KUMON LEVEL N TEST EXAM 2024 WITH 100% CORRECT ANSWERS

1. Given the fixed points F(0 ,√5) and F' (0 ,-√5), obtain the equation of the locus point P(x, y), such that PF +PF' = 8 Answer - √[x^2+(y-V5)^2]+√[x^2+(y+V5)^2]=8 x^2/11+y^2/16 = 1 2 Graph the region satisfying the given system of inequalities, and then state the range of values x and y x-y+1 ≥ 0 5-2x-y ≥ 0 Answer - all real numbers for x, y ≤ 7/3 3. Obtain the 30th term of an arithmetic sequence whose 5th term is -3 and whose 14th term is 24. Answer - a+4d=-3, a+13d=24, a=-15, d=3 72 4. Obtain the sum of the following arithmetic sequence. -15, -10, -5, 0, ... (through the 12th term) Answer - 150 5. 4 Σ (3k - 2) = k=1 Answer - 22 6. Obtain the 1st term and common ratio of a geometric sequence whose 5th term is 24 and 7th term is 96 First term: Common ratio: Answer - ar^4=24, ar^6=96 1st term: 3/2 Common ratio: ±2 7. Given a geometric sequence whose sum of the first 10 terms is 4, and whose sum of the terms from the 11th through 30th term is 48, obtain the sum of the terms from the 31st through the 60th term. Answer - S10=a(r^10-1)/r-1=4 S30=a(r^30-1)/r-1=52 r^10=3, r^60= 1456 1404 8. Given the sequence [an]: 2, 5, 10, 17, 26, 37, 50, ..., obtain the general term an, and the sum of the first n terms, Sn Answer - bn=2n+1, an=n^2+1 an = n^2+1 Sn = n/6(2n^2 +3n +7) 9. lim 2n^2 - 1/3n^2 +4 = Answer - 2-1/n^2/3+4/n^2 2/3 10. lim √3n/√2n+1 = Answer - √3/√2+1/n √6/2 11. lim 5^n + 2^n/3^2n +4^n = Answer - (5/9)^n+(2/9)^n/1+(4/9)^n 0 12. Evaluate the following limit when r>0 lim r^n+2/r^n + 1/r^n Answer - lim (1+2*1/rn)/(1+1/r2n), lim r2n+2rn/r2n+1 r>1, limit 1 r=1, limit 3/2 0<r<1, limit 0 13. Determine the range of values of x at which the given infinite geometric series diverges, and then find its sum. x + x^2(x-2) + x^3(x-2)^2 + ... Range of values: Sum: Answer - Range of values of x: 1-√2<x<1, 1<x<1+√2 S= x/1+2x - x^2 14.State whether the following series converges or diverges. If it converges, determine the sum. (1/1) + (1/1+2) +(1/1+2