Mathematics MCQ Quiz - Objective Question with Answer for Mathematics - Download Free PDF

Concept:

Solution:

Given:

If x, y, and z are the angles of a triangle and z = 135°

⇒ x + y + z = 180^o

⇒ x + y = 180^o - 135^o

⇒ x + y = 45o

⇒ tan (x + y) = tan (45o)

⇒ 

⇒ tan x + tan y = 1 - tan x tan y

Adding 1 both side,

⇒ 1 + tan x + tan y = 1 - tan x tan y + 1

⇒ 1 + tan x + tan y + tan x tan y =  2

⇒ 1 + tan x + tan y(1 + tan x) =  2

⇒ (1 + tan x) (1+ tan y) =  2

∴ The value of (1 + tan x) (1 + tan y) is 2

Calculation:

We can rewrite the terms as:

Using the standard trigonometric identities  and , we get:

Now, substitute the values:

Thus, we get:

Hence, the correct answer is Option 1.

Calculation:

In the word "ARRANGE", we have:

3 vowels: A, A, E

4 consonants: R, R, N, G

We need to preserve the relative order of vowels and consonants. Vowels can only occupy vowel positions and consonants can only occupy consonant positions.

Step 1: Arrangements of consonants.

We have 4 consonants (R, R, N, G). These can be arranged in:

Step 2: Arrangements of vowels.

We have 3 vowels (A, A, E). These can be arranged in:

Step 3: Total number of arrangements.

The total number of arrangements is the product of the two arrangements:

Therefore, the number of arrangements in which the relative order of vowels and consonants remains unchanged is 36.

Calculation:

Total number of letters in the word "ARRANGE" = 7

The word "ARRANGE" contains the following repeated letters:

• 2 A's
• 2 R's

Step 1: Total number of arrangements of the letters in "ARRANGE".

The total number of arrangements is given by the formula for permutations of multiset elements:

Substitute the values:

Step 2: Arrangements where the two A's are together.

If the two A's are together, we treat them as a "block", reducing the problem to arranging 6 entities: (AA), R, R, N, G, E. The total number of arrangements is:

Step 3: Arrangements where the two R's are together.

If the two R's are together, we treat them as a "block", reducing the problem to arranging 6 entities: (RR), A, A, N, G, E. The total number of arrangements is:

Step 4: Arrangements where both A's and R's are together.

If both the two A's and the two R's are treated as "blocks", we need to arrange 5 entities: (AA), (RR), N, G, E. The total number of arrangements is:

Step 5: Arrangements where neither the two A's nor the two R's are together.

The number of arrangements where neither the two A's nor the two R's are together is:

Therefore, the number of arrangements where neither the two A's nor the two R's are together is 660.

Calculation:

Total number of letters in the word "ARRANGE" = 7

The word "ARRANGE" contains the following repeated letters:

• 2 A's
• 2 R's

Step 1: Total number of arrangements of the letters in "ARRANGE".

The total number of arrangements is given by the formula for permutations of multiset elements:

Substitute the values:

Step 2: Arrangements where the two A's are together.

If the two A's are together, we treat them as a "block", reducing the problem to arranging 6 entities: (AA), R, R, N, G, E. The total number of arrangements is:

Step 3: Arrangements where the two R's are together.

If the two R's are together, we treat them as a "block", reducing the problem to arranging 5 entities: (AA), (RR), N, G, E. The total number of arrangements is:

Step 4: Arrangements where the two A's are together but the two R's are not together.

The number of arrangements where the two A's are together but the two R's are not together is:

Therefore, the number of arrangements in which the two A's are together but the two R's are not together is 240.

Concept:

sin (2nπ ± θ) = ±  sin θ

sin (90 + θ) = cos θ

Calculation:

Given: sin (1920°)

⇒ sin (1920°) = sin(360° × 5° + 120°) = sin (120°)

⇒ sin (120°) = sin (90° + 30°) = cos 30°  = √3 / 2

Concept:

Order: The order of a differential equation is the order of the highest derivative appearing in it.

Degree: The degree of a differential equation is the power of the highest derivative occurring in it, after the equation has been expressed in a form free from radicals as far as the derivatives are concerned.

Calculation:

Given:

For the given differential equation the highest order derivative is 1.

Now, the power of the highest order derivative is 3.

We know that the degree of a differential equation is the power of the highest derivative

Hence, the degree of the differential equation is 3.

Mistake PointsNote that, there is a term (dx/dy) which needs to convert into the dy/dx form before calculating the degree or order. 

Given:

The given data is 5, 10, 3, 6, 4, 8, 9, 3, 15, 2, 9, 4, 19, 11, 4

Concept used:

The mode is the value that appears most frequently in a data set

At the time of finding Median

First, arrange the given data in the ascending order and then find the term

Formula used:

Mean = Sum of all the terms/Total number of terms

Median = {(n + 1)/2}^th term when n is odd 

Median = 1/2[(n/2)^th term + {(n/2) + 1}^th] term when n is even

Range = Maximum value – Minimum value 

Calculation:

Arranging the given data in ascending order 

2, 3, 3, 4, 4, 4, 5, 6, 8, 9, 9, 10, 11, 15, 19

Here, Most frequent data is 4 so 

Mode = 4

Total terms in the given data, (n) = 15 (It is odd)

Median = {(n + 1)/2}th term when n is odd 

⇒ {(15 + 1)/2}^th term 

⇒ (8)^th term

⇒ 6 

Now, Range = Maximum value – Minimum value 

⇒ 19 – 2 = 17

Mean of Range, Mode and median = (Range + Mode + Median)/3

⇒ (17 + 4 + 6)/3 

⇒ 27/3 = 9

∴ The mean of the Range, Mode and Median is 9

Formula used:

The mean of grouped data is given by,

Where, 

Xi = mean of ith class

fi = frequency corresponding to ith class

Given:

Calculation:

Now, to calculate the mean of data will have to find ∑fiXi and ∑fi as below,

Then,

We know that, mean of grouped data is given by

= 

= 35.7

Hence, the mean of the grouped data is 35.7

Concept:

a² - b² = (a - b) (a + b)

sec x = 1/cos x and cosec x = 1/sin x

a³ + b³ = (a + b) (a² + b² - ab)

Calculation:

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

⇒ sin A

∴ The correct answer is option (1).

Concept:

• Rational numbers are those numbers that show the ratio of numbers or the number which we get after dividing it with any two integers.
• Irrational numbers are those numbers that we can not represent in the form of simple fractions a/b, and b is not equal to zero.
• When we add any two rational numbers then their sum will always remain rational.
• But if we add an irrational number with a rational number then the sum will always be an irrational number.

Explanation:

Case:1 Take two irrational numbers π and 1 - π

⇒ Sum =  π +1 - π = 1

Which is a rational number.

Case:2 Take two irrational numbers π and √2 

⇒ Sum =  π + √2

Which is an irrational number.

Hence, a sum of two irrational numbers may be a rational or an irrational number.

Given:

tan0° tan1° tan2° tan3° tan4° …… tan89°

Formula:

tan 0° = 0

Calculation:

tan0° × tan1° × tan2° × ……. × tan89°

⇒ 0 × tan1° × tan2° × ……. × tan89°

⇒ 0

Concept:

Let z = x + iy be a complex number.

• Modulus of z = 
• arg (z) = arg (x + iy) = 
• Conjugate of z =  = x – iy

Calculation:

Let z = (1 + i) ³

Using (a + b) ³ = a³ + b³ + 3a²b + 3ab²

⇒ z = 1³ + i³ + 3 × 1² × i + 3 × 1 × i²

= 1 – i + 3i – 3

= -2 + 2i

So, conjugate of (1 + i) ³ is -2 – 2i

NOTE:

The conjugate of a complex number is the other complex number having the same real part and opposite sign of the imaginary part.

Concept:

cosec² x – cot² x = 1

Calculation:

Given: p = cosec θ – cot θ and q = (cosec θ + cot θ)^-1

⇒ cosec θ + cot θ = 1/q

As we know that, cosec² x – cot² x = 1

⇒ (cosec θ + cot θ) × (cosec θ – cot θ) = 1

⇒ p = q

Concept:

sin² x + cos² x = 1

Calculation:

Given: sin θ + cos θ = 7/5 

By, squaring both sides of the above equation we get,

⇒ (sin θ + cos θ)² = 49/25

⇒ sin² θ + cos² θ + 2sin θ.cos θ = 49/25

As we know that, sin2 x + cos2 x = 1

⇒ 1 + 2sin θcos θ = 49/25

⇒ 2sin θcos θ = 24/25

∴ sin θcos θ = 12/25