Logarithmic Equations and Linear Inequalities
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Questions and Answers

Who is the child of K?

  • F (correct)
  • T
  • P
  • G

    • What is the relationship between R and M?

  • R is M's son.
  • R is M's brother.
  • R is M's father.
  • R is M's husband. (correct)

    • Which statement about the gender of G is true?

  • G is male.
  • The gender of G cannot be determined. (correct)
  • G is the sister of H.
  • G is female.

    • How is S related to U?

    <p>S is U's aunt.</p> Signup and view all the answers

    What can be inferred about the family structure?

    <p>There are two couples in the family.</p> Signup and view all the answers

    What is the solution to the equation log4(x2 + 1) - log4(x + 1) = 2?

    <p>x = 16</p> Signup and view all the answers

    Which property of logarithms is used to simplify loga x = logb x / logb a?

    <p>Change of Base Formula</p> Signup and view all the answers

    How much time will it take for a sum to become 8 times its original value if it doubles in 15 years?

    <p>45 years</p> Signup and view all the answers

    What is the principal amount if a sum of money amounts to ₹2784 in 4 years at simple interest?

    <p>₹2400</p> Signup and view all the answers

    In the system of inequalities 3x + 2y < 24, x + 2y < 16, and x + y > 10, which option represents the correct shaded region?

    <p>Option C</p> Signup and view all the answers

    At what interest rate does a sum of money double in 14.2 years under compound interest?

    <p>5% per annum</p> Signup and view all the answers

    What is the future value of an item worth ₹1000 after 2 years, compounded semi-annually at a rate of 22% per annum?

    <p>₹1518</p> Signup and view all the answers

    What is the interest earned in one year if a sum of money amounts to ₹2688 in 3 years and ₹2784 in 4 years?

    <p>₹96</p> Signup and view all the answers

    What is the main difference between simple interest and compound interest?

    <p>Simple interest is calculated on the original principal only, while compound interest is calculated on the principal and accumulated interest.</p> Signup and view all the answers

    How much will ₹25,000 amount to in two years with interest rates of 4% and 5% respectively for each year?

    <p>₹27,300</p> Signup and view all the answers

    What is the present value of furniture that depreciates at 10% per year, if its current value is ₹8,700?

    <p>₹30,000</p> Signup and view all the answers

    What does the effective rate of interest depend on?

    <p>The frequency of compounding the interest.</p> Signup and view all the answers

    If a person needs ₹10,000 at the end of 3 years with an interest rate of 6% compounded quarterly, how much do they need to deposit each quarter?

    <p>₹11.01</p> Signup and view all the answers

    What is the present value of an annuity that pays ₹200 at the end of each quarter for 10 years with a 5% interest rate compounded quarterly?

    <p>₹620</p> Signup and view all the answers

    What will be Mr. A's annual installment if he borrows ₹500,000 over 20 years at 10% interest?

    <p>₹58,729</p> Signup and view all the answers

    How much does a company need to invest annually in a sinking fund to accumulate ₹200,000 in 20 years at 5% interest?

    <p>₹6,499</p> Signup and view all the answers

    What is the present value of receiving ₹10,000 every year for 16 years, deposited in a bank with 8.5% interest compounded annually?

    <p>₹93,421</p> Signup and view all the answers

    What is the odd one out in the following set based on their properties: 295, 381, 552, 729?

    <p>552</p> Signup and view all the answers

    How is the word 'teacher' encoded as 'VGGCEJJ'?

    <p>By adding 2 to each letter</p> Signup and view all the answers

    If 'health' is coded as 'GISKJD', what will 'north' be coded as?

    <p>GSMQN</p> Signup and view all the answers

    Which word is coded as 'XVEC' using the pattern of adding two to each letter's order value?

    <p>CHAIR</p> Signup and view all the answers

    From the sequence 'CEHLK', 'MPTO', 'QT', and 'Y', which is the odd one out?

    <p>Y</p> Signup and view all the answers

    What direction does a person face after walking 75 meters North, then turning left and walking 25 meters followed by another left turn and walking 80 meters?

    <p>South West</p> Signup and view all the answers

    In the seating arrangement of P, Q, R, S, T, U, V, and W, who is sitting fourth to the left of P?

    <p>T</p> Signup and view all the answers

    Study Notes

    Logarithmic and Exponential Equations

    • The equation log4(x2 + 1) - log4(x + 1) = 2 can be simplified using the rule: log a - log b = log (a/b).
    • The equation then becomes log4(x2 + 1)/(x + 1) = 2.
    • Simplifying and applying the definition of logarithm, the solution x = 16 is obtained.
    • Similar questions involving logarithms and their properties are found in the module.

    Logarithmic Equations and Properties

    • The property loga x = logb x / logb a can be used to simplify logarithmic equations with different bases.
    • The expression log2412 + log3624 + log4836 + 1 can be simplified using this property.
    • This simplifies to 2 * log4824.
    • Option C matches the final expression.

    System of Linear Inequalities

    • A graph depicting a system of linear inequalities is provided.
    • The shaded regions on the graph show the solution regions for each inequality.
    • The system of inequalities includes 3x + 2y < 24, x + 2y < 16, and x + y > 10.
    • Option C provides the correct shading.

    Linear Inequalities

    • The graph of x + y ≤ 6 is shown by a line and the shaded area to its left.
    • This inequality represents the region to the left of the line x + y = 6.

    Simple Interest

    • The time taken for an amount to become 8 times its original value is 4 times the time needed to double.
    • The interest rate can be expressed as 1/t, where t is the time for the principal to double.
    • If an amount doubles in 15 years, it becomes 8 times its original value in 60 years.

    Compound Interest

    • A sum of money doubling in 15 years will need an additional 30 years to become 8 times its original amount, making a total of 45 years.

    Simple Interest

    • A sum of money increases to ₹2784 in 4 years and ₹2688 in 3 years, which means the interest earned in one year is ₹96.
    • The principal amount is ₹2400, and the interest rate is 4%.

    Compounding Interest

    • A sum will double in 14.2 years at a 5% annual compound interest rate.

    Future Value of an Investment

    • ₹1000 will grow to ₹1518 in 2 years with semi-annual compounding at a 22% annual rate.

    Difference Between Simple and Compound Interest

    • The difference between simple and compound interest on ₹10,000 at 10% for 4 years is ₹641.

    Future Value of an Investment (with Variable Interest Rates)

    • ₹25,000 will become ₹27,300 in two years with successive interest rates of 4% and 5%.

    Depreciation

    • Furniture depreciating at 10% per year will have a value of ₹30,000 three years prior to its current worth of ₹8700.

    Effective Rate of Interest

    • The effective interest rate is independent of the initial amount.

    Calculating Present Value

    • If p * A2 = 96 and r = 8% (compounded annually), then p = ₹15,000

    Quarterly Compounding

    • To reach ₹10,000 in 3 years with 6% quarterly compounding, a deposit of ₹11.01 is required at the end of each quarter.

    Present Value of an Annuity

    • The present value of an annuity paying ₹200 quarterly for 10 years at 5% compounded quarterly is ₹620.

    Loan Repayment

    • A ₹500,000 house loan repaid in equal annual installments over 20 years at 10% interest results in an annual installment of ₹58,729.

    Sinking Fund

    • To pay off a ₹200,000 debt in 20 years, a sinking fund requiring annual investments of ₹6,499 at 5% interest is needed.

    Present Value of an Annuity Due

    • The present value of a ₹10,000 annual gift (starting today) for 16 years at 8.5% annual compounding is ₹93,421.

    Combinations

    • Choosing 3 vertices from 12 points (5 non-collinear, 7 collinear) results in 185 possible triangles.

    Combinations with Constraints

    • A student with 3 library tickets and 8 books of interest (Mathematics Part 2 only if Part 1 is also borrowed) can choose 3 books in 41 ways.

    Exam Question Selection

    • An exam with 12 questions (7 in Part A, 5 in Part B) requires selecting 8 questions with at least 3 from each part. There are 420 possible selections.

    Odd Man Out

    • The odd man out in 295, 381, 552, 729 is 552, as it's the only even number.

    Coding & Decoding

    • "teacher" is coded as "VGGCEJJ" using a pattern of adding two to each alphabet's position.
    • "children" is coded as "FJEKNF" using the same pattern.

    Finding the Right Pattern

    • "health" coded as "GISKJD" uses a subtraction of one from each alphabet's position.
    • "north" is coded as "GSMQN" following the same pattern.

    Another Coding Pattern

    • "honey" coded as "QIVZ" uses an addition of two to each alphabet's position.
    • The word coded as "XVEC" is "CHAIR" using the same pattern.

    Odd One Out

    • In the sequence CEHLK, MPTO, QT, and Y, Y is the odd one out, as the other words follow a consistent pattern of adding 3 and 4 to the alphabetical positions.

    Direction, Distance and Position

    • Detailed descriptions of various movement scenarios are provided, with final directions specified.

    Seating Arrangement

    • A detailed seating arrangement of 8 individuals (P, Q, R, S, T, U, V, and W) on a bench facing North is described. This is further described in a way that can be used to solve similar arrangement problems.

    Arrangement on a Floor

    • Detailed arrangements of 6 apartments (P, Q, R, S, T, and U) on a floor in two rows are specified.

    Sitting Arrangement around a Circular Table

    • Six people (A, B, C, D, E, F) seated around a circular table are arranged based on positions relative to each other.

    Linear Arrangement

    • Five people (A, B, C, D, E) are arranged in a linear sequence based on their positions relative to each other.

    Blood Relation

    • Detailed descriptions of blood relationships and family structures are given as specific examples of problems that can be solved.

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    Description

    This quiz focuses on logarithmic equations, properties, and systems of linear inequalities. Test your knowledge on simplifying logarithmic expressions and solving inequalities through various properties. Perfect for students looking to deepen their understanding of these mathematical concepts.

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