16 Questions
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?
36
Is $3^4 = 27 + 3$ correct? If not, what is the correct equation?
No, $3^4 = 81$.
Find the 6th term in $(a + w)^{13}$.
$\binom{13}{5} a^8 w^5$
Find the 7th term in $(m + n)^{12}$.
$\binom{12}{6} m^6 n^6$
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
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)
Given $cos θ = 5/7$, what is the exact value of $sin θ$ if $θ$ is acute?
sin θ = (2√6)/7
Given $cos θ = 5/7$, what is the exact value of $sin θ$ if $θ$ is obtuse?
sin θ = - (2√6)/7
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
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
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.
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 $\beta - 1$.
$(x - (\alpha - 1))(x - (\beta - 1)) = 2x^2 - 5x + 3$
If $\alpha$ and $\beta$ are the roots of $2x^2 - 3x - 1 = 0$, find the equation whose roots are $\alpha^2$ and $\beta^2$.
$(x - \alpha^2)(x - \beta^2) = 2x^2 + 3x + 1$
Given that $2x$, 5 and $6 - x$ are the first three terms of an AP, what is the common difference $d$?
$d = 1$
The 35th term of an A.P. is 69. Find the sum of its 69 terms.
2405
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
Continuous Assessment Questions covering various math topics such as algebra, polynomial equations, and arithmetic progressions.
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