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 13. According to new CBSE Exam Pattern, MCQ Questions for Class 10 Maths Carries 20 Marks.
CBSE Sample Papers for Class 10 Maths Paper 13
Board | CBSE |
Class | X |
Subject | Maths |
Sample Paper Set | Paper 13 |
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 13 of Solved CBSE Sample Paper for Class 10 Maths is given below with free PDF download solutions.
Time Allowed: 3 hours
Max. 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 whether the rational number \(\frac { 7 }{ 25 } \) will have a terminating decimal expansion or a non-terminating repeating decimal expansion.
Question 2.
Find the value(s) of k, if the quadratic equation 3x2 – k\(\sqrt { 3 } \)x + 4 = 0 has equal roots.
Question 3.
Find the eleventh term from the last term of the AP: 27, 23, 19, …, – 65.
Question 4.
Find the coordinates of the point on y-axis which is nearest to the point (-2,5).
Question 5.
In the given figure, ST || RQ, PS = 3 cm and SR = 4 cm. Find the ratio of the area of ∆PST to the area of ∆PRQ.
Question 6.
If cos A = \(\frac { 5 }{ 5 } \) , find the value of 4 + 4 tan2A.
SECTION-B
Question 7.
If two positive integers p and q are written as p=a2b3 and q=a2b; a, b are prime numbers, then verify:
LCM (p, q) x HCF (p, q) = pq
Question 8.
The sum of first n terms of an AP is given by Sn = 2n2 + 3n. Find the sixteenth term of
the AP.
Question 9.
Find the value(s) of k for which the pair of linear equations kx + y = k2 and x + ky = 1 have infinitely many solutions.
Question 10.
If \(\left( 1,\frac { p }{ 3 } \right) \) is the mid-point of the line segment joining the points (2,0) and \(\left( 0,\frac { 2 }{ 9 } \right) \), then show that the line 5x + 3y + 2 = 0 passes through the point (-1,3p).
Question 11.
A box contains cards numbered 11 to 123. A card is drawn at random from the box. Find the probability that the number on the drawn card is
- a square number
- a multiple of 7.
Question 12.
A box contains 12 balls of which some are red in colour. If 6 more red balls are put in the box and a ball is drawn at random, the probability of drawing a red ball doubles than what it was before. Find the number of red balls in the bag.
SECTION-C
Question 13.
Show that exactly one of the numbers n, n + 2 or n + 4 is divisible by 3.
Question 14.
Find all the zeroes of the polynomial 3x4 + 6x3 – 2x2 – 10x – 5 if two of its zeros are \(\sqrt { \frac { 5 }{ 3 } } \) and – \(\sqrt { \frac { 5 }{ 3 } } \)
Question 15.
Seven times a two digit number is equal to four times the number obtained by reversing the order of its digits. If the difference of the digits is 3, determine the number.
Question 16.
In what ratio does the x-axis divide the line segment joining the points (-4,-6) and (-1, 7)? Find the co-ordinates of the point of division.
OR
The points A(4,-2), B(7,2), C(0,9) and D(-3,5) form a parallelogram. Find the length of the altitude of the parallelogram on the base AB.
Question 17.
Inthegiven figure ∠1= ∠2and ∆NSQ=∆MTR,
then prove that ∆PTS ~ ∆PRQ.
In an equilateral triangle ABC, D is a point on the side BC such that BD = \(\frac { 1 }{ 3 } \)BC. Prove that 9AD2 = 7AB2
Question 18.
In the given figure XY and X’Y’ are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersecting XY at A and X’Y’ at B. Prove that ∠AOB = 90°.
Question 19.
Evaluate:
OR
If sin θ + cos θ = \(\sqrt { 2 } \) , then evaluate tan θ + cot θ.
Question 20.
In the given figure ABPC is a quadrant of a circle of radius 14 cm and a semi circle is drawn with BC as diameter. Find the area of the shaded region.
Question 21.
Water in a 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?
OR
A cone of maximum size is carved out from a cube of edge 14 cm. Find the surface area of the remaining solid after the cone is carved out.
Question 22.
Find the mode of the following distribution of marks obtained by the students in an examination:
Marks obtained | 0-20 | 20-40 | 40-60 | 60-80 | 80-100 |
Number of students | 15 | 18 | 21 | 29 | 17 |
SECTION-D
Question 23.
A train travelling at a uniform speed for 360 km would have taken 48 minutes less to travel the same distance if its speed were 5 km/hour more. Find the original speed of the train.
OR
Check whether the equation 5x2 – 6x – 2 = 0 has real roots and if it has, find them by the method of completing the square. Also verify that roots obtained, satisfy the given equation.
Question 24.
An AP consists of 37 terms. The sum of the three middle most terms is 225 and the sum of the last three terms is 429. Find the AP.
Question 25.
Show that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
OR
Prove that the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides.
Question 26.
Draw a triangle ABC with side BC = 7 cm, ∠B = 45°, ∠A = 105°. Then, construct a triangle whose sides are \(\frac { 4 }{ 3 } \) times the corresponding sides of ∆ABC.
Question 27.
Prove that
Question 28.
The angles of depression of the top and bottom of a building 50 metres high as observed from the top of a tower are 30° and 60°, respectively. Find the height of the tower and also the horizontal distance between the building and the tower.
Question 29.
Two dairy owners A and B sell flavoured milk filled to capacity in mugs of negligible thickness, which are cylindrical in shape with a raised hemispherical bottom. The mugs are 14 cm high and have diameter of 7 cm as shown in the given figure. Both A and B sell flavoured milk at the rate of ₹80 per litre. The dairy owner A uses the formula arh to find the volume of milk in the mug and charges 343.12 for it. The dairy owner B is of the view that the price of actual quantity of milk should be charged. What according to him should be the price of one mug of milk? Which value is exhibited by the dairy owner B?’ \(\left( use\quad \Pi =\frac { 22 }{ 7 } \right) \)|
Question 30.
The following distribution shows the daily pocket allowance of children of a locality. The mean pocket allowance is ₹18. Find the missing frequency k.
Daily pocket allowance (in ₹) | 11-13 | 13-15 | 15-17 | 17-19 | 19-21 | 21-23 | 23-25 |
Number of children | 3 | 6 | 9 | 13 | k | 5 | 4 |
OR
The following frequency distribution shows the distance (in metres) thrown by 68 students in a javelin throw competition.
Distance (in m) | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 | 60-70 |
Number of students | 4 | 5 | 13 | 20 | 14 | 8 | 4 |
Answer
SECTION-A
Answer 1.
Since in rational number \(\frac { 7 }{ 75 } \),75 = 52 x 31 is not in the form of 2n 5m, so \(\frac { 7 }{ 75 } \) represent non-terminating repeating decimal.
Answer 2.
Since 3x2 – k \(\sqrt { 3 } \) x + 4 = 0 has equal roots.
Answer 3.
Given AP series is 27, 23, 19, …, – 65.
Here, last term l=-65 and d=-4
Now, 11th term from last = 1 – (11 – 1) xd
= -65 – 10 x (- 4) = -65 + 40 = -25
Answer 4.
(0,5)
Answer 5.
Here, ST||RQ
PS = 3 cm; SR = 4 cm
⇒ PR = 3 + 4 = 7 cm
Now, \(\frac { ar\left( \triangle PST \right) }{ ar\left( \triangle PQR \right) } =\frac { P{ S }^{ 2 } }{ P{ R }^{ 2 } } \)
\(=\frac { { 3 }^{ 2 } }{ { 7 }^{ 2 } } =\frac { 9 }{ 49 } \)
Answer 6.
Given, cos A =\(\frac { 2 }{ 5 } \)
SECTION-B
Answer 7.
Given, p = a2b3, q = a3b
LCM (p, q) = a3b3
HCF (p, q) = a2b2
Now, LCM (p, q) x HCF (p, q) = a3b3 x a2b = a5b4
= (a2b3)x (a3b) = pq
Answer 8.
Given, Sn = 2n2 + 3n
⇒ Sn-1 = 2 (n – 1)2 + 3(n – 1)
= 2(n2 – 2n + 1) + 3n – 3
= 2n2 – 4n + 2 + 3n – 3 = 2n2 – n-1
We have an = Sn-Sn-1
= 2n2 + 3n – (2n2 – 1 – 1)
= 2n2 + 3n – 2n2+n +1
an = 4n + 1
⇒ a16 = 4 x 16 + 1 = 65
Answer 9.
Given equation are
Answer 10.
Here, \(\left( 1,\frac { p }{ 3 } \right) \) is the mid point of line joining A(2, 0) and B\(\left( 0,\frac { 2 }{ 9 } \right) \)
Answer 11.
Total square numbers from 11 to 123 = 16, 25, 36, 49, 64, 81, 100, 121
∴ P (square number) = \(\frac { 8 }{ 113 } \)
Also, total number that are multiple of 7 = 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98, 105, 112, 119.
∴ P(multiple of 7) = \(\frac { 16 }{ 113 } \)
Answer 12.
Let the number of red balls in bag be x.
Now, P (drawing the red ball before addition of 6 red balls) = \(\frac { x }{ 12 } \)
And P (drawing the red ball after addition of 6 red balls) = \(\frac { x+6 }{ 18 } \)
SECTION-C
Answer 13.
Let n = 3k, 3k + 1 or 3k + 2, where k € N.
Case-I when n = 3k
n is divisible by 3.
Now, n + 2 = 3k + 2 ⇒ x + 2 is not divisible by 3
n + 4 = 3k + 4 = 3(k + 1) + 1 = 3m + 1 (Let (k + 1) = m)
⇒ n +4 is not divisible by 3.
Case-II when n = 3k + 1
n is not divisible by 3.
n+ 2 = 3k + 1 + 2 = 3k + 3 = 3(k +1) = 3m [Let (k + 1) = m]
⇒ n + 2 is divisible by 3. n
+ 4 = 3k + 1 + 4 = 3k + 5 = 3k + 3 + 2 = 3(k + 1) + 2 = 3m + 2
⇒ n +4 is not divisible by 3.
Case-III when n = 3k + 2
n is not divisible by 3.
n+ 2 = 3k + 2 + 2 = 3k + 4 = 3(k +1) + 1 = 3m + 1
⇒ n+ 2 is not divisible by 3.
n + 4 = 3k + 2 + 4 = 3k + 6 = 3 (k + 2) = 3m
⇒ n +4 is divisible by 3.
Hence, exactly one of the numbers n, n + 2 or n + 4 is divisible by 3.
Answer 14.
Since \(\sqrt { \frac { 5 }{ 3 } } \) and \(-\sqrt { \frac { 5 }{ 3 } } \) are the zeros of 3x4 +6x3 – 2x2 – 10x – 5
Answer 15.
Let the unit digit and tens digit of two digit number be x and y.
So, the number is 10y + x.
The number when digits are reversed is 10x + y.
From question,
Answer 16.
Let the x-axis divides the line segment joining (-4,-6) and (-1,7) at the point P in the ratio k: 1.
Answer 17.
Given, ∠1 = ∠2 and ∆NSQ = ∆MTR
Answer 18.
Construction: OC is joined
In ΔOPA and ΔOCA
Answer 19.
Answer 20.
Since, ABPC is quadrant
Answer 21.
Let the area that can be irrigated in 30 minute be A m2.
Water flowing in canal in 30 minutes =\(\left( 10,000x\frac { 1 }{ 2 } \right) m\)=5,000 m
Volume of water flowing out in 30 minutes = (5,000x6x1.5)m3 = 45,000 m3
Volume of water required to irrigate the field = \(Ax\frac { 8 }{ 100 } { m }^{ 3 }\)
Since, water flowing out in 30 minutes is used to irrigate the field of area A m2
∴ Ax \(\frac { 8 }{ 100 } \) = 45,000 ⇒ A = 5,62,500 m2.
OR
Since, a cone of maximum size is carved out from a cube
So, radius r of cone = \(\frac { 14 }{ 2 } \) = 7 cm.
And height of cone = 14 cm
Slant height of cone = \(\sqrt { { 7 }^{ 2 }+{ 14 }^{ 2 } } =\sqrt { 245 } \)
\(7\sqrt { 5 } \).
Surface area of remaining solid = 6l2 – πr2 + πrl2
= 6 x 14 x 14 – \(\frac { 22 }{ 7 } \) x7x7 + \(\frac { 22 }{ 7 } \) x7x \(7\sqrt { 5 } \)
= 1,176-154 + 154 \(\sqrt { 5 } \)
= (1,022 + 154 \(\sqrt { 5 } \) ) cm3
Answer 22.
Since, the maximum number of students have got marks in the interval 60–80, so the modal class is 60–80.
Therefore, the lower limit of modal class (1) = 60; Class size (h) = 20
The frequency of modal class f1 = 29
The frequency of the class preceding the modal class (f0) = 21,
The frequency of the class succeeding the modal class (f2) = 17
Now, using the formula, we have
So, the mode marks is 68.
Empirical relationship between the three measures of central tendency is:
3 Median = Mode + 2 Mean ⇒ 3 Median = 68 + 2 × 53 = 174
Median = 58 marks
SECTION-D
Answer 23.
Let original speed of the train be x km/h.
Answer 24.
Let the three middle most terms of the AP be a -d, a, a + d.
Answer 25.
Given: A right triangle ABC right angled at B.
We have to prove AC2 = AB2 + BC2
Construction: BD in drawn ⊥ to AC.
Answer 26.
Steps of Construction:
Answer 27.
LHS
Answer 28.
In figure TR is tower and BG is building of height 50 m.
Answer 29.
Height of Mug (h) = 14 cm
Radius of Mug (r) = \(\frac { 7 }{ 2 } \)
Answer 30.
Here, we take assumed mean a = 18 and class size (h) = 2
OR
To draw a less than type Ogive at first we make a less than type cumulative frequency table and then we draw a graph by ploting the points corresponding to the ordered pairs given by (upper limit, corresponding cumulative frequency) as given below:
Less than Cumuative Frequency Table
Distance (in m) | 10 | 20 | 30 | 40 | 50 | 60 | 70 |
No. of students | 4 | 9 | 22 | 42 | 56 | 64 | 68 |
Median distance is value of x the corresponds to cuculative freqency \(\frac { N }{ 2 } =\frac { 68 }{ 2 } =34\).
From this point 34, we draw a perpendicular to the x-axis. The point of intersection of this perpendicular with x-axis determines the median of the data. Therefore, median distance = 36 cm.
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