Integers Between 100 and 1000 With Digit Sum Equal to 14
Count the 3-digit integers between 100 and 1000 whose digits sum to 14, using stars-and-bars with inclusion-exclusion for the digit-range constraints.
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Ellipse E and Hyperbola H Share Focal Distance 2√3, a−A=2 — Sum of Latus Rectums
Ellipse E and hyperbola H have the same distance between their foci (2√3), with a−A=2 and eccentricity ratio 1/3. Find the sum of their latus rectum lengths.
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αx+βy=109 Is Chord of x²/9+y²/4=1 With Midpoint (5/2,1/2), Find α+β
Given αx+βy=109 is the equation of a chord of x²/9+y²/4=1 with midpoint (5/2,1/2), find α+β using the T=S₁ chord-of-midpoint formula.
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Direction Cosines of the Line 5x−3=15y+7=3−10z
Find the direction cosines of 5x−3=15y+7=3−10z and a point through which it passes, by converting to symmetric form with equal-magnitude denominators.
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Two Persons A and B Throw a Pair of Dice Alternately. A Wins if He Throws a Sum of 5 Before B Throws a Sum of 8. If A Throws First, the Probability That A Wins Is
A and B alternately throw a pair of dice; A wins by throwing sum 5 before B throws sum 8. Full solution using independent events and an infinite GP.
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Let the Common Tangents to the Curves 4(x² + y²) = 9 and y² = 4x Intersect at the Point Q. An Ellipse Centred at O Has Semi-Minor and Semi-Major Axes Equal to OQ and 6. If e and l Are Its Eccentricity and Latus Rectum, Then l/e² Is
The common tangents to 4(x²+y²)=9 and y²=4x meet at Q. An ellipse has semi-minor axis OQ and semi-major axis 6. Find l/e² for this ellipse.
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A Committee of 4 Students Is Selected at Random From a Group Consisting of 8 Boys and 4 Girls. Given That There Is at Least One Girl in the Committee, Find the Probability That There Are Exactly 2 Girls
A committee of 4 is chosen from 8 boys and 4 girls. Given at least one girl is selected, find the probability of exactly 2 girls, using the conditional probability formula.
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Circle Through A(2,−1), B(3,4), Center on (x−5)²+(y−1)²=13/2, Find r²
Circle C passes through A(2,−1) and B(3,4), AB not a diameter, and its centre lies on (x−5)²+(y−1)²=13/2. Find r² using the perpendicular-bisector property.
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If A and B Are the Points of Intersection of the Circle x² + y² − 8x = 0 and the Hyperbola x²/9 − y²/4 = 1, and a Point P Moves on the Line 2x − 3y + 4 = 0, Then the Centroid of Triangle PAB Lies on the Line
A and B are intersection points of a circle and hyperbola; P moves on 2x−3y+4=0. Find the line on which the centroid of triangle PAB always lies.
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Binomial (1+x)^(n+4): 5th, 6th, 7th Coefficients in AP, Find Largest Coefficient
For some n≠10, the 5th, 6th and 7th coefficients of (1+x)^(n+4) are in AP. Find n, then the largest coefficient of the expansion.