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 |

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.

**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.

a and b are two positive integers such that the least prime factor of a is 3 and the least prime factor of his 5. Then calculate the least prime factor of (a + b). **[2014]**

Question 2.

Find the centroid of the triangle whose vertices are (-1,1), (-3,4) and (8,-11)

Question 3.

If sin A= [latex]\frac { 1 }{ 2 } [/latex] (0° < A < 90°) then find the value of cos^{3} A – 3 cos A [2014,2015]

Question 4.

Write the nature of roots of quadratic equation 4x^{2} + 4 √3x + 3 = 0

Question 5.

In ∆ABC, if X and Y are points on AB and AC respectively such that [latex]\frac { AX }{ XB } [/latex] = [latex]\frac { 3 }{ 4 } [/latex], AY = 5cm and YC = 9cm, then state whether XYand BC parallel or not.** [2015,2016]**

Question 6.

If a_{n} = [latex]\frac { n(n-3) }{ n+4 } [/latex] then find 18^{th} term of this sequence.

**Section – B**

Question 7.

Express the number [latex]0.\overline { 3178 } [/latex] in the form of rational number [latex]\frac { a }{ b } [/latex].

Question 8.

A factory had 120 workers in January and 90 of them were female workers. In February, another 15 male workers are added. A worker is then picked at random. Calculate the probability of picking a female worker.

Question 9.

The sum of the digits of a two digit number is 8. The number obtained by reversing the digits exceeds the original number by 18. Find the given number.

Question 10.

If P (x,y) is any point on the line joining the points A (a, 0) andB (0, b), then show that [latex]\frac { x }{ a } [/latex] + [latex]\frac { y }{ b } [/latex] = 1

Question 11.

If p,q, r are in A.P. then find the value of p^{3} + r^{3} – 8q^{3}.

Question 12.

Ajar contains only green, white and yellow marbles. The probability of selecting a green marble and white marble randomly from a jar is 1/4 and 1/3 respectively. If this jar contains 10 yellow marbles, what is the total number of marbles in the jar ?

**Section – C**

Question 13.

Prove that 3 + 2 √5 is irrational.

Question 14.

On dividing x^{3 }– 3x^{2} + x + 2 by a polynomial g(x), the quotient and remainder were x – 2 and -2x + 4 respectively. Find g(x).

Question 15.

Question 16.

Question 17.

From given fig. express ‘x’ in terms of a, b, c.

In figure, two line segments AC and BD intersect each other at the point P such that PA = 6 cm, PB = 3 cm, PC = 2.5 cm, PD = 5 cm, ∠APB = 50° and ∠CDP = 30° Then, find the value of ∠PBA

Question 18.

In fig. A circle touches the side BC of ∆ABC at P and touches AB and AC produced at Q and R respectively. If AQ= 5 cm, find the perimeter of ∆ABC.

Question 19.

Solve the equations :

Question 20.

Find the area of a triangle with vertices (a, b + c), (b, c + a) and (c, a + b).

Question 21.

In the given figure, OACB is a quadrant of a circle with centre O and radius 3.5 an. If OD=2an, find the area of the shaded region. **[2017]
**

Question 22.

In figure, a tent is in the shape of a cylinder surmounted by a conical top of same diameter. If the height and diameter of cylindrical part are 2.1 m and 3 m respectively and the slant height of conical part is 2.8 m, find the cost of canvas needed to make the tent if the canvas is available at the rate of ₹ 500/sq. metre. ( Use π = [latex]\frac { 22 }{ 7 } [/latex] )** [2016]
**

**OR**

Prove that the largest possible sphere is carved out from a wooden solid cube of side 7 cm. Find the volume of the wood left. ( Use π = [latex]\frac { 22 }{ 7 } [/latex] )

**[2014]**

**Section – D**

Question 23.

In a triangle, if the square of one side is equal to the sum of the squares of the other two sides, then prove that the angle opposite to the first side is a right angle.

**OR**

Prove that if a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.

Question 24.

Aboy onhorizontal plane finds bird flying at a distance of 100 m from him at an elevation of30°. Agirl standing on the roofof20 metre high building, finds the angle of elevation ofthe same bird to be 45°. Both the boy and the girl are on opposite sides of the bird. Find the distance of bird from the girl.

**OR**

A 1.2 m tall girl spots a balloon moving with the wind in a horizontal line at a height of 88.2 m from the ground. The angle of elevation of the balloon from the eyes of the girl at= any instant is 60°. After sometime, the angle of elevation reduces to 30°. Find the distance travelled by the balloon during the interval.

Question 25.

If the p^{th} term of an A.P. is ( [latex]\frac { 1 }{ q } [/latex] ) and its q^{th} term is ( [latex]\frac { 1 }{ p } [/latex] ). show that the sume of its first pq term is [latex]\frac { 1 }{ 2 } [/latex] (pq + 1).

Question 26.

Roots of the quadratic equation 36x^{2} – 12ax + (a^{2} – b^{2}) = 0 are [latex]\frac { a+b }{ c } [/latex] and [latex]\frac { a-b }{ c } [/latex] then, find the value of c.

**OR**

Find the real roots of the equation x^{2/3} + x^{1/3} – 2 = 0

Question 27.

A farmer connects a pipe ofintemal diameter 20 cm from a canal into a cylindrical tank which is 10 m in

diameter and 2 m deep. If the water flows through the pipe at the rate of 4 km per hour, in how much time will the tank be filled completely? **[2014]**

Question 28.

The following table shows marks secures by 140 students in an examination :

Marks |
0-10 | 10-20 | 20-30 | 30-40 | 40-50 |

No. of Student |
20 | 24 | 40 | 36 | 20 |

Calculation of mean by Step-deviation method.

Question 29.

Question 30.

Construct a tangent to a circle of radius 4 cm from a point on the concentric circle of radius 6 cm and measure its length.

**Solutions**

Solution 1.

Least prime factor of (a + b) is 2. Because a + b= 8 and least prime factor of 8 is 2. Since,

Solution 2.

Solution 3.

Solution 4.

∵ 4x^{2} + 4√3 x + 3 = 0

So,determinant = b2 – 4ac = (4√3)^{2} -4 × 4 × 3 = 0 Hence,

Hence, roots are real and equal

Solution 5.

Solution 6.

Solution 7.

Let x = 0.31783178 …(i)

Multiply by 10000

10000 x = 3178.31783178 …..(ii)

Subtracting (i) from (ii)

10000x = 3178.3178 …..

x = 0.3178 ….

Solution 8.

Initial number of workers = 120

When 15 male workers are added, then the total number of workers = 120 + 15 = 135

Number of female workers = 90

Probability of female workers = [latex]\frac { 90 }{ 135 } [/latex] = [latex]\frac { 2 }{ 3 } [/latex]

Solution 9.

Let the unit digits be x

∴ Tens’ digits = 8 – x

∴ Required number = 10(8 – x) + x = 80 – 9x

Reverse number = 10x + (8 – x) = 9x + 8

∴ (9x + 8) – (80 – 9x) = 18

⇒ x = 5

∴ Tens digit = 8 – 5 = 3

∴ Required number = 35

Solution 10.

As the point P (x, y) lies on the line joining the points A (a, 0) and B (0, b), the points A, B and P are collinear

⇒ a(b – y) + 0 (y – 0) + x (0 – b) = 0

⇒ ab – ay – bx = 0 ⇒ bx + ay = ab

⇒ [latex]\frac { x }{ a } [/latex] + [latex]\frac { y }{ b } [/latex] = 1

Solution 11.

∵ 2q = p + r

∴ p + r – 2q = 0

so, p^{3} + r^{3} – 8q^{3} = 3 × p × r × (-2q) = -6 pqr

Solution 12.

Let the no. of green marbles = x

Let the no. of white marbles = y

∴ Total no. ofmarbles = x + y + 10

Solution 13.

Solution 14.

We know that, if p(x) and g(x) are any two polynomials with g(x) ≠ 0, then we can find polynomials q (x) and r (x) such that

Solution 15.

Solution 16.

Solution 17.

Solution 18.

To Find : Perimeter of ∆ABC

Let AQ = 5 cm

and

AQ = AR ….. (i)

BQ = BP ….. (ii)

CP = CR ….. (iii)

(Tangent drawn from an external points are equal)

∴ Perimeter of ∆ABC = AB + BC + CA

= AB + BP + PC + CA [∵ BC = BP + PC]

= (AB + BQ) + (CR + CA) from (ii) and (iii)

= AQ + AR [∵ AQ = AR from (i)]

=AQ+AQ = 2 AQ =2 × 5 = 10cm

∴ Perimeter of ∆ABC = 10 cm.

Solution 19.

Solution 20.

Solution 21.

Solution 22.

Canvas needed to make the tent = C.S.A of the conical part + C.S.A ofthe cylindrical part Given that

Radius of the conical part = Radius of the cylindrical part = [latex]\frac { 3 }{ 2 } [/latex] m

Slant height ofthe conical part = l = 2.8 m

Height of the cylindrical part = h = 2.1 m

Solution 23.

Solution 24.

Let P, B and A be the positions of the bird, boy and girl respectively.

Given PB = 100m, AC = 20m, Let PM = xm.

In right-angled ∆PBO, we have,

**OR**

Suppose P be the position of the balloon if its angle of elevation from the eyes of the girl is 60° and Q be the position if angle of elevation is 30°.

Solution 25.

Solution 26.

Solution 27.

Internal diameter of pipe = 20 cm

Internal radius of pipe = 10 cm

Radius of cylindrical tank = 5 m = 500 cm

Height of cyclindrical tank = 200 cm

Water flows in 1 hour = 400000 cm

Let we assume the height of pipe = 400000 cm

Now, volume of tank = πr^{2} h = π (500)^{2} (200) = 50000000 π cm^{3}

Volume of water flow in hour = π (20)^{2} (400000) = 160000000 π

Time taken = [latex]\frac { 5 }{ 16 } [/latex] hour = [latex]\frac { 5 }{ 16 } [/latex] × 60 min = [latex]\frac { 300 }{ 16 } [/latex] min = [latex]\frac { 75 }{ 4 } [/latex] × 60 sec = 1125 sec or 18.75 min

Solution 28.

Solution 29.

Solution 30.

**Steps of construction :**

- Draw two circles with common centre O and of radii 4cm and 6cm.
- Take any point P on the outer circle.
- Joint the point P to the centre O.
- Draw perpendicular bisector of PO, which intersects PO at point Q.
- With centre Q and radius PQ or QO, draw a circle, which intersects the inner circle at points R and S.
- Draw rays PR and PS.

Thus, PR and PS are the required tangents.

By measurement PR = 4.5 cm = PS.

**Verification by Actual Calculation :
**

Join O to RandS.

Since, OR and OS are the radius through the point of contact of the tangent to the circle.

∴ OR ⊥PR and OS ⊥ PS

PR= √OP^{2} – OR^{2}

= √36 – 16 = √20

= 4.5 cm (Approximately)

Since, PR and PS are two tangents from an exterior point to the same circle

∴ PR = PS = 4.5 cm (Approximately)

Hence verified.

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