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Mathematics Delhi Set 2 2018-2019 CBSE (English Medium) Class 10 Question Paper Solution

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Mathematics [Delhi Set 2]
Marks: 80 Academic Year: 2018-2019
Date: March 2019
Duration: 3h
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(i) All questions are compulsory.

(ii) The question paper consists of 30 questions divided into four sections – A, B, C and D.

(iii) Section A comprises 6 questions of 1 mark each. Section B contains 6 questions of marks each. Section C contains 10 questions of 3 marks each. Section D contains questions of 4 marks each.

(iv) There is no overall choice. However, an internal choice has been provided in two questions of mark, two questions of 2 marks, four questions of 3 marks each and three questions of 4 marks each. You have to attempt only one of the alternative in all such questions.

(v) Use of calculators is not permitted.


[1] 1

Find the coordinates of point A, where AB is a diameter of the circle with centre (–2, 2) and B is the point with coordinates (3, 4).

Concept: Coordinate Geometry
Chapter: [6.01] Lines (In Two-dimensions) [6.01] Lines (In Two-dimensions)
[1] 2

Find a rational number between `sqrt2` and `sqrt3`

Concept: Concept of Irrational Numbers
Chapter: [1.01] Real Numbers
[1] 3

How many two digits numbers are divisible by 3?

Concept: Algebraic Conditions for Number of Solutions
Chapter: [2.01] Pair of Linear Equations in Two Variables
[1] 4
[1] 4.1

Find If A If tan 2A = cot (A-24°)

Concept: Basic Ideas of Probability
Chapter: [5.01] Probability [5.01] Probability
OR
[1] 4.2

Find the value of (sin2 33 + sin2 57°)

Concept: Right-angled Triangles and Pythagoras Property
Chapter: [3.02] Triangles
[1] 5
[1] 5.1

For what value of k, the roots of the equation x2 + 4x + k = 0 are real?

Concept: Nature of Roots
Chapter: [2.03] Quadratic Equations
OR
[1] 5.2

Find the value of k for which the roots of the equation 3x2 - 10x + k = 0 are reciprocal of each other.

Concept: Nature of Roots
Chapter: [2.03] Quadratic Equations
[1] 6

In Fig. 1, DE || BC, AD = 1 cm and BD = 2 cm. What is the ratio of the ar (Δ ABC) to the ar (Δ ADE)?

Concept: Ratio of Sides of Triangle
Chapter: [3.02] Triangles
SECTION - B
[2] 7

Find the value of k for which the following pair of linear equations has infinitely many solutions. 
2x + 3y = 7, (k +1) x+ (2k -1) y = 4k + 1

Concept: Pair of Linear Equations in Two Variables
Chapter: [2.01] Pair of Linear Equations in Two Variables
[2] 8
[1] 8.1

A die is thrown once. Find the probability of getting a number which is a prime number

Concept: Basic Ideas of Probability
Chapter: [5.01] Probability [5.01] Probability
[1] 8.2

A die is thrown once. Find the probability of getting a number which
lies between 2 and 6.

Concept: Basic Ideas of Probability
Chapter: [5.01] Probability [5.01] Probability
[2] 9

Find the ratio in which the segment joining the points (1, –3) and (4, 5) is divided by the x-axis? Also, find the coordinates of this point on the x-axis.

Concept: Division of a Line Segment
Chapter: [3.03] Constructions
[2] 10

A game consists of tossing a coin 3 times and noting the outcome each time. If getting the same result in all the tosses is a success, find the probability of losing the game.

Concept: Basic Ideas of Probability
Chapter: [5.01] Probability [5.01] Probability
[2] 11
[2] 11.1

Which term of the AP 3, 15, 27, 39, .... will be 120 more than its 21st term?

Concept: Algebraic Conditions for Number of Solutions
Chapter: [2.01] Pair of Linear Equations in Two Variables
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OR
[2] 11.2

If Sn, the sum of first n terms of an AP is given by Sn = 3n2 - 4n, find the nth term.

Concept: Algebraic Conditions for Number of Solutions
Chapter: [2.01] Pair of Linear Equations in Two Variables
[2] 12
[2] 12.1

Find the HCF of 1260 and 7344 using Euclid's algorithm.

Concept: Real Numbers Examples and Solutions
Chapter: [1.01] Real Numbers
OR
[2] 12.2

Show that every positive odd integer is of the form (4q + 1) or (4q + 3), where q is some integer.

Concept: Euclid’s Division Lemma
Chapter: [1.01] Real Numbers
SECTION - C
[3] 13

The arithmetic mean of the following frequency distribution is 53. Find the value of k.

Class 0-20 20-40 40-60 60-80 80-100
Frequency 12 15 32 k 13
Concept: Graphical Representation of Cumulative Frequency Distribution
Chapter: [5.02] Statistics
[3] 14

Find the area of the segment shown in Fig. 2, if the radius of the circle is 21 cm and ∠AOB = 120° `( "Use"  π = (22)/(7))`

Concept: Area of Circle
Chapter: [7.01] Areas Related to Circles
[3] 15
[3] 15.1

In Fig. 3, ∠ACB = 90° and CD ⊥ AB, prove that CD2 = BD x AD.

Concept: Right-angled Triangles and Pythagoras Property
Chapter: [3.02] Triangles
OR
[3] 15.2

If P and Q are the points on side CA and CB respectively of ΔABC, right angled at C, prove that (AQ2 + BP2 ) = (AB2 + PQ2)

Concept: Right-angled Triangles and Pythagoras Property
Chapter: [3.02] Triangles
[3] 16

In Fig. 4, a circle is inscribed in a ΔABC having sides BC = 8 cm, AB = 10 cm and AC = 12 cm. Find the lengths BL, CM and AN.

Concept: Number of Tangents from a Point on a Circle
Chapter: [3.01] Circles
[3] 17

Water in a canal, 6 m wide and 1.5 m deep, is flowing with a speed of 10 km/hour. How much area will it irrigate in 30 minutes; if 8 cm standing water is needed?

Concept: Volume of a Combination of Solids
Chapter: [7.02] Surface Areas and Volumes
[3] 18

Prove that `sqrt2` is an irrational number.

Concept: Proofs of Irrationality
Chapter: [1.01] Real Numbers
[3] 19

Find the value of k such that the polynomial x2 − (k + 6)x + 2(2k −1) has sum of its zeros equal to half of their product.

Concept: Geometrical Meaning of the Zeroes of a Polynomial
Chapter: [2.04] Polynomials
[3] 20
[3] 20.1

Find the point on the y-axis which is equidistant from the points (5, −2) and (−3, 2).

Concept: Coordinate Geometry
Chapter: [6.01] Lines (In Two-dimensions) [6.01] Lines (In Two-dimensions)
OR
[3] 20.2

The line segment joining the points A(2, 1) and B(5, −8) is trisected at the points P and Q such that P is nearer to A. If P also lies on the line given by 2x − y + k = 0, find the value of k.

Concept: Constructions Examples and Solutions
Chapter: [3.03] Constructions
[3] 21
[3] 21.1

A father's age is three times the sum of the ages of his two children. After 5 years his age will be two times the sum of their ages. Find the present age of the father.

Concept: Algebraic Methods of Solving a Pair of Linear Equations - Cross - Multiplication Method
Chapter: [2.01] Pair of Linear Equations in Two Variables
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OR
[3] 21.2

A fraction becomes `(1)/(3)` when 2 is subtracted from the numerator and it becomes `(1)/(2)` when 1 is subtracted from the denominator. Find the fraction.

Concept: Algebraic Methods of Solving a Pair of Linear Equations - Elimination Method
Chapter: [2.01] Pair of Linear Equations in Two Variables
[3] 22
[3] 22.1

Prove that `(sin θ + cosec θ)^2 + (cos θ + sec θ)^2 = 7 + tan^2 θ + cot^2 θ`.

Concept: Right-angled Triangles and Pythagoras Property
Chapter: [3.02] Triangles
OR
[3] 22.2

Prove that (1 + cot A - cosec A ) (1 + tan A + sec A) = 2

Concept: Right-angled Triangles and Pythagoras Property
Chapter: [3.02] Triangles
SECTION - D
[4] 23

Prove that `(tan^2"A")/(tan^2 "A"-1) + (cosec^2"A")/(sec^2"A"-cosec^2"A") = (1)/(1-2 co^2 "A")`

Concept: Trigonometric Identities
Chapter: [4.02] Trigonometric Identities [4.03] Introduction to Trigonometry
[4] 24

The first term of an AP is 3, the last term is 83 and the sum of all its terms is 903. Find the number of terms and the common difference of the AP.

Concept: Algebraic Conditions for Number of Solutions
Chapter: [2.01] Pair of Linear Equations in Two Variables
[4] 25

Construct a triangle ABC with side BC = 6 cm, ∠B = 45°, ∠A = 105°. Then construct another triangle whose sides are `(3)/(4)` times the corresponding sides of the ΔABC.

Concept: Similarity Triangle Theorem
Chapter: [3.02] Triangles
[4] 26
[4] 26.1

If the median of the following frequency distribution is 32.5. Find the values of f1 and f2.

Class 0-10 10-20 20-30 30-40 40-50 50-60 60-70 Total
Frequency f1 5 9 12 f2 3 2 40
Concept: Graphical Representation of Cumulative Frequency Distribution
Chapter: [5.02] Statistics
OR
[4] 26.2

The marks obtained by 100 students of a class in an examination are given below.

Mark No. of Student
0 - 5 2
5 - 10 5
10 - 15 6
15 - 20 8
20 - 25 10
25 - 30 25
30 - 35 20
35 - 40 18
40 - 45 4
45 - 50 2

Draw 'a less than' type cumulative frequency curves (ogive). Hence find the median.

Concept: Graphical Representation of Cumulative Frequency Distribution
Chapter: [5.02] Statistics
[4] 27

Prove that in a right angle triangle, the square of the hypotenuse is equal to the sum of squares of the other two sides.

Concept: Right-angled Triangles and Pythagoras Property
Chapter: [3.02] Triangles
[4] 28

A bucket open at the top is in the form of a frustum of a cone with a capacity of 12308.8 cm3. The radii of the top and bottom of the circular ends of the bucket are 20 cm and 12 cm respectively. Find the height of the bucket and also the area of the metal sheet used in making it. (Use π = 3.14)

Concept: Surface Area of a Combination of Solids
Chapter: [7.02] Surface Areas and Volumes
[4] 29
[4] 29.1

Two water taps together can fill a tank in `1(7)/(8)`  hours. The tap with longer diameter takes 2 hours less than the tap with a smaller one to fill the tank separately. Find the time in which each tap can fill the tank separately.

Concept: Quadratic Equations Examples and Solutions
Chapter: [2.03] Quadratic Equations
OR
[4] 29.2

A boat goes 30 km upstream and 44 km downstream in 10 hours. In 13 hours, it can go 40 km upstream and 55 km downstream. Determine the speed of the stream and that of the boat in still water.

Concept: Quadratic Equations Examples and Solutions
Chapter: [2.03] Quadratic Equations
[4] 30
[4] 30.1

A man in a boat rowing away from a lighthouse 100 m high takes 2 minutes to change the angle of elevation of the top of the lighthouse from 60° to 30°.
Find the speed of the boat in metres per minute. [Use `sqrt(3` = 1.732]Use 3=1.732">

Concept: Heights and Distances
Chapter: [4.01] Heights and Distances
OR
[4] 30.2

Two poles of equal heights are standing opposite each other on either side of the road, which is 80 m wide. From a point between them on the road, the angles of elevation of the top of the poles are 60° and 30° respectively. Find the height of the poles and the distances of the point from the poles.

Concept: Heights and Distances
Chapter: [4.01] Heights and Distances
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