# Mathematics 30/1/3 2018-2019 CBSE (English Medium) Class 10 Question Paper Solution

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

Section A
 1

Two positive integers a and b can be written as a = x3yand b = xy3. x, y are prime numbers. Find LCM (a, b).

Concept: Real Numbers Examples and Solutions
Chapter: [1.01] Real Numbers
 2

How many two digits numbers are divisible by 3?

Concept: Algebraic Conditions for Number of Solutions
Chapter: [2.01] Pair of Linear Equations in Two Variables
 3

In Fig. 1, DE || BC, AD = 1 cm and BD = 2 cm. What is the ratio of the ar (Δ ABC) to the ar (Δ ADE)? Concept: Ratio of Sides of Triangle
Chapter: [3.02] Triangles
 4

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

Concept: Section Formula
Chapter: [6.01] Lines (In Two-dimensions)
 5
 5.a

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

Concept: Nature of Roots
OR
 5.b

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
 6
 6.a

Find A if tan 2A = cot (A-24°).

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

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

Concept: Right-angled Triangles and Pythagoras Property
Chapter: [3.02] Triangles
Section B
 7
 7.a

Find, how many two digit natural numbers are divisible by 7.

Concept: Real Numbers Examples and Solutions
Chapter: [1.01] Real Numbers
OR
 7.b

If the sum of first n terms of an AP is n2, then find its 10th term.

Concept: General Term of an Arithmetic Progression
Chapter: [2.02] Arithmetic Progressions
 8

A game consist of tossing a one-rupee coin 3 times and noting the outcome each time. Ramesh will win the game if all the tosses show the same result, (i.e. either all thee heads or all three tails) and loses the game otherwise. Find the probability that Ramesh will lose the game.

Concept: Basic Ideas of Probability
Chapter: [5.01] Probability [5.01] Probability
 9

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

Concept: Division of a Line Segment
Chapter: [3.03] Constructions
 10

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
 11

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
 12
 12.a

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

Concept: Real Numbers Examples and Solutions
Chapter: [1.01] Real Numbers
 12.b

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

Concept: Euclid’s Division Lemma
Chapter: [1.01] Real Numbers
Section C
 13

Find all zeros of the polynomial 3x3 + 10x2 − 9x − 4 if one of its zero is 1.

Concept: Division Algorithm for Polynomials
Chapter: [2.04] Polynomials
 14

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 lengths of TP and TQ. 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
 15

Prove that (2+sqrt3)/5 is an irrational number, given that sqrt 3 is an irrational number.

Concept: Concept of Irrational Numbers
Chapter: [1.01] Real Numbers
 16
 16.a

Prove that (sin θ + cosec θ)2 + (cos θ + sec θ)2 = 7 + tanθ + cotθ.

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

Prove that (1 + cot A - cosec A ) (1 + tan A + sec A) = 2

Concept: Right-angled Triangles and Pythagoras Property
Chapter: [3.02] Triangles
 17
 17.a

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: Algebraic Methods of Solving a Pair of Linear Equations - Cross - Multiplication Method
Chapter: [2.01] Pair of Linear Equations in Two Variables
OR
 17.b

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
 18
 18.a

Find the point on the y-axis which is equidistant from the points (5, −2) and (−3, 2).

Concept: Coordinate Geometry
Chapter: [6.01] Lines (In Two-dimensions) [6.01] Lines (In Two-dimensions)
OR
 18.b

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: Constructions Examples and Solutions
Chapter: [3.03] Constructions
 19

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
 20

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

Concept: Volume of a Combination of Solids
Chapter: [7.02] Surface Areas and Volumes
 21
 21.a

In Fig. 3, ∠ACB = 90° and CD ⊥ AB, prove that CD2 = BD x AD. Concept: Right-angled Triangles and Pythagoras Property
Chapter: [3.02] Triangles
OR
 21.b

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
 22

Find the area of the shaded region in the given figure, if ABCD is a rectangle with sides 8 cm and 6 cm and O is the centre of the circle. Concept: Area of Circle
Chapter: [7.01] Areas Related to Circles
Section D
 23

If sec θ = x + 1/(4"x"), x ≠ 0, find (sec θ + tan θ)

Concept: Trigonometric Identities
Chapter: [4.02] Trigonometric Identities [4.03] Introduction to Trigonometry
 24

Prove that “That ratio of areas of two similar triangles is equal to the square of the ratio of their corresponding sides.”

Concept: Basic Geometric Constructions
Chapter: [6.01] Lines (In Two-dimensions)
 25

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

 Daily income (in ₹) 200-220 220-240 240-260 260-280 280-300 Number of workers 12 14 8 6 10

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

Concept: Graphs of Linear Equations
Chapter: [6.01] Lines (In Two-dimensions)
 26

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
Chapter: [3.02] Triangles
 27

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: [4.01] Heights and Distances
 28
 28.a

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
OR
 28.b

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.

Concept: Heights and Distances
Chapter: [4.01] Heights and Distances
 29
 29.a

Two water taps together can fill a tank in 1(7)/(8)  hours. The tap with 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
 29.b

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
 30

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

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