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CBSE Class 10 Maths Standard 2022 Question Paper 30/1/3 (Term 2) with Solutions

All 18 questions from the CBSE Class 10 Mathematics Standard board paper, Set 30/1/3 (2022, Term 2), with answers and step-by-step solutions. Total 40 marks. Tap “Show answer & solution” under any question.

Set 30/1/1 Set 30/1/2 Set 30/1/3 Set 30/2/1 Set 30/2/2 Set 30/2/3 Set 30/3/1 Set 30/3/2 Set 30/3/3 Set 30/4/1 Set 30/4/2 Set 30/4/3

Q12 marksVery Short AnswerQuadratic Equations

Find the value of $m$ for which the quadratic equation
$(m - 1) x^{2} + 2 (m - 1) x + 1 = 0$
has two real and equal roots.

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Answer:

m = 2

For equal roots, $D = b^{2} - 4 a c = 0$ (and $m - 1 \neq = 0$ ).
$[2 (m - 1)]^{2} - 4 (m - 1) (1) = 0$
$4 (m - 1) (m - 1 - 1) = 0 \Rightarrow 4 (m - 1) (m - 2) = 0$
$m = 1$ is rejected since then the equation is not quadratic.
So $m = 2$ .

Also asked in: 2022 Standard 30/1/1, 2022 Standard 30/1/2

Q1 (OR) (OR)2 marksVery Short AnswerQuadratic Equations

Solve the following quadratic equation for x :
$3 x^{2} + 10 x + 73 = 0$

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Answer:

x = - 3

x = - \frac{7}{3} = - \frac{7 3}{3}

Split the middle term: $3 \times 73 = 21 = 3 \times 7$ and $3 + 7 = 10$ .
$3 x^{2} + 3 x + 7 x + 73 = 0$
$3 x (x + 3) + 7 (x + 3) = 0$
$(x + 3) (3 x + 7) = 0$
$x = - 3$ or $x = - \frac{7}{3} = - \frac{7 3}{3}$

Also asked in: 2022 Standard 30/1/1, 2022 Standard 30/1/2

Q22 marksVery Short AnswerQuadratic Equations

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.

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Answer: 6 years

Let the present age be $x$ years.
$(x - 5) (x + 7) = 2 x + 1$
$x^{2} + 2 x - 35 = 2 x + 1 \Rightarrow x^{2} = 36$
$x = 6$ (age cannot be negative).
Rehan's present age is 6 years.

Also asked in: 2022 Standard 30/1/1, 2022 Standard 30/1/2

Q32 marksVery Short AnswerArithmetic Progressions

Find the sum of the first twelve 2-digit numbers which are multiples of 6.

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Answer: 540

The 2-digit multiples of 6 are 12, 18, 24, ...; $a = 12$ , $d = 6$ .
$a_{12} = 12 + 11 \times 6 = 78$ .
$S_{12} = \frac{12}{2} (12 + 78) = 6 \times 90 = 540$

Q3 (OR) (OR)2 marksVery Short AnswerArithmetic Progressions

In an AP, if $a_{2} = 26$ and $a_{15} = - 26$ , then write the AP.

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Answer: 30, 26, 22, 18, ...

$a + d = 26$ and $a + 14 d = - 26$ .
Subtracting: $13 d = - 52 \Rightarrow d = - 4$ .
$a = 26 - (- 4) = 30$ .
The AP is 30, 26, 22, 18, ...

Q42 marksVery Short AnswerSurface Areas and VolumesNot in current syllabus

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.

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Answer: 126

Volume of sphere $= \frac{4}{3} π (10.5)^{3}$ .
Volume of one cone $= \frac{1}{3} π (3.5)^{2} (3)$ .
Number of cones $= \frac{\frac{4}{3} π ( 10.5 ) ^{3}}{\frac{1}{3} π ( 3.5 ) ^{2} \times 3} = \frac{4 \times 10.5 \times 10.5 \times 10.5}{3.5 \times 3.5 \times 3}$
$= \frac{4 \times 3 \times 3 \times 10.5}{3} = 126$

Also asked in: 2022 Standard 30/1/1

Q52 marksVery Short AnswerCircles

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.

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Answer:

27

Let the chord AB of the larger circle touch the smaller circle at M; O is the common centre.
OM $⊥$ AB (radius is perpendicular to tangent), and OM = 3 cm, OA = 4 cm.
AM $= O A^{2} - O M^{2} = 16 - 9 = 7$ cm.
The perpendicular from the centre bisects the chord, so AB $= 27$ cm.

Also asked in: 2022 Standard 30/1/1, 2022 Standard 30/1/2

Q62 marksVery Short AnswerStatistics

Find the mode of the following frequency distribution :
Class: 10–20, 20–30, 30–40, 40–50, 50–60
Frequency: 15, 10, 12, 17, 4

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Answer:

\frac{385}{9} \approx 42.78

Highest frequency is 17, so the modal class is 40–50.
$l = 40$ , $h = 10$ , $f_{1} = 17$ , $f_{0} = 12$ , $f_{2} = 4$ .
Mode $= l + \frac{f _{1} - f _{0}}{2 f _{1} - f _{0} - f _{2}} \times h = 40 + \frac{17 - 12}{34 - 12 - 4} \times 10$
$= 40 + \frac{50}{18} = 40 + 2.78 = 42.78$ (approx.)

Also asked in: 2022 Standard 30/1/1, 2022 Standard 30/1/2

Q73 marksShort AnswerSome Applications of Trigonometry

The angle of elevation of the top of a building from the foot of the tower is $3 0^{\circ}$ and the angle of elevation of the top of the tower from the foot of the building is $6 0^{\circ}$ . If the tower is 50 m high, then find the height of the building.

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Answer:

\frac{50}{3}

= 16 \frac{2}{3}

Let the tower AB = 50 m and the building CD = $h$ m, with feet B and D a distance $d$ apart.
From D: $tan 6 0^{\circ} = \frac{50}{d} \Rightarrow d = \frac{50}{3}$ m.
From B: $tan 3 0^{\circ} = \frac{h}{d} \Rightarrow h = \frac{d}{3} = \frac{50}{3}$ m.
Height of the building $= 16 \frac{2}{3}$ m (about 16.67 m).

Also asked in: 2022 Standard 30/1/1, 2022 Standard 30/1/2

Q7 (OR) (OR)3 marksShort AnswerSome Applications of Trigonometry

From a point on a bridge across a river, the angles of depression of the banks on opposite sides of the river are $3 0^{\circ}$ and $4 5^{\circ}$ respectively. If the bridge is at a height of 3 m from the banks, then find the width of the river.

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Answer:

3 (1 + 3)

\approx 8.19

Let P be the point on the bridge, 3 m above the banks, and A, B the banks on opposite sides; D is the point directly below P.
Angle of depression $3 0^{\circ}$ to A: $tan 3 0^{\circ} = \frac{3}{A D} \Rightarrow A D = 33$ m.
Angle of depression $4 5^{\circ}$ to B: $tan 4 5^{\circ} = \frac{3}{D B} \Rightarrow D B = 3$ m.
Width AB $= 33 + 3 = 3 (1 + 3)$ m $\approx 8.19$ m.

Also asked in: 2022 Standard 30/1/1, 2022 Standard 30/1/2

Q83 marksShort AnswerTrianglesNot in current syllabus

Draw a line segment AB = 7 cm. Divide it in the ratio 3 : 2.

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Answer: Construction: point P on AB with AP : PB = 3 : 2 (AP = 4.2 cm, PB = 2.8 cm).

Draw AB = 7 cm. Draw a ray AX making an acute angle with AB.
Mark 5 (= 3 + 2) equal arcs on AX: $A_{1}, A_{2}, A_{3}, A_{4}, A_{5}$ .
Join $A_{5} B$ . Through $A_{3}$ draw a line parallel to $A_{5} B$ , meeting AB at P.
By BPT, AP : PB = $A A_{3} : A_{3} A_{5}$ = 3 : 2.
Check: AP $= \frac{3}{5} \times 7 = 4.2$ cm, PB = 2.8 cm.

Q93 marksShort AnswerStatistics

For what value of x, is the median of the following frequency distribution 34.5 ?
Class: 0–10, 10–20, 20–30, 30–40, 40–50, 50–60, 60–70
Frequency: 3, 5, 11, 10, x, 3, 2

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Answer:

x = 13

Cumulative frequencies: 3, 8, 19, 29, $29 + x$ , $32 + x$ , $34 + x$ ; so $N = 34 + x$ .
Median 34.5 lies in the class 30–40: $l = 30$ , $c f = 19$ , $f = 10$ , $h = 10$ .
$34.5 = 30 + \frac{\frac{N}{2} - 19}{10} \times 10 = 11 + \frac{N}{2}$
$\frac{N}{2} = 23.5 \Rightarrow N = 47$
$34 + x = 47 \Rightarrow x = 13$

Also asked in: 2022 Standard 30/1/1

Q103 marksShort AnswerStatistics

The following table gives the production yield of wheat of farms of a village :
Production Yield (in kg/ha): 50–60, 60–70, 70–80, 80–90, 90–100
Number of Farms: 7, 12, 11, 8, 2
Find the mean production yield, using assumed mean method.

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Answer: 71.5 kg/ha

Class marks $x_{i}$ : 55, 65, 75, 85, 95. Take assumed mean $a = 75$ ; $d_{i} = x_{i} - 75$ : $- 20, - 10, 0, 10, 20$ .
$f_{i} d_{i}$ : $- 140, - 120, 0, 80, 40$ ; $Σ f_{i} d_{i} = - 140$ , $Σ f_{i} = 40$ .
Mean $= a + \frac{Σ f _{i} d _{i}}{Σ f _{i}} = 75 + \frac{- 140}{40} = 75 - 3.5 = 71.5$
Mean production yield is 71.5 kg/ha.

Q114 marksLong AnswerCircles

In Figure 1, a triangle ABC with $∠ B = 9 0^{\circ}$ 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.

Diagram for CBSE 2022 Class 10 Maths question 11

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Answer: Proved.

Let the tangent at P meet BC at Q. Join BP.
Since $∠ A B C = 9 0^{\circ}$ , BC $⊥$ AB at B, the end of the diameter, so BC is tangent to the circle at B.
QB and QP are tangents from Q, so QB = QP and hence $∠ QB P = ∠ QP B$ .
$∠ A P B = 9 0^{\circ}$ (angle in a semicircle), so $∠ B P C = 9 0^{\circ}$ .
In $△ B P C$ : $∠ QC P = 9 0^{\circ} - ∠ QB P$ and $∠ QP C = 9 0^{\circ} - ∠ QP B$ .
So $∠ QC P = ∠ QP C$ , giving QP = QC.
Hence QB = QP = QC, i.e. Q is the midpoint of BC: the tangent at P bisects BC.

Also asked in: 2022 Standard 30/1/1

Q124 marksLong AnswerSurface Areas and Volumes

In Figure 2, from a solid cube of side 7 cm, a cylinder of radius 2.1 cm and height 7 cm is scooped out. Find the total surface area of the remaining solid.

Diagram for CBSE 2022 Class 10 Maths question 12

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Answer: 358.68 cm²

TSA = surface of cube − 2 circular ends + curved surface of cylinder.
Surface of cube $= 6 \times 7^{2} = 294$ cm².
Two circular ends $= 2 π r^{2} = 2 \times \frac{22}{7} \times 2.1 \times 2.1 = 27.72$ cm².
Curved surface of cylinder $= 2 π r h = 2 \times \frac{22}{7} \times 2.1 \times 7 = 92.4$ cm².
TSA $= 294 - 27.72 + 92.4 = 358.68$ cm².

Q12 (OR) (OR)4 marksLong AnswerSurface Areas and VolumesNot in current syllabus

A well of diameter 5 m is dug 24 m deep. The earth taken out of it has been spread evenly all around it in the shape of a circular ring of width 3 m to form an embankment. Find the height of the embankment.

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Answer: 6.25 m

Volume of earth dug out $= π (2.5)^{2} \times 24 = 150 π$ m³.
Embankment: inner radius 2.5 m, outer radius $2.5 + 3 = 5.5$ m.
Area of ring $= π (5. 5^{2} - 2. 5^{2}) = π (30.25 - 6.25) = 24 π$ m².
Height $= \frac{150 π}{24 π} = 6.25$ m.

Q134 marksCase StudyArithmetic Progressions

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 $n^{t h}$ term of this AP. (2)
(b) Which figure has 61 matchsticks ? (2)

Diagram for CBSE 2022 Class 10 Maths question 13

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Answer: (a) 4, 6, 8, ...;

a_{n} = 2 n + 2

(b) Figure 8

(a) Triangles: Figure 1 has 4, Figure 2 has 6, Figure 3 has 8, so the AP is 4, 6, 8, ...
$a = 4$ , $d = 2$ : $a_{n} = 4 + (n - 1) 2 = 2 n + 2$ .
(b) Matchsticks: Figure 1 uses $4 + 4 \times 2 = 12$ , Figure 2 uses $7 + 6 \times 2 = 19$ , Figure 3 uses $10 + 8 \times 2 = 26$ .
AP 12, 19, 26, ... with $a = 12$ , $d = 7$ : $a_{n} = 12 + (n - 1) 7 = 7 n + 5$ .
$7 n + 5 = 61 \Rightarrow n = 8$ . Figure 8 has 61 matchsticks.

Also asked in: 2022 Standard 30/1/1, 2022 Standard 30/1/2

Q144 marksCase StudySome Applications of Trigonometry

Gadisar Lake is located in the Jaisalmer district of Rajasthan. It was built by the King of Jaisalmer and rebuilt by Gadsi Singh in 14 $^{t h}$ century. The lake has many Chhatris. 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 Chhatri (point B) is $4 5^{\circ}$ and angle of depression of its reflection in water (point C) is $6 0^{\circ}$ . If the height of Chhatri above water level is (approximately) 10 m, then
(a) draw a well-labelled figure based on the above information; (2)
(b) find the height ( $h$ ) of the point A above water level. (Use $3 = 1.73$ ) (2)

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Answer: (a) Figure: A at height

h

above water; B is 10 m above water and C is 10 m below water on the vertical through the Chhatri; elevation of B

4 5^{\circ}

, depression of C

6 0^{\circ}

. (b)

h = 20 - 103 \approx 2.7

(a) Draw the water line and the vertical line of the Chhatri meeting it at D, with B 10 m above D and its reflection C 10 m below D. A is a point $h$ m above the water; draw the horizontal AM to the vertical BC. Mark $∠ B A M = 4 5^{\circ}$ and $∠ M A C = 6 0^{\circ}$ .
(b) Let AM $= x$ . Then BM $= 10 - h$ and MC $= 10 + h$ .
$tan 4 5^{\circ} = \frac{10 - h}{x} \Rightarrow x = 10 - h$ .
$tan 6 0^{\circ} = \frac{10 + h}{x} \Rightarrow 10 + h = 3 (10 - h)$ .
$h (1 + 3) = 10 (3 - 1) \Rightarrow h = \frac{10 ( 3 - 1 )}{3 + 1} = 5 (3 - 1)^{2} = 20 - 103$
$h = 20 - 17.3 = 2.7$ m.

Also asked in: 2022 Standard 30/1/1, 2022 Standard 30/1/2

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