CBSE Sample Papers for Class 10 Maths Paper 3

These Sample papers are part of CBSE Sample Papers for Class 10 Maths. Here we have given CBSE Sample Papers for Class 10 Maths Paper 3. According to new CBSE Exam Pattern, MCQ Questions for Class 10 Maths Carries 20 Marks.

CBSE Sample Papers for Class 10 Maths Paper 3

Board	CBSE
Class	X
Subject	Maths
Sample Paper Set	Paper 3
Category	CBSE Sample Papers

Students who are going to appear for CBSE Class 10 Examinations are advised to practice the CBSE sample papers given here which is designed as per the latest Syllabus and marking scheme as prescribed by the CBSE is given here. Paper 3 of Solved CBSE Sample Paper for Class 10 Maths is given below with free PDF download solutions.

Time Allowed: 3 hours
Maximum Marks: 80

General Instructions:

• All questions are compulsory.
• The question paper consists of 30 questions divided into four sections A, B, C and D.
• Section A contains 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 8 questions of 4 marks each.
• There is no overall choice. However, an internal choice has been provided in four questions of 3 marks each and three questions of 4 marks each. You have to attempt only one of the alternatives in all such questions.
• Use of calculators is not permitted.

SECTION-A

Question 1.
Write 98 as product of its prime factors.

Question 2.
If tan A = \(\frac { 3 }{ 4 } \) and A + B = 90°, then what is the value of cot B?

Question 3.
Show that x = – 2 is a solution of 3x² + 13x + 14 = 0.

Question 4.
Find the number of natural numbers between 101 and 999 which are divisible by both 2 and 5.

Question 5.
Find distance between the points (0,5) and (- 5, 0).

Question 6.
If it is given that ∆ABC ~ ∆PQR with \(\frac { BC }{ QR } =\frac { 1 }{ 3 } \) then find \(\frac { ar\left( \triangle PQR \right) }{ ar\left( \triangle ABC \right) } \).

SECTION-B

Question 7.
If the 10th term of an AP is 47 and its first term is 2, find the sum of its first 15 terms.

Question 8.
Justify the statement: “Tossing a coin is a fair way of deciding which team should get the batting first at the beginning of a cricket game.”

Question 9.
Find the solution of the pair of equations:
\(\frac { 3 }{ x } +\frac { 8 }{ y } =-1\), \(\frac { 1 }{ x } +\frac { 2 }{ y } =2\), x,y≠0

Question 10.
The coordinates of the vertices of∆ABC are A(4,1), B(- 3, 2) and C(0, k). Given that the area of ∆ ABC is 12 unit², find the value of k.

Question 11.
Write a rational number between \(\sqrt { 3 } \)and \(\sqrt { 5 } \).

Question 12.
A child has a die whose six faces show the letters as given below:
A B C D E A
The die is thrown once. What is the probability of getting (z) A? (ii) D?

SECTION-C

Question 13.
Show that \(3+5\sqrt { 2 } \) is an irrational number.

Question 14.
Prove that: \(\frac { sin\theta }{ cot\theta +cosec\theta } =2+\frac { sin\theta }{ cot\theta -cosec\theta } \)

Question 15.
In figure, \(\frac { XP }{ PY } =\frac { XQ }{ QZ } =3\), if the area of ∆XYZ is 32 cm², then find the area of the quadrilateral PYZQ.

Question 16.
A circle touches the side BC of a ∆ABC at a point P and touches AB and AC when produced at Q and R respectively. Show that
AQ = \(\frac { 1 }{ 2 } \) (Perimeter of ∆ABC)

Question 17.
Find the ratio in which the line segment joining the points A (3, -6) and B (5, 3) is divided by x-axis. Also find the coordinates of the point of intersection.
                                                                                             OR
Find a relation between x and y such that the point P (x, y) is equidistant from the points A (2,5) and B (-3, 7).

Question 18.
In figure, OAPB is a sector of a circle of radius 3.5 cm with the centre at O and ∠AOB = 120°. Find the length of OAPBO.

OR
Find the area of the shaded region of figure if the diameter of the circle with centre O in 28cm and \(AQ=\frac { 1 }{ 2 } AB\)

Question 19.
Write a quadratic polynomial, sum of whose zeros is 2 V3 and their product is 2.
                                                                                OR
What are the quotient and the remainder, when 3x⁴ + 5x³ – 7x² + 2x + 2 is divided by x² + 3x + 1?

Question 20.
Form a pair of linear equations in two variables using the following information and solve it graphically.

Five years ago, Sagar was twice as old as Tiru. Ten years later Sagar’s age will be ten years more than Tiru’s age. Find their present ages. What was the age of Sagar when Tiru was born?

Question 21.
A toy is in the form of a cone of radius 3.5 cm mounted on a hemisphere of same radius. The total height of the toy is 15.5 cm. Find the total surface area of the toy.

Question 22.
The distribution below gives the weights of 30 students of a class. Find the median weight of the students.

Weight (in kg)	40-45	45-50	50-55	55-60	60-65	65-70	70-75
Number of students	2	3	8	6	6	3	2

SECTION-D

Question 23.
Prove that in a triangle, if the square of one side is equal to the sum of the squares of the other two sides, the angle opposite to the first side is a right angle. Using the converse of above, determine the length of an altitude of an equilateral triangle of side 2 cm.

Question 24.
From the top and foot of a tower 40m high, the angle of elevation of the top of a lighthouse is found to be 30° and 60° respectively. Find the height of the lighthouse. Also find the distance of the top of the lighthouse from the foot of the tower.

Question 25.
A solid is composed of a cylinder with hemispherical ends. If the whole length of the solid is 100cm and the diameter of the hemispherical ends is 28cm. Find the cost of polishing the surface of the solid at the rate of 5 paise per sq.cm.
                                                                                        OR
An open container made up of a metal sheet is in the form of a frustum of a cone of height 8cm with radii of its lower and upper ends as 4 cm and 10 cm respectively. Find the cost of oil which can completely fill the container at the rate of ₹50 per litre. Also, find the cost of metal used, if it costs ₹5 per 100 cm2. (Use π = 3.14)

Question 26.
The mean of the following frequency table is 53. But the frequencies f₁ and f₂ in the classes 20-0 and 60-80 are missing. Find the missing frequencies.

Age (in years)	0-20	20-0	40-60	60-80	80-100	Total
Number of People	15	f1	21	f2	17	100

Find the median of the following frequency distribution:

Marks	Frequency
0-100	2
100-200	5
200-300	9
300-400	12
400-500	17
500-600	20
600-700	15
700-800	9
800-900	7
900-1000	4

Question 27.
Find value of k so that the following quadratic equation has equal roots:
2x²-(k-2)x+1=0
                                                                                                       OR
300 oranges are distributed equally among a certain number of students. Had there been 10 more students, each would have received 1 orange less. Find the number of students.

Question 28.
In a school, students decided to plant trees in and around the school to reduce air pollution. It was decided that the number of trees, that each section of each class will plant, will be double of the class in which they are studying. If there are 1 to 12 classes in the school and each class has two sections, find how many trees were planted by the students. Which value is shown in this question?

Question 29.
Construct a circle whose radius is equal to 4 cm. Let P be a point whose distance from its centre is 6 cm. Construct two tangents to it from P.

Question 30.
If x sin³ θ+31 cos³ θ = sin θ cos θ and x sin θ = y cos θ, prove x² + y² = 1.

Answers

Answer 1.
The prime factorization of 98 is
98 = 2 x 7 x 7 = 2 x 7²

Answer 2.
In ∆ABC

Answer 3.
Put the value of X in the quadratic equation,

Answer 4.
Given: a₁ = 110, d = 10, a_n = 990

Answer 5.
Here x₁ = 0, y₁ = 5, x₂ = -5 and y₂ = 0

Answer 6.

SECTION-B

Answer 7.
Let a be the first term and d be the common difference of the AP.
Here a=2, a₁₀=47
a_n=a+(n-1)d

Answer 8.
The tossing of a coin has two outcomes head (H) and tail (T). The happening of both H and T are equally likely. So the result of an individual coin toss is completely unpredictable. Thus, both the teams get equal chance to bat first. Hence, the given statement is justified.

Answer 9.

Answer 10.

Answer 11.
A rational number between \(\sqrt { 3 } \) and \(\sqrt { 5 } \) is \(\sqrt { 3.24 } =1.8=\frac { 18 }{ 10 } =\frac { 9 }{ 5 } \)

Answer 12.
12. The total number of elementary events associated with random experiment of throwing a die is 6.

1. Let E be the event of getting a letter A.
   ∴ Favourable number of elementary event = 2
   ∴\(P\left( E \right) =\frac { 2 }{ 6 } =\frac { 1 }{ 3 } \)
2. Let E be the event of getting a letter D.
   ∴ Favourable number of elementary event = 1
   ∴\(P\left( E \right) =\frac { 1 }{ 6 } \)

Answer 13.
Let us assume the contrary that \(3+5\sqrt { 2 } \) is a rational number. Then there exists co-prime positive integers a and b such that

Answer 14.
We have,

Answer 15.

Answer 16.
Since tangents from an exterior point to a circle are equal in length.

Answer 17.
Let the required ratio be k : 1. Then the coordinates of the point of division are

Answer 18.

Answer 19.
Quadratic polynomial = x² – (Sum of the zeros) x + (Product of the zeros)
Here, sum of the zeros = \(2\sqrt { 3 } \) and product of the zeros = 2
So, quadratic polynomial is x² – \(2\sqrt { 3 } x\) + 2
                                                            OR

Answer 20.
Let the present age of Sagar be x years and present age of Tiru be y years.
Five years ago, the age of Sagar = (x – 5) years
Five years ago, the age of Tiru = (y – 5) years

Answer 21.
We have,
CD = 15.5 cm and OB = OD = 3.5 cm
Let r be the radius of the base of cone and h be the height of conical part of the toy.
Then, r = OB = 3.5 cm
h = OC – CD – OD = (15.5 – 3.5) cm = 12 cm

Answer 22.
Calculation of median

SECTION-D

Answer 23.
Given: A triangle ABC in which AC² = AB² + BC².
To Prove: ∠B = 90°. ,
Construction: We construct a ∆PQR right-angled at Q such that PQ = AB and QR = BC

Answer 24.
Let CD be the tower of height = 40 m and AB = ft m be the height of the light house. In ∆ ABC

Answer 25.

Answer 26.

Answer 27.
The given equation is 2x₂ – (k – 2) x + 1 = 0.
Here, a = 2, b = – (k – 2), c = 1
The given equation will have equal roots, if D = 0

Answer 28.
Number of plants planted by class 1 = 2 (class 1×2 sections) = 2xlx2 = 4 trees
Similarly, number of plants planted by class 2 = 2 (class 2×2 sections)
= 2x2x2 = 8 trees

Values:

• Environmental friendly
• Social awareness
• Sense of responsibility towards the society.

Answer 29.
Steps of Construction:

Step I: Draw a circle of radius 4 cm.
Step II: Mark a point P at a distance of 6 cm from the centre of the circle.
Step III: Join OP and bisect it. Let M be the mid-point of PO.
Step IV: Taking M as a centre and MO as radius, draw another circle. Let it intersect the given circle at the points Q and R.
Step V: Join PQ and PR.
Then PQ and PR are the two required tangents.

Answer 30.

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