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Standard XII

Mathematics

Inductive Step

Let a1, a2,...

Question

Let $a_{1}, a_{2}, a_{3}, a_{4}$ be real numbers such that and $a_{1}^{2} + a_{2}^{2} + a_{3}^{2} + a_{4}^{2} = 1$ . Then the smallest possible value of the expression $(a_{1} - a_{2})^{2} + (a_{2} - a_{2})^{2} + (a_{3} - a_{4})^{2} + (a_{4} - a_{1})^{2}$ lies in interval .

(0.15, 2)

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(- 1.5, 2.5)

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(2.5, 3)

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(3, 3.5)

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Solution

The correct option is B $(- 1.5, 2.5)$
Smallest value of $(a_{1} - a_{2})^{2} + (a_{2} - a_{3})^{2} + (a_{3} - a_{4})^{2} + (a_{4} - a_{1})^{2}$ is $0$ , when $a_{1} = a_{2} = a_{3} = a_{4} = \frac{1}{2}$ .

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Similar questions

Q. Let

a_{1}, a_{2}, a_{3}, a_{4}

be real numbers such that

a_{1} + a_{2} + a_{3} + a_{4} = 0

and

a_{1}^{2} + a_{2}^{2} + a_{3}^{2} + a_{4}^{2} = 1

. Then the smallest possible value of the expression

(a_{1} - a_{2})^{2} + (a_{2} - a_{3})^{2} + (a_{3} - a_{4})^{2} + (a_{4} - a_{1})^{2}

lies in the interval

Q. If

a_{1}

a_{2}

a_{3}

a_{4}

and

b

are real numbers such that

(a_{1}^{2} + a_{2}^{2} + a_{3}^{2}) b^{2} - 2 (a_{1} a_{2} + a_{2} a_{3} + a_{3} a_{4}) b + (a_{2}^{2} + a_{3}^{2} + a_{4}^{2}) \leq 0

then

a_{1}

a_{2}

a_{3}

a_{4}

are the terms of a/an

Q. If

a_{1}, a_{2}, a_{3}, \dots

are in A.P. and

a_{1}^{2} - a_{2}^{2} + a_{3}^{2} - a_{4}^{2} + \dots + a_{2 k - 1}^{2} - a_{2 k}^{2}

= M (a_{1}^{2} - a_{2 k}^{2})

. Then M =

Q. Let the sequence

a_{1}, a_{2}, a_{3} . . .

form an

A . P .

then the value of

a_{1}^{2} - a_{2}^{2} + a_{3}^{2} - a_{4}^{2} + \dots + a_{2 n - 1}^{2} - a_{2 n}^{2}

is equal to

Let the sequence $a_{1}, a_{2}, a_{3} . . . . . a_{n}$ form an A.P. Then
$a_{1}^{2} - a_{2}^{2} + a_{3}^{2} - a_{4}^{2} + . . . . . + a_{2 n - 1}^{2} - a_{2 n}^{2}$ is equal to

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Let a1,a2,a3,a4 be real numbers such that and a21+a22+a23+a24=1. Then the smallest possible value of the expression (a1−a2)2+(a2−a2)2+(a3−a4)2+(a4−a1)2 lies in interval .

The correct option is B (−1.5,2.5)Smallest value of (a1−a2)2+(a2−a3)2+(a3−a4)2+(a4−a1)2 is 0, when a1=a2=a3=a4=12.

Let $a_{1}, a_{2}, a_{3}, a_{4}$ be real numbers such that and $a_{1}^{2} + a_{2}^{2} + a_{3}^{2} + a_{4}^{2} = 1$ . Then the smallest possible value of the expression $(a_{1} - a_{2})^{2} + (a_{2} - a_{2})^{2} + (a_{3} - a_{4})^{2} + (a_{4} - a_{1})^{2}$ lies in interval .

The correct option is B $(- 1.5, 2.5)$
Smallest value of $(a_{1} - a_{2})^{2} + (a_{2} - a_{3})^{2} + (a_{3} - a_{4})^{2} + (a_{4} - a_{1})^{2}$ is $0$ , when $a_{1} = a_{2} = a_{3} = a_{4} = \frac{1}{2}$ .