/prove that:- Let G be agraph. Then X(G) 3 iff G has an odd cycle.

Find the area bounded by f(x) = sin x, g(x) = cos x in the first quadrant.

1. Matrix Operations Given: A = [ 33 ]A-[3-321] -3 B = [342]B-[3-41-2] (a) A² A2 Multiply A× A: -3 = (3 x 32x-3) (3 x 22 x 1) | = |[19–63 |-9-3 -6+21] = A² = 33 33 1-3×3+1x-3) (-3×2+1x1) [12]A2=[3-321][3-321]=[(3×3+2x-3)(-3×3+1x-3)(3×2+2×1)(-3×2+1×1)]=[9-6-9-36+2-6+1 ]=[3-128-5] (b) | A ||A| Determinant of A | A | (3 × 1) (2 x-3)=3+ 6 = 9|A|=(3×1)-(2x-3)=3+6=9 (c) Adjoint of A Swap diagonal elements and change sign of off-diagonals: A = [33], so adj (A) = |¯²]A=[3-321], so adj(A)=[13–23] -3 (d) B-¹B-1 First find | B ||B|: |B | (3x-2)- (1 × -4) = -6 + 4 = −2|B|=(3x-2)-(1x-4)=-6+4=-2 Then the adjoint of B: adj (B) = [² 3 adj(B)=[-24-13] Now, B-1 1 = |B| · adj (B) = 1 [²¯¯³¹³] = [2₂ B 0.5 |B-1=|B|1-adj(B)=-21[-24-13]=[1-20.5-1.5] 2. (a) Matrix Method: Solve (2x + 3y = 6 (2x-3y=14 {2x+3y=62x-3y=14 Matrix form: 22 33-22 = [223-3][xy]=[614] Find inverse of coefficient matrix: Determinant: | M | (2x-3) - (3 x 2) = -6 -6 = -12|M|=(2x-3)-(3×2)=-6-6=-12 Adjoint: adj(M) = [3]adj(M)-[-3-2-32] So…

Questions An insurance company's cumulative incurred claims for the last 5 accident years are given in the following table: Development Year Accident Year 0 2018 1 2 3 4 245 267 274 289 292 2019 255 276 288 294 2020 265 283 292 2021 263 278 2022 271 It can be assumed that claims are fully run off after 4 years. The premiums received for each year are: Accident Year Premium 2018 306 2019 312 2020 318 2021 326 2022 330 You do not need to make any allowance for inflation. 1. (a) Calculate the reserve at the end of 2022 using the basic chain ladder method. (b) Calculate the reserve at the end of 2022 using the Bornhuetter-Ferguson method. 2. Comment on the differences in the reserves produced by the methods in Part 1.

The numbered disks shown are placed in a box and one disk is selected at random. Find the probability of selecting a 4, given that a green disk is selected. Find the probability of selecting a 4, given that a green disk is selected. (Type an integer or a simplified fraction.) green blue green green green blue green blue

The table shows the distribution, by age, of a random sample of 3160 moviegoers ages 12-74. If one moviegoer is randomly selected from this population, find the probability, expressed as a simplified fraction, that the moviegoer is not in the 65-74 age range. The probability is (Type an integer or a simplified fraction.) Age Distribution of Moviegoers Ages Number 12-24 1090 25-44 860 45-64 890 65-74 320

Use the spinner shown. It is equally probable that the pointer will land on any one of the six regions. If the pointer lands on a borderline, spin again. If the pointer is spun twice, find the probability that it will land on yellow and then yellow. Find the probability that the spinner will land on yellow and then yellow. The probability is (Type an integer or a simplified fraction.) Green Red Gray Red Blue Yellow Q ☑

Use the spinner shown to answer the question. Assume that it is equally probable that the pointer will land on any one of the colored regions. If the pointer lands on a borderline, spin again. If the spinner is spun once, find the probability that the pointer lands in a region that is red or green. The probability that the pointer lands in a region that is red or green is (Type an integer or a simplified fraction.) green red green red yellow red

Let $f(x)$ be a continuous function on the interval $[0,1]$ such that $f(0) = f(1) = 0$. Prove that for any positive integer $n$, there exists a real number $x$ in $[0, 1 - \frac{1}{n}]$ such that $f(x) = f(x + \frac{1}{n})$.