Three quantities x, y, z are in the proportion 3:5:7. What is the value of $\\frac{2x-3y+4z}{-x-2y+2z}$?
A number y is chosen from the set {1, 2, 3, ..., 120}. The probability that $y + \\frac{120}{y} > 23$ is:
$y + 120/y > 23 \\implies y^2 - 23y + 120 > 0 \\implies (y-8)(y-15) > 0$. So $y < 8$ or $y > 15$. Favorable: y = 1-7 (7) and 16-120 (105) = 112. Probability = 112/120 = 0.9333.
Find the length of the chord defined by $y = 1 + 2x$ on the circle $x^2 + y^2 = 5$.
Substitute $y = 1+2x$ into $x^2 + y^2 = 5$: $x^2 + (1+2x)^2 = 5$, giving $5x^2 + 4x - 4 = 0$. Distance formula for chord length = $\\frac{4\\sqrt{5}}{30}$.
If x, y, z are positive integers such that $x^8 = y^6$, $x^3 = z^5$, and $y^{27} = z^m$, then what is m?
If $\\alpha$ and $\\beta$ are the roots of $px^2 + qx + r = 0$, then $(\\alpha^5\\beta^8 + \\alpha^8\\beta^5)$ equals:
$(\\alpha\\beta)^5(\\alpha^3 + \\beta^3) = (r/p)^5[(\\alpha+\\beta)^3 - 3\\alpha\\beta(\\alpha+\\beta)] = (r/p)^5[(-q/p)^3 - 3(r/p)(-q/p)] = \\frac{qr^5}{p^8}(3pr - q^2)$.
The value of k satisfying $|k-12| \\geq k^2 - 9k + 12$ are in the range [a, b]. Find a and b.
Solving $|k-12| \\geq k^2 - 9k + 12$ gives the range $[0, 8]$, so a=0, b=8.
A five-letter password is to be formed using letters from the alphabet exactly once. If the first letter is not Z, the last is not A, and the 4th letter is alphabetically next to the 1st letter, how many such passwords exist?
Find the mode of the class marks: 39,30,5,65,5,21,5,65,89,30,91,30,64,64,91,15,89,48,30,64,85,30,69,51,32,85,48,91,30,28,68,48,65,30,65,35,65,48,48,15,91,30,91,48,71,15,65,48,71,38.
Counting frequencies: 30 appears 8 times, 48 appears 9 times, 65 appears 7 times. Mode = 48 (highest frequency).
Find the median of the class marks.
Arranging 50 numbers in ascending order, median is average of 25th and 26th values. The 25th value is 48 and 26th is 65... Actually the median is 51.
Find the arithmetic mean of the class marks.
Sum of all marks divided by 50 = 2515/50 = 50.3.
Eight people A, B, C, D, E, F, G, H sit around a round table facing the center. If A does not sit adjacent to B, in how many ways can they sit?
If the roots of $x^2 + ax + b = 0$ are $\\tan 15^\\circ$ and $\\tan 30^\\circ$, then the value of $2 + b - a$ is:
$\\tan 15^\\circ = 2-\\sqrt{3}$, $\\tan 30^\\circ = 1/\\sqrt{3}$. Sum = $2-\\sqrt{3}+1/\\sqrt{3} = 2-2/\\sqrt{3} = -a$. Product = $(2-\\sqrt{3})/\\sqrt{3} = 2/\\sqrt{3} - 1 = b$. Then $a = 2/\\sqrt{3} - 2$, $b = 2/\\sqrt{3} - 1$. $2+b-a = 2 + (2/\\sqrt{3}-1) - (2/\\sqrt{3}-2) = 2-1+2 = 3$.
Consider a right circular cylinder $(x-1)^2 + (y-2)^2 = 16$. What is the volume of the portion bounded by $y=0$ and $x+y+z=10$?
The cylinder has radius 4 and is centered at (1,2). The plane $y=0$ and $x+y+z=10$ cut the cylinder. Volume = 96$\\pi$ cubic units.
Let A and B be two independent events. If $P(A \\cap B) = 1/30$ and $P(A' \\cap B') = 2/3$, then $P(A)$ and $P(B)$ are:
Let $P(A)=x$, $P(B)=y$. Since independent, $xy = 1/30$. Also $(1-x)(1-y) = 2/3$ → $1 - x - y + xy = 2/3$ → $x+y = 1/3 + 1/30 = 11/30$. Solving: $x,y = 1/5, 1/6$.
If $(1+\\lambda^2)x^2 + 2\\lambda cx + (c^2 - a^2) = 0$ has equal roots, then $a^2(1+\\lambda^2) = $
For equal roots, discriminant = 0: $(2\\lambda c)^2 - 4(1+\\lambda^2)(c^2-a^2) = 0$ → $4\\lambda^2c^2 - 4(1+\\lambda^2)(c^2-a^2) = 0$ → $\\lambda^2c^2 - (1+\\lambda^2)c^2 + (1+\\lambda^2)a^2 = 0$ → $-c^2 + (1+\\lambda^2)a^2 = 0$ → $a^2(1+\\lambda^2) = c^2$.
A five-digit number is to be formed using 1, 2, 3, 4, 5 each exactly once, such that the number is divisible by 4. How many such numbers are there?
What is the value of x for $4 + 4\\sqrt{x-1} = 2\\sqrt{x+1}$?
Which of the following points lie in the region of inequalities $x \\geq 1$, $y \\geq 1$, and $x + y \\leq 6$?
(4,2): $4\\geq1$, $2\\geq1$, $4+2=6\\leq6$ ✓. (2,5): $2\\geq1$, $5\\geq1$, $2+5=7>6$ ✗. (3,2): $3\\geq1$, $2\\geq1$, $3+2=5\\leq6$ ✓. Only (4,2) and (3,2) individually satisfy the inequalities. Since the question asks which lie in the region, both (4,2) and (3,2) are valid. The correct answer is (4,2), the first option.
The equation $(k+3)x^2 - 10(k+3)x + 15(k+9) = 0$, $k \\neq 3$, has equal roots. Find k.
Find the ratio of the average sale of all branches for the years 2020 and 2021.
From the table data, the ratio is 76.5:82.
A metallic solid cube of side 4cm is melted and made into a hollow sphere of internal radius 3cm. What is the external radius of the sphere?
Volume of cube = $4^3 = 64$ cm³. Volume of hollow sphere = $\\frac{4}{3}\\pi(R^3 - 27) = 64$ → $R^3 = 48/\\pi + 27$. So $R = (48/\\pi + 27)^{1/3}$.
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