Have a query? Ask a Tutor!!

Access to best educators and exclusive paper discussions

Offsite Campus Turito Championship

Can’t find what you’re looking for?

Our tutors are from top schools around the world, and they are waiting for you 24/7, anytime, anywhere.

Homework- One to One Solutions
Understand concepts to solve better
Get your doubts solved instantly

Maths-

General

Easy

Question

Let $R equals left parenthesis 2 plus square root of 3 right parenthesis to the power of 2 n end exponent$ and $f equals R minus left square bracket R right square bracket$ where $left square bracket bullet right square bracket$ denotes the greatest integer function, then $R open parentheses 1 minus f close parentheses$ is equal to

1
$2^{2 n}$
$2^{2 n} - 1$
$^{2 n} C_{n}$

The correct answer is: 1

Given that, $R equals left parenthesis 2 plus square root of 3 right parenthesis to the power of 2 n end exponent$ and $f equals R minus open square brackets R close square brackets$
As $0 less than 2 minus square root of 3 less than 1$ , we get $0 less than F equals left parenthesis 2 minus square root of 3 right parenthesis to the power of 2 n end exponent less than 1$
We have, $R plus F equals left parenthesis 2 plus square root of 3 right parenthesis to the power of 2 n end exponent plus left parenthesis 2 minus square root of 3 right parenthesis to the power of 2 n end exponent$
$equals 2 open square brackets blank to the power of 2 n end exponent C subscript 0 end subscript 2 to the power of 2 n end exponent plus blank to the power of 2 n end exponent C subscript 2 end subscript 2 to the power of 2 n minus 2 end exponent open parentheses square root of 3 close parentheses to the power of 2 end exponent plus blank to the power of 2 n end exponent C subscript 4 end subscript open parentheses 2 to the power of 2 n minus 4 end exponent close parentheses open parentheses square root of 3 close parentheses to the power of 4 end exponent plus... plus blank to the power of 2 n end exponent C subscript 2 n end subscript open parentheses square root of 3 close parentheses to the power of 2 n end exponent close square brackets$
$rightwards double arrow R plus F$ is an even integer
$rightwards double arrow open square brackets R close square brackets plus f plus F$ is an even integer
$rightwards double arrow f plus F$ is an integer
But, $0 less or equal than f less than 1 blank a n d blank 0 less than F less than 1$
$rightwards double arrow 0 less than f plus F blank less than 2$
But the only integer between 0 and 2 is 1. Thus,
$f plus F equals 1 blank rightwards double arrow blank 1 minus f equals F$
Now, $R open parentheses 1 minus f close parentheses equals R F equals left parenthesis 2 plus square root of 3 right parenthesis to the power of 2 n end exponent left parenthesis 2 minus square root of 3 right parenthesis to the power of 2 n end exponent$
$equals left parenthesis 4 minus 3 right parenthesis to the power of 2 n end exponent equals 1 to the power of 2 n end exponent equals 1$

Related Questions to study

If $left parenthesis 1 plus x plus x to the power of 2 end exponent right parenthesis to the power of n end exponent equals not stretchy sum from r equals 0 to 2 n of a subscript r end subscript x to the power of r end exponent comma blank t h e n blank a subscript 1 end subscript minus 2 a subscript 2 end subscript plus 3 a subscript 3 end subscript horizontal ellipsis minus 2 n a subscript 2 n end subscript$ is equal to

If $left parenthesis 1 plus x plus x to the power of 2 end exponent right parenthesis to the power of n end exponent equals not stretchy sum from r equals 0 to 2 n of a subscript r end subscript x to the power of r end exponent comma blank t h e n blank a subscript 1 end subscript minus 2 a subscript 2 end subscript plus 3 a subscript 3 end subscript horizontal ellipsis minus 2 n a subscript 2 n end subscript$ is equal to

Let $open square brackets x close square brackets$ denote the greatest integer less than or equal to $x$ . If $x equals open parentheses square root of 3 plus 1 close parentheses to the power of 5 end exponent$ , then $open square brackets x close square brackets$ is equal to

Let $open square brackets x close square brackets$ denote the greatest integer less than or equal to $x$ . If $x equals open parentheses square root of 3 plus 1 close parentheses to the power of 5 end exponent$ , then $open square brackets x close square brackets$ is equal to

$left parenthesis 2 to the power of 3 n end exponent minus 1 right parenthesis$ will be divisible by $left parenthesis for all n element of N right parenthesis$

$left parenthesis 2 to the power of 3 n end exponent minus 1 right parenthesis$ will be divisible by $left parenthesis for all n element of N right parenthesis$

parallel

The coefficient of $x to the power of 3 end exponent y to the power of 4 end exponent z to the power of 5 end exponent$ in the expansion of $left parenthesis x y plus y z plus x z right parenthesis to the power of 6 end exponent$ is

The coefficient of $x to the power of 3 end exponent y to the power of 4 end exponent z to the power of 5 end exponent$ in the expansion of $left parenthesis x y plus y z plus x z right parenthesis to the power of 6 end exponent$ is

Let $C subscript 1 end subscript comma blank C subscript 2 end subscript comma blank C subscript 3 end subscript$ are the usual binomial coefficients. Let $S equals C subscript 1 end subscript plus 2 C subscript 2 end subscript plus 3 C subscript 3 end subscript plus... plus n C subscript n end subscript$ , then $S$ equals

Let $C subscript 1 end subscript comma blank C subscript 2 end subscript comma blank C subscript 3 end subscript$ are the usual binomial coefficients. Let $S equals C subscript 1 end subscript plus 2 C subscript 2 end subscript plus 3 C subscript 3 end subscript plus... plus n C subscript n end subscript$ , then $S$ equals

The coefficient of $x$ in the expansion of $open parentheses 1 plus x close parentheses open parentheses 1 plus 2 x close parentheses open parentheses 1 plus 3 x close parentheses horizontal ellipsis horizontal ellipsis left parenthesis 1 plus 100 x right parenthesis$ is

The coefficient of $x$ in the expansion of $open parentheses 1 plus x close parentheses open parentheses 1 plus 2 x close parentheses open parentheses 1 plus 3 x close parentheses horizontal ellipsis horizontal ellipsis left parenthesis 1 plus 100 x right parenthesis$ is

parallel

$I f blank open vertical bar x close vertical bar less than fraction numerator 1 over denominator 2 end fraction comma blank t h e n blank t h e blank c o e f f i c i e n t blank o f blank x to the power of r end exponent blank i n blank t h e blank e x p a n s i o n blank o f fraction numerator 1 plus 2 x over denominator left parenthesis 1 minus 2 x right parenthesis to the power of 2 end exponent end fraction comma blank i s$

$I f blank open vertical bar x close vertical bar less than fraction numerator 1 over denominator 2 end fraction comma blank t h e n blank t h e blank c o e f f i c i e n t blank o f blank x to the power of r end exponent blank i n blank t h e blank e x p a n s i o n blank o f fraction numerator 1 plus 2 x over denominator left parenthesis 1 minus 2 x right parenthesis to the power of 2 end exponent end fraction comma blank i s$

In the expansion of $left parenthesis 1 plus x plus x to the power of 2 end exponent plus x to the power of 3 end exponent right parenthesis to the power of 6 end exponent comma$ then coefficient of $x to the power of 14 end exponent$ is

In the expansion of $left parenthesis 1 plus x plus x to the power of 2 end exponent plus x to the power of 3 end exponent right parenthesis to the power of 6 end exponent comma$ then coefficient of $x to the power of 14 end exponent$ is

A person can see clearly only upto 3m. What will be power of his spectacles lens, so that he can see clearly upto 12 m

A person can see clearly only upto 3m. What will be power of his spectacles lens, so that he can see clearly upto 12 m

Physics-General

parallel

The least distance of distinct vision for young adult with normal vision is about

The least distance of distinct vision for young adult with normal vision is about

Physics-General

A concave lens has a focal length of 20 cm. The power of lens will be

A concave lens has a focal length of 20 cm. The power of lens will be

Physics-General

The principle axis is also called ……………..of the lens.

The principle axis is also called ……………..of the lens.

Physics-General

parallel

A convex lens is also called a

A convex lens is also called a

Physics-General

An object 4 cm full is placed on the principle axis of a concave mirror of focal length 20 cm at a distance of 30 cm from it. The size of the image is

An object 4 cm full is placed on the principle axis of a concave mirror of focal length 20 cm at a distance of 30 cm from it. The size of the image is

Physics-General

The focal length of a convex mirror is 15 cm The radius of curvature of mirror is

The focal length of a convex mirror is 15 cm The radius of curvature of mirror is

Physics-General

parallel

card img

With Turito Academy.

card img

With Turito Foundation.

card img

Get an Expert Advice From Turito.

Turito Academy

card img

With Turito Academy.

Test Prep

card img

With Turito Foundation.