Mathematics Test 127

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Mathematics Test 127

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Question 1
4 / -1

There are 4 white and 3 black balls in a box. In another box, there are 3 white and 4 black balls. An unbiased dice is rolled. If it shows a number less than or equal to 3, then a ball is drawn from the first box, but if it shows a number more than 3, then a ball is drawn from the second box. If the ball drawn is black, then the probability that the ball was drawn from the first box is:

A
1/2

B
6/7

C
4/7

D
3/7

Solution

Let event A be the event of black ball chosen from the first box.

Let event B be the event that the chosen ball is black.

Now we need to find P(A/B)

Now the event A is basically the ball chosen from the first box ∩ chosen ball is black.

Probability of choosing the first box = 3/6 = 1/2

Probability of choosing the second box = 3/6 = 1/2

Now the Probability of choosing a black ball = (Probability of choosing the first bag × Probability of choosing black) + (Probability of choosing the second bag × Probability of choosing black)

Now the probability of choosing a black ball from the first box is 3/7 as there are total 7 balls and 3 of them are black,

Similarly, the probability of choosing a black ball from the second box is 4/7.

So, after putting all of them in the formula we will get:
Question 2
4 / -1

The number of binary operations on the set {a,b} are:

A
10

B
16

C
20

D
8

Solution

The binary operation is represented by ∗ and is defined by,

{a,b} × {a,b} → {a,b}

This relation shows that binary operation is a function from,

{(a,a), (a,b), (b,a), (b,b)} → {a,b}

The total number of binary operations on the set {a,b} is given by the number of elements in the set. There are two elements in the given set so, the total number of binary operation is 2⁴.

Thus, there are total of 16 binary operations on the given set.
Question 3
4 / -1

If be two perpendicular unit vectors such that is equal to:

A
1/√2

B
√2

C
√3

D
1

Solution

Now, we are given two vectors which are perpendicular to each other.

So, the angle between them is 90^∘.

So,we get

Now, from trigonometry we know that, cos 90^∘ = 0

As, is cross multiplied with itself, so the angle between them is 0^∘.
Question 4
4 / -1

If the system of linear equations

x+2ay+az=0

x+3by+bz=0

x+4cy+cz=0

has a non-zero solutions, then a,b,c are in:

A
A.P.

B
G.P.

C
H.P.

D
satisfy a + 2b + 3c = 0

Solution

Since, the system of linear equations has a non-zero solution, then:
Question 5
4 / -1

Let f,g:[−1,2]→R be continuous functions which are twice differentiable on the interval (-1,2). Let the values of f and g at the points -1,0 and 2 be as given in the following table:

In each of the intervals (-1,0) and (0,2) the function (f−3g) never vanishes. Then the correct statement(s) is (are):

A
f ′(x)−3g′(x)=0 has exactly three solutions in (-1,0)∪(0,2)

B
f′(x)−3g′(x)=0 has exactly one solution in (-1,0) and exactly one solutions in (0,2)

C
f′(x)−3g′(x)=0 has exactly two solutions in (-1,0) and exactly two solutions in (0,2)

D
None of the above

Solution

Therefore, by Rolle's theorem, h′(x) that is, F′(x)−3 g′(x)=0 has at least one root in (-1,0).
Question 6
4 / -1

A and B are two separate reservoirs of water. The capacity of reservoir A is double the capacity of reservoir B. Both the reservoirs are filled completely with water, their inlets are closed and then the water is released simultaneously from both the reservoirs. The rate of flow of water out of each reservoir at any instant of time is proportional to the quantity of water in the reservoir at the time.

One hour after the water is released, the quantity of water in reservoir A is 1.5 times the quantity of water in reservoir B. After how many hours do both the reservoirs have the same quantity of water?

A

B

C

D
None of these

Solution

Given: Volume of A is twice the volume of B at the start.

After 1 hour, the volume of A is 1.5 times the volume of B.

Let the water present at any time in A be x and in B be y.

For tank A,

Move dt to the right side and x to the denominator of the left side,

Cancel out exponential with log on the left side and factor the terms on the right side,

Move dt to the right side and y to the denominator of the left side,

As we know that, any number raised to the power of 0 equals 1.

Then,

Move variables on one side and constant on another side,

Move variables on one side and constant on another side,

Move (k₁−k₂) to the denominator of the right side and substitute the value from equation (4),
Question 7
4 / -1

If the vectors AB=3i + 4k and AC=5i−2j+4k are the sides of a △ABC, then the length of the median through A is:

A
√18

B
√72

C
√33

D
√45

Solution

We know that the sum of three vectors of a triangle is zero.
Question 8
4 / -1

The complex numbers z=x+ iy which satisfy the equation = 1, lie on:

A
the x− axis

B
straight line y = 5

C
a circle through the origin

D
None of these

Solution

Now, we are given a complex number z, where z=x+ iy. Now, we will apply the property of complex numbers. We will use the property:
Question 9
4 / -1

Find the particular solution of the differential equation dy/dx = 1 + x + y + xy, given that y=0 when x=1.

A

B

C

D

Solution
Question 10
4 / -1

At present, a firm is manufacturing 2000 items. It is estimated that the rate of change of production P w.r.t. additional number of workers x is given by dP/dx=100−12√x. If the firm employs 25 more workers, then the new level of production of items is:

A
3000

B
3500

C
4500

D
2500

Solution