Mathematics Chapter 2 Quiz

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

If $3^x + y = 243$ and $2^2x - 5y = 8$, then find $x$ and $y$.

x = 11, y = 210

What is $4 \times 3^2$ equal to?

Is $3^4 = 27 + 3$ correct? If not, what is the correct equation?

No, $3^4 = 81$.

Find the 6th term in $(a + w)^{13}$.

$\binom{13}{5} a^8 w^5$ Signup and view all the answers

Find the 7th term in $(m + n)^{12}$.

$\binom{12}{6} m^6 n^6$ Signup and view all the answers

How many ways are there to select 9 balls from 6 red balls, 5 white balls, and 5 blue balls if each selection consists of 3 balls of each color?

200 Signup and view all the answers

What are the values of $x$ in the interval $[0, 2 ext{π}]$ for which $4 ext{sin}^2x = 1$?

x = π/6, 5π/6, (7π/6,11π/6) Signup and view all the answers

Given $cos θ = 5/7$, what is the exact value of $sin θ$ if $θ$ is acute?

sin θ = (2√6)/7 Signup and view all the answers

Given $cos θ = 5/7$, what is the exact value of $sin θ$ if $θ$ is obtuse?

sin θ = - (2√6)/7 Signup and view all the answers

What is the ordinary argument ($ ext{arg} z$) and principal argument ($ ext{Arg} z$) of the complex number $z = (1-i)^{10}$?

arg z = -5π, Arg z = π Signup and view all the answers

What is the ordinary argument ($ ext{arg} z$) and principal argument ($ ext{Arg} z$) of the complex number $z = -2 + 2 ext{√3}i$?

arg z = 5π/6, Arg z = π/6 Signup and view all the answers

Using P.M.I prove that $6^n - 1$ is divisible by 5.

Base case: $P(1)$: $6^1 - 1 = 5$ which is divisible by 5. Induction step: Assume true for $n = k$, thus $6^k - 1$ is divisible by 5. For $n = k + 1$, $6^{k+1} - 1 = 6 imes 6^k - 1 = 6(6^k - 1) + 6 - 1$. Since $6^k - 1$ is divisible by 5, then $6(6^k - 1) + 5$ is also divisible by 5. Signup and view all the answers

If $\alpha$ and $\beta$ are the roots of the equation $2x^2 - 3x - 1 = 0$, find the equation with integer coefficients whose roots are $\alpha - 1$ and $\beta - 1$.

$(x - (\alpha - 1))(x - (\beta - 1)) = 2x^2 - 5x + 3$ Signup and view all the answers

If $\alpha$ and $\beta$ are the roots of $2x^2 - 3x - 1 = 0$, find the equation whose roots are $\alpha^2$ and $\beta^2$.

$(x - \alpha^2)(x - \beta^2) = 2x^2 + 3x + 1$ Signup and view all the answers

Given that $2x$, 5 and $6 - x$ are the first three terms of an AP, what is the common difference $d$?

$d = 1$ Signup and view all the answers

The 35th term of an A.P. is 69. Find the sum of its 69 terms.

2405 Signup and view all the answers

Study Notes

Continuous Assessment Questions

Proving Divisibility

Using P.M.I, prove that 6n - 1 is divisible by 5.

Roots of Equations

If α and β are the roots of the equation 22 - 3x - 1 = 0, then:
- The equation with integer coefficients whose roots are α-1 and β-1 can be found.
- The equation with integer coefficients whose roots are α² and β² can be found.
- The equation with integer coefficients whose roots are:
  - α³ and β³
  - α¹⁰ and β¹⁰
  - α⁸ + β⁸ and α⁸ + β⁸

Arithmetic Progressions

Given that 2x, 5, and 6 - x are the first three terms of an AP, the common difference d can be found.
The 35th term of an AP is 69, so the sum of its 69 terms can be found.

Geometric Progressions

The sum of a G.P. is given as 127, and the terms are 64, 32, 16, ..., so the number of terms can be found.

Combinatorics

The number of ways of selecting 9 balls from 6 red balls, 5 white balls, and 5 blue balls, with each selection consisting of 3 balls of each color, can be found.

Trigonometry

The values of x in the interval [0,2π] for which 4sin²x = 1 can be found.
The exact value of sin θ can be found if cos θ = 5/7, and θ is:
- Acute
- Obtuse

Complex Numbers

The ordinary argument (arg z) and the principal argument (Arg z) of the complex numbers:
- z = (1-i)¹⁰
- z = -2+2√3i

Linear Equations

The values of x and y can be found if 3x + y = 243 and 22x - 5y = 8.

Binomial Expansion

The indicated terms in the following binomial expansions can be found:
- (a + w)¹³ - 6th term
- (a - b)¹⁴ - 6th term
- (2a² - b)²⁰ - 7th term
- (m + n)¹² - 7th term
- (m - n)¹⁶ - 8th term
- (3a + b)⁷ - 4th term

Studying That Suits You

Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

Description

Continuous Assessment Questions covering various math topics such as algebra, polynomial equations, and arithmetic progressions.