Some identical balls are arranged in rows to form an equilateral … - Vidyadip

MCQ

Some identical balls are arranged in rows to form an equilateral triangle. The first row consists of one ball, the second row consists of two balls and so on. If $99$ more identical balls are added to the total number of balls used in forming the equilateral triangle, then all these balls can be arranged in a square whose each side contains exactly $2$ balls less than the number of balls each side of the triangle contains. Then the number of balls used to form the equilateral triangle is

✓
$190$
B
$262$
C
$225$
D
$157$

✓

Answer

Correct option: A.

$190$

a
$\frac{{n\left( {n + 1} \right)}}{2} + 99 = {\left( {n - 2} \right)^2}$

$ \Rightarrow {n^2} + n + 198 = 2{n^2} - 8n + 8$

$ \Rightarrow {n^2} - 9n - 190 = 0$

$ \Rightarrow \left( {n - 19} \right)\left( {n + 10} \right) = 0$

$ \Rightarrow n = 19$

$\therefore \,$ Number of ball is $\frac{{19 \times 20}}{2} = 190$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "MCQ" from 8. sequence and series→8. sequence and series — all question types→MATHS STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

The equation of a common tangent to the conics $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$ and $\frac{{{y^2}}}{{{a^2}}} - \frac{{{x^2}}}{{{b^2}}} = 1$ is

Two dice are thrown simultaneously. The probability of getting a pair of aces is

Let $y = f(x) = ax^2 + 2bx + c,$ (where $a, b, c \in R$ and $a \neq 0)$ if $f(x) = 0$ has imaginary roots and $4a + 4b + c < 0$ then which of the following is true

The equation of the common tangents to the two hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ and $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ are-

How many numbers greater than 10 lacs be formed from 2, 3, 0, 3, 4, 2, 3?

The number of solutions of $\sin ^{7} x+\cos ^{7}=1, x \in[0,4 \pi]$ is equal to :

$R$ is a relation from $\{11,12,13\}$ to $\{8,10,12\}$ defined by $y=x-3$. Then, $R^{-1}$ is:

If the coefficients of second, third and fourth term in the expansion of ${(1 + x)^{2n}}$ are in $A.P.$, then $2{n^2} - 9n + 7$ is equal to

Let $A=\left\{\theta \in R:\left(\frac{1}{3} \sin \theta+\frac{2}{3} \cos \theta\right)^2=\frac{1}{3} \sin ^2 \theta+\frac{2}{3} \cos ^2 \theta\right\}$.Then

If the sum of $1 + \frac{{1 + 2}}{2} + \frac{{1 + 2 + 3}}{3} + .....$ to $n$ terms is $S$, then $S$ is equal to