Sum of 1 + 6 + 9(1²+2²+3²)/7 + 12(1²+2²+3²+4²)/9 + … Up to 15 Terms
The sum of the series 1+6+9(1²+2²+3²)/7+12(1²+2²+3²+4²)/9+15(1²+…+5²)/11+… up to 15 terms is: (1)7820 (2)7520 (3)7830 (4)7510.
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If Four Distinct Points With Position Vectors a, b, c, d Are Coplanar, Then Which of the Following Is True
If four distinct points with position vectors a,b,c,d are coplanar, which scalar triple product identity must hold? A worked derivation from the coplanarity condition.
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The Remainder When (2021)^2023 Is Divided by 7 Is
Find the remainder when (2021)^2023 is divided by 7 — a modular arithmetic problem solved by reducing the base first, then finding a repeating power cycle.
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Ellipse E1 With Minor Axis Equal to Next Ellipse’s Major Axis, Sum of Areas
Let E1: x²/9+y²/4=1. Ellipses Ei share E1’s centre and eccentricity, with minor axis of Ei equal to major axis of Ei+1. Find (5/π)ΣAi.
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Is f(x)=(2^x−2^−x)/(2^x+2^−x) One-One, Onto, Both or Neither?
The function f:(−∞,∞)→(−∞,1), defined by f(x)=(2^x−2^−x)/(2^x+2^−x), is: neither one-one nor onto, onto but not one-one, both, or one-one but not onto?
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Find Vector d Perpendicular to Both b and c, With d·a = 21
Given vectors a=4i+5j−k, b=i−4j+5k, c=3i+j−k, find vector d perpendicular to both b and c such that d·a=21, by solving a system of three linear equations.
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G.P. of Increasing Positive Terms, a1a5=28 and a2+a4=29, Find a6
Let a1,a2,a3,… be a G.P. of increasing positive terms. If a1a5=28 and a2+a4=29, then a6 is equal to: (A) 628 (B) 812 (C) 526 (D) 784.
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Roots of x²−√2x−√3=0, Pn=αⁿ−βⁿ, Find (11√3−10√2)P10+(11√2+10)P11−11P12
Let α, β (α>β) be roots of x²−√2x−√3=0 and Pn=αⁿ−βⁿ. Find (11√3−10√2)P10+(11√2+10)P11−11P12 using the Newton recurrence for power sums.
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Seven Digit Numbers With Digit Sum 11, Using Only 1, 2 and 3
Let P be the set of seven digit numbers with sum of their digits equal to 11, formed using only digits 1, 2 and 3. Find the number of elements in P.
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The Sum to 10 Terms of the Series 1/(1+1²+1⁴) + 2/(1+2²+2⁴) + 3/(1+3²+3⁴) + … Is
The sum to 10 terms of the series 1/(1+1²+1⁴) + 2/(1+2²+2⁴) + 3/(1+3²+3⁴) + … is (A) 56/111 (B) 58/111 (C) 55/111 (D) 50/111.