OTHER QUESTIONS
1. A peacock is sitting on the top of a pillar, which is 9m high. From a point 27m away from the bottom of the pillar, a snake is coming to its hole at the base of the pillar. Seeing the snake the peacock pounce on it. If their speeds are equal, at what distance from the hole is the snake caught?Solution:
Let A be the point where peacock is sitting on the top of a pillar.
Let B be the point on the ground, where the snake is.
Let the distance from the hole, where the snake is caught be $x \:m$.
From the given data,
AD= 9 m, BD=27 m, CD=x, AC=BC=27-x
In $\triangle ACD$,
$AC^2=AD^2+CD^2$
$(27-x)^2=9^2+x^2$
$729-54x+x^2=81+x^2$
$54x=729-81$
$54x=648$
$x=\frac{648}{54}$
$x=12$
Therefore, the snake is caught at a distance of 12 m from the hole.
Aliter:
Let A be the point where peacock is sitting on the top of a pillar.
Let B be the point on the ground, where the snake is.
Let the distance covered by peacock and snake, before it is caught be x m.
Then the distance from the hole, where the snake is caught is $27-x$.
From the given data,
AD= 9 m, BD=27 m, AC=BC=x, CD=27-x
In $\triangle ACD$,
$AC^2=AD^2+CD^2$
$x^2=9^2+(27-x)^2$
$x^2=81+729-54x+x^2$
$54x=729+81$
$54x=810$
$x=\frac{810}{54}$
$x=15$
Therefore, the distance from the hole, where the snake is caught is $27-15=12 m$.
2. The angry Arjun carried some arrows for fighting with Bheeshm. With half the arrows, he cut down the arrows thrown by Bheeshm on him and with six other arrows he killed the rath driver of Bheeshm. With one arrow each he knocked down respectively the rath, flag and the bow of Bheeshm. Finally, with one more than four times the square root of arrows he laid Bheeshm unconscious on an arrow bed. Find the total number of arrows Arjun had.
Solution:
Let us consider Arjun had $x$ arrows.
Arrows used to cut down the arrows thrown by Bheeshm $=\frac{x}{2}$
Arrows used to kill the rath driver of Bheeshm $=6$
Arrows used to knock down rath, flag and bow of Bheeshm $=3$
Arrows used to lay down Bheeshm unconscious on the arrow bed $=4\sqrt{x}+1$
From the given data,
$\frac{x}{2}+6+3+4\sqrt{x}+1=x$
$\frac{x}{2}+4\sqrt{x}+10=x$
$4\sqrt{x}=x-\frac{x}{2}-10$
$4\sqrt{x}=\frac{2x-x-20}{2}$
$8\sqrt{x}=x-20$
Squaring on both sides, we get,
$(8\sqrt{x})^2=(x-20)^2$
$64x=x^2-40x+400$
$x^2-40x-60x+400=0$
$x^2-104x+400=0$
$(x-4)(x-100)=0$
$x-4=0, \implies x=4$, This is not possible.
$x-100=0, \implies x=100$.
Therefore, Arjun had 100 arrows.
3. One fourth of a herd of camels was seen in the forest. Twice the square root of the herd had gone to mountains and the remaining 15 camels were seen on the bank of a river. Find the total number of camels.
Solution:
Let the number of camels be $x$.
Number of camels seen in the forest $=\frac{x}{4}$
Number of camels gone to mountains $=2\sqrt{x}$
Number of camels seen on the bank of a river $=15$
$\frac{x}{4}+2\sqrt{x}+15=x$
$2\sqrt{x}=x-\frac{x}{4}-15$
$2\sqrt{x}=\frac{4x-x-60}{4}$
$8\sqrt{x}=3x-60$
Squaring on both sides, we get,
$(8\sqrt{x})^2=(3x-60)^2$
$64x=9x^2-360x+3600$
$9x^2-360x-64x+3600=0$
$9x^2-424x+3600=0$
$(x-36)(9x-100)=0$
$x-36=0, \implies x=36$
$9x-100=0, \implies x=\frac{100}{9}$, This is not possible.
Therefore, number of camels is 36.
4. A cottage industry produces a certain number of pottery articles in a day. It was observed on a particular day that the cost of production of each article (in rupees) was 3 more than twice the number of articles produced on that day. If the total cost of production on that day was ₹90, find the number of articles produced and the cost of each article.
Solution:
Let the number of articles produced be $x$.
Let the cost of each article be $2x+3$.
Total cost, $(x)(2x+3)=90$
$2x^2+3x=90$
$2x^2+3x-90=0$
$(x-6)(2x+15)=0$
$x-6=0, \implies x=6$
$2x+15=0, \implies x=\frac{-15}{2}$, This is not possible.
Therefore, the number of articels produced is 6
and the cost of each article is $2(6)+3=12+3=₹15$.