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# Mathematics 30/1/1 2018-2019 CBSE Class 10 Question Paper Solution

Course
Mathematics [30/1/1]
Date: March 2019
Duration: 3h

1. All questions are compulsory.

2. This question paper consists of 30 questions divided into four Section- A, B,C and D.

3. Section A contains 6 questions of 1 marks 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.

Section A
1

Find the coordinates if a point A, where AB is diameter of a circle whose centre is (2,-3) and B is the point (1,4).

Concept: Concepts of Coordinate Geometry
Chapter: [6.01] Lines (In Two-dimensions)
2
2.1

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

Concept: Nature of Roots
OR
2.2

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
3
3.1

Find A if tan 2A = cot (A-24°).

Concept: Trigonometric Identities
Chapter: [4.02] Trigonometric Identities [4.03] Introduction to Trigonometry
OR
3.2

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

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

How many two digits numbers are divisible by 3?

Concept: Real Numbers Examples and Solutions
Chapter: [1.01] Real Numbers
5

In fig.DE || BC ,AD = 1 cm and BD = 2 cm. what is the ratio of the ar(ΔABC) to the  ar (ΔADE)? Concept: Similarity Triangle Theorem
Chapter: [3.02] Triangles
6

Find a rational number between sqrt(2) " and sqrt(3) .

Concept: Revisiting Rational Numbers and Their Decimal Expansions
Chapter: [1.01] Real Numbers
Section B
7
7.1

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

Concept: Euclid’s Division Lemma
Chapter: [1.01] Real Numbers
OR
7.2

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

Concept: Division Algorithm for Polynomials
Chapter: [2.04] Polynomials
8
8.1

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

Concept: Sum of First n Terms of an AP
Chapter: [2.02] Arithmetic Progressions
OR
8.2

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

Concept: nth Term of an AP
Chapter: [2.02] Arithmetic Progressions
9

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

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

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

Concept: Probability Examples and Solutions
Chapter: [5.01] Probability
11

A die is thrown once. Find the probability of getting a number which (i) is a prime number (ii) lies between 2 and 6.

Concept: Probability Examples and Solutions
Chapter: [5.01] Probability
12

Find c if the system of equations cx+3y+(3 - c ) = 0, 12x + cy - c = 0 has infinitely many solutions?

Concept: Quadratic Equations Examples and Solutions
Section C
13

Prove that sqrt(2) is an irrational number.

Concept: Revisiting Irrational Numbers
Chapter: [1.01] Real Numbers
14

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

Concept: Geometrical Meaning of the Zeroes of a Polynomial
Chapter: [2.04] Polynomials
15
15.1

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: Probability Examples and Solutions
Chapter: [5.01] Probability
OR
15.2

A fraction becomes 1/2 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
16
16.1

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

Concept: Coordinate Geometry Examples and Solutions
Chapter: [6.01] Lines (In Two-dimensions)
OR
16.2

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: Coordinate Geometry Examples and Solutions
Chapter: [6.01] Lines (In Two-dimensions)
17
17.1

Prove that :(sinθ+cosecθ)2+(cosθ+ secθ)2 = 7 + tan2 θ+cotθ.

Concept: Trigonometric Identities
Chapter: [4.02] Trigonometric Identities [4.03] Introduction to Trigonometry
OR
17.2

Prove that: (1+cot A - cosecA)(1 + tan A+ secA) =2.

Concept: Trigonometric Identities
Chapter: [4.02] Trigonometric Identities [4.03] Introduction to Trigonometry
18

ln Figure, PQ is a chord of length 8 cm of a circle of radius 5cm and centre o. The tangents at P and Q intersect at point T. find the length of TP. Concept: Number of Tangents from a Point on a Circle
Chapter: [3.01] Circles
19
19.1

In the following Figure ∠ACB= 90° and CD ⊥ AB, prove that  CD2  = BD × AD Concept: Pythagoras Theorem
Chapter: [3.02] Triangles
OR
19.2

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: Pythagoras Theorem
Chapter: [3.02] Triangles
20

Find the area of the shaded region in the figure If ABCD is a rectangle with sides 8 cm and 6 cm and O is the centre of the circle. (Take π= 3.14) Concept: Circles Examples and Solutions
Chapter: [3.01] Circles
21

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

Concept: Conversion of Solid from One Shape to Another
Chapter: [7.02] Surface Areas and Volumes
22

Find the mode of the following frequency distribution.

 Class 0-10 10-20 20-30 30-40 40-50 50-60 60-70 Frequency 8 10 10 16 12 6 7
Concept: Graphical Representation of Cumulative Frequency Distribution
Chapter: [5.02] Statistics
Section D
23
23.1

Two water taps together can fill a tank in 1 7/8 hours. The tap with a 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
OR
23.2

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
24

If the sum of the first four terms of an AP is 40 and that of the first 14 terms is 280. Find the sum of its first n terms.

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

Prove that (sin "A" - cos "A" + 1)/(sin "A" + cos "A" - 1) = 1/(sec "A" - tan "A")

Concept: Introduction to Trigonometry
Chapter: [4.03] Introduction to Trigonometry
26
26.1

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 sqrt3 = 1.732]

Concept: Introduction to Trigonometry
Chapter: [4.03] Introduction to Trigonometry
26.2

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.

27

Construct a ΔABC in which CA = 6 cm, AB = 5 cm and ∠BAC = 45°. Then construct a triangle whose sides are 3/5 of the corresponding sides of ΔABC.

Concept: Similarity Examples and Solutions
Chapter: [3.02] Triangles
28

A bucket open at the top is in the form of a frustum of a cone with a capacity of 12308.8 cm. 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.

Concept: Surface Areas and Volumes Examples and Solutions
Chapter: [7.02] Surface Areas and Volumes
29

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: Areas of Similar Triangles
Chapter: [3.02] Triangles
30
30.1

If the median of the following frequency distribution is 32.5, find the values of f_1 and f_2.

 Class 0 – 10 10 – 20 20 – 30 30 -40 40 – 50 50 – 60 60 – 70 Total Frequency f_1 5 9 12 f_2 3 2 40

Concept: Median of Grouped Data
Chapter: [5.02] Statistics
30.2

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

 Marks No. of students 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 (orgive). Hence find median

Concept: Ogives (Cumulative Frequency Graphs)
Chapter: [5.02] Statistics

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