Mathematics Test

Result

Mathematics Test - 2

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Question 1
4 / -1

A
a/2

B
a

C
2a

D
a/6

Solution

Explanation:
Question 2
4 / -1

Find the distance of the point ( 2 , 3 , − 5 ) from the plane x + 2y − 2z − 9 = 0.

A
3

B
1

C
2

D
4

E
1

Solution

Explanation:

We know that the distance of the point (x1,y1,z1) from the plane zx + by + cz + d = 0 is given by

Given, the point is (2,3,−5).
Question 3
4 / -1

The shortest distance between the lines is equal to:

A

B

C

D

Solution

Given:

Substituting all the values in equation (1), we obtain:

Since distance is always non-negative, the distance between the given lines is units.
Question 4
4 / -1

A

B

C

D

Solution
Question 5
4 / -1

A
2912

B
2918

C
2920

D
2940

Solution
Question 6
4 / -1

The triangle formed by the three points whose position vectors are 2

A
An equilateral triangle

B
A right-angled triangle but not isosceles

C
An isosceles triangle but not a right-angled triangle

D
A right-angled isosceles triangle

Solution

We are given that the position vectors of the three points of the triangle are

Let us assume the points are

So, the triangle formed by them is triangle ABC.

Now let us find the sides of the triangle ABC.

First, let us find the side AB.

So, we can see that lengths of sides AB and C A are equal.

So, it is an isosceles triangle. Now, let us consider the Pythagoras theorem.

In a right-angled triangle, the square of the hypotenuse is equal to the sum of squares of the other two sides.

Here we can see that,

So, the triangle ABC is a right-angled isosceles triangle.
Question 7
4 / -1

The derivative of ln⁡(x + sin⁡ x) with respect to (x + cos⁡ x) is:

A

B

C

D

Solution

Explanation:

Differentiating both sides, we get:

The derivative of the sum of two functions is the sum of the derivatives of the two functions, that is

Therefore, the equation becomes:

Now, we will differentiate both sides of v=(x+cos⁡x) with respect to x.

Differentiating both sides, we get:
Question 8
4 / -1

If e₁ and e₂ are the roots of the equation x²− ax + 2 = 0 where e₁, e₂ are the eccentricities of an ellipse and a hyperbola respectively then the value of a belongs to:

A

B

C

D

Solution

Explanation:

Now it is given that, e₁ and e₂ are the roots of the equation x²− ax + 2 = 0, where e₁ and e₂ are the eccentricities of an ellipse and hyperbola.
Question 9
4 / -1

Find the number of 7-digit numbers possible whose sum of digits is 61.

A
28

B
26

C
24

D
23

Solution

Explanation:

As we know that if all gifts are maximum (i.e., 9) then the sum of digits is 63 which is 2 more than 61.

So, all digits cannot be 9.

But if all digits are the second-largest digit (i.e., 8 ) then the sum is 56 which is 5 less than 61.

So, out of seven digits, five digits must be equal to 9.

So, the sum of five digits will be equal to 5 × 9 = 45.

So, the sum of the remaining two digits must be equal to 61 − 45 = 16.

Now as we know that if the sum of two digits is 16 then these two digits must be 8 and 8 or 9 and 7.

Now as we know that according to the identity of permutation which states that if there is are r persons and total n identical items and all n items are distributed to r persons and let each of the i^tℎ person is given balls then the number of ways for this distribution is given as

And we know that if r is any positive integer than

So, therefore there are two possible cases.

Case 1:- (five digits are 9 and two digits are 8)
Question 10
4 / -1

A

B

C

D

Solution

Explanation:

We know that binomial is the algebraic expression involving two terms and each term with distinct variable. If x,y are the two terms of binomial with some positive integral power n then the binomial expansion is given by;

The above expression is called binomial formula or binomial identity.

We can express the binomial formula in summation form as

We know that the mean with frequenting data is sum of product of data values and frequencies divided by sum of frequencies.

If there are n data values say and with corresponding frequencies then the mean is given by

We are give in the question that the variable takes values 0,1,2,…,n with frequencies

So, let us find the sum of frequencies

We can write the summation in the left hand side as;

We use the binomial formula