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

Concept:

The slope of the tangent to a curve y = f(x) is m =

The slope of the normal = =

Calculation:

Given curve y = 2x³ - 5x² + x - 2

Differentiating the equation wrt x

 = 6x² - 10x + 1

Slope at the point (1, -1)

at x = 1 = 6(1)² - 10(1) + 1

 = 6 - 10 + 1

= -3

The slope of the normal (m') =

m' = = 

The slope of the normal to the curve at the point (1, -1) is 1/3.

∴ Option 2 is correct

Concept:

Principle of Inclusion-Exclusion (PIE):

• This principle is used to count the number of elements in the union of multiple sets when there is overlap among the sets.
• It corrects the overcounting by subtracting the sizes of pairwise intersections, adding triple-wise intersections, and so on.

Calculation:

Given,

Total number of students in a class = 100

n(A): Number who like Maths = 60

n(B): Number who like Physics = 50

n(C): Number who like Chemistry = 40

n(A ∩ B) = 30, n(B ∩ C) = 20, n(C ∩ A) = 25

n(A ∩ B ∩ C) = 15

We need to find the number of students who like at least one of the three subjects:

⇒ n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(B ∩ C) − n(C ∩ A) + n(A ∩ B ∩ C)

⇒ 60 + 50 + 40 − 30 − 20 − 25 + 15

⇒ 150 − 75 + 15 = 90

∴ The number of students who like at least one subject = 90

Concept:

Equivalence Relation and Partition of Sets:

• An equivalence relation on a set is a relation that is reflexive, symmetric, and transitive.
• Each equivalence relation on a set corresponds to a partition of that set.
• A partition divides the set into non-empty, disjoint subsets whose union is the entire set.
• In this question, we count only those partitions where each subset has at least two elements.

Important Terms:

• Partition: A way of dividing the set into disjoint non-empty subsets. Total number of partitions of a set of size n is given by the Bell number.
• Bell Number (Bₙ): The number of partitions of a set with n elements. Notation: Bₙ

Calculation:

Given,

Set S = {1, 2, 3, 4}

⇒ n = 4

We need the number of equivalence relations where no subset (block) has size 1

⇒ That means we need partitions where each block has size ≥ 2

⇒ We analyze possible valid partitions of 4 elements:

⇒ Case 1: One block of size 4 → Partition: {1,2,3,4} ⇒ Only 1 way

⇒ Case 2: Two blocks of size 2 each → Partitions like: { {1,2}, {3,4} } ⇒ Count = 3 ways

⇒ Case 3: One block of size 3 and one singleton → Rejected (singleton not allowed)

⇒ Case 4: Other combinations like {2,1,1}, {1,1,1,1} etc. → Rejected (contains singleton)

∴ The total number of such equivalence relations = 1 + 3 = 4

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.

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