Find the equation of the line passing through the point of intersection of the circles $(x-y)^2 + (y-3)^2 = 7$ and $(x-3)^2 + (y-2)^2 = 5$.
The radical axis of the two circles gives the line through their intersection points. Subtracting the equations yields $x - y = 1$.
Two complex numbers $\\alpha$ and $\\beta$ are such that $\\alpha + \\beta = 2$ and $\\alpha^4 + \\beta^4 = 272$. Then the quadratic equation whose roots are $\\alpha$ and $\\beta$ is:
Let $\\alpha\\beta = p$. Using $\\alpha^2 + \\beta^2 = (\\alpha+\\beta)^2 - 2\\alpha\\beta = 4 - 2p$ and $\\alpha^4+\\beta^4 = (\\alpha^2+\\beta^2)^2 - 2(\\alpha\\beta)^2$, we get $272 = (4-2p)^2 - 2p^2$, giving $p = -8$. The quadratic is $x^2 - 2x - 8 = 0$.
A pack of cards consists of 9 cards numbered 1 to 9. Three cards are drawn at random without replacement. The probability of getting 2 even and 1 odd numbered cards lies between:
Even numbers: {2,4,6,8} = 4, Odd numbers: {1,3,5,7,9} = 5. Ways to choose 2 even and 1 odd: $\\binom{4}{2} \\times \\binom{5}{1} = 6 \\times 5 = 30$. Total ways: $\\binom{9}{3} = 84$. Probability = $30/84 = 0.3571$, which lies between 0.34 and 0.36.
A right angle triangle is formed by two rectangular coordinate axes and the straight line whose intercepts on the x and y axes are 4 and 2 respectively. Which of the points P(1,1), Q(1,2), R(2,2) lie(s) outside the triangle?
The line equation is $x/4 + y/2 = 1$ or $x + 2y = 4$. The triangle is bounded by axes and this line. A point is inside if $x \\geq 0, y \\geq 0, x + 2y \\leq 4$. P(1,1): $1+2=3 \\leq 4$ (inside). Q(1,2): $1+4=5 > 4$ (outside). R(2,2): $2+4=6 > 4$ (outside). Both Q and R lie outside the triangle.
There are ten straight parallel roads in a city named X_i (1 ≤ i ≤ 10). There are another twelve straight roads perpendicular to each X_i, named Y_j (1 ≤ j ≤ 12). One person wants to travel from the crossing of X_1 and Y_1 to the crossing of X_10 and Y_12. How many different shortest paths are there?
graph LR S((Start X₁Y₁)) -->|9E, 11N| E((End X₁₀Y₁₂)) style S fill:#f9f,stroke:#333,stroke-width:2px style E fill:#bbf,stroke:#333,stroke-width:2px
Suppose M and N are two different units of measuring temperature. If 3°M = 14°N, -27°M = 4°N, and 6°M = t°N, then what is t?
Suppose $x, y$ are integers such that $0 \\le x, y \\le 10$. How many ordered pairs $(x, y)$ satisfy $$\\frac{x+y}{x-y} - \\frac{x-y}{x+y}=0$$
Category Employees (2013) Prod. 300 Acc. 350 Res. 400 Mktg. 200 Other 996
Based on the given table, what is the pooled average of the total number of employees of all categories in the year 2013?
Total = 300+350+400+200+996 = 2246; average = 2246/5 = **1246**.
What is the difference between the total number of ASST consultants added to the company and the total number of Associate consultants during the years 2012-2016?
The difference calculated from the table data is 237.
A number is chosen at random from the first 80 natural numbers. What is the probability that the number chosen is a multiple of 3, 5, or 7?
Multiples of 3: 26, multiples of 5: 16, multiples of 7: 11. Using inclusion-exclusion: multiples of LCM(3,5)=15: 5, LCM(3,7)=21: 3, LCM(5,7)=35: 2, LCM(3,5,7)=105: 0. Total = 26+16+11-5-3-2 = 43. Probability = 43/80 = 0.5375.
Which of the following statements is correct?
The equation $|x - |x-2|| = 6$ has exactly 2 solutions.
A solid glass cube of side length 2m is placed inside a spherical globe such that each corner of the cube touches the surface of the globe. What is the volume (in cu m) of the region inside the globe but outside the cube?
Cube diagonal = space diagonal = $2\\sqrt{3}$ m, which equals the sphere's diameter. Sphere radius = $\\sqrt{3}$ m. Volume of sphere = $4\\pi r^3/3 = 4\\pi(3\\sqrt{3})/3 = 4\\pi\\sqrt{3}$. Cube volume = 8. Volume difference = $4\\sqrt{3}(\\pi - 2/\\sqrt{3}) = 4(3\\pi - 2)$.
If $x + \frac{1}{x} = 3$, then find the value of $$\\sum_{i=1}^{4} \left(x^{i}+\frac{1}{x^{i}}\right)$$
Suppose *m* is the number of real solutions of $x=\sqrt{x}$ and *n* is the number of real solutions of $x^3-3x^2+3x-1=0$. Find the values of *m* and *n*.
A password of length five digits is to be created using digits from {1, 3, 5, 7} such that exactly one digit is repeated exactly two times in consecutive positions only, and no other digit is repeated. How many such passwords are possible?
In which of the following years was the value of deployment of bank credit for dyes as a percentage of the total deployment greater than 10%? (Refer to the industry-wise bank credit table)
Calculating dyes/total for each year: 2018: 16425/125647 = 13.07%, 2019: 19240/153620 = 12.52%, 2020: 20368/165888 = 12.28%, 2021: 22450/216522 = 10.37%. All four years exceed 10%.
The ratio of the deployment of bank credit to petrochemicals as a percentage of total bank credit as on 30 March 2018, to the same percentage as on 28 March 2021 is:
2018: 1864/125647 = 1.483%. 2021: 6258/216522 = 2.89%. Ratio = 1.48 : 2.89.
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