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WBBSE Class 9 Maths Multiple Choice Questions And Answers

Arithmetic

• WBBSE Class 9 Maths Arithmetic Chapter 1 Real Numbers Multiple Choice Questions
• WBBSE Class 9 Maths Arithmetic Chapter 2 Profit And Loss Multiple Choice Questions

Algebra

• WBBSE Class 9 Maths Algebra Chapter 1 Laws Of Indices Multiple Choice Questions
• WBBSE Class 9 Maths Algebra Chapter 2 Graph Multiple Choice Questions
• WBBSE Class 9 Maths Algebra Chapter 3 Linear Simultaneous Equations Multiple Choice Questions
• WBBSE Class 9 Maths Algebra Chapter 4 Polynomial Multiple Choice Questions
• WBBSE Class 9 Maths Algebra Chapter 5 Factorisation Multiple Choice Questions
• WBBSE Class 9 Maths Algebra Chapter 6 Logarithm Multiple Choice Questions

Geometry

• WBBSE Class 9 Maths Geometry Chapter 1 Properties Of Parallelogram Multiple Choice Questions
• WBBSE Class 9 Maths Geometry Chapter 2 Transversal And Mid Point Theorem Multiple Choice Questions
• WBBSE Class 9 Maths Geometry Chapter 3 Theorems On Area Multiple Choice Questions
• WBBSE Class 9 Maths Geometry Chapter 4 Theorems On Concurrence Multiple Choice Questions

Mensuration

• WBBSE Class 9 Maths Mensuration Chapter 1 Area And Perimeter Of Triangle And Quadrilateral Useful Information And Formulae Multiple Choice Questions
• WBBSE Class 9 Maths Mensuration Chapter 2 Circumference Of Circle Multiple Choice Questions
• WBBSE Class 9 Maths Mensuration Chapter 3 Area Of Circle Multiple Choice Questions

Coordinate Geometry

• WBBSE Class 9 Maths Coordinate Geometry Chapter 1 Distance Formulas Multiple Choice Questions
• WBBSE Class 9 Maths Coordinate Geometry Chapter 2 Internal And External Division Of Straight Line Segment Multiple Choice Questions
• WBBSE Class 9 Coordinate Geometry Chapter 3 Area Of Triangular Region Multiple Choice Questions

Statistics

• WBBSE Class 9 Maths Statistics Chapter 1 Statistics Multiple Choice Questions

WBBSE Class 9  Algebra Chapter 1 Laws Of Indices Multiple Choice Questions

Example 1. The value of (0.243)^0.2 x (10)^0.6 is

1. 0.3
2. 3
3. 0.9
4. 9

Solution: The correct answer is 2. 3

= \(\left(\frac{243}{1000}\right)^{\frac{2}{10}} \times(10)^{\frac{6}{10}}\)

= \(3^{8 \times \frac{2}{102}} \times 10^{\frac{2}{105}(-3)} \times 10^{\frac{3}{5}}\)

= \(3 \times 10^{-\frac{2}{5}+\frac{3}{5}}\)

= \(3 \times 10^0=3 \times 1=3\)

The value of (0.243)^0.2 x (10)^0.6 is = \(3 \times 10^0=3 \times 1=3\)

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Example 2. The value of \(2^{\frac{1}{2}} \times 2^{-\frac{1}{2}} \times(16)^{\frac{1}{2}}\) is

1. 1
2. 2
3. 4
4. \(\frac{1}{2}\)

Solution: The correct answer is 3. 4

⇒ \(2^{\frac{1}{2}} \times 2^{-\frac{1}{2}} \times 2^2 4 \times \frac{1}{2}=2^{\frac{1}{2}-\frac{1}{2}+2}=2^2=4\)

Example 3. If 4^x = 8³ then value of x is

1. \(\frac{3}{2}\)
2. \(\frac{9}{2}\)
3. 3
4. 9

Solution: The correct answer is 2. \(\frac{9}{2}\)

2^2x = 2^3×3

or, 2x = 9 or, x = \(\frac{9}{2}\)

Example 4. If 20^-x = \(\frac{1}{7}\) then the value of (20)^2x is

1. \(\frac{1}{49}\)
2. 7
3. 3
4. 9

Solution: 20^x = 7

or, (20)^2x = 7² = 49

∴ The correct answer is 3. 3

Example 5. If 4 x 5^x = 500 then value of x^x is

1. 8
2. 1
3. 64
4. 27

Solution: The correct answer is 4. 27

⇒ 5^x = 125

or 5^x = 5³, x = 3

∴ x^x = 3³ = 27

Example 6. Which one of the following is the value of \(2^{\frac{1}{2}} \times 4^{\frac{1}{4}} \times 16^{-\frac{1}{4}}\)?

1. 1
2. 2
3. 4
4. \(\frac{1}{2}\)

Solution: The correct answer is 1. 1

= \(2^{\frac{1}{2}} \times\left(2^2\right)^{\frac{1}{4}} \times\left(2^4\right)^{-\frac{1}{4}}=2^{\frac{1}{2}+\frac{2}{4}+\frac{-4}{4}}\)

= \(2^{\frac{1}{2}+\frac{1}{2}-1}=2^0=1\)

Example 7. If \(4^x=8^{\frac{1}{3}}\) then x = ?

1. \(\frac{2}{3}\)
2. \(\frac{3}{2}\)
3. 2
4. \(\frac{1}{2}\)

Solution: The correct answer is 4. \(\frac{1}{2}\)

⇒ \(4^x=8^{\frac{1}{3}}\)

⇒ or, \(\left(2^2\right)^x=\left(2^3\right)^{\frac{1}{3}} \quad \text { or, } 2^{2 x}=2^1\)

∴ 2x = 1 x = \(\frac{1}{2}\)

Example 8. If 2^x+1 +2^x-2 = 9, then x =?

1. 1
2. 2
3. 0
4. \(\frac{1}{2}\)

Solution: The correct answer is 2. 2

⇒ \(2^{x+1}+2^{x-2}=9\)

⇒ or, \(2^x\left(2^1+2^{-2}\right)=9\)

⇒ or, \(2^x\left(2+\frac{1}{2^2}\right)=9, \quad 2^x\left(\frac{9}{4}\right)=9\)

⇒ or, \(2^x=2^2\), x = 2

Example 9. \(\left(\frac{625}{81}\right)^{-\frac{3}{4}}=?\)

1. \(\frac{3}{5}\)
2. \(\frac{9}{25}\)
3. \(\frac{27}{125}\)
4. \(\frac{81}{625}\)

Solution: The correct answer is 3. \(\frac{27}{125}\)

= \(\frac{5^{-3}}{3^{-3}}=\left(\frac{3}{5}\right)^3=\frac{27}{125}\)

Example 10. \(\sqrt[3]{\left(\frac{1}{64}\right)^{\frac{1}{2}}}=?\)

1. 1
2. \(\frac{1}{2}\)
3. \(\frac{1}{3}\)
4. \(\frac{1}{6}\)

Solution: The correct answer is \(\frac{1}{2}\)

WBBSE Class 9 Maths  Algebra Chapter 2 Graph Multiple Choice Questions

Example 1. The graph of the equation 2x + 3 = 0 is

1. Parallel to X-axis
2. Parallel to Y-axis
3. Not parallel to any axis
4. Passing through origin

Solution: The correct answer is 2. parallel to Y-axis

⇒ x = constant is parallel to Y-axis.

The graph of the equation 2x + 3 = 0 is constant is parallel to Y-axis.

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Example 2. The graph of the equation ay + b = 0, (a, b are constants) is

1. Parallel to X-axis
2. Parallel to y-axis
3. Not parallel to any axis
4. Passing through origin

Solution: The correct answer is 1. parallel to X-axis

⇒ Y = const is parallel to X-axis.

The graph of the equation ay + b = 0, (a, b are constants) is const is parallel to X-axis.

Example 3. The graph of the equation 2x + 3y = 0 is

1. Parallel to X-axis
2. Parallel to Y-axis
3. Passing through the origin.
4. Passing through (2, 0)

Solution: The correct answer is 3. not parallel to any axis

⇒ ax + by = 0 is passing through origin as a·0 + b·0 = 0

The graph of the equation 2x + 3y = 0 is Passing through the origin.

Example 4. The graph of the equation cx + d = 0 (c & d are constants c ≠ 0) will be Y-axis when

1. d = -c
2. d = c
3. d = 0
4. d = 1

Solution: The correct answer is 3. d = 0

⇒ The equation of Y-axis is X = 0

Example 5. The graph of the equation ay + b = 0, (a, b are constants, a ≠ 0) will be X-axis, when

1. b = a
2. b = -a
3. b = 2
4. b = 0

Solution: The correct answer is 4. b = 0

⇒ The equation of the X axis is Y = 0

Example 6. (-2,-3) lies in the

1. 1st quadrant
2. 2nd quadrant
3. 3rd quadrant
4. 4th quadrant

Solution: The correct answer is 3. 3rd quadrant

⇒ (-,-) lies in the 3rd quadrant.

Example 7. The type of straight line is represented by y = 2 is

1. Parallel to X axis
2. Parallel to Y axis
3. Passing through origin
4. None of these

Solution: The correct answer is 1. parallel to X axis

Example 8. The value of abscissa of each point on the Y-axis

1. 0
2. 1
3. Constant
4. None of these

Solution: The correct answer is 1. 0

The value of abscissa of each point on the Y-axis 0.

Example 9. Distance of the point (q) from X axis is

1. q unit
2. p unit
3. \(\sqrt{p^2 \times q^2}\) unit
4. None of these

Solution: The correct answer is 1. q unit

⇒ Value of ordinate is the distance from X axis.

Distance of the point (q) from X axis is q unit

Example 10. The distance between the two points (4, 0) and (-6, 0) is

1. 10 units
2. 4 units
3. 6 units
4. None of these

Solution: The correct answer is 1. 10 units

⇒ Required distance (x₁ – x₂) units

= (4 – 6) units = 10 units

The distance between the two points (4, 0) and (-6, 0) is 10 units

WBBSE Class 9 Maths Algebra Chapter 3 Linear Simultaneous Equations Multiple Choice Questions

Example 1. The two equations 4x + 3y = 7 and 7x – 3y = 4 have

1. Only one solution
2. Infinite number of solutions
3. No solution
4. None of these

Solution: The correct answer is 1. Only one solution

Here \(\frac{4}{7} \neq \frac{3}{-3}\)

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Example 2. The two equations 3x + 6y = 15 and 6x + 12y = 30 have

1. Only one solution
2. Infinite number of solutions
3. No solution
4. None of these

Solution: The correct answer is 2. Infinite number of solutions

Here \(\frac{3}{6}=\frac{6}{12}=\frac{15}{30}\)

Example 3. The two equations 4x + 4y = 20 and 5x + 5y = 30 have solution (1,1)

1. Only one solution
2. Infinite no of solutions
3. No solution
4. None of these.

Solution: The correct answer is 3. No solution

Here \(\frac{4}{5}=\frac{4}{5} \neq \frac{20}{30}\)

Example 4. Which of the following equations have a solution (1, 1)

1. 2x + 3y = 9
2. 6x + 2y = 9
3. 3x + 2y = 5
4. 4x + 6y = 8

Solution: The correct answer is 3. 3x + 2y = 5

3.1 +2.1 = 5

Example 5. The two equations 4x + 3y = 25 and 5x – 2y = 14 have the solutions

1. x = 4, y = 3
2. x = 3, y = 4
3. x = 3, y = 3
4. x = 4, y = -3

Solution: The correct answer is 1. x = 4, y = 3

4x + 3y 25….. (1)

5x-2y= 14……..(2)

Multiplying equation (1) by 2 and equation (2) by 3

⇒ \(\begin{array}{r}
8 x+6 y=50 \\
15 x-6 y=42 \\
\hline 23 x=92
\end{array}\)

x = 4

∴ \(y=\frac{25-4 \times 4}{3}=3\)

Example 6. The solution of these equation x + y = 7 are

1. (1, 6), (3, 4)
2. (1, 6), (4, 3)
3. (1, 6) (4, 3)
4. (1, 6) (-4, 3)

Solution: The correct answer is 3. (1, 6) (4, 3)

1 + 6 = 7, 4 + 3 = 7

Example 7. The equations \(\frac{x}{4}\) + \(\frac{x}{2}\) + 6 = 0 and 8x + \(\frac{y}{k}\) + 2 = 0 can have no solution if the value of k is one of the following.

1. 2
2. ±\(\frac{1}{2}\)
3. ±2
4. 4

Solution: The correct answer is 2. ±\(\frac{1}{2}\)

or, \(\frac{1}{k^2}=4\)

k= +\(\frac{1}{2}\)

The value of k is +\(\frac{1}{2}\)

Example 8. The equations (1+k)x + \(\frac{y}{b}\)= 1 and 3x = \(\frac{y}{k}\) = 1 cannot be solves if k is

1. 1
2. 2
3. – 2
4. – 3

Solution: The correct answer is 3. -2

or, \(\frac{1+k}{3}=\frac{k}{6}\)

or, 6+ 6k 3k, 3k = -6,

k = -2

The value of k is -2

Example 9. 2x + 3t = 1, y = \(\frac{t}{3}\) + 1 can be solved

1. 2x + 9y = 10
2. 2x – 9y = 10
3. 2x – 9y = 0
4. 9x + 2y = 0

Solution: The correct answer is 1. 2x + 9y = 10

2x + 3 (3y – 3) = 1 or, 2x + 9y = 10.

WBBSE Class 9 Maths  Algebra Chapter 4 Polynomial Multiple Choice Questions

Example 1. Which of the following is a polynomial is one variable?

1. x + \(\frac{2}{x}\) + 3
2. 3√x + \(\frac{2}{\sqrt{x}}\) + 5
3. √2x² – √3x + 6
4. x¹⁰ + y⁵ + 8

Solution: The correct answer is 3. √2x² – √3x + 6

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Example 2. Which of the following is a polynomial?

1. x – 1
2. \(\frac{x-1}{x+1}\)
3. \(x^2-\frac{2}{x^2}+5\)
4. \(x^2+\frac{2 x^{\frac{3}{2}}}{\sqrt{x^2}}+6\)

Solution: The correct answer is 1. x – 1

Example 3. Which of the following is a linear polynomial?

1. x + x²
2. x + 1
3. 5x² – x + 3
4. x + \(\frac{1}{x}\)

Solution: The correct answer is 2. x + 1

⇒ (For linear polynomial degree will be 1).

Example 4. Which of the followings is a 2nd degree polynomial?

1. √x – 4
2. x³ + x
3. x³ + 2x + 6
4. x² + 5x + 6

Solution: The correct answer is 4. x² + 5x + 6

Example 5. The degree of the polynomial √3 is

1. \(\frac{1}{2}\)
2. 2
3. 1
4. 0

Solution: The correct answer is 4. 0

(√3 = √3 x x°)

The degree of the polynomial √3 is 0

Example 6. If the polynomial x³+ 6x² + 4x + k is devisible by (x + 2) then the value of k is

1. -6
2. -7
3. -8
4. -10

Solution: The correct answer is 3. -8

⇒ x + 2 = 0, x = -2

∴ f (-2) = 0

⇒ or, (-2)² + 6 (-2)² + 4 (-2) + k = 0

⇒ or, -8 + 24 – 8 + k = 0

⇒ or, k = -8

The value of k is -8

Example 7. In the polynomial f(x) if f(-\(\frac{1}{2}\)) =0 then the factor of f(x) will be

1. 2x – 1
2. 2x + 1
3. x – 1
4. x + 1

Solution: The correct answer is 2. 2x + 1

f(-\(\frac{1}{2}\))

∴ x = \(\frac{1}{2}\)

or, 2x + 1 = 0

The factor of f(x) will be 2x + 1.

Example 8. (x – 1) is a factor of the polynomial f(x) but it is not the factor of g(x). So (x – 1) will be a factor of

1. f(x) g(x)
2. – f(x) + g(x)
3. f(x) – g(x)
4. {f(x) + g(x)} g(x)

Solution: The correct answer is 1. f(x) g(x) (Product is the required answer)

Example 9. (x + 1) is a factor of xⁿ + 1 when

1. n is a positive odd integer
2. n is a positive even integer
3. n is a negative integer
4. n is a positive integer

Solution: The correct answer is 1. n is a positive odd integer

⇒ x + 1 = 0, x = -1

⇒ Now, (-1)ⁿ +1 = 0 n should be the positive odd integer.

Example 10. If n² – 1 is a factor of the polynomial an⁴ + bn³ + cn²+ dn + e then

1. a + c + e = b + d
2. a + b + e = c + d
3. a + b + c = d + e
4. b + c + d = a + e

Solution: The correct answer is a + c + e = b + d

⇒ n² = 1

⇒ n = ±1

∴ a (-1)⁴ + b (-1)³ + c (-1)² + d (-1) + e = 0

⇒ or, a – b + c – d + e = 0

⇒ or, a + c + e = b + d

Example 11. Which of the following is a polynomial?

1. \(\frac{x^2}{2}-\frac{2}{x^2}\)
2. \(\sqrt{2 x}-1\)
3. \(x^2+\frac{3 x^{\frac{3}{2}}}{\sqrt{x}}\)
4. \(\frac{x-1}{x+1}\)

Solution: The correct answer is 3. \(x^2+\frac{3 x^{\frac{3}{2}}}{\sqrt{x}}\)

Example 12. √2 is a polynomial of degree

1. 2
2. 0
3. 1
4. \(\frac{1}{2}\)

Solution: The correct answer is 2. 0

⇒ (√2 = √2x°)

Example 13. Degree of polynomial 4x⁴ + 0x³ + 0x⁵ + 5x + 7 is

1. 0
2. 1
3. Any natural no
4. Not defined

Solution: The correct answer is 3. Any natural no

Example 14. If p (x) = x² – 2√2x + 1, then p (2√2) is equal to

1. 0
2. 1
3. a√2
4. 8√2+1

Solution: The correct answer is 2. 1

⇒ P (2√2) = (2√2)² -2√2 – 2√2 + 1 = 8 – 8 + 1 = 1

Example 15. The value of the polynomial 5x – 4x² + 3, when x = -1 is

1. -6
2. 6
3. 2
4. -2

Solution: The correct answer is 1. -6

⇒ 5(-1) – 4(-1)² + 3 = -5 – 4 + 3 = -6

WBBSE Class 9 Maths Algebra Chapter 5 Factorisation Multiple Choice Questions

Example 1. If a² – b² = 11 × 9 and a & b are positive integers (a > b) then

1. a = 11, b = 9
2. a = 33, b = 3
3. a = 10, b = 1
4. a = 100, b = 1

Solution: The correct answer is 3. a = 100, b = 1

⇒ a² – b² = 11 x 9, (a + b) (a – b) = 11 × 9

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⇒ If a + b = 11, a – b = 9 then a + b + (a – b) = 11 + 9, 2a = 20

⇒ or, a = 10

∴ b = 11 – a = 11 – 10 = 1

a = 10, b = 1

Example 2. If \(\frac{a}{b}+\frac{b}{a}\) = 1 then one of the value of a³ + b³ is

1. 1
2. a
3. b
4. 0

Solution: The correct answer is 4. 0

\(\frac{a^2+b^2}{a b}\), a² + b² – ab = 0 then, a³ + b³ = (a + b) (a² + b² – ab) = (a + b) x 0 = 0

The value of a³ + b³ is  (a + b) (a² + b² – ab) = (a + b) x 0 = 0

Example 3. The value of 25² – 75² + 50² + 3 x 25 x 75 x 50 is

1. 150
2. 0
3. 25
4. 50

Solution: The correct answer is 2. 0

⇒ Here a + b + c = 25 + (-75) + 50 = 0

∴ a³ + b³ + c³ – 3abc = (a + b + c) (a² + b² + c² – ab – bc – ca) = 0

The value of 25² – 75² + 50² + 3 x 25 x 75 x 50 is 0.

Example 4. If a + b + c = 0 then value of \(\frac{a^2}{b c}+\frac{b^2}{c a}+\frac{c^2}{a b}\) is

1. 0
2. 1
3. -1
4. 3

Solution: The correct answer is 4. 3

\(\frac{a^2}{b c}+\frac{b^2}{c a}+\frac{c^2}{a b}=\frac{a^3+b^3+c^3}{a b c}=\frac{3 a b c}{a b c}\) as a + b + c = 0 = 3

Value of \(\frac{a^2}{b c}+\frac{b^2}{c a}+\frac{c^2}{a b}\) is 3.

Example 5. If x² – px + 12 = (x – 3)(x -a) is an identity then the value of a and p are respectively

1. a = 4, p = 7
2. a = 7, p = 4
3. a = 4, p = -7
4. a = -4, p = 7

Solution: The correct answer is 1. a = 4, p = 7

⇒ x² – px + 12 = x² – (3 + a) x + 3a

⇒ By comparing both sides as co-efficients of x and constant terms, we get

⇒ 3a = 12, a = 4

∴ p = 3 + a = 3 + 4 = 7

The value of a and p are respectively 4 and 7.

Example 6. If a^x.a^y.a^z then x³ + y³ + z³

1. 0
2. -1
3. 3
4. 3xyz

Solution: The correct answer is 4. 3xyz

⇒ a^x.a^y.a^z = a^x+y+z= a⁰, x + y + z = 0

∴ x³+ y³+ z³ = 3xyz

Example 7. \(\frac{(x-y)^3+(y-z)^3+(z-x)^3}{(x-y)(y-z)(z-x)}=\)

1. 3xyz
2. 3
3. xyz
4. 0

Solution: The correct answer is 2. 3

⇒ Let x – y = a, y – z = b, z – x = c

∴ x – y + y – z + z – x = 0,

∴ a + b + c = 0

∴ (x – y)³ + (y – z)³ + (z – x)³ = 3 (x − y)(y – z)(z – x)

⇒ or, \(\frac{3(x-y)(y-z)(z-x)}{(x-y)(y-z)(z-x)}\) = 3

Example 8. If a³ + b³ + c³ – 3abc = 0 and a + b + c ≠ 0 then

1. a = 2b + c
2. b = 2c + a
3. a = b = c
4. c = 2a + b

Solution: The correct answer is 3. a = b = c

⇒ a³ + b³ + c³ – 3abc = 0

⇒ or, \(\frac{1}{2}\)(a+b+c) {(a – b)² + (b – c)² + (c – a)²} = 0

⇒ but, a + b + c ≠ 0

∴ (a – b)² + (b – c)² + (c – a)² ≠ 0

∴ a – b = 0 = b – c = c – a

∴ a = b = c.

Example 9. If x² – px + 8 = (x – 2) (x – 4) be an identity then, p =?

1. 0
2. 2
3. 4
4. 6

Solution: The correct answer is 4. 6

⇒ x² – px + 8 = x² – (4 + 2)x + 8

⇒ By comparing, coefficients of x & constant terms, p = 4 + 2 = 6

p= 6.

Example 10. The number of factors of a⁶ – b⁶ is

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

Solution: The correct answer is 4. 4

⇒ a⁶ – b⁶ = (a³)² – (b³)² = (a³ + b³)(a³ – b³)

= (a + b) (a² – ab + b²) (a – b) (a² + ab + b²)

The number of factors of a⁶ – b⁶ is 4

WBBSE Class 9 Maths Algebra Chapter 6 Logarithm Multiple Choice Questions

Example 1. If log_√x0.25 = 4 then the value of x

1. 0.5
2. 0.25
3. 4
4. 16

Solution: The correct answer is 1. 0.5

⇒ (√x)⁴ = 0·25

⇒ \(x^{\frac{1}{2} \times 4}=(0 \cdot 5)^2\)

∴ x = 0.5

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The value of x = 0.5

Example 2. If log₁₀(7x – 5) = 2, then the value of x

1. 10
2. 12
3. 15
4. 18

Solution: The correct answer is 3. 15

⇒ 10² = 7x – 5

⇒ or, 7x = 100+ 5

⇒ or, x = 15

The value of x = 15

Example 3. If log₂³ = a, then log₈²⁷ =?

1. 3a
2. \(\frac{1}{a}\)
3. 2a
4. a

Solution: The correct answer is 4. a

⇒ \(\log _8 27=\frac{\log _2 27}{\log _2 8}\)

= \(\frac{3 \log _2 3}{3 \log _2 2}=\frac{a}{1}=a\)

log₈²⁷ = \(\frac{3 \log _2 3}{3 \log _2 2}=\frac{a}{1}=a\)

Example 4. If logx_√2 = a then, the value of logx_2√2 is

1. \(\frac{a}{3}\)
2. a
3. 2a
4. 3a

Solution: The correct answer is 1. \(\frac{a}{3}\)

⇒ x = (√2)^a

⇒ or, \(\log _{2 \sqrt{2}} x=\log _{2 \sqrt{2}}(\sqrt{2})^a=a \log _{2 \sqrt{2}} \sqrt{2}\)

= \(a \frac{\log _b \sqrt{2}}{\log _b 2 \sqrt{2}}(b \neq 0,1)=a \frac{\log _b(2)^{\frac{1}{2}}}{\log _b(2)^{\frac{3}{2}}}\)

= \(a \cdot \frac{\frac{1}{2} \log _b 2}{\frac{3}{2} \log _b 2}=\frac{a}{3}\)

The value of logx_2√2 = \(a \cdot \frac{\frac{1}{2} \log _b 2}{\frac{3}{2} \log _b 2}=\frac{a}{3}\)

Example 5. If \(\log _x \frac{1}{3}=-\frac{1}{3}\) then the value of x is

1. 27
2. 9
3. 3
4. \(\frac{1}{27}\)

Solution: The correct answer is 1. 27

⇒ \((x)^{-\frac{1}{3}}=\frac{1}{3}\)

⇒ or, \(x=\left(\frac{1}{3}\right)^{-3}=3^3=27\)

The value of x is \(x=\left(\frac{1}{3}\right)^{-3}=3^3=27\)

Example 6. \(\log _3\left(\frac{1}{81}\right)=?\)

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

Solution: The correct answer is 1. -1

⇒ Let \(\log _3\left(\frac{1}{81}\right)=x\)

∴ 3^x = 3^-4, x = -4

Example 7. If log_√7 = x then the value of x is

1. 3
2. 6
3. 9
4. 12

Solution: The correct answer is 2. 6

⇒ (√7)^x = 243 =7³

⇒ or, \(7^{\frac{x}{2}}=7^3\)

⇒ x = 6

The value of x is 6.

Example 8. State which of the following is the value of log_0.008√5

1. –\(\frac{1}{6}\)
2. –\(\frac{1}{3}\)
3. –\(\frac{1}{2}\)
4. –\(\frac{1}{8}\)

Solution: The correct answer is 1. –\(\frac{1}{6}\)

⇒ Let _0.008√5 = x

⇒ \((008)^x=\sqrt{5}\)

⇒ or, \(5^{-3 x}=5^{\frac{1}{2}} \quad x=-\frac{1}{6}\)

The value of log_0.008√5 \(5^{-3 x}=5^{\frac{1}{2}} \quad x=-\frac{1}{6}\)

Example 9. \(\log _4 2^{-8}=\)?

1. – 4
2. – 3
3. – 2
4. – 1

Solution: The correct answer is 1. -4

⇒ Let \(\log _4 2^{-8}=x\)

∴ \(4^x=2^{-8}\)

⇒ or, 2x = -8, x = -4

Example 10. If the logarithm of 5832 be 6, then the base is

1. 2√3
2. 3
3. 2
4. 3√2

Solution: The correct answer is 4. 3√2

⇒ Let log_x5832 = 6

⇒ x⁶ = 5832 = 3⁶, 2³

⇒ or, x⁶ =(3√2)⁶,

⇒ x = 3√2

The base is x = 3√2

WBBSE Class 9 Maths  Coordinate Geometry Chapter 2 Internal And External Division Of Straight Line Segment Multiple Choice Questions

Example 1. The midpoint of line segment joining two points (l, 2m), (-l + 2m, 2l – 2m) is

1. (l, m)
2. (2,- m)
3. (m, -l)
4. (m, l)

Solution: \(\left(\frac{l-l+2 m}{2}, \frac{2 m+2 l-2 m}{2}\right)\) = (m, l)

Example 2. The abscissa at the point P which divides the line segment joining two points A (1, 5), and B (-4, 7) internally in the ratio 2: 3 is

1. 1
2. 11
3. 1
4. -11

Solution: \(\left(\frac{2 \times(-4)+3 \times 1}{2+3}\right)=\frac{-8+3}{5}=-1\)

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Example 3. The coordinates of the end points of a diameter of a circle are (7, 9) and (-1,-3). The coordinates of the centre is

1. (3, 3)
2. (4, 6)
3. (3, -3)
4. (4, -6)

Solution: Centre = \(\left(\frac{7-1}{2}, \frac{9-3}{2}\right)\) = (3, 3)

Example 4. A point which divides the line segment joining two points (2, -5), (-3, -2) externally in 4: 3 The coordinates of point is

1. -18
2. -7
3. 18
4. 7

Solution: Co-ordinates = \(\left(\frac{-3 \times 4-3 \times 2}{4-3}, \frac{4(-2)-3(-5)}{4-3}\right)\) = (-18, 7)

Example 5. If the points P (1, 2), Q (4, 6), R (5, 7), and S (X, Y) are the vertices of a parallelogram PQRS then

1. x = 2, y = 4
2. x = 3, y = 4
3. x = 2, y = 3
4. x = 2, y = 5

Solution: Midpoint of PR = \(\left(\frac{5+1}{2}, \frac{7+2}{2}\right)=\left(3, \frac{9}{2}\right)\)

Midpoint of QS = \(=\left(\frac{4+x}{2}, \frac{6+y}{2}\right)\)

∴ \(\frac{4+y}{2}=3 \Rightarrow x=2, \quad \frac{9}{2}=\frac{6+y}{2}, \quad y=3\)

Example 6. The coordinates of the midpoint of the line segment joining the points (a + b, a- b) and (a – b, a + b) is

1. (a, b)
2. (a, a)
3. (b, b)
4. (2a, 2b)

Solution: \(\left(\frac{a+b+a-b}{2}, \frac{a-b+a+b}{2}\right)=(a, a)\)

Example 7. The coordinates of the midpoint of the line segment joining the points (1 – a, -2a), (2a – 2, a + 1) are

1. \(\left(\frac{a-1}{2}, \frac{1-a}{2}\right)\)
2. \(\left(\frac{2 a-1}{2}, \frac{1-2 a}{2}\right)\)
3. \(\left(\frac{a+3}{2}, \frac{-3 a+1}{2}\right)\)
4. \(\left(\frac{a+1}{2}, \frac{2 a+1}{2}\right)\)

Solution: \(\left(\frac{1-a+2 a-2}{2}, \frac{-2 a+a+1}{2}\right)=\left(\frac{a-1}{2}, \frac{1-a}{2}\right)\)

Example 8. The coordinates of the midpoint P of the line segment joining the points A (-m, 6), and B (4, n) are (-1, -1). The value of m and n are

1. 6, 8
2. -6, -8
3. 6, -8,
4. -6, 8

Solution: \(\frac{-m+4}{2}=-1, m=4+2=6, \quad \frac{6+n}{2}=-1, n=-8\)

WBBSE Class 9 Maths Arithmetic Chapter 1 Real Numbers Multiple Choice Questions

Example 1. The decimal expansion of √5 is

1. A terminating decimal
2. A terminating or recurring decimal
3. A non-terminating and non-recurring decimal
4. None of them

Solution:

√5 = 2.236067……….

⇒ Therefore the decimal expansion of √5 is a non-terminating and non-recurring decimal.

∴ So the correct answer is 3. A non-terminating and non-recurring decimal

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The decimal expansion of √5 is a non-terminating and non-recurring decimal.

Example 2. The product of two irrational numbers is

1. Always irrational number
2. Always rational number
3. Always an integer
4. Rational or irrational number

Solution: The product of two irrational number √18 and √2 is √18 x √2 = √36 = 6 [rational number]

⇒ Again √7 x √2 = √14 [irrational number]

⇒ Therefore product of two irrational number is rational or irrational number.

∴ So the correct answer is 4. None of them

The product of two irrational numbers is rational or irrational number

Example 3. π and \(\frac{22}{7}\)

1. Both are rational number
2. Both are always irrational number
3. π is rational and \(\frac{22}{7}\) is irrational
4. π is irrational and \(\frac{22}{7}\) is irrational

Solution: The ratio of perimeter and diameter of each circle is a fixed number and is denoted by π where π= (approx) or 3.14 (approx).

⇒ Therefore the value of π can not be expressed as the ratio of two integers so π is a irrational number.

⇒ As \(\frac{22}{7}\) is a ratio of two integers 22 and 7, therefore \(\frac{22}{7}\) is a rational number.

∴ So the correct answer is 4. π is irrational and \(\frac{22}{7}\) is irrational

π and \(\frac{22}{7}\) is irrational and \(\frac{22}{7}\) is irrational

Example 4. Between two rational numbers, there exist.

1. No rational number
2. Only one rational number
3. Infinite numbers of rational numbers.
4. No irrational number

Solution: If x and y are two rational numbers and x < y, then the rational numbers between x and y are (x + d), (x+2d),……, (x + nd).

⇒ where \(d=\frac{y-x}{x+1}\)

⇒ It is possible to take the value of n as large as we like, the number of rational numbers lying between x and y is infinite.

∴ So the correct answer is 3. Infinite numbers of rational numbers.

Between two rational numbers, infinite numbers of rational numbers exist.

Example 5. Between two irrational numbers, there exists

1. No rational number
2. Only one irrational number
3. Infinite numbers of irrational
4. No irrational number

Solution: There are infinite numbers of irrational numbers between two irrational numbers.

⇒ √2 = 1.4142103………

⇒ √3 = 1.732050807……..

⇒ The irrational numbers between √2 and √3 are 1.41421030030003……, 1.4142126122612226…..

∴ So the correct answer is 3. Infinite numbers of irrational

Between two irrational numbers, there exists Infinite numbers of irrational

Example 6. The number 0 is

1. Whole number but not an integer
2. Integer but not rational
3. Rational but not a real number
4. Whole numbers, integers, rational, and real numbers but not irrational

Solution: 0 = \(\frac{0}{1}\) = \(\frac{0}{2}\) = \(\frac{0}{17}\)

⇒ So 0 is a rational number,

⇒ Again 0 is a whole number and 0 is an integer that is neither positive nor negative.

∴ So the correct answer is 4. Whole numbers, integers, rational, and real numbers but not irrational

The number 0 is Whole numbers, integers, rational, and real numbers but not irrational

Example 7. The sum of two rational numbers is

1. Always rational
2. Always irrational
3. Always integer
4. None of them

Solution: The sum of two rational numbers will always be rational.

⇒ e.g.\(\frac{2}{3}\) + \(\frac{5}{6}\) = \(\frac{9}{6}\), √25 + √16 = \(\frac{9}{1}\)

∴ So the correct answer is 1. Always rational

The sum of two rational numbers is Always rational

Example 8. Which is not an irrational number from the following number?

1. 25
2. √7 – √3
3. 4 + 2√25
4. 9

Solution: 4 + √25 = 4 + 5 = \(\frac{9}{1}\) (rational number)

⇒ √3 x √2 = √6 (irrational number)

⇒ 2√5 (irrational number)

⇒(√7-√3) is a irrational number

∴ So the correct answer is 3. 4 + 2√25

Example 9. Which is not a rational number from the following number?

1. √0.4
2. 3.06
3. \(\sqrt{1 \frac{9}{16}}\)
4. √8

Solution: √0.4 = \(\sqrt{\frac{4}{9}}\) = \(\frac{2}{3}\) (rational number)

⇒ 3.06 = 3\(\frac{6}{90}\) (rational number) = 3\(\frac{1}{5}\) = \(\frac{46}{15}\) (rational number)

⇒ \(\sqrt{1 \frac{9}{16}}=\sqrt{\frac{25}{16}}=\frac{5}{4}\) (rational number)

⇒ √8 = \(\sqrt{4 \times 2}\) = 2√2 (irrational number)

∴ So the correct answer is 4. √8

Example 10. Which of the following number is a recurring decimal?

1. \(\frac{5}{8}\)
2. \(\frac{11}{25}\)
3. \(\frac{13}{80}\)
4. \(\frac{19}{24}\)

Solution: If the rational numbers of the form \(\frac{p}{q}\) where q has the prime factors 2 and 5 only be expressed into decimals, it will be terminating decimal numbers.

⇒ But if the rational numbers of the form be expressed into decimals, it will be recurring decimal numbers, where q has prime factors other than 2 and 5.

1. The denominator of \(\frac{5}{8}\) is 8 and 8 = 2³; 8 has no prime factor except 2.

∴ a terminating decimal number will be found if \(\frac{5}{8}\) be expressed into decimal.

2. The denominator of \(\frac{11}{25}\) is 25 and 25 = 5²; 25 has no prime factor except 5.

∴ a terminating decimal number will be found if \(\frac{11}{25}\) is expressed into decimal.

3. The denominator of \(\frac{13}{80}\) is 80 and 80 = 2⁴ x 5;

⇒ 80 has no prime factor except 2 and 5.

∴ a terminating decimal number will be found if \(\frac{13}{80}\) is expressed into decimal.

4. The denominator of \(\frac{19}{24}\) is 24 and 24 = 2³ x 3;

⇒ 24 has a prime factor 3 other than 2

∴ The decimal form of \(\frac{19}{24}\) will not be terminating. It will be recurring.

∴ So the correct answer is 4. \(\frac{19}{24}\)

WBBSE Class 9 Maths Arithmetic Chapter 2 Profit And Loss Multiple Choice Questions

Example 1. The ratio of cost price and selling price is 10: 11, the profit percentage is

1. 9
2. 11
3. 10\(\frac{1}{9}\)
4. 10

Solution: The ratio of cost price and selling price is 10: 11

⇒ Let cost price is ₹10x and selling price is ₹11x [x is common multiple and x > 0]

⇒ Profit = ₹(11x – 10x) = ₹x

Read and Learn More WBBSE Class 9 Maths Multiple Choice Questions

∴ Profit percentage = \(\frac{\text { Total profit }}{\text { Cost price }} \times 100\)

= \(\frac{x}{10 x} \times 100\) = 10

∴ The correct answer is 4. 10

The profit percentage is 10

Example 2. Buying a book at 40 and selling it at 60, the profit percentage will be

1. 50
2. 33\(\frac{1}{3}\)
3. 20
4. 30

Solution: C. P. = ₹40, S. P. = ₹60

Profit = ₹(60-40) = ₹20

Profit percentage = \(\frac{20}{40}\) x 100 = 50

∴ The correct answer is 1. 50

The profit percentage will be 50

Example 3. A shirt is sold at 360 and there is a loss of 10%. The cost price of the shirt is

1. ₹380
2. ₹400
3. ₹420
4. ₹450

Solution: Let the cost price of the shirt is ₹x

S.P. = ₹360

⇒ Loss percentage = \(₹ \frac{x-360}{x} \times 100\)

⇒ According to condition, \(\frac{x-360}{x} \times 100=10\)

⇒ \(\frac{x-360}{x}=\frac{10}{100}=\frac{1}{10}\)

⇒10x – 3600 = x

⇒ 9x = 3600

⇒ x = \(\frac{3600}{9}\) = 400

∴ The cost price is 400

∴ The correct answer is 2. ₹400

The cost price of the shirt is ₹400

Example 4. After 20% discount, the selling price of a geometry box becomes ₹48. The marked price of the geometry box is

1. 60
2. 75
3. 48
4. 50

Solution: Let the market price of the geometry box is ₹x [x > 0]

⇒ The selling price of the box after 20% discount is \(₹\left(x-x \times \frac{20}{100}\right)=₹ \frac{4 x}{5}\)

⇒ As per question, \(\frac{4 x}{5}\) = 48

⇒ x = \(\frac{5 \times 48}{4}\)

⇒ x = 60

∴ The marked price is 60

∴ The correct answer is 60

The marked price of the geometry box is 60

Example 5. A retailer buys medicine at 20% discount on marked price and sells to buyers at marked price. The retailer makes a profit percentage.

1. 20
2. 25
3. 10
4. 30

Solution: Let the marked price of medicine is ₹x [x > 0]

⇒ A retailer buys medicine at 20% discount on marked price

⇒ Cost price of medicine is \(₹\left(x-x \times \frac{20}{100}\right)=₹ \frac{4 x}{5}\)

⇒ Selling price of medicine is ₹x

∴ Profit = \(₹\left(x-\frac{4 x}{5}\right)=₹ \frac{x}{5}\)

∴ Profit percentage = \(\frac{\frac{x}{5}}{\frac{4 x}{5}} \times 100\)

= \(\frac{x}{5} \times \frac{5}{4 x} \times 100=25\)

∴ The correct answer is 2. 25

The retailer makes a profit percentage 25

Example 6. If a person sells an article for 300, gaining \(\frac{1}{4}\)th of its cost price, then gain percentage

1. 15
2. 20
3. 25
4. 30

Solution: Let the C.P. of the article be ₹x [x > 0]

⇒ gain = ₹\(\frac{x}{4}\)

⇒ S. P. = \(₹\left(x+\frac{x}{4}\right)=₹ \frac{5 x}{4}\)

⇒ \(x=\frac{300 \times 4}{5}=240\)

∴ C.P. = ₹240

∴ gain = ₹(300 – 240) = ₹60

⇒ gain percentage = \(\frac{6}{240}\) x 100 = 25

∴ The correct answer is 3. 25

The gain percentage 25

Example 7. If a% loss is on cost price then loss percentage on selling price is

1. \(₹ \frac{100 a}{100-a}\)
2. \(₹ \frac{100-a}{100 a}\)
3. \(₹ \frac{100 a}{100+a}\)
4. \(₹ \frac{100+a}{100 a}\)

Solution: If cost price is ₹100 then selling price is ₹(100 – a)

⇒ If selling price is ₹(100 a) then loss is ₹a

⇒ If selling price is ₹1 then loss is \(₹ \frac{a}{100-a}\)

⇒ If selling price is ₹100 then loss is \(₹ \frac{100 a}{100-a}\)

∴ Loss is \(₹ \frac{100 a}{100-a}\)

∴ So the correct answer is 1. \(₹ \frac{100 a}{100-a}\)

Loss percentage on selling price is \(₹ \frac{100 a}{100-a}\)

Example 8. If the ratio of cost price and selling price x: y [0 < x < y], the profit percentage is

1. \(\frac{y-x}{100 x}\)
2. \(\frac{100(y-x)}{x}\)
3. \(\frac{100(x-y)}{y}\)
4. \(\frac{100(x+y)}{y}\)

Solution: Let cost price is ₹ax and selling price is ₹ay.

Profit = ₹(ay – ax) = ₹ a (y – x)

∴ Profit percentage = \(\frac{a(y-x)}{a x} \times 100=\frac{100(y-x)}{x}\)

∴ So the correct answer is 2. \(\frac{100(y-x)}{x}\)

The profit percentage is \(\frac{100(y-x)}{x}\)