Madhyamik Class 10 Mathematics Solved Paper 2022

Mathematics Question Paper

Madhyamik Class 10 Mathematics Solved Paper 202

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]

Figures in the margin indicate full marks for each question
Special credits will be given for answers which are brief and to the point.
Marks will be deducted for spelling mistakes, untidiness and bad handwriting.

Question – 1

Choose the correct answer from the following questions: [1 × 6 = 6]

(i) At present the population of a village is P and if rate of increase of
population per year be r%, after n years the population will be:

(a) P (1 + $\text{r}\over100$ )ⁿ

(b) P (1 + $\text{r}\over50$ )ⁿ

(d) P (1 – $\text{r}\over100$ )ⁿ

Answer

(a) P (1 + $\text{r}\over100$ )ⁿ

(ii) Fatima, Shreya, and Smita started a business by investing total ₹ 6,000. After a year, Fatima, Shreya, and Smita get profit shares of ₹ 50, ₹ 100, and ₹ 150 respectively. Smita invested in this business:

(a) ₹ 1,000

(b) ₹ 2,000

(d) ₹ 4,000

Answer

Explanation:

Ratio of capital = 50 : 100 : 150

= 1 : 2 : 3

Fatima’s capital = 1x

Shreya’s capital = 2x

Smita’s capital = 3x

ATP: 1x + 2x + 3x = ₹ 6,000

or, 6x = ₹ 6,000

or, x = ₹ 1000

∴ Smita’s capital = 3x = 3 × ₹ 1000 = ₹ 3,000

(iii) If A : B = 2 : 3, B : C = 5 : 8, C : D = 6 : 7, then A : D =

(a) 2 : 7

(b) 7 : 2

(d) 5 : 14

Answer

(d) 5 : 14

Explanation:

$\text{A}\over \text{B}$ × $\text{B}\over \text{C}$ × $\text{C}\over \text{D}$ = $\frac{2 × 5 × 6}{3 × 8 × 7}$ = 5 : 14

(iv) ‘O’ is the center of a circle and PQ is a diameter. R is a point on the circle such that PR = RQ, then the value of ∠RPQ:

(a) 30°

(b) 90°

(d) 45°

Answer

(d) 45°

Explanation:

Since PR = RQ, triangle PRQ is isosceles, and

∠PRQ = 90° (because PQ is the diameter).

Using the angle sum property in triangle PRQ:

∠RPQ = 45°

(v) If two circles do not intersect or touch each other, then the maximum number of common tangents is/are:

(a) 2

(b) 1

(d) 4

Answer

(d) 4

Explanation:

If two circles do not intersect or touch each other, they are in an external position relative to each other. The maximum number of common tangents between two such circles is 4. These consist of:

2 external tangents
2 internal tangents

(vi) The volume of a solid sphere having radius 2r units is:

(a) $32\over 3$ r³ cm³

(b) $16\over 3$ r³ cm³

(d) $64\over 3$ r³ cm³

Answer

(a) $32\over 3$ r³ cm³

Question – 2

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

(i) The annual rate of compound interest is r% and if the first year principal is P, then the 2nd year principal is ___.

Answer

P(1 + r/100)

(ii) $711$ is an ___ number.

Answer

irrational number

(iii) If radius of a sphere is r and volume v, then ___.

Answer

r³

(iv) Two triangles are similar if their corresponding sides are ___.

Answer

in proportional

(v) If the opposite angles of a quadrilateral be supplementary, then the vertices of the quadrilateral will be ___.

Answer

cyclic quadrilateral

(vi) If the length, breadth, and height of a rectangular parallelopiped are equal, then the special name of this solid is ___.

Answer

cube

Question – 3

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

(i) At least 3 persons are needed in partnership business.

Answer

False

Explanation:

A partnership can be formed between two or more persons. There is no requirement for at least three persons in a partnership business.

(ii) The relation between principal and amount is principal < amount.

Answer

True

Explanation:

In compound interest or simple interest, the amount is always greater than the principal because interest is added to the principal over time. Hence, principal < amount.

(iii) The two roots of the equation $\pm 10$ .

Answer

True

Explanation:

The equation x² = 100 has two solutions, x = +10 and x = -10, so the two roots are ±10.

(iv) If a and b are in inverse variation, then a/b = constant.

Answer

True

(v) Two concentric circles have only one common tangent.

Answer

False

Explanation:

Two concentric circles (circles with the same center) have two common tangents, not just one. There are two external tangents between them.

(vi) The height, radius, and slant height of a right circular cone are always the three sides of a right-angled triangle.

Answer

True

Explanation:

In a right circular cone, the height, radius, and slant height form a right-angled triangle, with the slant height as the hypotenuse, the radius as one leg, and the height as the other leg.

Question – 4

Answer any ten questions: [2 x 10 = 20]

(i) The annual interest is $\frac{1}{16}$ part of its principal, then determine the interest of ₹ 690. The annual interest is for 8 months.

Answer

Principal (P) = ₹ 690

Simple interest = (1/16) × 690 = ₹ 43.125

Interest for 8 months = (8/12) × 43.125

= ₹ (2/3) × 43.125

= ₹ 28.75

(ii) The present population is 13,310. If the population be 17,280 after 3 years, what will be the rate of increase?

Answer

Future population (V) = 17,280

Initial population (V_o) = 13,310

rate (r) = r %

Time (n) = 3 years

A = P × (1 + r/100)ⁿ

or, 17,280 = 13,310 × (1 + r/100)³

or, $17280\over 13310$ = (1 + r/100)³

or, $1728\over 1331$ = (1 + r/100)³

or, ( $12\over 11$ )³ = (1 + r/100)³

or, 1 + r/100 = $12\over 11$

or, r/100 = $12\over 11$ – 1

or, r = $1\over 11$ × 100 = 9 $1\over 11$ %

(iii) The ratio of capitals of A, B, C is $1\over \text{x}[\katex] : 1\over \text{y}[\katex] : 1\over \text{z}[\katex], after a year there was a loss of Rs. z. Calculate the loss of C. Answer <div style="background-color: #e0e4fa; border-radius: 3px; padding: 5px; margin-bottom: 10px; color: #010013;"> The ratio of capitals of A, B and C = 1\over \text{x}[\katex] : 1\over \text{y}[\katex] : 1\over \text{z}[\katex] = 1\over \text{x}[\katex] × xyz : 1\over \text{y}[\katex] × xyz : 1\over \text{z}[\katex] × xyz = yz + xz + xy The loss of C = \text{xy} \over \text{yz + xz + xy}$ × z

= $\text{xyz} \over \text{yz + xz + xy}$

(iv) Find out the ratio of the sum and the product of two roots of the equation

Answer

Sum of roots (α + β) = $66\over 7$

Produt of root (α β) = $27\over 7$

Ratio of the sum and the product of two roots

= $66\over 7$ : $27\over 7$

= 66 : 27

(v) Rationalize the surds of the denominator: $\frac{12}{\sqrt{15} - 3}$ .

Answer

$\frac{12}{\sqrt{15} - 3}$

$\frac{12(\sqrt{15} + 3)}{(\sqrt{15})² - 3²}$

= $\frac{12(\sqrt{15} + 3)}{15 - 9}$

= $\frac{12(\sqrt{15} + 3)}{6}$

= 2√15 + 3

(vi) The radius of the circle with the center 'O' is 13 cm and a chord AB with the length of 10 cm on it. Calculate the distance of the chord AB from the center of the circle.

Answer

Radius (r) = 13 cm

BM = $10\over 2$ = 5 cm

Distance from the centre (OM) : OM² = 13² - 5²

or, OM² = 13² - 5² = 169 - 25 = 144

or, OM = √144 = 12 cm

(vii) AOB is a diameter of a circle whose center is O. The point C lies on the circle. If , find the value of $∠ OC A$ .

Answer

Given:

AOB is a diameter of the circle with center O.
Point C lies on the circle.
∠OBC = 60°.

∠ACB = 90° (angle subtended by a diameter at any point on the circle is a right angle)

Triangle OBC is isosceles (OB = OC, both are radii of the circle)

Since ∠OBC = 60°, we also have ∠OCB = 60°.

∠OCA + ∠OCB + ∠ACB = 180° (The sum of angles in any triangle is 180°)

or, ∠OCA + 150° = 180°.

or, ∠OCA = 180° - 150° = 30°.

(viii) A circle with the center 'O'. A point P is 26 cm away from the center of the circle, and the length of the tangent drawn from the point P to the circle is 10 cm. Calculate the length of the radius of the circle.

Answer

Let the radius of the circle be r.

The line from the center O to the point of tangency T (where the tangent meets the circle) is perpendicular to the tangent line PT at the point of tangency. Thus, we have a right-angled triangle OTP, where:

OT = r (the radius of the circle),
PT = 10 cm (the length of the tangent),
OP = 26 cm (the distance from the point P to the center O).

Using the Pythagorean theorem:

OP² = OT² + PT²

Substitute the known values:

26² = r² + 10²

Simplify the equation:

676 = r² + 100

Now, solve for r²:

r² = 676 - 100 = 576

Finally, take the square root of both sides:

r = √576 = 24 cm

Thus, the length of the radius of the circle is 24 cm.

(ix) DE || BC of Δ $A BC$ where D and E are two points on AB and AC, respectively. If AD = 5 cm, DB = 6 cm, and AE = 7.5 cm, calculate the length of AC.

Answer

Given

AD = 5 cm
DB = 6 cm
AE = 7.5 cm

AB = AD + DB

or, AB = 5 cm + 6 cm

or, AB = 11 cm

(AD / AB) = (AE / AC) [Since DE || BC]

or, (5 / 11) = (7.5 / AC)

or, 5 × AC = 7.5 × 11

or, AC = (7.5 × 11) / 5

or, AC = 82.5 / 5

or, AC = 16.5 cm

(x) If the height of two right circular cylinders is in the ratio of 1 : 2, and the perimeters of the base are in the ratio of 3 : 4, find the ratio of their volumes.

Answer

Given

h₁ : h₂ = 1 : 2 --- (i)

Perimeters of the base = 3 : 4

C₁ : C₂ = 3 : 4

or, 2πr1 : 2πr2 = 3 : 4

or, r₁ : r₂ = 3 : 4 --- (ii)

Ratio of their volumes = V₁ : V₂

= πr₁²h₁ : πr₂²h₂

= r₁²h₁ : r₂²h₂

= ${r_1}²\over {r_1}²$ × ${h_1}\over {h_2}$

= $({3\over 4})² × {1\over 2}$

= ${9\over 16} × {1\over 2}$

= 9 : 32

(xi) If the length of the radius of a sphere is increased by 50%, find how much percent will be increased of its curved surface area.

Answer

(xii) The length of the diagonal of a cube is $4\sqrt 3$ cm. Calculate the total surface area of the cube.

Answer

Question - 5

At the same rate of simple interest in percent per annum, if a principal becomes the amount of Rs. 7,100 in 7 years and Rs. 6,200 in 4 years, determine the principal and rate of simple interest in percent per annum.

Answer

ii) Three friends have started a business by investing Rs. 8,000, Rs. 10,000, and Rs. 12,000 respectively. They also took an amount as a bank loan. At the end of one year, they made a profit of Rs. 13,400. After paying the annual bank installment of Rs. 5,000, they divided the remaining money of the profit among themselves in the ratio of their capitals. Calculate the profit share of each.

Answer

iii) Calculate the difference between compound interest and simple interest on Rs. 20,000 for 2 years at 5% per annum.

Answer

Question - 6

6. Answer any two questions: [3x2=6]

(i) Solve: $\frac{1}{\text{a+b+x}}$ = $\frac{1}{\text{a}} + \frac{1}{\text{b}} + \frac{1}{\text{x}}$ , x ≠ 0, -(a + b)

Answer

(ii) Form the quadratic equation whose roots are -4 and 3.

Answer

(iii) If m + , then find the value of

(a) m² +

(b) m³ +

Answer

Question - 7

Answer any two questions: [3 × 2 = 6]

(i) Find the simplest value of:

Answer

(ii) If a =

Answer

If 15 farmers can cultivate 18 bighas of land in 5 days, determine by using the theory of variation the number of days required by 10 farmers to cultivate 12 bighas of land.

Answer

Question - 8

(i) If a : b = b : c, prove that $\frac{\text{abc(a + b + c)³}}{\text{(ab + bc + ca)³}}$

Answer

(ii) If $\frac{\text{a}}{\text{1 - a}} + \frac{\text{b}}{\text{1 - b}} + \frac{\text{c}}{\text{1 - c}} = 1, then find the value of \frac{\text{1}}{\text{1 - a}} + \frac{\text{1}}{\text{1 - b}} + \frac{\text{1}}{\text{1 - c}}.$

Answer

Question - 9

Answer any one question: [5]

(i) Prove that the opposite angles of a cyclic quadrilateral are supplementary.

Answer

(ii) Prove that the perpendicular drawn to a chord which is not a diameter, from the center of the circle, bisects the chord.

Answer

Question - 10

(i) ABCD is a cyclic quadrilateral. Chord DE is the external bisector of $∠ B D C$ . Prove that AE (or produced AE) is the external bisector of $∠BAC.$

Answer

(ii) Two chords, AB and CD of a circle with center O, when produced, intersect each other at the point P. Prove that ∠AOC−∠BOD = 2 × ∠BPC.

Answer

Question - 11

Answer any one question: [5]

(i) Draw a right-angled triangle having two sides 4 cm and 8 cm length respectively. Containing right angle. Then draw the circumcircle of the right-angled triangle. (Only traces of construction are required.)

Answer

(ii) Draw a circle with a radius of 2.6 cm and draw a tangent on this circle from an external point at a distance of 6 cm from the center of the circle.

Answer

Question - 12

Answer any four questions: [4x4=16]

(i) Half of a cuboidal water tank with length of 2.1 m and breadth of 1.5 m is filled with water. If 630 liters of water is poured into the tank, then calculate the increased height of water.

Answer

(ii) The height of a right circular cylinder is twice its radius. If the height would be 6 times its radius, then the volume of the cylinder would be greater by 539 cubic dm. Calculate the height of the cylinder.

Answer

(iii) In a right circular conical tent, 11 persons can stay. For each person, 4 sq. m space in the base and 20 cu.m. air are necessary. Determine the height of the tent put up exactly for 11 persons.

Answer

(iv) Calculate how many spherical marbles with 1 cm radius each may be formed by melting a solid sphere of iron having 8 cm of radius.

Answer

(v) The inner length, breadth, and height of a tea box are 7.5 dm, 6 dm, and 5.4 dm respectively. If the weight of the box filled with tea is 52 kg 350 gm, but in the empty state, its weight is 3.75 kg, then calculate the weight of 1 cubic dm of tea.

Answer

[Alternative Question for Sightless Candidates]

Question - 11

Answer any one question: [5]

i) The lengths of sides containing the right angle of a right-angled triangle are given. Describe the procedure of construction of the circumcircle of the triangle.

Answer

(ii) Describe the process of drawing one tangent to a circle from an external point.

Answer

[Additional Question for External Candidates]

13. (a) Answer any three questions: [2x3=6]

(i) Find the value of $a$ , if one root of the equation is 1.

Answer

(ii) What is the rate of simple interest per annum, when the interest of some money in 10 years will be $\frac{2}{5}$ part of its amount?

Answer

(iii) Which one of √8, is not a similar surd?

Answer

(iv) Find the total surface area of a cone whose diameter of the base is 20 cm and slant height is 25 cm.