New chat PYQ Library Doubts Rank report

Sign in to see Recents

Your guest activity stays on this device

Sign in to save progress →

Binomial Theorem question

2023 · 10 Apr · Shift 2 · Q23

Guest · filters and generic practice availableBrowsing as a guest · PYQ filters and generic practice are available. Sign in only for personalised features and saved progress.

PYQ Library
JEE Main
Mathematics
Binomial Theorem
2023 · 10 Apr · Shift 2 · Q23

Binomial Theorem question

2023 · 10 Apr · Shift 2 · Q23

JEE MainMathematicsBinomial TheoremMCQ+4 / −1

Let the number

(22)^{2022}+(2022)^{22}

leave the remainder

\alpha

when divided by 3 and

\beta

when divided by 7. Then

\left(\alpha^{2}+\beta^{2}\right)

is equal to :

A
13
B
10
C
20
D
5

View written solutionFree

Correct answer: D

Let $N=(22)^{2022}+(2022)^{22}.$ We need the remainder of $N$ when divided by $3$ and by $7$ .
Find $\alpha$ : remainder when $N$ is divided by $3$ .

Since $22 \equiv 1 \pmod{3}$ , $22^{2022} \equiv 1^{2022}=1 \pmod{3}.$
Also, $2022 \equiv 0 \pmod{3}$ , $2022^{22} \equiv 0^{22}=0 \pmod{3}.$

Therefore, $N \equiv 1+0=1 \pmod{3}.$ So, $\alpha=1.$

Find $\beta$ : remainder when $N$ is divided by $7$ .

Since $22 \equiv 1 \pmod{7}$ , $22^{2022} \equiv 1^{2022}=1 \pmod{7}.$
Also, $2022 \equiv 6 \pmod{7}$ , so $2022^{22} \equiv 6^{22} \pmod{7}.$ Now $6 \equiv -1 \pmod{7}$ , hence $6^{22} \equiv (-1)^{22}=1 \pmod{7}.$

Therefore, $N \equiv 1+1=2 \pmod{7}.$ So, $\beta=2.$

Compute $\alpha^2+\beta^2=1^2+2^2=1+4=5.$
Compare with options:

A: $13$
B: $10$
C: $20$
D: $5$

Hence the correct option is $\boxed{\text{D: }5}.$

More from Binomial Theorem

If the coefficients of $x$ and $x^{2}$ in $(1 + x)^{p} (1 - x)^{q}$ are 4 and $-$ 5 respectively, then $2 p + 3 q$ is equal to :2023 · MCQ
The mean of the coefficients of $x, x^{2}, \dots, x^{7}$ in the binomial expansion of $(2 + x)^{9}$ is .2023 · Numerical
The number of integral terms in the expansion of $(3^{\frac{1}{2}} + 5^{\frac{1}{4}})^{680}$ is equal to .2023 · Numerical
The sum of the coefficients of three consecutive terms in the binomial expansion of $(1 + x)^{n + 2}$ , which are in the ratio $1 : 3 : 5$ , is equal to :2023 · MCQ
If the $101 1^{th}$ term from the end in the binominal expansion of $(\frac{4 x}{5} - \frac{5}{2 x})^{2022}$ is 1024 times $101 1^{th}$ R term from the beginning, then $∣ x ∣$ is equal to2023 · MCQ
Fractional part of the number $\frac{4 ^{2022}}{15}$ is equal to2023 · MCQ
Let $α$ be the constant term in the binomial expansion of $(x - \frac{6}{x ^{\frac{3}{2}}})^{n}, n \leq 15$ . If the sum of the coefficients of the remaining terms in the expansion is 649 and the coefficient of…2023 · Numerical
The coefficient of $x^{5}$ in the expansion of $(2 x^{3} - \frac{1}{3 x ^{2}})^{5}$ is :2023 · MCQ