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

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Mathematics [Abroad Set(2)]
Marks: 80 Academic Year: 2018-2019
Date: March 2019
Duration: 3h
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Section - A
[1] 1

For what values of k does the quadratic equation 4x2 − 12x − k = 0 has no real roots?

Concept: Quadratic Equations
Chapter: [2.03] Quadratic Equations
[1] 2

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

Concept: Heights and Distances
Chapter: [4.01] Heights and Distances
[1] 3
[1] 3.a

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

Concept: Revisiting Rational Numbers and Their Decimal Expansions
Chapter: [1.01] Real Numbers
OR
[1] 3.b

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] 4

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

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

Evaluate:

`(tan 65^circ)/(cot 25^circ)`

Concept: Trigonometric Identities
Chapter: [4.02] Trigonometric Identities [4.03] Introduction to Trigonometry
OR
[1] 5.b

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] 6

Find the number of terms in the A.P.: 18, `15 1/2, 13, ...., -47.`

Concept: nth Term of an AP
Chapter: [2.02] Arithmetic Progressions
Section - B
[2] 7

A bag contains 15 balls, out of which some are white and the others are black. If the probability of drawing a black ball at random from the bag is `2/3`, then find how many white balls are there in the bag.

Concept: Basic Ideas of Probability
Chapter: [5.01] Probability [5.01] Probability
[2] 8

A card is drawn at random from a pack of 52 playing cards. Find the probability of drawing a card that is neither a spade nor a king.

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

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
OR
[2] 9.b

Find the value (s) of k for which the pair of equations

`{(kx,+,2y,=,3),(3x,-,6y,=,10):}" has a unique solution"`
Concept: Linear Equations in Two Variables
Chapter: [2.01] Pair of Linear Equations in Two Variables
[2] 10
[1] 10.a

How many multiples of 4 lie between 10 and 205?

Concept: General Term of an Arithmetic Progression
Chapter: [2.02] Arithmetic Progressions
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OR
[1] 10.b

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

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

Concept: Euclid’s Division Lemma
Chapter: [1.01] Real Numbers
[2] 12

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)
Section - C
[3] 13
[3] 13.a

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] 13.b

Prove that:

`sqrt((sectheta - 1)/(sec theta + 1)) + sqrt((sectheta + 1)/(sectheta - 1)) = 2cosectheta`

Concept: Trigonometric Identities
Chapter: [4.02] Trigonometric Identities [4.03] Introduction to Trigonometry
[3] 14
[3] 14.a

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)
OR
[3] 14.b

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] 15

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: Concept of Circle - Centre, Radius, Diameter, Arc, Sector, Chord, Segment, Semicircle, Circumference, Interior and Exterior, Concentric Circles
Chapter: [3.01] Circles [3.01] Circles
[3] 16
[3] 16.a

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
OR
[3] 16.b

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] 17

In Figure 2, two concentric circles with centre O, have radii 21 cm and 42 cm. If ∠AOB = 60°, find the area of the shaded region.

Concept: Circumference of a Circle
Chapter: [7.01] Areas Related to Circles [7.01] Areas Related to Circles
[3] 18

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] 19
[3] 19.a

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
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OR
[3] 19.b

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] 20

Prove that `2 + 3sqrt(3)` is an irrational number when it is given that `sqrt(3)` is an irrational number.

Concept: Concept of Irrational Numbers
Chapter: [1.01] Real Numbers
[3] 21

Sum of the areas of two squares is 157 m2. If the sum of their perimeters is 68 m, find the sides of the two squares.

Concept: Circumference of a Circle
Chapter: [7.01] Areas Related to Circles [7.01] Areas Related to Circles
[3] 22

Find the quadratic polynomial, sum, and product of whose zeroes are −1 and −20 respectively. Also, find the zeroes of the polynomial so obtained.

Concept: Concept of Polynomials
Chapter: [2.04] Polynomials [2.04] Polynomials
Section - D
[4] 23
[4] 23.a

A plane left 30 minutes later than the scheduled time and in order to reach its destination 1500 km away on time, it has to increase its speed by 250 km/hr from its usual speed. Find the usual speed of the plane.

Concept: Areas of Combinations of Plane Figures
Chapter: [7.01] Areas Related to Circles
OR
[4] 23.b

Find the dimensions of a rectangular park whose perimeter is 60 m and area 200 m2.

Concept: Circumference of a Circle
Chapter: [7.01] Areas Related to Circles [7.01] Areas Related to Circles
[4] 24

Find the value of x, when in the A.P. given below 2 + 6 + 10 + ... + x = 1800.

Concept: Sum of First n Terms of an AP
Chapter: [2.02] Arithmetic Progressions
[4] 25

If sec θ + tan θ = m, show that `(m^2 - 1)/(m^2 + 1) = sin theta`

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

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

Concept: Right-angled Triangles and Pythagoras Property
Chapter: [3.02] Triangles
[4] 27
[4] 27.a

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
[4] 27.b

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] 28

Construct a triangle with sides 5 cm, 6 cm, and 7 cm and then another triangle whose sides are `3/5` of the corresponding sides of the first triangle.

Concept: Similarity of Triangles
Chapter: [3.02] Triangles
[4] 29
[4] 29.a

Calculate the mean of the following frequency distribution :

Class: 10-30 30-50 50-70 70-90 90-110 110-130
Frequency: 5 8 12 20 3 2
Concept: Graphical Representation of Cumulative Frequency Distribution
Chapter: [5.02] Statistics
OR
[4] 29.b

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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