Remainder Theorem

Q. If ⁿC₁₂ = ⁿC₈ , then n =
(a) 20
(b) 12
(c) 6
(d) 30

Q. If we divide $(10x^4+17x^3-62x^2+30x-3)$ by $(2x^2+7x-1)$, then the quotient is __________.

1. $6x^3+21x^2-3x+5$
2. $5x^2-9x+3$
3. $5x^2-3x+9$
4. $5x^3-2x^2+9x-3$

Q. Write the remainder obtained when 1! + 2! + 3! + ... + 200! is divided by 14.

Q.

The set P = {x:x is a two digit number and sum of whose digits is 6}, written in roster form is:

1. P = {15, 24, 33, 42, 51, 60}

2. P = {24, 42}

3. P ={60, 61, 62 ----- 69}

4. P ={15, 51}

Q. ${\int }_0^3\left[x\right]dx=$ ______, where $\left[x\right]$ is greatest integer function.

1. $3$
2. $0$
3. $2$
4. $1$

Q.

Writethe following intervals in set-builder form:

(i) (–3, 0)

(ii) [6, 12]

(iii) (6, 12]

(iv) [–23, 5)

Q. If the given polynomial $f\left(x\right)=x^4+a{x}^3-5x^2+7x-6,$ is divided by $x-3$ and leaves remainder as $-3$ then the value of $a$ is

Q. Q-1 Find the remainder when 10^6 i s divided by 143 ? Q-2 Find the remainder When 5^100 is divided by 31 ?

Q. Let $r\left(x\right)$ be the remainder when the polynomial $x^{135}+x^{125}–x^{115}+x^5+1$ is divided by $x^3–x$. Then

1. degree of $r\left(x\right)$ is two
2. $r\left(x\right)$ is the zero polynomial
3. degree of $r\left(x\right)$ is one
4. $r\left(x\right)$ is a nonzero constant

Q. An $n-$digit number is a positive number with exactly $n-$digits. Nine hundred distinct $n-$ digit numbers are to be formed using only the three digits $2,5\text{and}7$. The smallest value of $n$ for which this is possible is

1. $6$
2. $9$
3. $8$
4. $7$

Q. Polynomial $P\left(x\right)$ contains only terms of odd degree. When $P\left(x\right)$ is divided by $(x-3)$, the remainder is $6$. If $P\left(x\right)$ is divided by $(x^2-9)$, then the remainder is $g\left(x\right)$. Then the value of $g\left(2\right)$ is

Q. The remainder when $4x^3+2x^2-5x+7$ is divided by $x-2$ is

1. $-8$
2. $8$
3. $37$
4. $-37$

Q. Let $x^2+x+1$ is divisible by $3$. If $x$ is divided by $3$, the remainder will be

Q. Simplify: - ( root of -7/4) -( root of -1/7)Ans given: [ -9( root of 7 ) i ]/14

Q. Let $x^2+x+1$ is divisible by 3. If $x$ is divided by 3, the remainder will be

1. 1
2. None of these
3. 0
4. 2

Q. If $p\left(x\right)$ is a polynomial of degree greater than $2$ such that $p\left(x\right)$ leaves remainder $a$ and $-a$ when divided by $x+a$ and $x-a$ respectively. If $p\left(x\right)$ is divided by $x^2-a^2$ then remainder is

1. $-x$
2. $-2x$
3. $2x$
4. $x$

Q. What is the remainder when 17to the power 5 is divided by 9.

Q. Let $P\left(x\right)=x^2+bx+c$, where $b$ and $c$ are integers. If $P\left(x\right)$ is a facter of both $x^4+6x^2+25$ and $3x^4+4x^2+28x+5$, then value of $P\left(1\right)$ is -

1. $12$
2. $8$
3. $10$
4. $4$

Q.

Write the following intervals in set-builder form:

(i) (–3, 0)

(ii) [6, 12]

(iii) (6, 12]

(iv) [–23, 5)

Q.

If $x^2+px+1$ is a factor of the expression $a{x}^3+bx+c$, then

1. $a^2+c^2-ab=0$

2. $a^2-c^2=ab$

3. $a^2+c^2=-ab$

4. $a^2-c^2=-ab$

Q. If a polynomial $p\left(x\right)$ is divided by $(x-a)$, then the remainder obtained is $p\left(a\right).$

1. True
2. False