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Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10 PDF Download

Class X
Mathematics – Standard
Sample Question Paper 2020-21
Max. Marks : 80
Duration : 3 hrs.

General Instructions : 
1. This question paper contains two parts A and B.
2. Both Part A and Part B have internal choices.

Part – A : 
1. It consists two sections - I and II.
2. Section I has 16 questions of 1 mark each. Internal choice is provided in 5 questions.
3. Section II has 4 questions on case study. Each case study has 5 case-based sub-parts. An examinee is to attempt any 4 out of 5 sub-parts.

Part – B : 
1. Section III, Question No. 21 to 26 are Very short answer Type questions of 2 marks each.
2. Section IV, Question No. 27 to 33 are Short Answer Type questions of 3 marks each.
3. Section V, Question No. 34 to 36 are Long Answer Type questions of 5 marks each.
4. Internal choice is provided in 2 questions of 2 marks, 2 questions of 3 marks and 1 question of 5 marks.

Part - A

Sections - I

Section I has 16 questions of 1 mark each. Internal choice is provided in 5 questions.

Q.1. Calculate the largest number which divides 70 and 125, leaves remainders 5 and 8, respectively.  (1 Mark)
OR
If p is a prime number, then find LCM of p, p2 and p3.  (1 Mark)
Ans. Required largest number = HCF of (70 – 5) and (125 – 8)   = HCF of 65 and 117 = 13
OR
p2 = p × p
p = p × 1
p3 = p × p × p
Required LCM = p × p × p = p3.

Q.2. Explain why 13233343563715 is a composite number?  (1 Mark)
OR
A number is chosen at random from the numbers – 3, – 2, – 1, 0, 1, 2, 3. What will be the probability that square of this number is less than or equal to 1.  (1 Mark)

Ans. Since, the given number ends in 5. Hence, it is a multiple of 5. Therefore, it is a composite number.
OR
No. of all possible outcomes = 7
No. of favourable outcomes = 3
The numbers whose square is ≤ 1 = – 1, 0, 1
∴ Required probability = 3 / 7

Q.3. How many polynomials can be formed with – 2 and 5 as zeroes?  (1 Mark)
Ans. We know that if we divide or multiply a polynomial by any constant (real number), then the zeroes of polynomial remains same.
Here, a  = −2 and b = +5
∴ α + β = −2 + 5 = 3 and
αβ = −2 × 5 = −10
So, required polynomial is x2 – (a + b)x + ab= x2 – 3x – 10
If we multiply this polynomial by any real number, let 5 and 2, we get 5x2 – 15x – 50 and 2x2 – 6x – 20 which are different polynomials but having same zeroes –2 and 5. So, we can obtain so many (infinite polynomials) from two given zeroes.

Q.4. Graphically, the pair of equations :  (1 Mark)
6x – 3y + 10 = 0
2x – y + 9 = 0
Represents what kind of lines.

Ans. Here,
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
So, the system of linear equations is inconsistent (no solution) and graph will be a pair of parallel lines.

Q.5. Is the equation (√2x + √3)2 + x2 = 3x2 - 5x quadratic? Justify.  (1 Mark)
Ans. (√2x)+ (√3)2 + 2 x √2x x √3 + x2 = 3x2 - 5x
2x2 + 3 + 2√6x + x2 = 3x2 - 5x
3x2 + 2√6x + 3 = 3x2 - 5x
x(5 + 2√6) + 3 = 0
It is not of the form of ax2 +bx +c = 0.
So it is not a quadratic equation.

Q.6. Find the 30th term of the A.P., : 10, 7, 4 ................  (1 Mark)
Ans. In the given AP, a = 10 and d = 7 – 10 = –3
Thus, the 30th term is t30 = 10 + (30 − 1) (−3) = −77

Q.7. Find the distance of the point (– 3, – 4) from the x-axis (in units).  (1 Mark)
Ans. The distance of the point (– 3, – 4) from x-axis = | – 4| = 4 units.

Q.8. In the given figure, if ∠AOB = 125°, then find ∠COD.  (1 Mark)
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
OR
In figure, O is the centre of a circle. PT and PQ are tangents to the circle from an external point P. If ∠TPQ = 70°, then find ∠TRQ.  (1 Mark)
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10Ans. Since, quadrilateral circumscribing a circle subtends supplementary angles at the centre of the circle.
∴ ∠AOB + ∠COD = 180°

125° + ∠COD = 180°

∠COD = 180° – 125° = 55°
OR
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10∠TOQ = 180° – 70° = 110°
(angle of supplementary)
Then, ∠TRQ = 1 / 2 ∠TOQ
(angle at the circumference of the circle by same arc)
= 1 / 2 x 110° = 55°.

Q.9. If the circumference of a circle and the perimeter of a square are equal, then find the relation between area of circle and area of square.  (1 Mark)
Ans. According to question,
Circumference of a circle = Perimeter of square
Let ‘r’ and ‘a’ be the radius of circle and side of square respectively.
2πr = 4a
22 / 7 r = 2a
11r = 7a
r = 7a / 11 ..(i)
Area of circle, A1 = πr2
From equation (i), we have
A1 = π(7a / 11)2
= 22 / 7(49a2 / 121)
= 14a2 / 11  ...(ii)
Area of circle, A2 =a2 ...(iii)
From equation (ii) and (iii), we have
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10

A1 > A2
Area of circle is greater than the area of square.

Q.10. During conversion of a solid from one shape to another, what will be the volume of new shape?  (1 Mark)
Ans. During reshaping a solid, the volume of new solid will be equal to old one or remains unaltered.

Q.11. In the figure of DABC, the points D and E are onthe sides CA, CB respectively such that DE || AB, AD = 2x, DC = x + 3, BE = 2x – 1 and CE = x. Then, the value of x is ...................... .  (1 Mark)
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10Ans. 
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
Using basic proportionality theorem
or, 5x = 3 or, x = 3 / 5
Detailed Answer :
In AB C, DE || AB
Then, CD / CA = CE / CB
or,Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
or,Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
or,Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
or, (x + 3)(3x – 1) = x(3x + 3)
or, 3x2 – x + 9x – 3 = 3x2 + 3x
or, 8x – 3 = 3x
or, 8x – 3x = 3
or, 5x = 3
∴ x = 3 / 5

Q.12. Find the value of sin260° + 2tan 45° – cos230°.  (1 Mark)
OR
If sin A = 3 / 4, then find value of sec A.  (1 Mark)
Ans. 
sin260º  + 2 tan 45º – cos230º
(√3 / 2)2 + 2(1) - (√3 / 2)2
= 2
OR
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
∴ sec A = 4 / √7

Q.13. Find the area (in cm2) of the circle that can be inscribed in a square of side 8 cm.  (1 Mark)
Ans. 
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10Side of square = diameter of circle = 8 cm
∴ Radius of circle, r = 8 / 2 = 4 cm
Area of circle = πr2
= π × 4 × 4 = 16πcm2

Q.14. The curved surface area of a cylinder is 264 m2 and its volume is 924 m3. Find the ratio of its height to its diameter.  (1 Mark)
Ans. Curved Surface area of cylinder = 2πrh Volume of cylinder = πr2h
πr2h / 2πrh = 924 / 264 ⇒ r / 2 = 7 / 2
∴ r = 7 m
2πrh = 264
or, 2 x 22 / 7 x 7 x h = 264
or, h = 6 m
∴ h / 2r = 6 / 14 = 3 / 7
Hence, h : d = 3 : 7

Q.15. If the distance between the points (4, k) and (1, 0) is 5, then what can be the possible values of k?  (1 Mark)
OR
Write the co-ordinates of a point P on x-axis which is equidistant from the points A(– 2, 0) and B (6, 0).  (1 Mark)

Ans. Using distance formula,
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
or 32 + k2 = 25
k = ± 4
OR
Using distance formula :
Point on x-axis is (2, 0)
Detailed Answer.
Let (x, 0) be equidistant from (– 2, 0) and (6, 0) by distant formula :
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
On squaring both sides,
(x + 2)2 = (x – 6)2
⇒ x2 + 4 + 4x = x2 + 36 – 12x
⇒ 4x + 12x = 36 – 4
⇒ 16x = 32
⇒ x = 2
Hence, the required point P(x, 0) = (2, 0)

Q.16. For the following distribution :  (1 Mark)
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
Find the sum of lower limits of median class and modal class.

Ans. 
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
The modal class is the class having the maximum frequency.
The maximum frequency 20 belongs to class (15 – 20).
Here, n  = 66
So, n / 2 = 66 / 2 = 33
33 lies in the class 10–15.
Therefore, 10 – 15 is the median class.
So, sum of lower limits of (15 – 20) and (10 – 15) is (15 + 10) = 25.

Section-II

Case study based questions are compulsory. Attempt any four sub parts of each question. Each subpart carries 1 mark

Q.17. Case Study based - 1 :
In a room, 4 friends are seated at the points A, B, C and D as shown in figure. Reeta and Meeta walk into the room and after observing for a few minutes Reeta asks Meeta.
Asks Meeta
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10

(a) What is the position of A?  (1 Mark)
(i) (4, 3)
(ii) (3, 3)
(iii) (3, 4)
(iv) None of these

Ans. (iii)
Solution. Point A lies it x – 3, y = 4
A = (3, 4)

(b) What is the middle position of B and C?  (1 Mark)
(i) (15 / 2, 11 / 2)
(ii) (2 / 15, 11 / 2)
(iii) (1 / 2, 1 / 2)
(iv) None of these
Ans. (i)
Solution. Mid point of B and C :
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
[∵ Co-ordinates of B = (6, 7) and C = (9 , 4)]

(c) What is the position of D?  (1 Mark)
(i) (6, 0)   
(ii) (0, 6) 
(iii) (6, 1) 
(iv) (1, 6)
Ans. (iii)
Solution. Point D lies at x = 6 and y = 1
D = (6, 1)

(d) What is the distance between A and B?  (1 Mark)
(i) 3√2 
(ii) 2√3 
(iii) 2√2 
(iv) 3√3
Ans. (i)
Solution. Since, A = (3, 4) and B = (6, 7)
Using distance formula
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10 
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
= √18 = 3√2 unit

(e) What is the equation of line CD?  (1 Mark)
(i) x – y – 5 = 0 
(ii) x + y – 5 = 0 
(iii) x + y + 5 = 0 
(iv) x – y + 5 = 0
Ans. (i)
Solution. Equation of line CD = Equation of line through C(9, 4) and D(6, 1)
i.e., Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
or, y – 4 = (x – 9)
or, x – y – 5 = 0

Q.18. Case Study based - 2 : 
Seema placed a lighedbulb at point O on the ceiling and directly below it placed a table. Now, she put a cardboard of shape ABCD between table and  lighted bulb. Then a shadow of ABCD is casted on the table as A'B'C'D' (see figure). Quadrilateral A'B'C'D' in an enlargement of ABCD with scale factor 1 : 2, Also, AB = 1.5 cm, BC = 25 cm, CD = 2.4 cm and AD = 2.1 cm; ∠A = 105°, ∠B = 100°, ∠C = 70° and ∠D = 85°.
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10

(a) What is the measurment of angle A'?  (1 Mark)
(i) 105° 
(ii) 100° 
(iii) 70° 
(iv) 80°
Ans. (i)
Solution. Quadrilateral A'B'C'D' is similar to ABCD.
∴ ∠A' = ∠A

⇒ ∠A' = 105°

(b) What is the length of A'B'?  (1 Mark)
(i) 1.5 cm 
(ii) 3 cm 
(iii) 5 cm 
(iv) 2.5 cm
Ans. (ii)
Solution. Given scale factor is 1 : 2
∴ A'B' = 2 AB
⇒ A'B' = 2 × 1.5 = 3 cm

(c) What is the sum of angles of quadrilateral A'B'C'D'?  (1 Mark)
(i) 180° 
(ii) 360° 
(iii) 270° 
(iv) None of these
Ans. (ii)
Solution. Sum of the angles of quadrilateral A'B'C'D' is 360°

(d) What is the ratio of sides A'B' and A'D'?  (1 Mark)
(i) 5 : 7  
(ii) 7 : 5 
(iii) 1 : 1 
(iv) 1 : 2
Ans. (i)
Solution. A'B' = 3 cm and A'D' = 2 AD
= 2 × 2.1 = 4.2 cm
∴ A'B' / A'D' = 3 / 4.2 = 30 / 42
= 5 / 7 = 5 : 7

(e) What is the sum of angles of C' and D'?  (1 Mark)
(i) 105° 
(ii) 100° 
(iii) 155° 
(iv) 140°
Ans. (iii)
Solution. C' = ∠C = 70°

and ∠D' = ∠D = 85°
∴ ∠C' + ∠D' = 70° + 85° = 155°

Q.19. Case Study based - 3 : 
An electrician has to repaired and electric fault on the pole of height 5 cm. She needs to reach a point 1.3 m below the top of the pole to undertake the repair work (see figure)
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10

(a) What is the length of BD?   (1 Mark)
(i) 1.3 m
(ii) 5 m
(iii) 3.7 m
(iv) None of these

Ans. (iii)
Solution. From figure, the electrician is required to reach at the point B on the pole AD.
So, BD = AD – AB = (5 – 1.3) m = 3.7 m

(b) What should be the length of Ladder, when inclined at an angle of 60° to the harizontal?   (1 Mark)
(i) 4.28 m
(ii) 3.7 / √3 m

(iii) 3.7 m
(iv) 7.4 m

Ans. (i)
Solution. In ΔADC,
∴ sin 60° = BD / BC
⇒ √3 / 2 = 3.7 / BC
⇒ BC =Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
⇒ BC = 4.28 m (approx.)

(c) How far from the foot of pole should she place the foot of the ladder?  (1 Mark)
(i) 3.7 
(ii) 2.14 
(iii) 1 / √3
(iv) None of these
Ans. (ii)
Solution. In DBDC,
∴ cot 60° = DC / BD
⇒ 1 / √3 = DC / 3.7
⇒ DC =  3.7 / √3
⇒ DC = 2.14 m (approx.)

(d) If the horizontal angle is changed to 30°, then what should be the length of the ladder?  (1 Mark)
(i) 7.4 m 
(ii) 3.7 m 
(iii) 1.3 m 
(iv) 5 m
Ans. (i)
Solution. In ΔBDC,
∴ sin 60° = BD / BC

⇒ 1 / 2 = 3.7 / BC
⇒ BC = 3.7 × 2 = 7.4 m

(e) What is the value of ∠B?  (1 Mark)
(i) 60° 
(ii) 90° 
(iii) 30° 
(iv) 180°
Ans. (iii)
Solution. In ΔADC, angle D is 90°.
So, by angle sum property.
∠B + ∠D + ∠C = 180°
or, ∠B = 180° – (90° + 60°)
= 30°

Q.20. Case Study based - 4 :
The weights (in kg) of 50 wrestlers are recorded in the following table :
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10

(a) What is the upper limit of modal class.  (1 Mark)
(i) 120 
(ii) 130 
(iii) 100 
(iv) 150
Ans. (ii)
Solution. Modal Class = 120 – 130
Upper Unit = 130

(b) What is the mode of the given data   (1 Mark)
(i) 21
(ii) 50
(iii) 25
(iv) 80

Ans. (i)
Solution. Mode of the given data is 21.

(c) How many wrestlers weights have more than 120 kg weight?  (1 Mark)
(i) 32 
(ii) 50 
(iii) 16 
(iv) 21
Ans. (i)
Solution. No. of wrestlers with more than 120 kg weight = 21 + 8 + 3 = 32

(d) What is the class mark for class 130 – 140?  (1 Mark)
(i) 105
(ii) 125
(iii) 135
(iv) 145
Ans. 
(iii)
Solution. 
For class mark of 130 – 140,
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
= 270 / 2 = 135

(e) Which method is more suitable to find the mean of the above data?  (1 Mark)
(i) Direct method
(ii) Assumed mean method
(iii) Step-Deviation method
(iv) None of these
Ans. (iii)

Part - B

Section - III

All questions are compulsory. In case of internal choices, attempt any one.

Q.21. Write the denominator of the rational number 257 / 500 in the form 2m × 5n, where m and n are non-negative integers. Hence write its decimal expansion without actual division.  (2 Mark)
Ans. Denominator = 500
= 22 × 53
Decimal expansion =Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
= 0.514

Q.22. In a rectangle ABCD, E is a point on AB such that AE = 2 / 3 AB. If AB = 6 km and AD = 3 km, then find DE.  (2 Mark)
Ans. In ΔCAB, ∠A = ∠B (Given)
∴ AC = CB
(By isosceles triangle property)
But, AD = BE (Given) ...(i)
Or, AC – AD = BC – BE
∴ CD = CE ...(ii)
Dividing equation (ii) by (i),
CD / AD = CE / BE
By converse of BPT,
DE || AB.
OR
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10Given,  AE = 2 / 3 AB = 2 / 3 x 6 = 4 km

In right angled triangle ADE,
DE2 = (3)2 + (4)2
or, DE2 = 25
∴ DE = 5 km.

Q.23. If sin (A + B) = 1 and sin (A – B) = 1 / 2, 0 ≤ A + B = 90° and A > B, then find A and B.  (2 Mark)
OR
Express : sin A and tan A in terms of sec A.  (2 Mark)

Ans. Here, sin (A + B) = 1 = sin 90º
or, A + B = 90º ...(i)
sin (A – B) = 1 / 2 = sin 30º
or, A – B = 30º ...(ii)
Solving eqns. (i) and (ii),
A = 60º and B = 30º
OR
(i) sin A =Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
(ii) tan2A = secA – 1
or,Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10

Q.24. An observer 1.5 m tall is 28.5 m away from a tower 30 m high. Find the angle of elevation of the top of the tower from his eye.  (2 Mark)
Ans. 
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10AE be observer = 1.5 m
BD is the tower = 30 m
∠BAC = θ, BC = 30 – 1.5 = 28.5 m
In ΔBAC, BC / AC = tan θ
⇒ 28.5 / 28.5 = tan θ

⇒ tan θ = 1 = tan  45°
⇒ θ = 45°
Hence, the angle of elevation is 45°.
OR
Let BD = x m and  DC = y m.
From ΔABD, ∠D = 90°
7√3 / x = tan 30°
⇒ 7√3 / x = 1 / √3
⇒  7√3 x √3
= 21 m
From ΔADC, ∠D = 90°
7√3 / y = tan 60°
⇒  7√3 = y√3
⇒ y = 7 m,
BC = BD +  DC
= 21 + 7 = 28 m.
Hence, the value of BC = 28 m.

Q.25. A child prepares a poster on "save water" on a square sheet whose each side measures 50 cm. At each corner of the sheet, she draws a quadrant of radius 15 cm in which she shows the ways to save water. At the centre, she draws a circle of diameter 21 cm and writes a slogan save water in it. Find the area of the remaining sheet.  (2 Mark)
Ans. Side of square= 50 cm
∴ Area of square = 50 × 50 = 2500 cm2 
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10

Radius of quadrant = 15 cm.
Area of 4 quadrants = 4 x 1 / 4 πr2 = πr2
= π × 15 × 15
= 22 / 7 x 225
= 707.14 cm2
Area of circle = πr2
= 22 / 7 x (21 / 2)2
= 22 / 7 x 21 / 2 x 21 x 2
= 346.5 cm2
Area of remaining sheet = Area of square – 4(area of quadrant) – Area of circle
= 2500 – 707.14 – 346.5

= 1446.36 sq. cm

Q.26. A teacher took a surprise test of maths. He observes the marks of five students of class. He observes the median is 45.5 and mode is 50.5. So find the mean of the marks of five students using an empirical formula.  (2 Mark)
OR
Find the mode of the following frequency distribution  (2 Mark)

Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
Ans. Given, Mode = 50.5
Median = 45.5
3 Median = Mode + 2 Mean
⇒ 3 × 45.5 = 50.5 + 2 Mean
⇒ Mean =Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
Hence, Mean = 43.
OR
Maximum frequency = 16,
Modal class is 30 – 40
Mode =Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
= 36

Section - IV

All questions are compulsory. In case of internal choices, attempt any one.

Q.27. Sum of the ages of a father and the son is 40 years. If father ’s age is three times that of his son, then find their respective ages.  (3 Mark)
OR
A part of monthly hostel charge is fixed and the remaining depends on the number of days one has taken food in the mess. When Swati takes food for 20 days, she has to pay ₹ 3,000 as hostel charges whereas Mansi who takes food for 25 days has to pay ₹ 3,500 as hostel charges. Find the fixed charges and the cost of food per day.  (3 Mark)

Ans. Let age of father and son be x and y respectively.
Then, x + y = 40 ...(i)
and x = 3y ...(ii)
By solving eqns. (i) and (ii), we get
x = 30 and y = 10
Thus, the ages of father and son are 30 years and 10 years.
OR
Let fixed charge be x and per day food cost be y
Then, x + 20y = 3000 ...(i)
and x + 25y = 3500 ...(ii)
Subtracting (i) from (ii), we get
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
y = 100 Substituting this value of y in (i), we get
x + 20(100) = 3000 x = 1000

∴ x = 1000 and y = 100
Hence, fixed charge and cost of food per day are ₹ 1,000 and ₹ 100.

Q.28. The ninth term of an A.P. is equal to seven times the second term and twelfth term exceeds five times the third term by 2. Find the first term and the common difference.  (3 Mark)
OR
The sum of first n terms of three arithmetic progressions are S1, S2 and S3 respectively. The first term of each A.P. is 1 and common differences are 1, 2 and 3 respectively. Prove that S1 + S3 = 2S2.  (3 Mark)

Ans. Let the first term of A.P. be a and common difference be d.
Given, a9 = 7a2 
or, a + 8d = 7(a + d) ...(i)
a + 8d = 7a + 7d – 6a + d = 0 ...(iii)
and a12 = 5a3 + 2
Again, a + 11d = 5(a + 2d) + 2 ...(ii)
a + 11d = 5a + 10d + 2 – 4a + d = 2 ...(iv)
Subtracting (iv) from (iii), we get – 2a = – 2
or, a = 1
From (iii),
– 6 + d = 0
d = 6
Hence, first term = 1 and common difference = 6
OR
Since, S1 = 1 + 2 + 3 + .... + n.
S2 = 1 + 3 + 5 + ...upto n terms
and S3 = 1 + 4 + 7 + ...upto n terms
or, S1 = n(n + 1) / 2
Also,Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
= n / 2[2n] = n2
andClass 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
= n(3n - 1) / 2
Now, Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
= n[4n] / 2
= 2n2 = 2S2
Hence Proved.

Q.29. A 1.2 m tall girl spots a balloon moving with the wind in a horizontal line at a height of 88.2 m from the ground. The angle of elevation of the balloon form the edges of the girl at any instant of 60°. After sometime, the angle of elevation reduces to 30°.  (3 Mark)
(i) Find the distance travelled by the balloon during the interval.
(ii) Which mathematical concept is used in the above problem ?

Ans. (i) Let P be the position of the balloon when its angle of elevation from the eyes of the girl is 60° and Q be the position when angle of elevation is 30°.
 In ΔOLP, tan 60° = PL / OL
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10⇒ √3 = 87 / OL

⇒ OL = 87 / √3
In ΔOMQ, tan 30° = QM / OM
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
⇒ OM = 87√3
∴ Distance travelled by the balloon,
PQ = LM = OM – OL
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
(ii) Height and distance.

Q.30. If (x2 + y2)(a2 + b2) = (ax + by)2. Prove that x / a + y / b.  (3 Mark)
Ans. 
Given, (x2 + y2)(a2 + b2) = (ax + by)2
⇒ x2a2 + x2b2 + y2a2 + y2b2 = a2x2 + b2y2 + 2abxy
⇒ x2b2 + y2a2 – 2abxy = 0
⇒ (xb – ya)2 = 0
[∵ (a - b)2 = a2 + b2 - 2ab]
⇒ xb = ya
∴ x / a + y / b
Hence proved

Q.31. A bag contains 18 balls out of which x balls are red.  (3 Mark)
(i) If one ball is drawn at random from the bag, what is the probability that it is not red ?
(ii) If 2 more red balls are put in the bag, the probability of drawing a red ball will be 9 / 8 times the probability of drawing a red ball in the first case. Find the value of x.

Ans. P(red ball) = x / 18
(i) P(no red ball) =Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10 
(ii) Now, Total number of balls = 18 + 2 = 20
Total number of Red balls = x + 2
P(red balls) =Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
Now, According to the question,
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
⇒ 180x = 144x + 288
⇒ 36x = 288
⇒ x = 288 / 36 = 8

Q.32. In the given figure, OP is equal to the diameter of a circle with centre O and PA and PB are tangents. Prove that ABP is an equilateral triangle.  (3 Mark)
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10Ans. Construction : Join A to B.
We have,
OP = diameter
⇒ OQ + QP = diameter
⇒ Radius + QP = diameter
⇒ OQ = PQ = radius
Thus, OP is the hypotenuse of right angled ∆AOP.
So, In ∆AOP, sin θ = AO / OP = 1 / 2
θ = 30°
Hence, ∠APB = 60°
Now, in ∆ABP, AP = PB
So, ∠PAB = ∠PBA = 60°
∴ ∆APB is an equilateral triangle.

Q.33. If bcos θ = a, then prove that cosec θ + cot θ =Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10(3 Mark)
Ans. 
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10Given, cos θ = a / b
AC2 = AB2 – BC2
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
Hence proved.

Section - V

All questions are compulsory. In case of internal choices, attempt any one.

Q.34. SolveClass 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10where a + b ≠ 0. (5 Mark)
OR
Check whether the equation 5x2 – 6x – 2 = 0 has real roots and if it has, find them by the method of completing the square. Also, verify that roots obtained satisfy the given equation.  (5 Mark)

Ans. 
Given,Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
⇒ x(a + b+ x) = – ab
⇒ x2 + (a + b)x + ab = 0
⇒ (x + a)(x + b) = 0
⇒ x = – a or x = – b
OR
Discriminant ⇒ b2 – 4ac.

Here, a = 5, b = (– 6) and c = (– 2)
Then, b2 – 4ac = (– 6)2 – 4 × 5 × – 2
= 36 + 40 = 76 > 0
So the equation has real and two distinct roots.
Again, 5x– 6x = 2 (dividing both the sides by 5)
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
On adding square of the half of coefficient of x
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
Verification :
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
Similarly,
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
Hence, Verified.

Q.35. In the given figure, D and E trisect BC. Prove that 8AE2 = 3AC2 + 5AD2.  (5 Mark)
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
Ans. Let BD = DE = EC be x
BE = 2x
and BC = 3x
Now, in ΔABE,
AE2 = AB2 + BE2

= AB2 + 4x2, ...(i)
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
In ΔABC,
AC2 = AB2 + BC2 = AB2 + 9x2
In ΔADB, and AD2 = AB2 + BD2 = AB2 + x2
Now, on multiplying (i) by 8,
8AE2 = 8AB2 + 32x2 ...(ii)
and 3AC2 + 5AD2 = 3(AB2 + 9x2) + 5 (AB2 + x2)
= 3AB2 + 27x2 + 5AB2 + 5x2
= 8AB2 + 32x2 = 8 (AB2 + 4x2)
∴ 3AC2 + 5AD2 = 8AE2. [From eqns. (i)]
Hence proved.

Q.36. A solid is in the form of a cylinder with hemispherical ends. The total height of the solid is 20 cm and the diameter of the cylinder is 7 cm. Find the total volume of the solid. (π = 22 / 7)  (5 Mark)
Ans. Height of cylinder= 20 – 7 = 13 cm.
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10Total volume = Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10 
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
Detailed Solution :
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
Height of the cylinder(h)= (20 – 7) cm = 13 cm
Radius of circular part(r) = 7 / 2 cm
Volume of solid = Volume of cylinder + 2 × Volume of hemisphere
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10
= 77 / 2(53 / 3)cm3
= 680.17 cm3

The document Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 | CBSE Sample Papers For Class 10 is a part of the Class 10 Course CBSE Sample Papers For Class 10.
All you need of Class 10 at this link: Class 10
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FAQs on Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Standard) - 2 - CBSE Sample Papers For Class 10

1. How can I prepare for the CBSE Class 10 Mathematics exam?
Ans. To prepare for the CBSE Class 10 Mathematics exam, you can follow these steps: 1. Understand the syllabus: Go through the syllabus provided by CBSE and make a note of all the topics and subtopics that you need to cover. 2. Create a study schedule: Plan your study time effectively by allocating specific time slots for each topic. This will help you stay organized and cover all the important concepts. 3. Practice regularly: Mathematics requires practice, so solve as many sample papers, previous year question papers, and practice exercises as possible. This will help you improve your problem-solving skills and time management. 4. Seek help when needed: If you come across any doubts or difficulties, don't hesitate to seek help from your teachers, classmates, or online resources. Clarifying your doubts will ensure a better understanding of the concepts. 5. Revise thoroughly: Allocate sufficient time for revision before the exam. Review all the important formulas, concepts, and solve additional questions to reinforce your learning. Remember to stay focused, stay positive, and believe in your abilities. With consistent effort and a systematic approach, you can excel in your CBSE Class 10 Mathematics exam.
2. Are there any important topics that I should focus on for the CBSE Class 10 Mathematics exam?
Ans. Yes, there are a few important topics that you should focus on for the CBSE Class 10 Mathematics exam. These topics carry significant weightage in the exam and mastering them can help you score well. Some of these important topics include: 1. Real Numbers: Understand the properties of rational and irrational numbers, Euclid's division lemma, and prime factorization. 2. Polynomials: Learn about the factor theorem, remainder theorem, and division algorithm for polynomials. 3. Pair of Linear Equations in Two Variables: Practice solving linear equations using graphical methods, substitution, and elimination method. 4. Quadratic Equations: Understand the nature of roots, discriminant, and methods of solving quadratic equations. 5. Trigonometry: Focus on trigonometric ratios, trigonometric identities, and applications of trigonometry. 6. Coordinate Geometry: Study the concepts of distance formula, section formula, and slope of a line. 7. Statistics and Probability: Learn to calculate mean, median, mode, and standard deviation. Also, understand the basics of probability. While these topics are important, it is essential to have a good understanding of all the concepts covered in the syllabus to perform well in the exam.
3. Is it necessary to solve previous year question papers for the CBSE Class 10 Mathematics exam?
Ans. Yes, it is highly recommended to solve previous year question papers for the CBSE Class 10 Mathematics exam. Solving these papers will provide you with several benefits: 1. Familiarity with exam pattern: By solving previous year question papers, you will become familiar with the exam pattern, marking scheme, and types of questions asked. This will help you understand the structure of the exam and prepare accordingly. 2. Time management practice: The CBSE Class 10 Mathematics exam has a time limit, and solving previous year question papers will help you practice time management. You will learn to allocate the right amount of time to each section and develop strategies to solve the questions efficiently. 3. Identify weak areas: By analyzing your performance in previous year question papers, you can identify your weak areas and focus on improving them. This will allow you to allocate more time and effort to topics that need additional attention. 4. Gain confidence: Solving previous year question papers will boost your confidence as you become familiar with the type of questions and the level of difficulty. This will help you approach the actual exam with a positive mindset. Remember to solve the previous year question papers in a timed manner, just like the actual exam, to get the most benefit out of it.
4. What are some effective tips to score well in the CBSE Class 10 Mathematics exam?
Ans. Scoring well in the CBSE Class 10 Mathematics exam requires a combination of understanding concepts, practicing regularly, and implementing effective strategies. Here are some tips to help you score well: 1. Understand the concepts: Focus on understanding the concepts rather than memorizing formulas. Clear your doubts and seek help if needed. 2. Practice regularly: Mathematics requires practice, so solve as many problems as possible. Work on sample papers, previous year question papers, and additional practice exercises to improve your problem-solving skills. 3. Time management: Practice solving questions within the given time limit. Allocate specific time slots for each question or section to ensure you complete the paper on time. 4. Revise and review: Allocate sufficient time for revision before the exam. Review all the important formulas, concepts, and solve additional questions to reinforce your learning. 5. Solve mock tests: Take mock tests to simulate the exam environment and evaluate your performance. Analyze your mistakes and work on improving them. 6. Stay calm and focused: Maintain a positive mindset, stay calm during the exam, and read the questions carefully. Avoid unnecessary stress or panic. 7. Write legibly and organize your answers: Neat and organized presentation of answers can earn you extra marks. Clearly label each step and provide necessary explanations. By following these tips and consistently putting in effort, you can score well in the CBSE Class 10 Mathematics exam.
5. Are there any recommended books or study materials for the CBSE Class 10 Mathematics exam preparation?
Ans. Yes, there are several recommended books and study materials for the CBSE Class 10 Mathematics exam preparation. Some of the popular ones include: 1. Mathematics - Textbook for Class X by NCERT: This is the official textbook prescribed by CBSE. It covers all the topics and concepts in a comprehensive manner. 2. RD Sharma Class 10 Mathematics: This book is highly recommended for additional practice and problem-solving. It provides a wide range of questions with varying difficulty levels. 3. Mathematics for Class 10 by R.D. Aggarwal: This book is also a popular choice among students and covers the entire CBSE syllabus with detailed explanations and examples. 4. Oswaal CBSE Question Bank Class 10 Mathematics: This book contains a variety of questions, including previous year question papers, sample papers, and practice questions. It also provides solutions and marking schemes. Apart from these books, you can also refer to online resources, video tutorials, and practice apps for additional study materials and practice. Choose the study materials that suit your learning style and preferences.
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