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Question Paper Solutions for Mathematics All India Set 2 2017-2018 CBSE Class 10

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Mathematics
All India Set 2
2017-2018 March
Marks: 90

[1]1

Given `triangle ABC ~ triangle PQR`, if `(AB)/(PQ) = 1/3`, then find `(ar  triangle ABC)/(ar triangle PQR)`

Chapter: [3.01] Triangles
Concept: Similarity of Triangles
[1]2

What is the value of (cos2 67° – sin2 23°)?

Chapter: [4.02] Trigonometric Identities
Concept: Trigonometric Ratios of Complementary Angles
[1]3

Find the distance of a point P(xy) from the origin.

Chapter: [6.01] Lines (In Two-dimensions)
Concept: Distance Formula
[1]4

If x = 3 is one root of the quadratic equation x2 – 2kx – 6 = 0, then find the value of k.

Chapter: [2.01] Polynomials
Concept: Polynomials Examples and Solutions
[1]5

What is the HCF of the smallest prime number and the smallest composite number?

Chapter: [1.01] Real Numbers
Concept: Fundamental Theorem of Arithmetic
[1]6

In an AP, if the common difference (d) = –4, and the seventh term (a7) is 4, then find the first term.

Chapter: [2.04] Arithmetic Progressions
Concept: nth Term of an AP
[2]7

An integer is chosen at random between 1 and 100. Find the probability that it is

1) divisible by 8.

2) not divisible by 8.

Chapter: [5.02] Probability
Concept: Probability Examples and Solutions
[2]8

Two different dice are tossed together. Find the probability :

1) of getting a doublet

2) of getting a sum 10, of the numbers on the two dice.

Chapter: [5.02] Probability
Concept: Probability Examples and Solutions
[2]9

Find the ratio in which P(4, m) divides the line segment joining the points A(2, 3) and B(6, –3). Hence find m.

Chapter: [6.01] Lines (In Two-dimensions)
Concept: Section Formula
[2]10

Given that `sqrt2` is irrational prove that `(5 + 3sqrt2)`  is an irrational number

Chapter: [1.01] Real Numbers
Concept: Revisiting Irrational Numbers
[2]11

In Fig. 1, ABCD is a rectangle. Find the value of x and y.

Chapter: [2.02] Pair of Linear Equations in Two Variables
Concept: Equations Reducible to a Pair of Linear Equations in Two Variables
[2]12

Find the sum of first 8 multiples of 3

Chapter: [2.04] Arithmetic Progressions
Concept: Sum of First n Terms of an AP
[3]13

A plane left 30 minutes late than its scheduled time and in order to reach the destination 1500 km away in time, it had to increase its speed by 100 km/h from the usual speed. Find its usual speed.

Chapter: [2.01] Polynomials
Concept: Polynomials Examples and Solutions
[3]14 | Attempt Any One
[4]14.1

Prove that the area of an equilateral triangle described on one side of a square is equal to half the area of the equilateral triangle described on one of its diagonals

Chapter: [3.01] Triangles
Concept: Areas of Similar Triangles
[4]14.2

If the areas of two similar triangles are equal, prove that they are congruent

Chapter: [3.01] Triangles
Concept: Areas of Similar Triangles
[3]15

Prove that the lengths of the tangents drawn from an external point to a circle are equal.

Chapter: [3.02] Circles
Concept: Number of Tangents from a Point on a Circle
[3]16 | Attempt Any One
[3]16.1

A wooden article was made by scooping out a hemisphere from each end of a solid cylinder, as shown in given figure. If the height of the cylinder is 10 cm, and its base is of radius 3.5 cm, find the total surface area of the article [Use π = 22/7]

Chapter: [7.02] Surface Areas and Volumes
Concept: Surface Area of a Combination of Solids
[3]16.2

A heap of rice is in the form of a cone of base diameter 24 m and height 3.5 m. Find the volume of the rice. How much is canvas cloth required to just cover the heap?

Chapter: [7.02] Surface Areas and Volumes
Concept: Volume of a Combination of Solids
[3]17

The table below shows the salaries of 280 persons :

Salary (In thousand Rs) No. of Persons
5 – 10 49
10 – 15 133
15 – 20 63
20 – 25 15
25 – 30 6
30 – 35 7
35 – 40 4
40 – 45 2
45 – 50 1

Calculate the median salary of the data.

Chapter: [5.01] Statistics
Concept: Median of Grouped Data
[3]18 | Attempt Any One
[3]18.1

If 4 tan θ = 3, evaluate `((4sin theta - cos theta + 1)/(4sin theta + cos theta - 1))`

Chapter: [4.01] Introduction to Trigonometry
Concept: Trigonometric Ratios
[3]18.2

If tan 2A = cot (A – 18°), where 2A is an acute angle, find the value of A

Chapter: [4.02] Trigonometric Identities
Concept: Trigonometric Ratios of Complementary Angles
[3]19

Find the area of the shaded region in Fig. 3, where arcs drawn with centres A, B, C and D intersect in pairs at mid-points P, Q, R and S of the sides AB, BC, CD and DA respectively of a square ABCD of side 12 cm. [Use π = 3.14]

Chapter: [7.02] Surface Areas and Volumes
Concept: Surface Area of a Combination of Solids
[3]20 | Attempt Any One
[3]20.1

If A(–2, 1), B(a, 0), C(4, b) and D(1, 2) are the vertices of a parallelogram ABCD, find the values of a and b. Hence find the lengths of its sides

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

If A(–5, 7), B(–4, –5), C(–1, –6) and D(4, 5) are the vertices of a quadrilateral, find the area of the quadrilateral ABCD

Chapter: [6.01] Lines (In Two-dimensions)
Concept: Area of a Triangle
[3]21

Find HCF and LCM of 404 and 96 and verify that HCF × LCM = Product of the two given numbers.

Chapter: [1.01] Real Numbers
Concept: Fundamental Theorem of Arithmetic
[3]22

Find all zeroes of the polynomial `(2x^4 - 9x^3 + 5x^2 + 3x - 1)` if two of its zeroes are `(2 + sqrt3)`  and `(2 - sqrt3)`

Chapter: [2.01] Polynomials
Concept: Relationship Between Zeroes and Coefficients of a Polynomial
[4]23

Draw a triangle ABC with BC = 6 cm, AB = 5 cm and ∠ABC = 60°. Then construct a triangle whose sides are `3/4` of the corresponding sides of the ∆ABC.

Chapter: [3.03] Constructions
Concept: Constructions Examples and Solutions
[4]24

The sum of four consecutive numbers in an AP is 32 and the ratio of the product of the first and the last term to the product of two middle terms is 7: 15. Find the numbers.

Chapter: [2.04] Arithmetic Progressions
Concept: Arithmetic Progression
[4]25 | Attempt Any One
[4]25.1
 
 

In an equilateral triangle ABC, D is a point on side BC such that BD = `1/3BC` . Prove that 9 AD2 = 7 AB2

 
 
Chapter: [3.01] Triangles
Concept: Pythagoras Theorem
[4]25.2

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

Chapter: [3.01] Triangles
Concept: Similarity of Triangles
[4]26 | Attempt Any One
[4]26.1

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.

Chapter: [2.01] Polynomials
Concept: Polynomials Examples and Solutions
[4]26.2

A train travels at a certain average speed for a distance of 63 km and then travels at a distance of 72 km at an average speed of 6 km/hr more than its original speed. If it takes 3 hours to complete the total journey, what is the original average speed?

Chapter: [2.03] Quadratic Equations
Concept: Quadratic Equations
[4]27

As observed from the top of a 100 m high lighthouse from the sea-level, the angles of depression of two ships are 30° and 45°. If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships. [Use `sqrt3`=1.732]

Chapter: [4.03] Heights and Distances
Concept: Heights and Distances
[4]28

The diameters of the lower and upper ends of a bucket in the form of a frustum of a cone are 10 cm and 30 cm respectively. If its height is 24 cm, find:

1) The area of the metal sheet used to make the bucket.

2) Why we should avoid the bucket made by ordinary plastic? [Use π = 3.14]

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

The mean of the following distribution is 18. Find the frequency f of class 19 – 21.

Class 11-13 13-15 15-17 17-19 19-21 21-23 23-25
Frequency 3 6 9 13 f 5 4

 

Chapter: [5.01] Statistics
Concept: Mean of Grouped Data
[4]29.2

The following distribution gives the daily income of 50 workers of a factory.

Daily income (in Rs 100 − 120 120 − 140 140 − 160 160 − 180 180 − 200
Number of workers 12 14 8 6 10

Convert the distribution above to a less than type cumulative frequency distribution, and draw its ogive.

Chapter: [5.01] Statistics
Concept: Graphical Representation of Cumulative Frequency Distribution
[4]30

Prove that `(sin A - 2sin^3 A)/(2cos^3 A - cos A) = tan A`

Chapter: [4.01] Introduction to Trigonometry
Concept: Trigonometric Ratios

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