CBSE Class 10 Term 2 Mathematics Standard 2022, Solution: Paper Code-30/1/1, Set 1

CBSE Class 10 Term 2 Mathematics Standard 2022, Solution: Paper Code-30/1/1, Set 1

Section A

Q.1 Find the sum of first 30 terms of AP: –30, –24, –18,/..

Solution: a= –30, d= 6

S30 = 15 [–60 + 29 (6)]

= 15 [–60+174]

= 15 [114]

= 1710 

OR

Q:  If an AP is Sn = n(4n+1), then find the AP

Solution: S1 = T1 = 5

S2= T1 + T2= 18

S2–S1 = T2= 13

AP= 5, 13, 21, 29, …

Q.2: A solid metallic sphere of radius 10.5 cm is melted and recast into a number of smaller cones, each of radius 3.5 cm and height 3 cm. Find the number of cones so formed. 

Solution: 

r1 = 10.5 cm (sphere) and 

r2 (cone)= 3.5 cm, 

h2= 3cm

4/3πr³ = n x 1/3πr2² h

4r³ = nr2² h

4 x (105/10)³= n x 35/10 x 35/10 x 3

21 x 6= n 

Number of cones formed= 126

Q. 3 (a) Find the value of m for which the quadratic equation 

(m–1) x² + 2(m–1)x + 1 = 0

Solution: D= 0

4 (m–1)² – 4(m–1) = 0

4 (m–1) [(m–1)–1]= 0

(m–1) (m–2)= 0

m= 1 and m= 2

m = 1, 2

(b) Solve the following quadratic equation for x: √3x² + 10x + 7√3= 0

Solution: √3x² + 3x + 7x + 7√3= 0

√3x (x+ √3) + 7 (x + √3)= 0

(x+ √3) (√3x + 7)= 0

(x+ √3)  = 0 and (√3x + 7) = 0

x= – √3 and x= –7/√3

x= – √3, –7/√3

Q.4: Find the mode of the following frequency distribution: 

l = 40, 

f1= 17, 

f2= 4, 

f0= 12

Mode= l +[( f1–f0) / (2f1 – fo – f2)]

= 40 + (17–12 )/ (34–12–4 x 10)

= 40 + 5/18 x 10

= 40 + 25/9

= 40 + 2.78

= 42.78

Q.5: The product of Rehan’s age (in years) 5 years ago and his age 7 years from now, is one more than twice his present age. Find his present age. 

Solution: 

Past Age	Present Age	Future Age
x-5	x	x+7

(x-5) (x+7) = 2x + 1

x² + 2x -35 = 2x+1

x² – 36 = 0

x² = 36 

x= ± 6

Present Age 6 years

Q.6 Two concentric circles are of radii 4 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.

Solution:

Section B

Q.7: For what value of x, is the median of the following frequency distribution 34:5? 

Class	Frequency.     c.f
0-10	3
10-20	5
20-30	11
30-40	10
40-50	x
50-60	3
60-70	2

Solution:

Q.8: Draw a circle of radius 3 cm. Take two points P and Q on one of its extended diameter each at a distance of 7 cm from its centre. Construct tangents to the circle from these two points P and Q.

Solution:

Q.9: (a) The angle of elevation of the top of a building from the foot of the tower is 30° and the angle of elevation of the top of the tower from the foot of the building is 60°. If the tower is 50 m high, then find the height of the building.

Solution:

tan 60° = 50/b

b = 50/ tan 60°

b = 50/ √3

b= 50√3 / 3

tan 30° = h/(50√3/3)

h= 50√3/3 x 1/√3

= 50/3 m= 16.67 metres

OR

Q.9:(b) From a point on a bridge across a river, the angles of depression of the banks on opposite sides of the river are 30° and 45° respectively. If the bridge is at a height of 3 m from the banks,  then find the width of the river.

Solution:

Q.10: Following is the daily expenditure on lunch by 30 employees of a company, Find the mean daily expenditure of the employees.

Solution:

Q.11: From a solid cylinder of the height of 30 cm and radius of 7 cm, a conical cavity of height of 24 cm and same radius is hollowed out. Find the total surface area of the remaining solid. 

Solution: H= 30 cm

R= 7 cm

Total Surface Area of Cylinder = Curved Surface Area of Cylinder + Curved Surface Area of Cone + πr²

TSA = 2 πrh +  πrl + πr²

= πr (2h+ l+ r)

= 22/7 x 7 (60+25 + 7)

= 22 x 92 

TSA= 2024 cm 

Q.12: In figure 1, a traingle ABC with angle B = 90 degrees is shown. Taking AB as diameter, a circle has been drawn intersecting AC at point P. Prove that the tangent drawn at point P bisects BC. 

Solution:

Case Study:1

Q.13: In Mathematics, relations can be expressed in various ways. The matchstick patterns are based on linear relations. Different strategies can be used to calculate the number of matchsticks used in different figures. 

One such pattern is shown below. Observe the pattern and answer the following questions using Arithmetic Progression:

(a)  Write the AP for the number of triangles used in the figures. Also, write the nth term of this AP. Which figure has 61 matchsticks? 

Solution- AP = 4, 6, 8, 10

a= 4, d= 2

Tn = 4 + 2(n-1)

Tn= = 4 + 2n – 2

Tn= 2n +2

(b) Which figures has 61 matchstickes?

Solution: 12, 19, 26, …

Tn = 61

a + (n-1) d= 61

12 + 7(n-1) = 61

7(n-1) = 49

n-1 = 7

n= 8

8th figure has 61 matchsticks

Q.14 Gadisagar lake is located in the jaisalmer district of Rajasthan. It was built by king of jaisalmer ad rebuilt by Gadsi Singh in 14th Century. The lake has many chattris. One of them is shown below: 

Observe the picture. From a point A h m above from water level, the angle of elevation of top of chattari (point B) is 45 degrees and angle of depression of its reflection in water(Point C) is 60 degrees. If the height of chhattri above the water level is 10m, then

a) Draw a well-labelled figure based on the above information;

b) Find the height of the point A above the water level. 

(Use √3 = 1.73)

Solution: