Mathematics 30/1/1 2018-2019 CBSE (English Medium) Class 10 Question Paper Solution

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] 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: Coordinate Geometry
Chapter: [6.01] Lines (In Two-dimensions) [6.01] Lines (In Two-dimensions)
[1] 2
[1] 2.1

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

Concept: Nature of Roots of a Quadratic Equation
OR
[1] 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 of a Quadratic Equation
[1] 3
[1] 3.1

Find A if tan 2A = cot (A-24°).

Concept: Trigonometric Identities
Chapter: [4.02] Trigonometric Identities [4.03] Introduction to Trigonometry
OR
[1] 3.2

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

Concept: Trigonometric Identities
Chapter: [4.02] Trigonometric Identities [4.03] Introduction to Trigonometry
[1] 4

How many two digits numbers are divisible by 3?

Concept: Real Numbers Examples and Solutions
Chapter: [1.01] Real Numbers
[1] 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 of Triangles
Chapter: [3.02] Triangles [3.02] Triangles [3.02] Triangles
[1] 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
[12] 7
[2] 7.1

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

Concept: Euclid’s Division Lemma
Chapter: [1.01] Real Numbers
OR
[2] 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
[2] 8
[2] 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 A.P.
Chapter: [2.02] Arithmetic Progressions
OR
[2] 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
[2] 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)
[2] 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: Basic Ideas of Probability
Chapter: [5.01] Probability [5.01] Probability
[2] 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: Basic Ideas of Probability
Chapter: [5.01] Probability [5.01] Probability
[2] 12

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

Section C
[3] 13

Prove that sqrt(2) is an irrational number.

Concept: Concept of Irrational Numbers
Chapter: [1.01] Real Numbers
[3] 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
[3] 15
[3] 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: Basic Ideas of Probability
Chapter: [5.01] Probability [5.01] Probability
OR
[3] 15.2

A fraction becomes 1/3 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
[3] 16
[3] 16.1

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

Concept: Coordinate Geometry
Chapter: [6.01] Lines (In Two-dimensions) [6.01] Lines (In Two-dimensions)
OR
[3] 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
Chapter: [6.01] Lines (In Two-dimensions) [6.01] Lines (In Two-dimensions)
[3] 17
[3] 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
[3] 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
[3] 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
[3] 19
[3] 19.1

In the following Figure ∠ACB= 90° and CD ⊥ AB, prove that  CD2  = BD × AD

Concept: Right-angled Triangles and Pythagoras Property
Chapter: [3.02] Triangles
OR
[3] 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: Right-angled Triangles and Pythagoras Property
Chapter: [3.02] Triangles
[3] 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: 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] 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
[3] 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
[4] 23
[4] 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.

OR
[4] 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.

[4] 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 A.P.
Chapter: [2.02] Arithmetic Progressions
[4] 25

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

Concept: Trigonometry
Chapter: [4.03] Introduction to Trigonometry [4.03] Introduction to Trigonometry
[4] 26
[4] 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: Trigonometry
Chapter: [4.03] Introduction to Trigonometry [4.03] Introduction to Trigonometry
[4] 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.

[4] 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 of Triangles
Chapter: [3.02] Triangles [3.02] Triangles [3.02] Triangles
[4] 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: Concept of Surface Area, Volume, and Capacity
Chapter: [7.02] Surface Areas and Volumes [7.02] Surface Areas and Volumes
[4] 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
[4] 30
[4] 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
[4] 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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