Consider the relations $R_{1}$ and $R_{2}$ defined as $aR_{1}b \Leftrightarrow a^{2} + b^{2} = 1$ for all $a,b \in R$ and $(a,b)R_{2}(c,d)$ $\Leftrightarrow a + d = b + c$ for all $(a,b),(c,d) \in N \times N$. Then

Let $Z$ be the set of all integers, $A = \left\{ (x,y) \in Z \times Z:(x - 2)^{2} + y^{2} \leq 4 \right\}$ and\\ If the total number of relations from to is , then the value of is

Let $R$ be a relation on $Z \times Z$ defined by $(a,b)R(c,d)$ if and only if is divisible by 5 . Then is

Let a relation $R$ on $N \times N$ be defined as: $\left( x_{1},y_{1} \right)R\left( x_{2},y_{2} \right)$ if and only if or . Consider the two statements :\\ (I) is reflexive but not symmetric. (II) is transitive Then which one of the following is true?

Let $R$ be the set of real numbers. \end{enumerate} Statement-1: $A = \{(x,y) \in R \times R:y - x$ is an integer is an equivalence . relation on .\\ Statement-2: for some rational number \} is . an equivalence relation on .

Let the sum of the first three terms of an A.P. be 39 and the sum of\n its last four terms be 178 . If the first term of this A.P. is 10 ,\n then the median of the A.P. is

Let $s_{1},s_{2},s_{3},\ldots\ldots..s_{10}$ respectively be the sum to 12 terms of 10 A.P. whose first terms are . 10 and the common difference are respectively. Then is equal to

Let $A_{1},G_{1}$ and denote the arithmetic, geometric and harmonic means . respectively of two distinct positive numbers. For let and have. arithmetic, geometric and harmonic means as respectively Which of the following statements is correct?

The sum of the first 20 terms of the series $5 + 11 + 19)\n (+ 29 + 41 + ....... is

Let $a_{1},a_{2},a_{3},\ldots a_{n},\ldots$ be in A.P. If , then the sum of its first 17 terms is equal to

Given an A.P. whose terms are all positive integers. The sum of its first nine terms is greater than 200 and less than 220 . If the second term in it is 12 , then its $4^{\text{th}\text{~}}$ term is

The number of common terms in the progressions $4,9,14,19,\ldots\ldots$, up to $25^{\text{th}\text{~}}$ term and . . . ., up to term is :

Let the sum of an infinite G.P., whose first term is a and the common ratio is $r$, be 5 . Let the sum of its first five terms be $98/25$. Then the sum of the first 21 terms of an A.P., whose first term is $10ar,n^{\text{th}\text{~}}$ term is and the common difference is , is equal to

Let $AP(a;d)$ denote the set of all the terms of an infinite arithmetic progression with first term $a$ and common difference $d > 0$. If , then equals

If $a_{n} = \frac{3}{4} - \left( \frac{3}{4} \right)^{2} + \left( \frac{3}{4} \right)^{3} + \ldots( - 1)^{n - 1}\left( \frac{3}{4} \right)^{n}$\n and , then find the least natural number\n such that

Different A.P. are constructed with the first term 100 , the last term 199, and integral common differences. The sum of the common differences of all such A.P. having at least 3 terms and at most 33 terms is

Let $x$ be the arithmetic mean and $y,z$ be the two geometric\n means between any two positive numbers. Then\n $\frac{y^{3} + z^{3}}{xyz} =$

If the $m^{\text{th}\text{~}},n^{\text{th}\text{~}}$ and terms of an A.P and G.P. are equal and are then prove that

A man saves ₹ 200 in each of the first three months of his service. In each of the subsequent months his saving increases by ₹ 40 more than the saving of immediately previous month. His total saving from the start of service will be ₹ 11040 after

The sum of the squares of three distinct real numbers, which are in G.P. is $s^{2}$. n If their sum is $\alpha s$, show that $\alpha^{2} \in \left\lbrack \frac{1}{3},1 \right) \cup (1,3\rbrack$