Arithmetic progression

Let $a,b,c$ be positive integers in arithmetic progression such that the equation
$$ax^2+bx+c=0$$
has only integer solutions.

Then which of the following statements is (are) TRUE ?

Let $A$ be the set of first 101 terms of an A.P., whose first term is 1 and the common difference is 5 and let $B$ be the set of first 71 terms of an A.P., whose first term is 9 and the common difference is 7. Then the number of elements in $A\cap B$, which are divisible by 3, is :

Let $a_1,a_2,a_3,\dots$ be an A.P. and $g_1=a_1,g_2,g_3,\dots$ be an increasing G.P. If $a_1=a_2+g_2=1$ and $a_3+g_3=4$, then $a_{10}+g_5$ is equal to :

The first term of an A.P. of 30 non-negative terms is $\dfrac{10}3$. If the sum of this A.P. is the cube of its last term, then its common difference is:

From a month of 31 days, 3 different dates are selected at random. If the probability that these dates are in an increasing A.P. is equal to $\dfrac ab$, where $a,b\in\mathbb N$ and $\gcd(a,b)=1$, then $a+b$ is equal to ________.

Let the sum of the first $n$ terms of an A.P. be $3n^2+5n$. Then the sum of squares of the first 10 terms of the A.P. is:

Let $A_1,A_2,A_3,\dots,A_{39}$ be 39 arithmetic means between the numbers 59 and 159. Then the mean of $A_{25},A_{28},A_{31}$ and $A_{36}$ is equal to :

The sum of the first ten terms of an A.P. is 160 and the sum of the first two terms of a G.P. is 8. If the first term of the A.P. is equal to the common ratio of the G.P. and the first term of the G.P. is equal to common difference of the A.P., then the sum of all possible values of the first term of the G.P. is:

Let $\alpha=3+4+8+9+13+14+\dots$ upto 40 terms. If $(\tan\beta)^{\frac\alpha{1020}}$ is a root of the equation $x^2+x-2=0$, $\beta\in\left(0,\dfrac\pi2\right)$, then $\sin^2\beta+3\cos^2\beta$ is equal to :

Let $\mathbb{R}$ denote the set of all real numbers. Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that $f(x) > 0$ for all $x \in \mathbb{R}$, and $f(x+y) = f(x)f(y)$ for all $x, y \in \mathbb{R}$.

Let the real numbers $a_1, a_2, \ldots, a_{50}$ be in an arithmetic progression. If $f(a_{31}) = 64f(a_{25})$, and

$$\sum_{i=1}^{50} f(a_i) = 3\left(2^{25} + 1\right),$$

then the value of

$$\sum_{i=6}^{30} f(a_i)$$

is _____.

Let $l_1, l_2, \ldots, l_{100}$ be consecutive terms of an arithmetic progression with common difference $d_1$, and let $w_1, w_2, \ldots, w_{100}$ be consecutive terms of another arithmetic progression with common difference $d_2$, where $d_1d_2 = 10$. For each $i = 1, 2, \ldots, 100$, let $R_i$ be a rectangle with length $l_i$, width $w_i$ and area $A_i$. If $A_{51} - A_{50} = 1000$, then the value of $A_{100} - A_{90}$ is _____.

Let $a_1, a_2, a_3, \ldots$ be an arithmetic progression with $a_1 = 7$ and common difference 8. Let $T_1, T_2, T_3, \ldots$ be such that $T_1 = 3$ and $T_{n+1} - T_n = a_n$ for $n \ge 1$. Then, which of the following is/are TRUE?

Let $S=\{\sqrt n:1\le n\le50$ and $n$ is odd$\}$.

Let $a\in S$ and $A=\begin{bmatrix}1&0&a\\-1&1&0\\-a&0&1\end{bmatrix}$.

If $\displaystyle\sum_{a\in S}\det(\operatorname{adj}A)=100\lambda$, then $\lambda$ is equal to :

If $\{a_i\}_{i=1}^n$, where $n$ is an even integer, is an arithmetic progression with common difference 1, and $\displaystyle\sum_{i=1}^na_i=192$, $\displaystyle\sum_{i=1}^{n/2}a_{2i}=120$, then $n$ is equal to :

The sum of all the elements of the set $\{\alpha\in\{1,2,\dots,100\}:\mathrm{HCF}(\alpha,24)=1\}$ is ______.

If $b_n=\displaystyle\int_0^{\frac\pi2}\frac{\cos^2nx}{\sin x}dx$, $n\in\mathbb N$, then

Let $A=\{n\in\mathbb N:\text{H.C.F.}(n,45)=1\}$ and Let $B=\{2k:k\in\{1,2,\dots,100\}\}$. Then the sum of all the elements of $A\cap B$ is ______.

If $x=\displaystyle\sum_{n=0}^\infty a^n$, $y=\displaystyle\sum_{n=0}^\infty b^n$, $z=\displaystyle\sum_{n=0}^\infty c^n$, where $a,b,c$ are in A.P. and $|a|<1$, $|b|<1$, $|c|<1$, $abc\ne0$, then :

If $a_1,a_2,a_3,\dots$ and $b_1,b_2,b_3,\dots$ are A.P., and $a_1=2$, $a_{10}=3$, $a_1b_1=1=a_{10}b_{10}$, then $a_4b_4$ is equal to -

Let $A=\{1,a_1,a_2,\dots,a_{18},77\}$ be a set of integers with $1<a_1<a_2<\dots<a_{18}<77$. Let the set $A+A=\{x+y:x,y\in A\}$ contain exactly 39 elements. Then, the value of $a_1+a_2+\dots+a_{18}$ is equal to ______.