# Mathematics All India (Set 2) 2018-2019 English Medium Class 10 Question Paper Solution

Mathematics [All India (Set 2)]
Date: March 2019
Duration: 3h

(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 8questions of 4 marks each.

(iv) There is no overall choice. However, an internal choice has been provided in two questions of 1mark, 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.

Section A
 1

Two concentric circles of radii a and b (a > b) are given. Find the length of the chord of the larger circle which touches the smaller circle.

Concept: Number of Tangents from a Point on a Circle
Chapter: [0.031] Circles
 2
 2.1 In Figure 1, PS = 3 cm, QS = 4 cm, ∠PRQ = θ, ∠PSQ = 90°, PQ ⊥ RQ and RQ = 9 cm. Evaluate tan θ.

Concept: Area of a Triangle
Chapter: [0.061] Lines (In Two-dimensions)
OR
 2.2

If tan α =5/12 find the value of sec α.

Concept: Derivation of the n th Term
Chapter: [0.022000000000000002] Arithmetic Progressions
 3

Write the discriminant of the quadratic equation (x + 5)2 = 2 (5x − 3).

Concept: Nature of Roots of a Quadratic Equation
 4
 4.1

Find after how many places of decimal the decimal form of the number 27/(2^3. 5^4. 3^2) will terminate.

Concept: Rational Numbers and Their Decimal Expansions
Chapter: [0.011000000000000001] Real Numbers
OR
 4.2

Express 429 as a product of its prime factors.

Concept: Euclid’s Division Lemma
Chapter: [0.011000000000000001] Real Numbers
 5

Find the sum of the first 10 multiples of 6.

Concept: Sum of First n Terms of an A.P.
Chapter: [0.022000000000000002] Arithmetic Progressions
 6

Find the positive value of m for which the distance between the points A(5, −3) and B(13, m) is 10 units.

Concept: Heights and Distances
Chapter: [0.040999999999999995] Heights and Distances
Section B
 7

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

Concept: Basic Ideas of Probability
Chapter: [0.051] Probability [0.051] Probability

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

Concept: Probability - A Theoretical Approach
Chapter: [0.051] Probability
 8

Cards numbered 7 to 40 were put in a box. Poonam selects a card at random. What is the probability that Poonam selects a card which is a multiple of 7?

Concept: Basic Ideas of Probability
Chapter: [0.051] Probability [0.051] Probability
 9
 9.1

Points A(3, 1), B(5, 1), C(a, b) and D(4, 3) are vertices of a parallelogram ABCD. Find the values of a and b.

Concept: Similarity of Triangles
Chapter: [0.032] Triangles [0.032] Triangles [0.032] Triangles
OR
 9.2

Points P and Q trisect the line segment joining the points A(−2, 0) and B(0, 8) such that P is near to A. Find the coordinates of points P and Q.

Concept: Division of a Line Segment
Chapter: [0.033] Constructions
 10

Solve the following pair of linear equations:

3x − 5y = 4
2y + 7 = 9x

Concept: Pair of Linear Equations in Two Variables
Chapter: [0.021] Pair of Linear Equations in Two Variables
 11
 11.1

If HCF of 65 and 117 is expressible in the form 65n − 117, then find the value of n.

Concept: Euclid’s Division Lemma
Chapter: [0.011000000000000001] Real Numbers
OR
 11.2

On a morning walk, three persons step out together and their steps measure 30 cm, 36 cm, and 40 cm respectively. What is the minimum distance each should walk so that each can cover the same distance in complete steps?

Concept: Heights and Distances
Chapter: [0.040999999999999995] Heights and Distances
 12

In the quadratic equation kx2 − 6x − 1 = 0, determine the values of k for which the equation does not have any real root.

Concept: Nature of Roots of a Quadratic Equation
Section C
 13
 13.1
[1.5] 13.1.1

A, B and C are interior angles of a triangle ABC. Show that

sin (("B"+"C")/2) = cos  "A"/2

Concept: Trigonometric Ratios of Complementary Angles
Chapter: [0.042] Trigonometric Identities [0.043] Introduction to Trigonometry
[1.5] 13.1.2

A, B and C are interior angles of a triangle ABC. Show that

If ∠A = 90°, then find the value of tan(("B+C")/2)

Concept: Trigonometric Ratios of Complementary Angles
Chapter: [0.042] Trigonometric Identities [0.043] Introduction to Trigonometry
OR
 13.2

If tan (A + B) = 1 and tan(A-B)=1/sqrt3 , 0° < A + B < 90°, A > B, then find the values of A and B.

Concept: Trigonometric Ratios of Some Special Angles
Chapter: [0.043] Introduction to Trigonometry
 14
 14.1

In Figure 2, PQ is a chord of length 8 cm of a circle of radius 5 cm. The tangents at P and Q intersect at a point T. Find the length TP. Concept: Construction of Tangents to a Circle
Chapter: [0.033] Constructions
OR
 14.2

Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.

Concept: Number of Tangents from a Point on a Circle
Chapter: [0.031] Circles
 15

A class teacher has the following absentee record of 40 students of a class for the whole term. Find the mean number of days a student was absent.

 Number of days: 0-6 6-12 12-18 18-24 24-30 30-36 36-42 Number of students: 10 11 7 4 4 3 1
Concept: Mean of Grouped Data
Chapter: [0.052000000000000005] Statistics
 16

A car has two wipers that do not overlap. Each wiper has a blade of length 21 cm sweeping through an angle of 120°. Find the total area cleaned at each sweep of the blades. ("Take" π =22/7)

Concept: Areas of Sector and Segment of a Circle
Chapter: [0.071] Areas Related to Circles
 17
 17.1

The perpendicular from A on side BC of a Δ ABC meets BC at D such that DB = 3CD. Prove that 2AB2 = 2AC+ BC2.

Concept: Areas of Similar Triangles
Chapter: [0.032] Triangles
OR
 17.2

AD and PM are medians of triangles ABC and PQR, respectively where ΔABC ~ ΔPQR, prove that ("AB")/("PQ") = ("AD")/("PM")

Concept: Criteria for Similarity of Triangles
Chapter: [0.032] Triangles
 18

Check whether g(x) is a factor of p(x) by dividing polynomial p(x) by polynomial g(x),
where p(x) = x5 − 4x3 + x2 + 3x +1, g(x) = x3 − 3x + 1

Concept: Relationship Between Zeroes and Coefficients of a Polynomial
Chapter: [0.024] Polynomials
 19
 19.1

Prove that is sqrt3 irrational number.

Concept: Proofs of Irrationality
Chapter: [0.011000000000000001] Real Numbers
OR
 19.2

Find the largest number which on dividing 1251, 9377 and 15628 leave remainders 1, 2 and 3 respectively.

Concept: Euclid’s Division Lemma
Chapter: [0.011000000000000001] Real Numbers
 20

Find the area of the triangle ABC with the coordinates of A as (1, −4) and the coordinates of the mid-points of sides AB and AC respectively are (2, −1) and (0, −1).

Concept: Similarity of Triangles
Chapter: [0.032] Triangles [0.032] Triangles [0.032] Triangles
 21

Two numbers are in the ratio of 5: 6. If 7 is subtracted from each of the numbers, the ratio becomes 4: 5. Find the numbers.

Concept: Real Numbers Examples and Solutions
Chapter: [0.011000000000000001] Real Numbers
 22

Water is flowing at the rate of 2.52 km/h through a cylindrical pipe into a cylindrical tank, the radius of whose base is 40 cm. If the increase in the level of water in the tank, in half an hour is 3.15 m, find the internal diameter of the pipe.

Concept: Concept of Surface Area, Volume, and Capacity
Chapter: [0.07200000000000001] Surface Areas and Volumes [0.07200000000000001] Surface Areas and Volumes
Section D
 23
 23.1 In Figure 3, a decorative block is shown which is made of two solids, a cube, and a hemisphere. The base of the block is a cube with an edge 6 cm and the hemisphere fixed on the top has a diameter of 4⋅2 cm. Find

(a) the total surface area of the block.
(b) the volume of the block formed. ("Take"  pi = 22/7)

Concept: Concept of Surface Area, Volume, and Capacity
Chapter: [0.07200000000000001] Surface Areas and Volumes [0.07200000000000001] Surface Areas and Volumes
OR
 23.2

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 circular ends are 20 cm and 12 cm, respectively. Find the height of the bucket and the area of metal sheet used in making the bucket. (use π = 3.14)

Concept: Heights and Distances
Chapter: [0.040999999999999995] Heights and Distances
 24
 24.1

If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, prove that the other two sides are divided in the same ratio

Concept: Similarity of Triangles
Chapter: [0.032] Triangles [0.032] Triangles [0.032] Triangles
OR
 24.2

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

Concept: Areas of Similar Triangles
Chapter: [0.032] Triangles
 25

Change the following distribution to a 'more than type' distribution. Hence draw the 'more than type' ogive for this distribution.

 Class interval: 20−30 30−40 40−50 50−60 60−70 70−80 80−90 Frequency: 10 8 12 24 6 25 15
Concept: Graphical Representation of Cumulative Frequency Distribution
Chapter: [0.052000000000000005] Statistics
 26

The shadow of a tower standing on level ground is found to be 40 m longer when the Sun's altitude is 30° than when it was 60°. Find the height of the tower. (Given sqrt3 = 1.732)

Concept: Heights and Distances
Chapter: [0.040999999999999995] Heights and Distances
 27
 27.1

If m times the mth term of an Arithmetic Progression is equal to n times its nth term and m ≠ n, show that the (m + n)th term of the A.P. is zero.

Concept: nth Term of an AP
Chapter: [0.022000000000000002] Arithmetic Progressions
OR
 27.2

The sum of the first three numbers in an Arithmetic Progression is 18. If the product of the first and the third term is 5 times the common difference, find the three numbers.

Concept: Sum of First n Terms of an A.P.
Chapter: [0.022000000000000002] Arithmetic Progressions
 28

A shopkeeper buys some books for Rs 80. If he had bought 4 more books for the same amount, each book would have cost Rs 1 less. Find the number of books he bought.

Concept: Real Numbers Examples and Solutions
Chapter: [0.011000000000000001] Real Numbers
 29

Construct a pair of tangents to a circle of radius 4 cm from a point which is at a distance of 6 cm from its centre.

Concept: Number of Tangents from a Point on a Circle
Chapter: [0.031] Circles
 30

Prove the following:

1/(1+sin^2theta) + 1/(1+cos^2theta) + 1/(1+sec^2theta) + 1/(1+cosec^2theta) = 2

Concept: Trigonometry
Chapter: [0.043] Introduction to Trigonometry [0.043] Introduction to Trigonometry

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