the properties of angles in a triangle and a quadrilateral to find the unknown angles and their difference.
Let the quadrilateral be PQRS. The sum of the interior angles of a quadrilateral is 360°.
\[ \angle P + \angle Q + \angle R + \angle S = 360° \]In \( \triangle TRS \), T is the intersection of the bisectors of \( \angle R \) and \( \angle S \). Therefore, \( \angle TRS = \frac{1}{2} \angle R \) and \( \angle TSR = \frac{1}{2} \angle S \).
The sum of angles in \( \triangle TRS \) is 180°.
\[ \angle STR + \angle TRS + \angle TSR = 180° \]\[ 87.5° + \frac{1}{2} \angle R + \frac{1}{2} \angle S = 180° \]Simplify the equation from step 3:
\[ \frac{1}{2} (\angle R + \angle S) = 180° - 87.5° \]\[ \frac{1}{2} (\angle R + \angle S) = 92.5° \]\[ \angle R + \angle S = 2 \times 92.5° \]\[ \angle R + \angle S = 185° \]Substitute this value into the quadrilateral angle sum equation from step 1:
\[ \angle P + \angle Q + 185° = 360° \]\[ \angle P + \angle Q = 360° - 185° \]\[ \angle P + \angle Q = 175° \]We are given that the ratio of \( \angle P \) to \( \angle Q \) is 2 : 3. Let \( \angle P = 2x \) and \( \angle Q = 3x \).
Substitute these into the equation from step 5:
\[ 2x + 3x = 175° \]\[ 5x = 175° \]\[ x = \frac{175°}{5} \]\[ x = 35° \]Now find the measures of \( \angle P \) and \( \angle Q \):
\[ \angle P = 2x = 2 \times 35° = 70° \]\[ \angle Q = 3x = 3 \times 35° = 105° \]Finally, find the difference between the measures of \( \angle Q \) and \( \angle P \):
\[ \angle Q - \angle P = 105° - 70° = 35° \]C) The difference between \( \angle Q \) and \( \angle P \) is 35°.
Five friends C, D, E, F and G are seated around a round table facing the centre. Only D is seated between G and E. D is seated 2nd to the right of F, who is not seated at either side of G. Where is C seated?
To the immediate left of G
To the immediate right of E
To the immediate right of F
To the immediate left of F
This problem involves arranging five friends around a round table based on given positional clues. We will use a step-by-step approach to deduce the seating arrangement and then identify the position of C.
D) To the immediate left of F
As per the final arrangement (F, E, D, G, C), C is immediately to the left of F.
The question asks about the number of new jobs proposed to be created under the Production Linked Incentive (PLI) scheme across 14 sectors, a key initiative for achieving the vision of Atmanirbhar Bharat. This is a factual question requiring knowledge of government schemes and their stated targets.
C) 60 lakh — The Production Linked Incentive (PLI) scheme, covering 14 sectors, is projected to create 60 lakh (6 million) new jobs over the next five years, contributing significantly to the Atmanirbhar Bharat vision.
To solve this problem, we need to follow the order of operations, often remembered by the acronym BODMAS/PEMDAS. This stands for Brackets/Parentheses, Orders/Exponents, Division and Multiplication (from left to right), and Addition and Subtraction (from left to right).
Thus, the value of the expression \(5 + 12 \div 4 - 2 \times (1 + 3)\) is \(0\).
B) 0 is the correct answer, as calculated by following the order of operations.
This question asks about the solubility of barium sulfate (\(\text{BaSO}_4\)) which is formed from the reaction of sulfate ions (\(\text{SO}_4^{2-}\)) and barium ions (\(\text{Ba}^{2+}\)). We need to recall the general solubility rules for ionic compounds, particularly those involving sulfates and alkaline earth metals.
D) Water insoluble — Barium sulfate (\(\text{BaSO}_4\)) is a classic example of an insoluble ionic compound. Its insolubility is a key characteristic and is often used in qualitative analysis to test for sulfate ions or barium ions.
This is a syllogism problem. We will use Venn diagrams to represent the relationships given in the statements and then evaluate each conclusion based on these diagrams.
Only conclusion II follows. "All cloths are green" and "All green are leaves" together imply "All cloths are leaves." If all cloths are leaves, then it is necessarily true that some leaves are cloths.
This problem involves analyzing statements and conclusions based on the principles of syllogism, often visualized using Venn diagrams. We need to determine which conclusions logically follow from the given statements, assuming the statements are true.
Correct Option: C) Only conclusion II follows.