SRCC GBO 2021

This is a classic question from boats and streams, where you need to apply the equation of 'speed = distance/time'.

Given that the speed of the boat in still water is a kmph, and the speed of the current is b kmph.

In the instance where the boat goes against the stream (resultant speed is a-b): 0.5x = $$\ \frac{\ 3x}{(a-b)}$$

From this, (a-b) = 6    ........(1)

In the instance where the boat goes along with the stream (resultant speed is a+b): 0.3x= $$\frac{\ 2.4x}{(a+b)}$$

From this, (a+b)= 8    ............(2)

Using the two equations: a=7 and b=1

Hence, the value of $$(a^2 + b^2)$$ is 50.

The total number of possibilities the two dice can produce together is 36.

To have 9 as the sum on the two dice, the possible combinations are: (3,6)(6,3)(4,5)(5,4)

Hence, the probability is 4/36, which is 1/9.

To know the nature of roots, we need to find the discriminant of the equation.

The discriminant for this equation is $$(-7)^2$$-4(2)(2), which is 49-16= 35.

Now, the discriminant is greater than zero, which means that the roots will be real and distinct. However, $$\sqrt{35\ }$$ is an irrational number, and hence the roots shall be real, unique, and irrational. 

Here, we need to check the options and see which one has one element from A and the other from B in each pair, without having any pair with repetitions or elements from the same set. 

Option A has (1,1) contravenes the principle discussed above, as it has two elements from set A.

Option B is correct, as it has one element each from the two sets, and there is no repetition as such. 

Option C is incorrect, as the element '4' is repeated twice.

Option D is incorrect because of the same reason that was pointed out for Option A.  

The ratio of milk and water in A is 4:5, and in B is 3:1

Let us assume that in A, the volume of milk and water is 4x and 5x (total volume is 9x).

For B, the volume can be assumed as 3y and y (total volume is 4y).

After mixing A and B, we get the ratio of milk and water as 3:2

This means, $$\frac{\ (4x+3y)\ }{(5x+y)}$$ = $$\ \frac{\ 3}{2}$$

From the above equation, we get$$\ \frac{\ x}{y}$$as $$\ \frac{\ 3}{7}$$

Hence, the volume of A becomes 27, and that of B becomes 28. 

This makes the ratio of A and B as $$\ \frac{\ 27}{28}$$.

It is said that two opposite sides of the quadrilateral are parallel.

So, it can be Square, Rhombus, Parallelogram, or Rectangle.

It is not square and rhombus because two adjacent sides are unequal(given).

It is not rectangle, because if it is rectangle the diagonal length will be $$\sqrt{\ 3^2+4^2}=5^{ }$$

So, it is a parallelogram. 

We know that for a parallelogram $$d_1^2+d_2^2=2\left(a^2+b^2\right)$$.

$$d_1=1$$

a=3

b =4

$$1^2+d_2^2=2\left(3^2+4^2\right)$$.

$$1^2+d_2^2=2\left(3^2+4^2\right)$$.$$1+d_2^2=50\ \Rightarrow d_2^2=49\ \Rightarrow\ d_2=7$$.

The value of the expression $$(800 + 7n - n^2)$$ is $$(800 + 7*7- 7^2)$$=800+49-49=800

$$\sqrt{\ 16\ +\ 2\sqrt{\ 55}}$$ can be seen as the square root of 16 + $$2\sqrt{\ 55}$$

Now on observing 16 + $$2\sqrt{\ 55}$$, we can see it as an expansion of $$(a+b)^2$$: a being $$\sqrt{11\ }$$ and b being $$\sqrt{5\ }$$.

Hence, we can rewrite the equation as $$\left(\sqrt{\ 11}+\sqrt{\ 5}\right)^2$$ , whose square root is $$\sqrt{\ 11}$$+$$\sqrt{\ 5}$$.

Let the sides of the triangle be AB, BC, and AC.

Each of the sides is made of radii of two circles. Let us assume that the radius of the three circles are x, y, and z.

Hence, 

 x+y= 21 ..............(1)

 y+z= 22 ...............(2)

 z+x= 23 ................(3)

Now from equation 3, z= 23-x ........... (4)

Substituting equation 4 in equation 2, we get x-y= 1 ...............(5)

Using equation 1 and 5, we get x= 11, y= 10, and z= 12.

The smallest radius is 10 cm, and the area of that circle is $$\pi10^2\ $$, which is approximately 314$$cm^2\ $$

Let us take terms of GP as $$a,ar,ar^2$$

then the terms in AP will be $$ar,ar^2,a$$

In that case, $$ar^2=\frac{(a+ar)}{2}$$

$$2ar^2=a+ar$$

On simplifying the equation, we get r =  or r = -1/2

r can not be 1 as that will make each term of the GP same which will not make it an infinite GP, so we get r = -1/2

⇒ Also given that sum to infinite terms of the GP = 36

$$\frac{a}{1-r\ }=36$$

plugging r=-1/2 we get

a = 54.

For the common roots, the first approach shall be to find the common points of the two equations and see if it is the root for both of them. If not, then solve the equations independently and check for the common root. 

Using the first approach, $$x^2 + 10x + 24$$ = $$x^2 + 14x + 48$$, we get x as -6 for the common point. 

On substituting this in each of the equations, we find that it is a root to both the equations, and hence the answer.