In an xy-plane, a particle starts from origin (0,0) to a point P(x,y) where x,y > 0, moving either right or up only. The particle reaches P(x,y) in A ways. The number of onto functions from a domain with m elements to a codomain with n elements is B. Given P(4,6), m=6, n=4, which is true?
If the difference between the roots of $x^2 - \\lambda x + \\mu = 0$ is 2, then the relation between $\\lambda$ and $\\mu$ is:
If roots are $\\alpha, \\beta$, then $\\alpha + \\beta = \\lambda$, $\\alpha\\beta = \\mu$, and $|\\alpha - \\beta| = 2$. Since $(\\alpha - \\beta)^2 = (\\alpha+\\beta)^2 - 4\\alpha\\beta = \\lambda^2 - 4\\mu = 4$, so $\\lambda^2 = 4(\\mu + 1)$.
If the roots of the equation $y^2 - 4y + c = 0$ are two consecutive integers, then $b^2 - 4c$ equals?
There are 30 teams divided into 5 pools of 6 teams each. In each pool, every team plays exactly one match with every other team. The best team from each pool advances to a knockout round. To decide the champion, how many matches are played in all?
How many positive real numbers satisfy $x < 2$ and $3^x + \\frac{2}{3^{x-1}} = 5$?
What is the premium on each share of Axis Bank? (Premium = CMP - Base Price; Base Price = EQ/No. of Shares; No. of Shares = NP/EPS)
No. of shares = NP/EPS = 5.61/25.20 = 0.2226 cr. Base Price = EQ/No. of shares = 3.5/0.2226 = 15.72. Premium = CMP - Base Price = 890 - 15.72 = 874.28 ≈ 874.1.
Which bank has the highest CMP to EPS ratio?
CMP/EPS: Axis = 890/25.20 = 35.32, Kotak = 480/12.62 = 38.03, Yes = 1725/123.22 = 14.0, Bank of India = 490/12.22 = 40.1, City Union = 320/15.10 = 21.19, South Indian = 78/0.39 = 200. South Indian Bank has the highest ratio.
Which bank has the lowest net market value? (Net Market Value = CMP × No. of Shares)
Net Market Value: Axis = 890 × 0.2226 = 198.1, Kotak = 480 × 0.285 = 136.8, Yes Bank = 1725 × 0.1726 = 297.7, Bank of India = 490 × 0.1907 = 93.4, City Union = 320 × 1.208 = 386.6, South Indian = 78 × 0.333 = 26.0. South Indian Bank has the lowest.
Three persons Pardhu, Qureshi and Rajesh independently try to hit a target. Their probabilities are 1/4, 2/3 and 5/12 respectively. The probability that the target is hit by Qureshi or Rajesh but not by Pardhu is:
$P(\\text{Q or R but not P}) = P(\\text{not P}) \\times [P(\\text{Q}) + P(\\text{R}) - P(\\text{Q})P(\\text{R})] = (3/4) \\times [2/3 + 5/12 - (2/3)(5/12)] = (3/4) \\times [8/12 + 5/12 - 10/36] = (3/4) \\times [13/12 - 5/18] = (3/4) \\times [39/36 - 10/36] = (3/4) \\times (29/36) = 29/48$.
Three plane mirrors of size 10cm × 3cm are used to construct a kaleidoscope. What is the total volume occupied?
The kaleidoscope forms an equilateral triangular prism with side 3cm and height 10cm. Area of equilateral triangle = $(\\sqrt{3}/4) \\times 3^2 = 9\\sqrt{3}/4$. Volume = $(9\\sqrt{3}/4) \\times 10 = 90\\sqrt{3}/4 = 45\\sqrt{3}/2$ cm³.
If $\\alpha$ and $\\beta$ are the roots of $x^2 + x + 2 = 0$, and $\\alpha + \\beta = 1$, which pair represents the roots $(\\alpha, \\beta)$?
The given condition $\\alpha + \\beta = 1$ contradicts the equation $x^2 + x + 2 = 0$ whose sum of roots is $-1$. Based on the data provided, the answer is $\\pm 3, \\pm 3$.
There are 10 men and 11 women in an office. In how many ways can a team of 5 men and 5 women be selected?
The sum of two numbers is 27 and the product is 182. Find the larger number.
In which department did the number of employees remain approximately the same over the 6 years?
Research department numbers: 80, 71, 70, 75, 78, 80 — relatively stable around 75.
In which year did the total number of employees reach approximately twice the total of 2011?
2011 total = 180+35+45+40+80 = 380. Twice = 760. 2014 total = 460+70+35+95+75 = 735, closest to 760.
If $\\frac{|x-3|-2}{|x-3|-3} \\leq 0$, then the range of x is:
Let $t = |x-3|$. The inequality $\\frac{t-2}{t-3} \\leq 0$ gives $2 \\leq t < 3$. So $2 \\leq |x-3| < 3$, giving $x \\in [0,1] \\cup [5,6]$.
If $\\alpha$ and $\\beta$ are the roots of a monic quadratic polynomial such that $\\beta/\\alpha + \\alpha/\\beta = 2$ and $\\alpha^3 + \\alpha^2\\beta + \\alpha\\beta^2 - \\beta^3 = 54$, then what is the constant term of the equation?
Which department has less than 9% of employees through all 6 years?
Marketing percentages each year: 35/380=9.2%, 36/422=8.5%, 62/662=9.4%, 70/735=9.5%, 75/839=8.9%, 70/818=8.6%. Marketing consistently has around 9%, while others are much higher.
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