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Odisha BSE Madhyama Mathematics study materials
Odisha BSE Madhyama Mathematics study resources

Odisha BSE Madhyama Mathematics practice questions

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Question 1Madhyama Class IX Syllabus

In the Odisha Madhyama Class IX course, the value of (√5 + √3)(√5 − √3) is:

  1. √2
  2. 2 Correct answer.
  3. 8
  4. 2√15

Explanation

This is a difference of squares: (a+b)(a−b)=a²−b². Here a²=5 and b²=3, so the product equals 5−3=2. The cross terms +√15 and −√15 cancel each other.

Question 2Madhyama Class IX Syllabus

The rationalising factor of 1/(√7 − 2) is:

  1. √7 + 2 Correct answer.
  2. 2 − √7
  3. √7 − 2
  4. 7 − 4

Explanation

To remove the surd from the denominator, multiply numerator and denominator by the conjugate √7 + 2. The denominator then becomes (√7)² − 2² = 7 − 4 = 3, a rational number.

Question 3Madhyama Class IX Syllabus

If 2x + 3y = 12 and 2x − 3y = 0, then the value of x is:

  1. 6
  2. 4
  3. 2
  4. 3 Correct answer.

Explanation

Adding the two equations eliminates y: 4x = 12, so x = 3. Substituting into 2x − 3y = 0 gives 6 = 3y, hence y = 2, which checks in the first equation.

Question 4Madhyama Class IX Syllabus

The degree of the polynomial 4x³ − 2x² + 7x − 9 is:

  1. 2
  2. 3 Correct answer.
  3. 1
  4. 4

Explanation

The degree of a polynomial in one variable is the highest power of the variable with a non-zero coefficient. In 4x³ − 2x² + 7x − 9 the highest such power is 3, so the degree is 3.

Question 5Madhyama Class IX Syllabus

If the angles of a triangle are in the ratio 2 : 3 : 4, the largest angle measures:

  1. 90°
  2. 40°
  3. 80° Correct answer.
  4. 60°

Explanation

The angle sum of a triangle is 180°, and the ratio parts total 2+3+4=9. Each part equals 180°/9 = 20°, so the largest angle is 4 × 20° = 80°.

Question 6Madhyama Class IX Syllabus

The mean of the observations 12, 15, 18, 21, 24 is:

  1. 15
  2. 18 Correct answer.
  3. 17
  4. 20

Explanation

The mean equals the sum of observations divided by their number. The sum is 12+15+18+21+24 = 90, and 90 ÷ 5 = 18.

Question 7Madhyama Class IX Syllabus

The point (−4, 0) in the Cartesian plane lies on:

  1. the negative x-axis Correct answer.
  2. the origin
  3. the positive x-axis
  4. the negative y-axis

Explanation

A point (x, 0) lies on the x-axis because its y-coordinate is zero. Since x = −4 is negative, the point lies on the negative side of the x-axis.

Question 8Madhyama Class IX Syllabus

The area of a triangle whose base is 10 cm and height 6 cm is:

  1. 36 cm²
  2. 30 cm² Correct answer.
  3. 60 cm²
  4. 16 cm²

Explanation

The area of a triangle equals half the product of base and height. Here ½ × 10 × 6 = 30, so the area is 30 cm².

Question 9Madhyama Class IX Syllabus

Factorising x² − 7x + 12 gives:

  1. (x + 3)(x + 4)
  2. (x − 1)(x − 12)
  3. (x − 2)(x − 6)
  4. (x − 3)(x − 4) Correct answer.

Explanation

We need two numbers whose product is +12 and whose sum is −7; these are −3 and −4. Hence x² − 7x + 12 = (x − 3)(x − 4), which expands correctly.

Question 10Madhyama Class IX Syllabus

If x + 1/x = 5, then x² + 1/x² equals:

  1. 27
  2. 25
  3. 23 Correct answer.
  4. 24

Explanation

Squaring both sides gives x² + 2 + 1/x² = 25. Subtracting the middle term 2 yields x² + 1/x² = 23.

Question 11Madhyama Class IX Syllabus

The sum of the first 20 natural numbers is:

  1. 190
  2. 200
  3. 210 Correct answer.
  4. 420

Explanation

The sum of the first n natural numbers is n(n+1)/2. For n = 20 this gives 20 × 21 ÷ 2 = 210.

Question 12Madhyama Class IX Syllabus

In a right triangle, if one acute angle is 35°, the other acute angle is:

  1. 45°
  2. 55° Correct answer.
  3. 35°
  4. 65°

Explanation

The two acute angles of a right triangle are complementary because the third angle is 90°. Thus the other acute angle is 90° − 35° = 55°.

Question 13Madhyama Class IX Syllabus

A chord of length 8 cm is at a distance of 3 cm from the centre of a circle. The radius of the circle is:

  1. √55 cm
  2. 5 cm Correct answer.
  3. 4 cm
  4. 7 cm

Explanation

The perpendicular from the centre bisects the chord, giving a right triangle with legs 4 cm and 3 cm. The radius is the hypotenuse: √(16+9) = √25 = 5 cm.

Question 14Madhyama Class IX Syllabus

If tan θ = 3/4 and θ is acute, then sin θ equals:

  1. 4/5
  2. 3/4
  3. 5/3
  4. 3/5 Correct answer.

Explanation

With tan θ = opposite/adjacent = 3/4, the hypotenuse is √(9+16)=5. Hence sin θ = opposite/hypotenuse = 3/5.

Question 15Madhyama Class IX Syllabus

The median of the data 7, 3, 9, 5, 11, 13 is:

  1. 6
  2. 7
  3. 9
  4. 8 Correct answer.

Explanation

Arranging in order gives 3, 5, 7, 9, 11, 13. With six observations the median is the mean of the third and fourth values: (7+9)/2 = 8.

Question 16Madhyama Class IX Syllabus

The simple interest on ₹5,000 at 8% per annum for 3 years is:

  1. ₹400
  2. ₹1,500
  3. ₹1,200 Correct answer.
  4. ₹1,000

Explanation

Simple interest equals P × R × T ÷ 100. Substituting gives 5000 × 8 × 3 ÷ 100 = ₹1,200.

Question 17Madhyama Class IX Syllabus

If the diagonal of a square is 10 cm, its area is:

  1. 20 cm²
  2. 100 cm²
  3. 25 cm²
  4. 50 cm² Correct answer.

Explanation

For a square of side a, the diagonal is a√2, so a = 10/√2 and a² = 100/2 = 50. Since the area is a², it equals 50 cm².

Question 18Madhyama Class IX Syllabus

The quadratic equation x² − 5x + 6 = 0 has roots:

  1. 2 and 3 Correct answer.
  2. −2 and −3
  3. 1 and 6
  4. −1 and −6

Explanation

We need two numbers with product 6 and sum 5, namely 2 and 3. Thus (x−2)(x−3)=0 gives x = 2 or x = 3.

Question 19Madhyama Class IX Syllabus

If a transversal cuts two parallel lines, and one pair of corresponding angles measures (3x + 10)° and (5x − 30)°, then x equals:

  1. 15
  2. 25
  3. 10
  4. 20 Correct answer.

Explanation

Corresponding angles formed by a transversal on parallel lines are equal. So 3x + 10 = 5x − 30, giving 40 = 2x and x = 20. Checking: both angles then measure 70°.

Question 20Madhyama Class IX Syllabus

The value of sin 30° cos 60° + cos 30° sin 60° is:

  1. 1/2
  2. √3/2
  3. 1 Correct answer.
  4. 0

Explanation

The expression matches the identity sin(A+B) = sin A cos B + cos A sin B with A = 30° and B = 60°. Hence it equals sin 90° = 1.

Question 21Madhyama Class IX Syllabus

If the polynomial p(x) = x³ − 4x² + x + 6 is divided by (x − 2), the remainder is:

  1. 0 Correct answer.
  2. −4
  3. 6
  4. 2

Explanation

By the remainder theorem, the remainder on division by (x − 2) is p(2). Computing: 8 − 16 + 2 + 6 = 0, so (x − 2) is a factor.

Question 22Madhyama Class IX Syllabus

Two circles of radii 5 cm and 3 cm touch externally. The length of their common external tangent is:

  1. √34 cm
  2. 4 cm
  3. 8 cm
  4. 2√15 cm Correct answer.

Explanation

For externally touching circles, the direct common tangent has length √(d² − (r₁ − r₂)²) with d = r₁ + r₂ = 8. Thus √(64 − 4) = √60 = 2√15 cm.

Question 23Madhyama Class IX Syllabus

If the roots of x² − px + 8 = 0 differ by 2, then the positive value of p is:

  1. 4
  2. 8
  3. 2
  4. 6 Correct answer.

Explanation

Let the roots be α and α+2. Their product α(α+2)=8 gives α²+2α−8=0, so α=2 or α=−4. Their sum is 2α+2, which equals p; with α=2, p = 6.

Question 24Madhyama Class IX Syllabus

A sum of money doubles itself in 8 years at simple interest. In how many years will it become four times itself at the same rate?

  1. 16 years
  2. 32 years
  3. 20 years
  4. 24 years Correct answer.

Explanation

Doubling means the interest earned in 8 years equals the principal, so the rate is 100/8 = 12.5% per annum. Becoming four times requires interest of 3P, which takes 3 × 8 = 24 years.

Question 25Madhyama Class IX Syllabus

The mean of 10 observations is 20. If one observation 30 is removed, the mean of the remaining observations is:

  1. 18
  2. 19
  3. 21
  4. 170/9 Correct answer.

Explanation

The total of the 10 observations is 10 × 20 = 200. Removing 30 leaves a total of 170 over 9 observations, so the new mean is 170/9 ≈ 18.89.

Question 26Madhyama Class IX Syllabus

In the Madhyama syllabus, the value of (√5 + √3)(√5 − √3) is:

  1. 2 Correct answer.
  2. 2√15
  3. 4
  4. 8

Explanation

This is a difference of squares: (a + b)(a − b) = a² − b². So (√5)² − (√3)² = 5 − 3 = 2. The surd terms cancel entirely.

Question 27Madhyama Class IX Syllabus

The compound interest on ₹8,000 for 2 years at 10% per annum, compounded annually, is:

  1. ₹1,600.50
  2. ₹1,600
  3. ₹1,680 Correct answer.
  4. ₹1,760

Explanation

Amount = 8000 × (1.1)² = 8000 × 1.21 = ₹9,680. Compound interest = 9680 − 8000 = ₹1,680, which exceeds the simple interest of ₹1,600 because interest earns interest in year two.

Question 28Madhyama Class IX Syllabus

In a circle, an arc subtends an angle of 72° at the centre. If the radius is 7 cm, the length of the arc is (take π = 22/7):

  1. 22/5 cm
  2. 11/5 cm
  3. 88/5 cm
  4. 44/5 cm Correct answer.

Explanation

Arc length = (θ/360) × 2πr = (72/360) × 2 × (22/7) × 7. This simplifies to (1/5) × 44 = 44/5 cm, that is 8.8 cm.

Question 29Madhyama Class IX Syllabus

The point of intersection of the lines x + y = 7 and x − y = 3 is:

  1. (3, 4)
  2. (4, 3)
  3. (2, 5)
  4. (5, 2) Correct answer.

Explanation

Adding the two equations eliminates y: 2x = 10, so x = 5. Substituting into x + y = 7 gives y = 2, so the intersection is (5, 2), which satisfies both equations.

Question 30Madhyama Class IX Syllabus

In a cyclic quadrilateral ABCD, angle A = (2x + 10)° and angle C = (3x − 20)°. The value of x is:

  1. 28
  2. 38 Correct answer.
  3. 40
  4. 30

Explanation

Opposite angles of a cyclic quadrilateral are supplementary, so (2x + 10) + (3x − 20) = 180. This gives 5x − 10 = 180, hence 5x = 190 and x = 38.

Question 31Madhyama Class IX Syllabus

The mode of the data 4, 6, 4, 7, 9, 4, 6, 8 is:

  1. 4 Correct answer.
  2. 7
  3. 9
  4. 6

Explanation

The mode is the observation occurring most frequently. Here 4 appears three times, 6 appears twice, and 7 and 9 appear once each, so the mode is 4.

Question 32Madhyama Class IX Syllabus

If the perimeter of a rectangle is 40 cm and its length is 12 cm, then its area is:

  1. 96 cm² Correct answer.
  2. 48 cm²
  3. 120 cm²
  4. 192 cm²

Explanation

Perimeter = 2(l + b) = 40, so l + b = 20 and b = 20 − 12 = 8 cm. Area = l × b = 12 × 8 = 96 cm².

Question 33Madhyama Class IX Syllabus

The rationalising factor of 1/(√7 − √5) is:

  1. √7 − √5
  2. √7 + √5 divided by 2
  3. √7 + √5 Correct answer.
  4. 7 − 5

Explanation

To remove surds from the denominator of 1/(√7 − √5), multiply numerator and denominator by the conjugate √7 + √5. The denominator becomes 7 − 5 = 2, a rational number.

Question 34Madhyama Class IX Syllabus

The sum of the interior angles of a regular polygon is 1080°. The number of sides of the polygon is:

  1. 10
  2. 6
  3. 8 Correct answer.
  4. 12

Explanation

For an n-sided polygon the interior angle sum is (n − 2) × 180°. Setting (n − 2) × 180 = 1080 gives n − 2 = 6, so n = 8.

Question 35Madhyama Class IX Syllabus

If log₁₀ 2 = 0.3010, then the value of log₁₀ 8 is:

  1. 0.1505
  2. 0.6020
  3. 1.2040
  4. 0.9030 Correct answer.

Explanation

Since 8 = 2³, log₁₀ 8 = 3 log₁₀ 2 = 3 × 0.3010 = 0.9030. This uses the power rule for logarithms.

Question 36Madhyama Class IX Syllabus

The HCF of the polynomials x² − 9 and x² + 6x + 9 is:

  1. x + 3 Correct answer.
  2. x − 3
  3. x + 9
  4. x² − 9

Explanation

Factorising gives x² − 9 = (x + 3)(x − 3) and x² + 6x + 9 = (x + 3)². The highest common factor is therefore (x + 3).

Question 37Madhyama Class IX Syllabus

The distance between the points (3, 4) and (0, 0) is:

  1. 7 units
  2. 5 units Correct answer.
  3. 1 unit
  4. 25 units

Explanation

Using the distance formula, d = √[(3 − 0)² + (4 − 0)²] = √(9 + 16) = √25 = 5. This is the standard 3-4-5 right triangle.

Question 38Madhyama Class IX Syllabus

A cone has base radius 6 cm and slant height 10 cm. Its total surface area is (take π = 3.14):

  1. 188.40 cm²
  2. 376.80 cm²
  3. 301.44 cm² Correct answer.
  4. 113.04 cm²

Explanation

Total surface area of a cone = πr(l + r) = 3.14 × 6 × (10 + 6) = 3.14 × 96 = 301.44 cm². This includes both the curved surface and the base.

Question 39Madhyama Class IX Syllabus

If sin θ = 5/13 and θ is acute, then the value of cos θ is:

  1. 5/12
  2. 13/12
  3. 12/13 Correct answer.
  4. 13/5

Explanation

For an acute angle, sin²θ + cos²θ = 1, so cos²θ = 1 − 25/169 = 144/169. Since θ is acute, cos θ = 12/13, giving the 5-12-13 triangle.

Question 40Madhyama Class IX Syllabus

The 10th term of the arithmetic progression 3, 7, 11, 15, ... is:

  1. 43
  2. 35
  3. 40
  4. 39 Correct answer.

Explanation

Here a = 3 and d = 4, so the nth term is a + (n − 1)d. For n = 10, the term is 3 + 9 × 4 = 3 + 36 = 39.

Question 41Madhyama Class IX Syllabus

If a and b are the roots of 2x² − 7x + 3 = 0, then the value of a + b is:

  1. 3/2
  2. 2/7
  3. −7/2
  4. 7/2 Correct answer.

Explanation

For ax² + bx + c = 0, the sum of roots is −b/a. Here a = 2 and b = −7, so the sum is 7/2. The product would be 3/2.

Question 42Madhyama Class IX Syllabus

In a triangle ABC, DE is parallel to BC, with D on AB and E on AC. If AD = 3 cm, DB = 5 cm and AE = 4.5 cm, then EC is:

  1. 6 cm
  2. 7.5 cm Correct answer.
  3. 4 cm
  4. 9 cm

Explanation

By the basic proportionality theorem, AD/DB = AE/EC. So 3/5 = 4.5/EC, giving EC = (5 × 4.5)/3 = 7.5 cm.

Question 43Madhyama Class IX Syllabus

The probability of getting a sum of 9 when two fair dice are thrown together is:

  1. 1/9 Correct answer.
  2. 1/6
  3. 1/12
  4. 5/36

Explanation

There are 36 equally likely outcomes. Sums of 9 occur as (3,6), (4,5), (5,4), (6,3), giving 4 favourable outcomes. Probability = 4/36 = 1/9.

Question 44Madhyama Class IX Syllabus

The value of (sin 45° + cos 45°)² is:

  1. √2
  2. 4
  3. 1
  4. 2 Correct answer.

Explanation

Since sin 45° = cos 45° = 1/√2, their sum is 2/√2 = √2. Squaring gives 2, which also follows from sin²θ + cos²θ + 2 sin θ cos θ = 1 + 1 = 2.

Question 45Madhyama Class IX Syllabus

A shopkeeper marks an article at ₹1,200 and allows a discount of 15%. If his cost price is ₹900, his profit percentage is:

  1. 15%
  2. 20%
  3. 33⅓%
  4. 13⅓% Correct answer.

Explanation

Selling price = 1200 × 0.85 = ₹1,020. Profit = 1020 − 900 = ₹120, so profit percentage = (120/900) × 100 = 13⅓%.

Question 46Madhyama Class IX Syllabus

If x − 1/x = 3, then the value of x³ − 1/x³ is:

  1. 36 Correct answer.
  2. 18
  3. 27
  4. 45

Explanation

Use the identity a³ − b³ = (a − b)³ + 3ab(a − b) with a = x and b = 1/x, so ab = 1. Then x³ − 1/x³ = 27 + 3 × 1 × 3 = 36.

Question 47Madhyama Class IX Syllabus

A man rows downstream 32 km and upstream 14 km, taking 4 hours each time. The speed of the stream is:

  1. 2 km/h
  2. 2.25 km/h Correct answer.
  3. 1.5 km/h
  4. 3 km/h

Explanation

Downstream speed = 32/4 = 8 km/h and upstream speed = 14/4 = 3.5 km/h. Stream speed = (8 − 3.5)/2 = 2.25 km/h.

Question 48Madhyama Class IX Syllabus

The roots of the equation x² − 2x + 5 = 0 are:

  1. 1 ± i
  2. 2 ± i
  3. −1 ± 2i
  4. 1 ± 2i Correct answer.

Explanation

Using the quadratic formula, x = [2 ± √(4 − 20)]/2 = [2 ± √(−16)]/2 = [2 ± 4i]/2 = 1 ± 2i. The discriminant is negative, so the roots are complex conjugates.

Question 49Madhyama Class IX Syllabus

If the ratio of the volumes of two spheres is 27 : 64, then the ratio of their surface areas is:

  1. 81 : 256
  2. 9 : 16 Correct answer.
  3. 27 : 64
  4. 3 : 4

Explanation

Volumes vary as the cube of the radius, so the ratio of radii is the cube root of 27 : 64, namely 3 : 4. Surface areas vary as the square of the radius, giving 9 : 16.

Question 50Madhyama Class IX Syllabus

A ladder 13 m long rests against a vertical wall with its foot 5 m from the wall. If the foot is moved 2 m further away from the wall, the top of the ladder slides down by:

  1. 1 m Correct answer.
  2. 3 m
  3. 0.5 m
  4. 2 m

Explanation

Initially the top is at √(169 − 25) = 12 m. After moving the foot to 7 m, the top is at √(169 − 49) = √120 ≈ 10.95 m. The slide is about 1.05 m, so of the choices given, 1 m is the correct answer.