Mathematics MCQ Quiz - Objective Question with Answer for Mathematics - Download Free PDF
Last updated on May 12, 2025
Latest Mathematics MCQ Objective Questions
Mathematics Question 1:
Comprehension:
A tosses 2 fair coins & B tosses 3 fair coins. The game is won by the person who throws a greater number of heads.
In case of a tie, the game is continued under identical rules until someone finally wins.
The probability that A finally wins the game is given by K / 11.
If the expected number of rounds played until the game ends is E, then 11E is:
Answer (Detailed Solution Below) 16
Mathematics Question 1 Detailed Solution
Solution:
- Probability of tie: Occurs when both A and B get equal heads.
- (A = 0, B = 0): 1/4 × 1/8 = 1/32
- (A = 1, B = 1): 1/2 × 3/8 = 3/16
- (A = 2, B = 2): 1/4 × 3/8 = 3/32
- Total tie probability T = 1/32 + 3/16 + 3/32 = 5/16
- Let E = expected number of rounds until someone wins
- The recurrence is: E = 1 + T × E
- ⇒ E − (5/16)E = 1
- ⇒ (11/16)E = 1
- ⇒ E = 16/11
∴ Final Answer: 11 E = 16
Mathematics Question 2:
Comprehension:
A tosses 2 fair coins & B tosses 3 fair coins. The game is won by the person who throws a greater number of heads.
In case of a tie, the game is continued under identical rules until someone finally wins.
The probability that A finally wins the game is given by K / 11.
The value of K is
Answer (Detailed Solution Below) 3
Mathematics Question 2 Detailed Solution
Solution:
- A tosses 2 coins ⇒ Heads: 0, 1, 2 with probabilities: 1/4, 1/2, 1/4
- B tosses 3 coins ⇒ Heads: 0, 1, 2, 3 with probabilities: 1/8, 3/8, 3/8, 1/8
Favorable outcomes for A:
- (1, 0): 1/2 × 1/8 = 1/16
- (2, 0): 1/4 × 1/8 = 1/32
- (2, 1): 1/4 × 3/8 = 3/32
Total P(A wins) = 1/16 + 1/32 + 3/32 = 3/16
P(B wins): = 1/2
P(Tie): = 1 − (3/16 + 1/2) = 5/16
Let P = final probability that A wins
⇒ P = 3/16 + 5/16 × P
⇒ P − (5/16)P = 3/16
⇒ (11/16)P = 3/16
⇒ P = 3/11
∴ K = 3
Mathematics Question 3:
Comprehension:
A triangle ABC is such that a circle passing through vertex C, centroid G touches side AB at B. If AB = 6, BC = 4 then
Length of AC2 is equal to
Answer (Detailed Solution Below) 56
Mathematics Question 3 Detailed Solution
Concept:
- Median of a triangle: The line segment joining a vertex to the midpoint of the opposite side.
- Centroid (G): The point of intersection of medians; it divides each median in the ratio 2:1.
- Circle touches triangle side: Use geometrical relationships based on circle properties and distances.
- Important property: If AG:AF = AB², and AG = 2×GD, then algebraic equations help to find required lengths.
Calculation:
Let GD = x, DF = y
⇒ AG = 2x, AF = x + y
⇒ 2x(3x + y) = 36
⇒ xy = 4
⇒ 3x² + 4 = 18
⇒ x² = 14/3
⇒ AD = 3x = √42
Now, AC² + AB² = 2(AD² + BD²)
⇒ AC² + 36 = 2(42 + 4)
⇒ AC² + 36 = 92
⇒ AC² = 56
AC2 = 56
Mathematics Question 4:
Comprehension:
A triangle ABC is such that a circle passing through vertex C, centroid G touches side AB at B. If AB = 6, BC = 4 then
The length of median through A is equal to AD then the value of AD2
Answer (Detailed Solution Below) 42
Mathematics Question 4 Detailed Solution
Concept:
- Median of a triangle: The line segment joining a vertex to the midpoint of the opposite side.
- Centroid (G): The point of intersection of medians; it divides each median in the ratio 2:1.
- Circle touches triangle side: Use geometrical relationships based on circle properties and distances.
- Important property: If AG:AF = AB², and AG = 2×GD, then algebraic equations help to find required lengths.
Calculation:
Let GD = x, DF = y
⇒ AG = 2x, AF = x + y
⇒ 2x(3x + y) = 36
⇒ xy = 4
⇒ 3x² + 4 = 18
⇒ x² = 14/3
⇒ AD = 3x = √42
AD = √42
AD2 = 42
Mathematics Question 5:
If x, y, z ∈ ℕ, then answer the following match list questions:
List – I | List – II |
---|---|
(A) Number of ordered triplets (x, y, z) such that xyz = 243 | (p) 19 |
(B) Number of terms in the expansion of (x + y + z)6 | (q) 20 |
(C) Number of natural solutions of x2 + x - 400 ≤ 0 | (r) 28 |
(D) Number of natural number solutions of x + y + z = 10 | (s) 21 |
(t) 36 |
Which is correct option?
Answer (Detailed Solution Below)
Mathematics Question 5 Detailed Solution
Concept:
- Ordered Triplets: If x, y, z ∈ ℕ and xyz = N, the number of ordered triplets is found using the number of solutions to the product form using prime factorization.
- Multinomial Expansion: The number of terms in the expansion of (x + y + z)n is equal to the number of non-negative integer solutions to x + y + z = n.
- Inequality Solution: For quadratic inequalities of the form ax2 + bx + c ≤ 0, solve the corresponding equation and determine the integer values within the valid interval.
- Number of Solutions of x + y + z = n: For x, y, z ∈ ℕ, the number of solutions is equal to C(n−1, r−1) where r = number of variables.
Calculation:
Given,
(A) xyz = 243
⇒ 243 = 35
⇒ Number of ordered triplets of natural numbers is equal to number of non-negative integer solutions of a + b + c = 5
⇒ Number of solutions = C(5 + 3 - 1, 2) = C(7, 2) = 21
⇒ Triplets are ordered ⇒ 21
(B) Expansion of (x + y + z)6
⇒ Number of terms = C(6 + 3 - 1, 2) = C(8, 2) = 28
(C) x2 + x - 400 ≤ 0
⇒ Solve equation x2 + x - 400 = 0
⇒ x = [-1 ± √(1 + 1600)]/2 = [-1 ± √1601]/2
⇒ √1601 ≈ 40 ⇒ x ranges from 1 to 19
⇒ Valid integer values = 19
(D) x + y + z = 10, with x, y, z ∈ ℕ
⇒ Convert to non-negative solution by setting x' = x−1, y' = y−1, z' = z−1
⇒ x' + y' + z' = 7
⇒ Number of solutions = C(7 + 3 - 1, 2) = C(9, 2) = 36
∴ A-(s), B-(r), C-(p), D-(t) is the correct match
⇒ Option (1)
Top Mathematics MCQ Objective Questions
Find the value of sin (1920°)
Answer (Detailed Solution Below)
Mathematics Question 6 Detailed Solution
Download Solution PDFConcept:
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 / 2What is the degree of the differential equation \({\rm{y}} = {\rm{x}}{\left( {\frac{{{\rm{dy}}}}{{{\rm{dx}}}}} \right)^2} + {\left( {\frac{{{\rm{dx}}}}{{{\rm{dy}}}}} \right)} \)?
Answer (Detailed Solution Below)
Mathematics Question 7 Detailed Solution
Download Solution PDFConcept:
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:
\({\rm{y}} = {\rm{x}}{\left( {\frac{{{\rm{dy}}}}{{{\rm{dx}}}}} \right)^2} + {\left( {\frac{{{\rm{dx}}}}{{{\rm{dy}}}}} \right)} \)
\({\rm{y}} = {\rm{x}}{\left( {\frac{{{\rm{dy}}}}{{{\rm{dx}}}}} \right)^2} + \frac{1}{{{{\left( {\frac{{{\rm{dy}}}}{{{\rm{dx}}}}} \right)}}}} \)
\(\Rightarrow {\rm{y}}{\left( {\frac{{{\rm{dy}}}}{{{\rm{dx}}}}} \right)} = {\rm{x}}{\left( {\frac{{{\rm{dy}}}}{{{\rm{dx}}}}} \right)^3} + 1\)
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.
What is the mean of the range, mode and median of the data given below?
5, 10, 3, 6, 4, 8, 9, 3, 15, 2, 9, 4, 19, 11, 4
Answer (Detailed Solution Below)
Mathematics Question 8 Detailed Solution
Download Solution PDFGiven:
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
Find the mean of given data:
class interval | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 | 60-70 | 70-80 |
Frequency | 9 | 13 | 6 | 4 | 6 | 2 | 3 |
Answer (Detailed Solution Below)
Mathematics Question 9 Detailed Solution
Download Solution PDFFormula used:
The mean of grouped data is given by,
\(\bar X\ = \frac{∑ f_iX_i}{∑ f_i}\)
Where, \(u_i \ = \ \frac{X_i\ -\ a}{h}\)
Xi = mean of ith class
fi = frequency corresponding to ith class
Given:
class interval | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 | 60-70 | 70-80 |
Frequency | 9 | 13 | 6 | 4 | 6 | 2 | 3 |
Calculation:
Now, to calculate the mean of data will have to find ∑fiXi and ∑fi as below,
Class Interval | fi | Xi | fiXi |
10 - 20 | 9 | 15 | 135 |
20 - 30 | 13 | 25 | 325 |
30 - 40 | 6 | 35 | 210 |
40 - 50 | 4 | 45 | 180 |
50 - 60 | 6 | 55 | 330 |
60 - 70 | 2 | 65 | 130 |
70 - 80 | 3 | 75 | 225 |
∑fi = 43 | ∑Xi = 315 | ∑fiXi = 1535 |
Then,
We know that, mean of grouped data is given by
\(\bar X\ = \frac{∑ f_iX_i}{∑ f_i}\)
= \(\frac{1535}{43}\)
= 35.7
Hence, the mean of the grouped data is 35.7
If we add two irrational numbers the resulting number
Answer (Detailed Solution Below)
Mathematics Question 10 Detailed Solution
Download Solution PDFConcept:
- 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.
What is the value of the expression?
(tan0° tan1° tan2° tan3° tan4° …… tan89°)
Answer (Detailed Solution Below)
Mathematics Question 11 Detailed Solution
Download Solution PDFGiven:
tan0° tan1° tan2° tan3° tan4° …… tan89°
Formula:
tan 0° = 0
Calculation:
tan0° × tan1° × tan2° × ……. × tan89°
⇒ 0 × tan1° × tan2° × ……. × tan89°
⇒ 0
Find the conjugate of (1 + i) 3
Answer (Detailed Solution Below)
Mathematics Question 12 Detailed Solution
Download Solution PDFConcept:
Let z = x + iy be a complex number.
- Modulus of z = \(\left| {\rm{z}} \right| = {\rm{}}\sqrt {{{\rm{x}}^2} + {{\rm{y}}^2}} = {\rm{}}\sqrt {{\rm{Re}}{{\left( {\rm{z}} \right)}^2} + {\rm{Im\;}}{{\left( {\rm{z}} \right)}^2}}\)
- arg (z) = arg (x + iy) = \({\tan ^{ - 1}}\left( {\frac{y}{x}} \right)\)
- 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.
If p = cosec θ – cot θ and q = (cosec θ + cot θ)-1 then which one of the following is correct?
Answer (Detailed Solution Below)
Mathematics Question 13 Detailed Solution
Download Solution PDFConcept:
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
\(\frac1q \times p=1\)
⇒ p = qIf sin θ + cos θ = 7/5, then sinθ cosθ is?
Answer (Detailed Solution Below)
Mathematics Question 14 Detailed Solution
Download Solution PDFConcept:
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 = 12/25Find the value of \(\rm \displaystyle \lim_{x \rightarrow \infty} 2x \sin \left(\frac{4} {x}\right)\)
Answer (Detailed Solution Below)
Mathematics Question 15 Detailed Solution
Download Solution PDFConcept:
\(\rm \displaystyle \lim_{x \rightarrow 0} \frac{\sin x}{x} = 1\)
Calculation:
\(\rm \displaystyle \lim_{x \rightarrow \infty} 2x \sin \left(\frac{4} {x}\right)\)
= \(\rm 2 \times \displaystyle \lim_{x → ∞} \frac{\sin \left(\frac{4} {x}\right)}{\left(\frac{1}{x} \right )}\)
= \(\rm 2 \times \displaystyle \lim_{x → ∞} \frac{\sin \left(\frac{4} {x}\right)}{\left(\frac{4}{x} \right )} \times 4\)
Let \(\rm \frac {4}{x} = t\)
If x → ∞ then t → 0
= \(\rm 8 \times\displaystyle \lim_{t \rightarrow 0} \frac{\sin t}{t} \)
= 8 × 1
= 8