Madhyamik Class 10 Mathematics Solved Paper 2024

2024

Mathematics

Time – 3 Hours 15 Minutes

(First 15 minutes for reading the question paper)

Full\ Marks =\begin{cases} \ \ 90\ - For\ Regular\ Candidates\\ 100\ - For\ External\ Candidates\end{cases}

Special credit will be given for answers which are brief and to the point.

Marks will be deducted for spelling mistakes, untidiness and bad handwriting.

[The answers of the question Nos.1, 2, 3, 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 on the answer script. Tables and Calculators of any type are not allowed. Approximate value of π may be taken as 22/7, if required. Graph paper will be supplied with question paper. Arithmetic problems may be solved by algebraic method.]

[Alternative Question No. 11 is given for sightless candidates on Page No.15]

[Additional Question No. 16 is only for external candidates on Page No.15]

1. Choose the correct option in each case from the following: 1×6=6

(i) If the rate of simple interest and compound interest are both 10% per annum, then the ratio of simple interest and compound interest in second year of a certain principal will be –

(a) 20 : 21

(b) 10 : 11

(c) 5 : 6

(d) 1 : 1

Solution

(ii) If one root of the equation ax²+abcx+bc = 0 (a≠0) is reciprocal to the other, then

(a) abc = 1

(b) b = ac

(c) bc = 1

(d) a = bc

Solution

(iii) Two circles of diameter 5 cm and 7 cm touch each other internally, the distance between their centres will be –

(a) 1 cm

(b) 2 cm

(c) 3 cm

(d) 4 cm

Solution

(iv) The minimum value of tanθ + cotθ is

(a) 0

(c) -2

(b) 2

(d) 1

Solution

(v) If the height and base area of a solid hemisphere and a solid cylinder are same, then the ratio of their volume is

(a) 1 : 3

(b) 1 : 2

(c) 2 : 3

(d) 3 : 4

Solution

(vi) If A and M be the arithmetic mean and median of first ten natural numbers, then relation between them will be

(a) A > M

(b) A < M

(c) A = 1/M

(d) A = M

Solution

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

(i) The equation (P – 3) x² + 5x + 10=0 will not be a quadratic equation for P= ___________ .

(ii) The sum of Principal and Compound interest in certain time is called _________ .

(iii) The corresponding sides of two similar triangles are _______ .

(iv) If sin (θ – 30°) = 1/2, then cos θ= _________ .

(v) If V be the volume, R be the radius of the base and H be the height of a right circular cone, then H = _________ .

(vi) The numbers 8, 9, 12, 17, x+2, x+4, 30, 34, 39 are in increasing order and if the median of these numbers is 24, then x = _______ .

Solution

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

(i) In a partnership business, the ratio of the capitals of three members is a:b:c and the ratio of time of investment of their capitals is x:y:z. The ratio of their profit will be ax:by:cz.

Solution

(ii) If a ∝ b, b ∝ 1/c, and c ∝  d, then a ∝ 1/d.

Solution

(iii) If two chords of a circle are at an equal distance from the centre of that circle, then those two chords will be always parallel.

Solution

(iv) The hour hand of a clock rotates through an angle radian in 2 hours.

Solution

(v) The ratio of whole surfaces of a solid sphere and a solid hemisphere of equal radius is 2 : 1.

Solution

(vi) The average of n numbers is x̄. If the sum of first (n-1) numbers is K, then n^th number is (n-1) x̄ +K.

Solution

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

(i) In how many years will Rs. 500 get Rs. 105 compound interest at 10% per annum compound interest?

Solution

(ii) In a partnership business, ratio of capitals of Ila, Rahima and Bela is 3 : 8 : 5. If profit of Ila is Rs. 600 less than profit of Bela, Find the total profit in the business.

Solution

(iii) If the roots of the equation: x² – 22x + 105 = 0 are α and β, then find the value of {1\over \alpha} + {1\over \beta}.

Solution

(iv) If (3x-2y) : (3x+2y) = 4 : 5 then what is the value of (x+y) : (x-y)?

Solution

(v) BOC is a diameter of a circle whose centre is O. ABCD is a cyclic quadrilateral. If ∠ADC =110°, find ∠ACB.

Solution

(vi) In trapezium ABCD, BC||AD and AD=4 cm, two diagonals AC & BD meet at O in such a way that {AO \over OC}={DO \over OB}={1\over2},then find the length of BC.

Solution

(vii) In △ABC, ∠ABC=90°, AB=6 cm, BC=8cm, then find the circum-radius of the triangle ABC.

Solution

(viii) If r cosθ = 2√3 and r sinθ = 2, then find the value of r and θ when 0⁰<θ<90°

Solution

(ix) If sin (A+B) = 1 and cos (A-B) = 1, then find the value of cot 2A, 0⁰ ≤ (A+B) ≤ 90° and A ≥ B

Solution

(x) How much percent of the curved surface area of a sphere will be increased if the radius is doubled.

Solution

(xi) The length of the diagonal of each face of a cube is 6 √2 cm. Find the total surface area of the cube.

Solution

(xii) The mean of a frequency distribution is 7 and Σfixi = 140, then find the value of Σfi.

Solution

5. Answer any one question: 5

i) After retirement from service, Gobinda Babu got Rs. 5,00,000. He deposited a part of it in post office at 7.2% simple interest per annum and he deposited other part in a bank at 6% simple interest per annum. Every year he got Rs. 33,600 in total as interest from bank and post office. Find the amounts he deposited in bank and post office separately.

Solution

ii) Aman has taken a loan of Rs. 25,000 for 3 years, in such a way that the compound interest for 1^st year, 2^nd year and 3^rd year is 4%, 5% and 6% p.a. respectively. Find the amount be deposited after 3 years.

Solution

6. Answer any one question: 3

i) Speed of A is 1m/sec more than that of B. A reaches 2 seconds earlier than B in 180m sprint competition. Find the speed of B in m/sec.

Solution

ii) Solve : (2x + 1) + {3\over (2x + 1)}=4, {(x \neq - {1\over 2} )}

Solution

7. Answer any one question:

i) If (√a + √b) ∝ (√a-√b) then show that (a + b) ∝ \sqrt{ab}  

Solution

ii) If x = √3 + √2 and y= 1/x then find {(x+{1\over x})^2}+({1\over y}-y)^2

Solution

8. Answer any one question: 3

i) If {x\over y+z}= {y\over z+x}= {z\over x+y}, then show that each ratio is 1/2 or -1.

Solution

ii) If a, b, c are in continued proportion, then prove that {1\over b}= {1\over b-a}+ {1\over b-c}

Solution

9. Answer any one question: 5

i) Prove that in any circle, angles in the same segment are equal.

Solution

ii) If two tangents are drawn to a circle from a point outside it, then prove that the line segments joining the point of contact and the exterior point are equal and they subtend equal angles at the centre.

Solution

10. Answer any one question: 3

i) ABCD is a circumscribed quadrilateral of a circle with centre O. Show that AB + CD = AD + BC.

Solution

ii) In a triangle PQR, ∠P = 90° and PS is perpendicular to QR.

Then prove that {1\over PS^2}- {1\over PQ^2}= {1\over PR^2}.

Solution

11. Answer any one question: 5

i) Draw a circle of radius 4 cm. Draw a tangent on this circle from an external point at a distance 9 cm from the centre of the circle.

Solution

ii) Draw a right angled triangle whose two adjacent sides of the right angle are 4 cm and 5 cm. Construct the circumcircle of this triangle.

Solution

12.Answer any two questions: 3×2=6

i) The difference of acute angles of a right angled triangle is 72°. Determine these acute angles in circular measure.

Solution

ii) If 5 sin² θ + 4 cos² θ = 9/2, then from this relation find the value of tan θ.

Solution

iii) If sin 17° = x/y then show that sec 17° – sin 73° = x^2\over y \sqrt{y^2-x^2} 

Solution

13. Answer any one question: 5

i) The angle of elevation of the top of a pillar of height h from two points in the same horizontal line through the foot of the pillar and in the same side of the pillar are θ and φ respectively. Find the distance between two points.

Solution

ii) Two pillars of equal heights are at the point A and B on the opposite side of the road which is 120m wide. From a point C on the line joining the foots of the pillars, the angle of elevation of the top of the pillar at A and B are 60° and 30° respectively. Find AC.

Solution

14. Answer any two questions: 4×2=8

i) The lower part of an ice-cream is conical and upper part is hemispheric with same circular base. If the height of the cone is 9cm and radius of the base 2.5 cm. Find the volume of the ice-cream.

Solution

ii) Difference of outer and inner curved surface area of a hollow cylindrical pipe is 44 sq.cm and length is 14cm. If the Volume of the material of the pipe be 99 cubic cm. Find inner and outer radius.

Solution

iii) If 75 buckets of water of equal measure are taken out from the water of a cubical water-filled-tank, then 2/5^th of the water remains in the tank. If the length of one edge of the tank is 1.5 metres, then calculate the quantity of water in litre that can be held in each bucket.

Solution

15. Answer any two questions: 2×4=8

i) Find the mode from the following data.

Class Interval	0-5	5-10	10-15	15-20	20-25	25-30	30-35	35-40
Frequency	2	6	10	16	22	11	8	5

 

Solution

ii) Find the mean by any suitable method from the following frequency distribution table.

Class Interval	85-105	105-125	125-145	145-165	165-185	185-205
Frequency	3	12	18	10	5	2

 

Solution

iii) Find the median of the data from the following distribution.

Marks Obtained	Less than  10	Less than 20	Less than 30	Less than 40	Less than 50	Less than 60
No. of Students	8	15	29	42	60	70

 

Solution

[Alternative Question for Sightless Candidates]

11. Answer any one question: 5

i) Describe the process of drawing a tangent to a circle from an external point.

Solution

ii) Describe the process of construction of circumcircle of a right angled triangle.

Solution

[Additional Questions for External Candidates]

16. a) Answer any three questions:

i) 0.6 of income of A = 75% of income of B. Find the ratio of income of A and B.

Solution

ii) Length of diagonal of a cube is 3 √3 cm. Find the volume of the cube.

Solution

iii) If sinθ = √3 cosθ , Find the value of tanθ+cotθ

Solution

iv) Find the ratio of the cost price and selling price of an article which is sold at 25% profit.

Solution

b) Answer any four questions: 4×1=4

i) If the sum of the roots of the equation ax2 + bx + ac = 0 ( a ≠ 0) is zero, then find b.

Solution

ii) Find the number of direct common tangents of two circles which touch each other externally.

Solution

iii) Express 22°30′ in radians.

Solution

iv) The angle in the segment of a circle which is greater than a semicircle is acute or obtuse?

Solution

v) write the magnitude of each angle of a cyclic parallelogram.

Solution