14) (x + 2)(x – 3)(x - 6) < 0 +++++ ) -10-9 -8-7-6 -5 -4 -3 -2 -1 01 2 3 45 6789 10

Q: 1. Prove that 7(x – 1) 2.

A: Let, f(x)= ex-7x+7 ,for x≥2. Differentiating both sides with respect to x, f'(x)= ex-7≥0 . This…

Q: How many positive integers less than 10000 are there which contain at least one 7 or at least one 9…

A:

Q: 2. Prove that " The sum of two odd numbers is even: using direct proof.

A:

Q: How many 6-digit numbers are there where no three con- secutive digits are the same? (For values…

A:

Q: Prove 3 | 17k+1-2*5k+3+11k+3 for k>=0 (nonnegative integer)

A: Here in this problem we have to prove that for k≥0 i.e k is a non negative integer…

Q: Expand and simplify. (8h +3) (7h² – 4h+1) - O 56h3 – 32h² + 8h + 3 - O 56h3 – 11h? 4h + 3 O 56h3 +…

A:

Q: 13. Prove 7 + √2 irrational

A: We need to prove that 7+2 is irrational number.  On the contrary, assume that 7+2 is a rational…

Q: (a) In (x - 3) = 7 (b) In (x + 4) + In (x- 4) = 1

A: Given(a) In(x-3) = 7(b) In(x+4) +In(x-4) =1solve for x(a) In(x-3) = 7In(x-3) =7take Ine  both…

Q: 8. How many positive integers less than 1000 a) have exactly three decimal digits? b) have an odd…

A: According to our guideliness we can answer only the first 3 subparts. Positive integers less than…

Q: How would you prove for any integer n, n(n² - n)is proof by counter example proof by cases since n…

A: We need to select the correct way of proving that for any integer n, n(n2-n) is even.

Q: Select all the expressions that are greater than 3. Be sure to show your work 1.67 + 1.4 2.97 +…

A: We have to select all the expressions that are greater than 3. (1) 1.67 + 1.4  = 3.07 &gt; 3

Q: Suppose we have L-shaped tiles made up of three squares: Prove that for all n > 1, it's possible to…

A: Suppose we have L shaped tiles Prove that for all n&gt;=1 it's possible to lay these tiles down to…

Q: -, 1+2+2 '+.+ 2 -1 for all natural numbers n.

A: We will use the principle of mathematical induction to prove the result.

Q: -9-8-7-6 -5 -4 -3 -2 1 2 3 4 5 6789 -2- -3 -4 -5- -6- -7 -8 -9 87 # 00N1

A: Given that, There is a graph in which a streight line is passes. We have to find the equation of the…

Q: How many integers greater than 5400 and less than 10000 have both of the following properties: -…

A: To count the number of positive integers satisfying all the required properties

Q: If 2x³ - 4x² + x + 1 = 0 so x = 4x² - 2x³ - 1 then |x5-x4] < €, X4 = 1.36603, x5 = 1.366024, is the…

A: As per the question we are given an equation and the approx root after 4th and 5th iteration of the…

Q: Prove that an odd number multiplied by an odd number is odd.

A: The objective is to prove that an odd number multiplied by an odd number is odd.

Q: 2) Express 100C50 as a sum of two terms from Pascal's triangle.

A: Let us find thatThe  100C50 as a sum of two terms from Pascal's triangle.we know that the formula of…

Q: 4. Write in a simple form (2 In (x + 3) + In x – In (x² – 1)]

A: As per our guidelines we are supposed to answer only one question.* *Kindly repost other question as…

Q: Prove that for any 9 consecutive natural numbers, there exists one number that is prime to the other…

A:

Q: 10 (Σ υ)- (Σ=--) 100 k=1 k=1

A:

Q: What two nonnegative real numbers with a sum of 23 have thelargest possible product?

A: we have to find the two non negative real number with a sum of 23 which will have largest possible…

Q: (i) Deduce that the difference in the number of integers in the numerator and denominator of (* 2N N…

A: Properties of floor functions will be used.

Q: 21. How many positive integers x are there such that 3x has 3 digits and 4x has four digits?

A:

Q: prove that "The Sum of two odd numbers is even: using direct. Note:- 1 want the ansawer on a paper.

A: The sum of two odd numbers is even. Proof :          Properties odd and even integer.          If…

Q: Prove 3√3  is irrational

A: See the attachment.

Q: Show that there are infinitely many prime numbers.  0 /∈ N. (zero is not an element of natural…

A:

Q: If 2x³-4x²+x+1=0 so x = 4x² - 2x³ - 1 then x5-x4|< €, X4 = 1.36603, x5 = 1.366024, is the error…

A:

Q: 3. How many zeros at the end does the number 2022! have? (n! is a product of all positive integers…

A: Given,Number = 2022!To find: Number of zeros at the end (trailing zeros) of the given number  As per…

Q: Prove that if the sum of the digits is subtracted from a natural number, then the result is…

A: let us assume the number to be three digit number xyz therefore, xyz=100x+10y+z sum of digits will…

Q: : a number is even  if

A:

Q: Prove that a four-digit number is a multiple of 8 whenever the number formed by the first three…

A: The solution is given by using the expanded form of number as follows

Q: How many positive odd integers less than 10,000 can be written using the digits 3, 4, 6, 8, and 0 if…

A:

Q: How many positive integers not exceeding 3000 are divisible by exactly two of 4, 10, 12? (Steps…

A:

Q: - 1, 1, 9; degree: 3 omial with integer coeffi

A: Is an expression consisting of variables (also called indeterminants) and coefficients, that…

Q: (a) A team of 5 debaters of IUB is to be chosen from a section where we have 6 male and 4 female…

A: As per our guidelines we can answer only 1 question so kindly repost the remaining questions…

Q: Prove that 1 1 1 + 9.11 - 8 1.3 5.7

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Q: If x is an odd integer then 5x – 7 is an even integer. (use direct proof)

A: Given that “x” is an odd integer then “5x-7” is an even integer.

Q: 13 of 20 limz→-6 746 a+6 Does not exist 8.

A: Let's find.

Q: Which term of the expansion of (x + 7)° contains x° Term number contains x°.

A:

Q: sum of two even integers is even

A:

Q: Prove that a natural number (an-1an-2a1ao)10 is divisible by 9 if and only if the sum of the digits…

A:

Q: 3. There are seven students who always compare their CS test grades. They discovered that either the…

A: It is given that there are seven students who always compare their CS test grades. They discovered…

Q: 7, Evaluate the integral Express numbers in exact form.use symbolic notation and fractions where…

A:

Q: How many 5-tuples (x1, x2, x3, x4, X5) of positive integers from 1 to n are there? How many 5-tuples…

A: Since you have asked multiple questions, we will solve the Question Q1 (a) and (b) and (c) for you.…

Q: How many natural number that are less or equal to 1000 and that are not multiplea of 3,5or7 exist

A: To find: Number of natural numbers ≤1000 that are not multiple of 3 or 5 or 7.

Q: If n is an even integer then 7-n3 is sometimes even and sometimes odd. O is an odd integer is an…

A: We have to state whether the 7-n3 is an even integer, odd integer, or sometimes even and sometimes…

Q: 15 of 20 -722 –8x =? limz→00 3–72²+6x+6 O Does not exist O 00 O 7

A:

Q: Suppose that n students take a quiz and their scores sum to 150. What is the smallest number n to…

A: Pigeonhole Principle: The pigeonhole principle states that if n items are put into m containers with…