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

Last updated on May 26, 2025

Latest Mathematics MCQ Objective Questions

Mathematics Question 1:

If the four distinct points (4, 6), (–1, 5), (0, 0) and (k, 3k) lie on a circle of radius r, then 10k + r2 is equal to 

  1. 32 
  2. 33 
  3. 34 
  4. 35

Answer (Detailed Solution Below)

Option 4 : 35

Mathematics Question 1 Detailed Solution

Calculation: 

qImage6825a7b87646fb13ad1ad8bc

m1m2 = – 1 so right angle equation circle is

⇒ (x – 4) (x – 0) + (y – 6) (y – 0) = 0

⇒ x2 + y2 – 4x – 6y = 0

⇒ (k,3k) lies on it so

⇒ k2 + 9k2 – 4k – 18k = 0

⇒ 10k2 – 22k = 0 

⇒ k = 0, \(\frac{11}{5}\)

k = 0 is not possible so k = \(\frac{11}{5}\)

also r = \(\sqrt{4+9}=\sqrt{13}\)

so 10k + r2\(\text { 10. } \frac{11}{5}+(\sqrt{13})^{2}=35\)

Hence, the Correct answer is Option 4.

Mathematics Question 2:

The absolute difference between the squares of the radii of the two circles passing through the point (–9, 4) and touching the lines x + y = 3 and x – y = 3, is equal to _____.
 

Answer (Detailed Solution Below) 1 - 768

Mathematics Question 2 Detailed Solution

Concept:

Circle Equation and Radius:

  • The radius of a circle touching a line can be found using the perpendicular distance from the center to the line.
  • The circle equation is \( (x - a)^2 + (y - b)^2 = r^2 \), where \( (a,b) \) is the center and \( r \) is the radius.

qImage6821fd23e5d4ff708bfe7854

Calculation:

Center of circle is \( (a, 0) \).

Radius \( r \) is distance from center to line \( x + y = 3 \):

\( r = \left| \frac{a - 0 - 3}{\sqrt{2}} \right| = \left| \frac{a - 3}{\sqrt{2}} \right| \)

Equation of circle:

\( (x - a)^2 + y^2 = \left( \frac{a - 3}{\sqrt{2}} \right)^2 \)

Circle passes through point \( (-9,4) \):

\( (-9 - a)^2 + 4^2 = \frac{(a - 3)^2}{2} \)

Expanding and simplifying:

\( (a + 9)^2 + 16 = \frac{(a - 3)^2}{2} \)

\( 2(a^2 + 18a + 81 + 8) = a^2 - 6a + 9 \)

\( 2a^2 + 36a + 194 = a^2 - 6a + 9 \)

\( a^2 + 42a + 185 = 0 \)

Factorizing:

\( (a + 37)(a + 5) = 0 \implies a = -37, -5 \)

Calculating radii:

\( r_1 = \left| \frac{-37 - 3}{\sqrt{2}} \right| = \frac{40}{\sqrt{2}} = 20\sqrt{2} \)

\( r_2 = \left| \frac{-5 - 3}{\sqrt{2}} \right| = \frac{8}{\sqrt{2}} = 4\sqrt{2} \)

Absolute difference between squares of radii:

\( |r_1^2 - r_2^2| = |(20\sqrt{2})^2 - (4\sqrt{2})^2| = |800 - 32| = 768 \)

Hence, the correct answer is 768.

Mathematics Question 3:

Comprehension:

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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.

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If the expected number of rounds played until the game ends is E, then 11E is:

Answer (Detailed Solution Below) 16

Mathematics Question 3 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 4:

Comprehension:

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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 4 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 5:

Comprehension:

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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 5 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

Top Mathematics MCQ Objective Questions

Find the value of sin (1920°)

  1. 1 / 2
  2. 1 / √2
  3. √3 / 2
  4. 1 / 3

Answer (Detailed Solution Below)

Option 3 : √3 / 2

Mathematics Question 6 Detailed Solution

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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

What is the degree of the differential equation \({\rm{y}} = {\rm{x}}{\left( {\frac{{{\rm{dy}}}}{{{\rm{dx}}}}} \right)^2} + {\left( {\frac{{{\rm{dx}}}}{{{\rm{dy}}}}} \right)} \)?

  1. 1
  2. 2
  3. 3
  4. 4

Answer (Detailed Solution Below)

Option 3 : 3

Mathematics Question 7 Detailed Solution

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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:

\({\rm{y}} = {\rm{x}}{\left( {\frac{{{\rm{dy}}}}{{{\rm{dx}}}}} \right)^2} + {\left( {\frac{{{\rm{dx}}}}{{{\rm{dy}}}}} \right)} \)

\({\rm{y}} = {\rm{x}}{\left( {\frac{{{\rm{dy}}}}{{{\rm{dx}}}}} \right)^2} + \frac{1}{{{{\left( {\frac{{{\rm{dy}}}}{{{\rm{dx}}}}} \right)}}}} \)

\(\Rightarrow {\rm{y}}{\left( {\frac{{{\rm{dy}}}}{{{\rm{dx}}}}} \right)} = {\rm{x}}{\left( {\frac{{{\rm{dy}}}}{{{\rm{dx}}}}} \right)^3} + 1\)

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

  1. 10
  2. 12
  3. 8
  4. 9

Answer (Detailed Solution Below)

Option 4 : 9

Mathematics Question 8 Detailed Solution

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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

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

  1. 39.95
  2. 35.70
  3. 43.95
  4. 23.95

Answer (Detailed Solution Below)

Option 2 : 35.70

Mathematics Question 9 Detailed Solution

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Formula 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

f= 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 ∑X = 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

  1. Is always an rational number 
  2. Is always an irrational number
  3. May be a rational or an irrational number
  4. Always an integer

Answer (Detailed Solution Below)

Option 3 : May be a rational or an irrational number

Mathematics Question 10 Detailed Solution

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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.

Simplify \(\frac{{\left( {1 - {\rm{sinAcosA}}} \right)\left( {{\rm{si}}{{\rm{n}}^2}{\rm{A}} - {\rm{co}}{{\rm{s}}^2}{\rm{A}}} \right)}}{{{\rm{cosA}}\left( {{\rm{secA}} - {\rm{cosecA}}} \right)\left( {{\rm{si}}{{\rm{n}}^3}{\rm{A}} + {\rm{co}}{{\rm{s}}^3}{\rm{A}}} \right)}}\)

  1. sin A
  2. cos A
  3. sec A
  4. cosec A

Answer (Detailed Solution Below)

Option 1 : sin A

Mathematics Question 11 Detailed Solution

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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:

 \(\frac{{\left( {1 - {\rm{sinAcosA}}} \right)\left( {{\rm{si}}{{\rm{n}}^2}{\rm{A}} - {\rm{co}}{{\rm{s}}^2}{\rm{A}}} \right)}}{{{\rm{cosA}}\left( {{\rm{secA}} - {\rm{cosecA}}} \right)\left( {{\rm{si}}{{\rm{n}}^3}{\rm{A}} + {\rm{co}}{{\rm{s}}^3}{\rm{A}}} \right)}}\) 

⇒ \( \frac{{\left( {{\rm{1}} - {\rm{sinAcosA}}} \right)\left( {{\rm{sinA}} + {\rm{cosA}}} \right)\left( {{\rm{sinA}} - {\rm{cosA}}} \right)}}{{{\rm{cosA}}\left[ {\frac{1}{{{\rm{cosA}}}} - \frac{1}{{{\rm{sinA}}}}} \right]\left( {{\rm{sinA}} + {\rm{cosA}}} \right)\left( {{\rm{si}}{{\rm{n}}^2}{\rm{A}} + {\rm{co}}{{\rm{s}}^2}{\rm{A}} - {\rm{sinAcosA}}} \right)}}\)

⇒ \( \frac{{\left( {1 - {\rm{sinAcosA}}} \right)\left( {{\rm{sinA}} + {\rm{cosA}}} \right)\left( {{\rm{sinA}} - {\rm{cosA}}} \right)}}{{{\rm{cosA}}\left[ {\frac{{{\rm{sinA}} - {\rm{cosA}}}}{{{\rm{sinA}}.{\rm{cosA}}}}} \right]\left( {{\rm{sinA}} + {\rm{cosA}}} \right)\left( {1 - {\rm{sinAcosA}}} \right)}}\)

⇒ \(\frac{sinA - cosA}{cosA[\frac{sinA - cosA}{sinA.cosA}]}\)

⇒ \(\frac{(sinA - cosA)\times sinA.cosA}{cosA[sinA - cosA]}\)

⇒ \(\frac{ sinA.cosA}{cosA}\)

⇒ sin A

∴ The correct answer is option (1).

What is the value of the expression?

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

  1. 1
  2. 1/2
  3. 0
  4. 2

Answer (Detailed Solution Below)

Option 3 : 0

Mathematics Question 12 Detailed Solution

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Given:

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

  1. -2 + 2i
  2. -2 – 2i
  3. 1 - i
  4. 1 – 3i

Answer (Detailed Solution Below)

Option 2 : -2 – 2i

Mathematics Question 13 Detailed Solution

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Concept:

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?

  1. p - q = 1
  2. p = q 
  3. p + q = 1
  4. p + q = 0

Answer (Detailed Solution Below)

Option 2 : p = q 

Mathematics Question 14 Detailed Solution

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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

\(\frac1q \times p=1\)

⇒ p = q

If sin θ + cos θ = 7/5, then sinθ cosθ is?

  1. 11/25
  2. 12/25
  3. 13/25
  4. 14/25

Answer (Detailed Solution Below)

Option 2 : 12/25

Mathematics Question 15 Detailed Solution

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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
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