**Madhyamik Class 10 Mathematics Solved Paper 2023**

**MATHEMATICS**

**Time: 3 Hours 15 Minutes**

*(First 15 Minutes for reading the question paper only, **3 Hours for writing) *

**Full Marks: For Regular Candidates – 90**

**For External Candidates – 100**

*[The answers to questions nos. 1, 2, 3, and 4 are to be written at the beginning of the answer- script mentioning the question numbers in the serial order. Necessary calculation and drawing must be given on the right-hand side by drawing margins on the first few pages of the answer script. Tables and calculators of any type are not allowed. Approximate value of π may be taken as \frac{22}{7} if necessary. Graph paper will be supplied if required. Arithmetic problems may be solved by the algebraic method.] *

*[Alternative question no. 11 is given for visually impaired candidates on page no. 15] *

*[Additional Question No. 16 is only for external candidates on page no. 16]*

**1. Choose the correct answer from the following questions: 1×6=6**

(i) Three friends *A, B, and C* started a business with capitals Rs. *x*, 2*x* and y respectively, at the end of the term profit is Rs. *z*, then the share of the profit of *A* is

- Rs. \frac{xz}{3x+y}
- Rs. \frac{2xz}{3x+y}
- Rs. \frac{z}{2x+y}
- Rs. \frac{xyz}{3x+y}

(ii) Number of solutions of equation *x*^{2} = *x* is

- 1
- 2
- 0
- 3

(iii) If two circles touch each other internally, then the number of common tangents of the circles are

- 1
- 2
- 3
- 4

(iv) For any value of θ the maximum value of 5 + 4 sin θ is

- 9
- 1
- 0
- 5

(v) If the ratio of the volumes of two solid spheres is 27 : 8, then the ratio of their curved surface area is

- 1 : 2
- 9 : 4
- 1 : 8
- 1 : 16

(vi) Three values of a variable are 4, 5 and 7, if their frequencies are p – 2, p + 1 and p – 1 respectively and the Mean is 5.4, then the value of p is:

- 1
- 2
- 3
- 4

**2. Fill up the blanks ( any five): [1×5=5]**

- If the amount of Rs. 180 after one year will be Rs. 198, then the rate of simple interest is _____.
- If mean proportional of (
*a*) and (4^{2}bc*bc*) is*x*, then the value of*x*is _____. - If tan θ cos 60° = \frac{\sqrt{3}}{2} then the value of sin(θ – 15°) is _____.
- If
*∠A*and*∠B*are complementary then ∠*A*+ ∠*B*= _____. - The median of the numbers 8, 15, 10, 11, 7, 9, 11, 13 and 16 is _____.
- The shape of a pencil with one end sharpened is the combination of a and a _____.

**3. Write True or False ( any five): [1×5=5]**

- In the compound interest if the rate of interest in the first three years is
*r*_{1},%,*r*_{2}%, 2*r*_{3}% respectively, then the amount for the principal*P*at the end of three years is*P*\big(1+ \frac{ r_{1} }{100}) \big(1+ \frac{ r_{2} }{100}) \big(1+ \frac{ r_{3}}{100}) - The values of cos 36° and sin 54° are equal.
- One tangent can be drawn on a circle from an external point.
- The compound ratio of 2
*ab*:*c*²,*bc*:*a*² and*ca*: 2*b*^{2}is 1 : 1. - If the numerical values of the curved surface area and volume of a sphere are equal, then the radius will be 3 units.
- The Mode of the data 5, 2, 4, 3, 5, 2, 5, 2, 5, 2 is 2.

**4. Answer the following questions ( any ten): [2×10=20]**

- Find the rate of simple interest per annum when the interest of some money in 5 years will be \frac{2}{5} part of its principal.
- In a business capitals of A and B are in the ratio \frac{1}{7} : \frac{1}{4} If they make a profit of Rs. 11,000 at the end of the year, calculate the share of their profit.
- If the sum of the roots of the equation x
^{² }– x = K(2x – 1) is 2, then find the value of K. - If b ∝ a
^{²}and a increase in the ratio of 2 : 3, then find in what ratio b will be cc increased. - AB and CD are two chords of a circle. If we extend BA and DC, they intersect each other at point P. Prove that
**∠**PCB =**∠**PAD. - In ΔABC, L and M are two points on the sides AC and BC respectively such that LM || AB and AL are (x – 2) units, AC = 2x + 3 units, BM = (x – 3) units and BC = 2x units. Determine the value of x.
- Two circles touch each other externally at point C. A direct common tangent AB touches the two circles at points A and B. Find the value of
**∠**ACB. - If tan 2A = cot(A – 30°), then find the value of sec(A + 20°).
- If tan θ = \frac{8}{15} find the value of sin θ.
- If the volume of a right circular cone is V cubic unit, the base area is A sq. unit and the height is H unit, then find the value of \frac{AH}{3V}
- Find the ratio of the volumes of a solid right circular cylinder and a solid right circular cone of equal radii and equal heights.
- If 6, 8, 10, 12, 13, x are in increasing order and their mean and median are equal, then find the value of x.

**5. Answer any one question: [5]**

- The number of smokers is decreasing at the rate of 6 \frac{1}{4} % per year due to publicity of anti-smoking. If at present the number of smokers in a town is 22500, find the number of smokers of that town 2 years ago.
- In a partnership business, the ratio of the capital of three friends is 6: 4 : 3. After 4 months 1
^{st}friend withdraws his half of the capital and after 8 more months total profit is Rs. 61,050. Find the share of the profit of three friends.

**6. Answer any one question : [3]**

- Solve: \frac{x-3}{x+3} – \frac{x+3}{x-3} + 6 \frac{6}{7} = 0 . (x ≠ 3, -3)
- If the price of 1 dozen pens is reduced by Rs. 6, then 3 more pens will be got for Rs. 30. Calculate the price of 1 dozen pens before the reduction of price.

**7. Answer any one question: [3]**

- If x = \frac{1}{2-\sqrt{3}} and y = \frac{1}{2+ \sqrt{3} }, then find the value of \frac{1}{x+1} + \frac{1}{y+1}
- If
*x*∝*y*and*y*∝*z*, then show that \frac{x}{yz} + \frac{y}{zx} + \frac{z}{xy} \propto \frac{1}{x} + \frac{1}{y} + \frac{1}{z}.

**8. Answer any one question : [3]**

- If \frac{a^{2}}{b+c} =\frac{b^{2}}{c+a} = \frac{c^{2}}{a+b} = 1, then show that \frac{1}{1+a} + \frac{1}{1+b} + \frac{1}{1+c} = 1
- If the fourth and fifth of the five numbers in continued proportion are 54 and 162 respectively, find the first number.

**9. Answer any one question: [5]**

- Prove that in a cyclic quadrilateral opposite angle are supplementary.
- Prove that the tangent to a circle at any point on it is perpendicular to the radius that passes through the point of contact.

**10. Answer any one question: [3]**

- ABCD is a cyclic quadrilateral. Bisectors of ∠DAB and ∠BCD intersect the circle at X and Y respectively. If O be the centre of the circle, find ∠XOY.
- Prove that a cyclic trapezium is an isosceles trapezium.

**11. Answer any one question: [5]**

- Draw a right-angled triangle of which two sides containing the right angle have the lengths 5 cm and 6 cm. Now draw an incircle of the triangle.
- Construct a square of the equal area of an equilateral triangle of side 7 cm.

**12. Answer any two questions: [3×2=6]**

- If cos θ = \frac{x}{\sqrt{x^2+y^2}} , then prove that x sin θ = y cos θ
- Radius of a circle is 7 cm. Find the angle in radians which is subtended by an arc of this circle of length 5.5 cm at the centre of the circle.
- show that \frac{tan\theta+sec \theta-1}{tan\theta-sec\theta+1} = \frac{1+sec\theta }{cos\theta}

**13. Answer any one question: [5]**

- Angle of elevation of the top of an incomplete tower from a point at a distance 50 m from its foot is 30°. How much should the height of the tower be increased so that the angle of elevation of the top will be 45° from that point?
- From the roof of the building the angle of depression of the top and foot of the lamp post is 30° and 60° respectively. Find the ratio of the heights of the building and the lamp post.

**14. Answer any two questions: [4×2=8]**

- Two solid spheres with radii of 1 cm and 6 cm lengths are melted and a hollow sphere with an outer radius of 9 cm is made. Determine the inner radius of the new hollow sphere.
- The height of a right circular cone is twice the radius of the base. If the height were seven times the diameter of the base then the volume of the cone would have been 539 cu cm more. Find the height of the cone.
- The curved surface area of a right circular cylindrical wooden log of uniform density is 440 sq. decimeters. The weight of 1 cubic decimeter of wood is 3 kg and the weight of a log is 18.48 quintals. Find the diameter of the log.

**15. Answer any two questions: [4×2=8]**

(i) If the arithmetic mean and total frequency of the following distribution are 50 and 120 respectively, then find the value of f_{1} and f_{2}:

Class |
Frequency |

0 - 20 | 17 |

20 - 40 | f_{1} |

40 - 60 | 32 |

60 - 80 | f_{2} |

80 - 100 | 19 |

(ii) Construct the table of cumulative frequency (greater than type) and draw the ogive from the following frequency distribution :

Class |
Frequency |

0 - 10 | 7 |

10 - 20 | 10 |

20 - 30 | 23 |

30 - 40 | 50 |

40 - 50 | 6 |

50 - 60 | 4 |

(iii) Find the mode of the following frequency distribution :

Class |
Frequency |

50 - 59 | 5 |

60 - 69 | 20 |

70 - 79 | 40 |

80 - 89 | 50 |

90 - 99 | 30 |

100 - 109 | 6 |

**[Alternative Question for Sightless Candidates]**

**11. Answer any one question :**

- Describe the process of drawing an incircle of a right-angled triangle.
- Describe the method of construction of a square of the equal area of an equilateral triangle.

**[Additional Question for External Candidates]**

**16. (a) Answer any three questions: [2×3=6]**

- If x ∝ y, y ∝ z and z ∝ x, then find the relation between the constants of variations.
- In a partnership business, the capital of A is 1\frac{1}{2} times that of B. At the end of the year if B gets Rs. 1,500 as a share of the profit, find the share of A.
- If x+\sqrt{x^2-9} = 9 then find the value of x-\sqrt{x^2-9}.
- The numerical value of the volume of a sphere is twice the numerical value of its surface area. Find the radius of the sphere.

**(b) Answer any four questions: [1×4=4]**

- Which one is greater \sqrt{7} - \sqrt{2} or \sqrt{8} - \sqrt{3}?
- Under which condition the quadratic equation ax
^{2}+ bx + c = 0 (a ≠ 0) have one zero roots. - If the lengths of three sides of two triangles are in proportion, then which type of triangle is this?
- In how many years a sum of money at 6 \frac{1}{4}% simple interest per annum would be 4 double?
- Fill up the blank :

The front angle formed at the center of a circle by an arc is the ____ of the angle formed by the same arc at any point on the circle.