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Mathematics 30/4/3 2018-2019 CBSE (English Medium) Class 10 Question Paper Solution

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Mathematics [30/4/3]
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 2 marks each. Section C contains 10 questions of 3 marks each. Section D contains 8 questions of 4 marks each.
(iv) There is no overall choice. However, an internal choice has been provided in two questions of 1 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 alternatives in all such questions.
(v) The use of calculators is not permitted.


SECTION A
[1] 1 | Question numbers 1 to 6 carry 1 marks each.

Which term of the A.P. −4, −1, 2, ... is 101?

Concept: nth Term of an AP
Chapter: [2.02] Arithmetic Progressions
[1] 2

Evaluate:
`(tan 65°)/(cot 25°)`

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

Express (sin 67° + cos 75°) in terms of trigonometric ratios of the angle between 0° and 45°.

Concept: Trigonometric Identities
Chapter: [4.02] Trigonometric Identities [4.03] Introduction to Trigonometry
[1] 3

Find the value of k for which the quadratic equation kx (x − 2) + 6 = 0 has two equal roots.

Concept: Quadratic Equations Examples and Solutions
Chapter: [2.03] Quadratic Equations
[1] 4

Find a rational number between `sqrt(2)  "and" sqrt(7)`.

Concept: Revisiting Rational Numbers and Their Decimal Expansions
Chapter: [1.01] Real Numbers

Write the number of zeroes in the end of a number whose prime factorization is 2× 53 × 32 × 17.

Concept: Solutions of Quadratic Equations by Factorization
Chapter: [2.03] Quadratic Equations
[1] 5

Find the distance between the points (a, b) and (−a, −b).

Concept: Distance Formula
Chapter: [6.01] Lines (In Two-dimensions)
[1] 6

Let ∆ ABC ∽ ∆ DEF and their areas be respectively, 64 cm2 and 121 cm2. If EF = 15⋅4 cm, find BC.

Concept: Similarity of Triangles
Chapter: [3.02] Triangles
SECTION B
[2] 7 | Question numbers 7 to 12 carry 2 marks each.

Find the solution of the pair of the equation :
`3/x + 8/y = - 1; 1/x - 2/y = 2`, x, y ≠ 0

Concept: Algebraic Methods of Solving a Pair of Linear Equations - Substitution Method
Chapter: [2.01] Pair of Linear Equations in Two Variables

Find the value(s) of k for which the pair of equations
kx 2y = 3
3x + 6y = 10
 has a unique solution.

Concept: Nature of Roots
Chapter: [2.03] Quadratic Equations
[2] 8

Use Euclid's division algorithm to find the HCF of 255 and 867.

Concept: Euclid’s Division Lemma
Chapter: [1.01] Real Numbers
[2] 9

The point R divides the line segment AB, where A(−4, 0) and B(0, 6) such that AR=34AB.">AR = `3/4`AB. Find the coordinates of R.

Concept: Coordinate Geometry
Chapter: [6.01] Lines (In Two-dimensions) [6.01] Lines (In Two-dimensions)
[2] 10

How many multiples of 4 lie between 10 and 205?

Concept: General Term of an Arithmetic Progression
Chapter: [2.02] Arithmetic Progressions

Determine the A.P. whose third term is 16 and 7th term exceeds the 5th by 12.

Concept: nth Term of an AP
Chapter: [2.02] Arithmetic Progressions
[2] 11

Three different coins are tossed simultaneously. Find the probability of getting exactly one head.

Concept: Basic Ideas of Probability
Chapter: [5.01] Probability [5.01] Probability
[2] 12

A die is thrown once. Find the probability of getting.
(a) a prime number.
(b) an odd number

Concept: Basic Ideas of Probability
Chapter: [5.01] Probability [5.01] Probability
SECTION C
[3] 13 | Question numbers 13 to 22 carry 3 marks each.
[3] 13.1

In the given figure, BL and CM are medians of a ∆ABC right-angled at A. Prove that 4 (BL2 + CM2) = 5 BC2.

Concept: Right-angled Triangles and Pythagoras Property
Chapter: [3.02] Triangles
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OR
[3] 13.2

Prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals

Concept: Right-angled Triangles and Pythagoras Property
Chapter: [3.02] Triangles
[3] 14

In Figure , two concentric circles with centre O, have radii 21cm and 42 cm. If ∠ AOB = 60°, find the area of the shaded region. [use π=22/7]

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

A cone of height 24 cm and radius of base 6 cm is made up of modelling clay. A child reshapes it in the form of a sphere. Find the radius of the sphere and hence find the surface area of this sphere.

Concept: Surface Area of a Combination of Solids
Chapter: [7.02] Surface Areas and Volumes
OR
[3] 15.2

A farmer connects a pipe of internal diameter 20 cm form a canal into a cylindrical tank in his field, which is 10 m in diameter and 2 m deep. If water flows through the pipe at the rate of 3 km/h, in how much time will the tank be filled?

Concept: Conversion of Solid from One Shape to Another
Chapter: [7.02] Surface Areas and Volumes
[3] 16

Calculate the mode of the following distribution:

Class 10 − 15 15 − 20 20 − 25 25 − 30 30 − 35
Frequency 4 7 20 8 1
Concept: Mode of Grouped Data
Chapter: [5.02] Statistics
[3] 17

Show that `(2+3√2)/7` is not a rational number, given that √2 is an irrational number.

Concept: Concept of Irrational Numbers
Chapter: [1.01] Real Numbers
[3] 18

Obtain all the zeroes of the polynomial 2x4 − 5x3 − 11x+ 20x + 12 when 2 and − 2 are two zeroes of the above polynomial.

Concept: Concept of Polynomials
Chapter: [2.04] Polynomials [2.04] Polynomials
[3] 19

A motorboat whose speed is 18 km/hr in still water takes 1 hr more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream.

Concept: Concept of Polynomials
Chapter: [2.04] Polynomials [2.04] Polynomials
[3] 20
[3] 20.1

Prove that:
(sin θ + 1 + cos θ) (sin θ − 1 + cos θ) . sec θ cosec θ = 2

Concept: Trigonometric Ratios of Complementary Angles
Chapter: [4.02] Trigonometric Identities [4.03] Introduction to Trigonometry
OR
[3] 20.2

Prove that:
`sqrt(( secθ - 1)/(secθ + 1)) + sqrt((secθ + 1)/(secθ - 1)) = 2cosecθ`

Concept: Trigonometric Identities
Chapter: [4.02] Trigonometric Identities [4.03] Introduction to Trigonometry
[3] 21
[3] 21.1

In what ratio does the point P(−4, y) divides the line segment joining the points A(−6, 10) and B(3, −8)? Hence find the value of y.

Concept: Distance Formula
Chapter: [6.01] Lines (In Two-dimensions)
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OR
[3] 21.2

Find the value of p for which the points (−5, 1), (1, p) and (4, −2) are collinear.

Concept: Area of a Triangle
Chapter: [6.01] Lines (In Two-dimensions)
[3] 22

ABC is a right triangle in which ∠B = 90°.  If AB = 8 cm and BC = 6 cm, find the diameter of the circle inscribed in the triangle.

Concept: Areas of Similar Triangles
Chapter: [3.02] Triangles
SECTION D
[4] 23 | Question numbers 23 to 30 carry 4 marks each.

In an AP, the first term is -4, the last term is 29 and the sum of all its terms is 150. Find its common difference.

Concept: Arithmetic Progressions Examples and Solutions
Chapter: [2.02] Arithmetic Progressions
[4] 24

Draw a circle of radius 4 cm. From a point 6 cm away from its centre, construct a pair of tangents to the circle and measure their lengths.

Concept: Construction of Tangents to a Circle
Chapter: [3.03] Constructions
[4] 25

Prove that :
2(sin6 θ + cos6 θ) − 3 (sin4 θ + cos4 θ) + 1 = 0

Concept: Trigonometric Identities
Chapter: [4.02] Trigonometric Identities [4.03] Introduction to Trigonometry
[4] 26
[4] 26.1

Solve for x : `1/(2a + b + 2x) =1/(2a) + 1/b + 1/(2x); x ≠ 0, x ≠ (−2a −b)/2`, a, b ≠ 0

Concept: Quadratic Equations Examples and Solutions
Chapter: [2.03] Quadratic Equations
OR
[4] 26.2

The sum of the areas of two squares is `640m^2` . If the difference in their perimeter be 64m, find the sides of the two square 

Concept: Solutions of Quadratic Equations by Completing the Square
Chapter: [2.03] Quadratic Equations
[4] 27

In ∆ ABC, AD ⊥ BC.
Prove that  AC2 = AB2 +BC2 − 2BC x BD

Concept: Right-angled Triangles and Pythagoras Property
Chapter: [3.02] Triangles
[4] 28
[4] 28.1

A moving boat is observed from the top of a 150 m high cliff moving away from the cliff. The angle of depression of the boat changes from 60° to 45° in 2 minutes. Find the speed of the boat in m/min.

Concept: Trigonometric Identities
Chapter: [4.02] Trigonometric Identities [4.03] Introduction to Trigonometry
OR
[4] 28.2

There are two poles, one each on either bank of a river just opposite to each other. One pole is 60 m high. From the top of this pole, the angle of depression of the top and foot of the other pole are 30° and 60° respectively. Find the width of the river and height of the other pole.

Concept: Trigonometric Identities
Chapter: [4.02] Trigonometric Identities [4.03] Introduction to Trigonometry
[4] 29
[4] 29.1

Find the mean of each of the following frequency distributions

Class interval 10 - 30 30 - 50 50 - 70 70 - 90 90 - 110 110 - 130
Frequency 5 8 12 20 3 2
Concept: Mean of Grouped Data
Chapter: [5.02] Statistics
OR
[4] 29.2

The following table gives production yield in kg per hectare of wheat of 100 farms of a village :

Production yield
(kg/hectare) :
40−45 45−50 50−55 55−60 60−65 65−70
Number of farms 4 6 16 20 30 24

Change the distribution to a 'more than type' distribution, and draw its ogive.

Concept: Graphs of Linear Equations
Chapter: [6.01] Lines (In Two-dimensions)
[4] 30

A container opened at the top and made up of a metal sheet, is in the form of a frustum of a cone of height 16 cm with radii of its lower and upper ends as 8 cm and 20 cm respectively. Find the cost of milk which can completely fill the container, at the rate of ₹ 50 per litre. Also find the cost of metal sheet used to make the container, if it costs ₹ 10 per 100 cm2. (Take π = 3⋅14)

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

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