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 10.

## CBSE Sample Papers for Class 10 Maths Paper 10

Board | CBSE |

Class | X |

Subject | Maths |

Sample Paper Set | Paper 10 |

Time Allowed | 3 hours |

Maximum Marks | 80 |

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 10 of Solved CBSE Sample Paper for Class 10 Maths is given below with free PDF download solutions.

**General Instructions**

All questions are compulsory.

- This questions paper consists of 30 questions, distributed in four sections – A, B, C and D.
- Section A contains six questions of 1 mark each; Section B contains six questions of 2 marks each; Section C contains ten question of 3 marks each; and Section D contains eight question of 4 marks each.
- There is no overall choice. However, an internal choice has been provided in four questions of 3 marks each; and in three questions of 4 marks each. You have to attempt only one of the two alternatives given in these questions.
- Use of calculators is not permitted.

**SECTION – A**

**Question 1.**

Write an irrational number between \(\sqrt { 2 }\) and \(\sqrt { 3 }\) **(1)**

**Question 2.**

Check if x = – 1 is a zero of x^{3} + 3x^{2} + 3x + 2. **(1)**

**Question 3.**

Solve for x: x^{2} – 5x = 0 **(1)**

**Question 4.**

If cos (20° + x) = sin 30°, find the value of x: **(1)**

**Question 5.**

Volume of a sphere is 38808 cu. cm. Find its radius and hence its surface area. **(1)**

**Question 6.**

Find the perimeter of the sector of a circle of radius 9 cm with central angle 35°. **(1)**

CBSE Class 10 Previous Year Question Papers

**SECTION – B**

**Question 7.**

Represent \(\sqrt { 2 }\) + 1 on the number line. **(2)**

**Question 8.**

Write any four solutions of the equation: x – 4y = 5. **(2)**

**Question 9.**

Which term of the AP: 5, 2, -1, ……………… is 22? **(2)**

**Question 10.**

Write the sum of the first 12 terms of the

AP: 11, 16, 21, 26,……….. **(2)**

**Question 11.**

In Fig. 1, show that:

AC^{2} = AB^{2} + BC^{2} + 2BC.BD **(2) **

**Question 12.**

Draw a circle of radius 3 cm. Take a point A on the circle. At A, draw a tangent to the circle, by using the centre of the circle. **(2)**

**SECTION – C**

**Question 13.**

Show that the square of any positive integer cannot be of the form 5q + 2 or 5q + 3 for any integer q. **(3)**

**Question 14.**

If one zero of the polynomial p(x) = 3x^{2} – 8x + 2k – 1 is seven times the other zero, then find the zeros and the value of k. **(3)**

**OR**

If the zeros of the polynomial x^{3} – 3x^{2} + x + 1 are p – q, p,p + q, find p and q.

**Question 15.**

Solve the following system of equations graphically:

x – 2y = 0

3x + 4y = 20 **(3)**

**OR**

Asha is 5 years older than Sushila. Five years ago, Asha was twice as old as Sushila was then. Find their present ages.

**Question 16.**

The sides of a right-angled triangle are x – 1, x and x + 1. Find the value of x and hence, the sides of the triangle. **(3)**

**Question 17.**

Each term of an AP whose first term is ‘a’ and common difference is ‘d’ is doubled. Is the resulting sequence an AP? If so, find its first term and the common difference. **(3)**

**OR**

Find the sum of all natural numbers upto 100 which are divisible by 3.

**Question 18.**

If (6, 1), (8, 2), (9, 4) and (p, 3) are the vertices of a parallelogram taken in order, find the value of p **(3)**

**OR**

A (10, 5), B (6, -3) and C (2, 1) are the vertices of ∆ABC. L is the mid-point of AB, and M is the mid-point of AC. Write down the coordinates of L and M. Show that 2 LM = BC.

**Question 19.**

Find the area of ∆ABC shown in the figure. **(3) **

**Question 20.**

Prove that, in two concentric circles, the chord of the larger circle, which touches the smaller circle, is bisected at the point of contact. **(3)**

**Question 21.**

Prove that:

**Question 22.**

The following table gives production yield per hectare of wheat of 100 farms of a village.

Change the distribution to a “more then type” cumulative distribution and draw its ogive. **(3)**

**Question 23.**

A group consisting of 12 persons, of which 3 are extremely patient, other 6 are extremely honest and the rest are extremely kind. A person from the group is selected at random. Assuming that each person is equally likely to be selected, find the probability of selecting a person who is (i) extremely patient (ii) extremely kind or honest. **(4)**

**Question 24.**

The distribution below gives the weights of 30 students of a class. Find the mean and median weights of the students. **(4) **

**Question 25.**

A car has two wipers which do not overlap. Each wiper has a blade of length 25 cm sweeping through an angle of 115°. Find the total area cleared at each sweep of the blades. **(4)**

**OR**

Find the area of the segment AYB (shaded region) shown in the figure, if theradiusof the circle is 21 cm and ∠AOB = 120°. (Use π = \(\frac { 22 }{ 7 } \))

**Question 26.**

A cistern, internally measuring 150 cm × 120 cm × 110 cm, has 1,29,600 cu cm of water in it. Porous bricks are placed in the water until the cistern is full to the brim. Each brick absorbs one-seventeenth of its own volume of water. How many bricks can be put in without overflowing the water, each brick being 22.5 cm × 7.5 cm × 6.5 cm? **(4)**

**Question 27.**

Two sides and median bisecting one of the sides of a triangle are respectively proportional to the two sides and the corresponding median of the other triangle. Prove that the triangles are similar. **(4)**

**OR**

Prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals.

**Question 28.**

Construct a ∆ABC in which AB = 6.5 cm, ∠B = 60°, and BC = 5.5 cm. Also, construct a triangle A’BC’ similar to ∆ABC, whose each side is \(\frac { 2 }{ 3 } \) rd the corresponding side of the ∆ABC. **(4)**

**Question 29.**

Evaluate:

**OR**

If A,B,C are the interior angles of a ∆ABC, show that cosec ( \(\frac { A+B }{ 2 } \) = sec \(\frac { C }{ 2 } \)

**Question 30.**

The angle of elevation of a cloud from a point 60 m above a lake is 30° and the angle of depression of its reflection in the lake is 60°. Find the height of the cloud. **(4)**

**Solutions**

**SECTION – A**

**Solution 1.**

The irrational number \(\frac { \sqrt { 2 } +\sqrt { 3 } }{ 2 } \) lies between \(\sqrt { 2 }\) and \(\sqrt { 3 }\) **(1)**

**Solution 2.**

Let p(x) = x^{3} + 3x^{2} + 3x + 2

Then p(-1) = (-1)^{3} + 3(-1)^{2} + 3(-1) + 2

= -1 + 3 – 3 + 2

= 1 ≠ 0

Hence, x = -1 is not a zero of x^{3} + 3x^{2} + 3x + 2. **(1)**

**Solution 3.**

Here,

x^{2} – 5x = 0 (gives)

x(x – 5) = 0

⇒ x = 0 or x – 5 = 0

i.e. x = 0 or x = 5 **(1)**

**Solution 4.**

cos (20° + x) = sin 30° gives

cos (20° + x) = cos (90° – 30°) = cos 60°

⇒ 20° + x = 60°

⇒ x = 40° **(1)**

**Solution 5.**

If ‘r’ cm is the radius of the sphere, then

**Solution 6. **

**SECTION – B**

**Solution 7. **

**Solution 8.**

x – 4y = 5

⇒ x = 5 + 4y

So, its four solutions could be

x=5, y = 0; x = 9, y = 1

x = 1, y = -1; x = 13, y = 2 **(2)**

**Solution 9.**

Let an be -22. Then

**Solution 10. **

**Solution 11. **

**Solution 12. **

**SECTION – C**

**Solution 13.**

Let x be a positive integer and m = 5. Then, by Euclid’s Division Lemma, we have

x = 5m + r, where m ≥ 0 and ≤r < 5.

⇒ x = 5m, 5m + 1, 5m + 2, 5m + 3, 5m + 4

If x = 5m + 1, then x^{2} = 25m^{2} = 5(5m^{2}) = 5q, for some integer q

If x = 5m + 1, then x^{2} = 25m^{2} + 10m + 5 = 5 (5m^{2} + 2m) + 4

= 5q + 1, for some integer q

If x = 5m + 2, then x^{2} – 25m^{2} + 20m + 4 = 5(5m^{2} + 4m) + 4

= 5q + 4, for some integer q

If x = 5m + 3, then x^{2} = 25m^{2} + 30m + 9 = 5(5m^{2} + 6m + 1) + 4

= Sq + 4, for some integer q

If x = 5m + 4, then x^{2} = 25m^{2} + 40m + 16 = 5(5m^{2} + 8m + 3) + 1

= 5q + 1, for some integer q

Thus, the square of any positive integer is of the form 5q, 5q + 1 or 5q + 4, and it is not of the form 5q + 2 or 5q + 3, for some integer q. **(3)**

**Solution 14. **

**Solution 15.**

Here, are tables of values for the given equations: x – 2y = 0 and 3x + 4y = 20

**Solution 16. **

**Solution 17.**

The given AP is

a, a + d, a + 2d, ……………………

On doubling each term, we have the following sequence:

2a, 2a + 2d, 2a + 4d, …………………… (i)

Since

t_{2} – t_{1} = (2a + 2d) – 2a = 2d

t_{3} – t_{2} = (2a + 4d) – (2a + 2d) = 2d

The sequence (i) is an AP, whose first term is ‘2d’ and the common difference is 2d **(3)**

**OR **

**Solution 18. **

**Solution 19. **

**Solution 20**.

Let AB be the chord of the larger circle C_{2} which touches the smaller circle C_{1} at the point, say P.

We need to show that AP = BP.

Join OP. Then, AB is a tangent to C_{1} at P and OP is its radius

⇒ OP ⊥ AB

Now, AB is a chord of the circle C_{2} and OP ⊥ AB.

Therefore, OP is the bisector of the chord AB, as the perpendicular from the centre bisects the chord. **(3)**

i.e AP = BP

**Solution 21. **

**Solution 22.**

The ‘more than type’ cumulative distribution, corresponding to the given distribution is:

Now we draw the ogive by plotting the points (50, 100) (55, 98), (60, 90), (70, 54) and (75, 16) **(3) **

**SECTION – D**

**Solution 23.**

i. P(anextremelypatient) = \(\frac { 3 }{ 12 } \) = \(\frac { 1 }{ 4 } \)

ii. P(an extremely kind or honest) = \(\frac { 6+3 }{ 12 } \) = \(\frac { 9 }{ 12 } \) or \(\frac { 3 }{ 4 } \) **(4)**

**Solution 24.**

For Mean

Let the assumed Mean be 57.5

**Solution 25.**

Here, one blade of a wiper sweeps a sector area of a circle of radius 25 cm, th sector angle being equal to 115°.

**Solution 26.**

Here,

Volume of cistern = 150 × 120 × 110 cu-cm = 1980000 cu-cm

Volume of brick = 22.5 × 7.5 × 6.5 cu-cm

= 1096.875 cu.cm

Amount of water present in the cistern = 1,29,600 cu-cm

Let ‘n’ bricks can be put in the vessel, without overflowing the water, then

**Solution 27.**

Let ∆ABC and ∆DEF be two triangles in which AP and DQ are the medians such that

**Solution 28. **

**Solution 29. **

**Solution 30.**

Let AB represent the lake, C the cloud and its image is represented by D. **(3) **

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