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Definition of Relations

100.Given tha...

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100.Given that x, a1, a2, y are in A.P. and x, b1, b2, b3, y are also in A.P., then the value of (a2-a1) /(b2-b3) if (x≠ y) (A) 1/2 (B) 3/4 (C) 4/3 (D) 1

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Q. Assertion :Suppose four distinct positive numbers

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

b_{1} = a_{1}, b_{2} = b_{1} + a_{2}, b_{3} = b_{2} + a_{3}

b_{4} = b_{3} + a_{4} .

STATEMENT- 1: The numbers

b_{1}, b_{2}, b_{3}, b_{4}

are neither in AP nor in GP, and Reason: STATEMENT- 1: The numbers

b_{1}, b_{2}, b_{3}, b_{4}

2 y - x = 8; 5 y - x = 14, y - 2 x = 1

(a_{1}, b_{1}), (a_{2}, b_{2}), (a_{3}, b_{3})

(a_{1} + a_{2} + a_{3} + b_{1} + b_{2} + b_{3})

Q. Suppose four distinct positive numbers

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

G . P

b_{1} = a_{1}, b_{2} = b_{1} + a_{2}, b_{3} = b_{2} + a_{3}

b_{4} = b_{3} + a_{4}

.
STATEMENT-1 : The numbers

b_{1}, b_{2}, b_{3}, b_{4}

A . P

G . P

.
STATEMENT-2 : The numbers

b_{1}, b_{2}, b_{3}, b_{4}

H . P

.

Q. Suppose four distinct positive number

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

are in G.P. Let

b_{1} = a_{1}, b_{2} = b_{1} + a_{2}, a_{3} = b_{2} + a_{3}

b_{4} = b_{3} + a_{4}

Statement 1 - The numbers

b_{1}, b_{2}, b_{3}, b_{4}

are neither in A.P. nor in G.P.
Statement 2 - The numaber

b_{1}, b_{2}, b_{3}, b_{4}

a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3} \in R

and are such that

a_{i} b_{j} \neq 1

1 \leq i, j \leq 3

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} \frac{1 - a_{1}^{3} b_{1}^{3}}{1 - a_{1} b_{1}} & \frac{1 - a_{1}^{3} b_{2}^{3}}{1 - a_{1} b_{2}} & \frac{1 - a_{1}^{3} b_{3}^{3}}{1 - a_{1} b_{3}} \frac{1 - a_{2}^{3} b_{1}^{3}}{1 - a_{2} b_{1}} & \frac{1 - a_{2}^{3} b_{2}^{3}}{1 - a_{2} b_{2}} & \frac{1 - a_{2}^{3} b_{3}^{3}}{1 - a_{2} b_{3}} \frac{1 - a_{3}^{3} b_{1}^{3}}{1 - a_{3} b_{1}} & \frac{1 - a_{3}^{3} b_{2}^{3}}{1 - a_{3} b_{2}} & \frac{1 - a_{3}^{3} b_{3}^{3}}{1 - a_{3} b_{3}} \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

> 0 Provided either

a_{1} < a_{2} < a_{3}

b_{1} < b_{2} < b_{3}

a_{1} > a_{2} a_{3}

b_{1} > b_{2} > b_{3}

(a_{1} - a_{2}) (a_{2} - a_{3}) (a_{3} - a_{1}) (b_{1} - b_{2}) (b_{2} - b_{3}) (b_{3} - b_{1}) < 0

,

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