Logarithmic Equations and Linear Inequalities

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 log_a x = log_b x / log_b 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.