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 7.

## CBSE Sample Papers for Class 10 Maths Paper 7

Board |
CBSE |

Class |
X |

Subject |
Maths |

Sample Paper Set |
Paper 7 |

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 7 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 questons. - Use of calculators is not permitted.

**SECTION-A**

**Question 1.**

The decimal expansion of the rational number will terminate after how many places of decimals?

**Question 2.**

For what value of p, 2p – 1, 7 and 3p are three consecutive terms of an AP ?

**Question 3.**

In the figure, ∠M = ∠N = 46°. Express x in terms of a, b and c where a, b and c are lengths of LM, MN and NK respectively.

**Question 4.**

If sin θ = , then find the value of (2 cot^{2} θ + 2).

**Question 5.**

Find the value of a so that the point (3, a) lies on the line represented by 2x – 3y = 5.

**Question 6.**

If a and b are the roots of the equation x^{2} + ax – b = 0, then find a and b.

**SECTION-B**

**Question 7.**

Find the value(s) of k for which the pair of linear equations kx + 3y = k – 2 and 12x + ky = k has no solution.

**Question 8.**

If S_{n}, the sum of first n terms of an AP is given by S_{n} = 3n^{2} – 4n, then find its nth term.

**Question 9.**

Explain why 3 x 5 x 7 + 7 is a composite number.

**Question 10.**

Check whether (5, – 2), (6,4) and (7, – 2) are the vertices of an isosceles triangle.

**Question 11.**

The king, queen and jack of clubs are removed from a deck of 52 playing cards and remaining cards are shuffled. A card is drawn from the remaining cards. Find the probability of getting a card of

- heart
- queen.

**Question 12.**

Two players, Sangeeta and Reshma, play a tennis match. It is known that the probability of Sangeeta’s winning the match is 0.62. What is the probability of Reshma’s winning the match?

**SECTION-C**

**Question 13.**

Prove that is an irrational number.

**Question 14.**

Solve the following pair of equations:

**Question 15.**

In the figure, ∆ABC is right angled at C and DE ⊥ AB. Prove that AABC ~ AADE and hence find the lengths of AE and DE.

** OR**

In the figure, DEFG is a square and ∠BAC = 90°. Show that DE^{2} = BD x EC.

**Question 16.**

Find the value of sin 30° geometrically.

** OR**

Without using trigonometrical tables, evaluate:

**Question 17.**

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

** OR**

The line segment joining the points A (2,1) and B (5, -8) is trisected at the point 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.

**Question 18.**

In the figure, PQ = 24 cm, PR -7 cm and O is the centre of the circle. Find the area of shaded region. (Take π = 3.14)

**Question 19.**

If the polynomial 6x^{4} + 8x^{3} + 17x^{2} + 21x + 7 is divided by another polynomial 3x^{2} + 4x + 1, the remainder comes out to be (ax + b), find a and b.

**Question 20.**

Two tangents PA and PB are drawn to a circle with centre O from an external point P. Prove that ∠APB = 2∠OAB in the given figure.

** OR**

Prove that the parallelogram circumscribing a circle is a rhombus.

**Question 21.**

A hemispherical bowl of internal diameter 36 cm contains liquid. This liquid is filled into 72 cylindrical bottles of diameter 6 cm. Find the height of the each bottle, if 10% liquid is wasted in this transfer.

**Question 22.**

The following table gives the daily income of 50 workers of a factory:

Daily income (in ?) |
100 -120 | 120 -140 | 140 -160 | 160 -180 | 180 – 200 |

Number of workers |
12 | 14 | 8 | 6 | 10 |

Find the mean of the above data.

**SECTION—D**

**Question 23.**

The sum of the squares of two consecutive odd numbers is 394. Find the numbers.

** OR**

Places A and B are 100 km apart on a highway. One car starts from A and another from B at the same time. If the cars travel in the same direction at different speeds, they meet in 5 hours. If they travel towards each other, they meet in 1 hour. What are the speeds of the two cars?

**Question 24.**

Prove that, if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.

Using the above result, do the following:

Rajeev prepared a poster on ‘National Integration’ for decoration on Independence day on a triangular sheet (say ∆ABC), where DE||BC and BD = CE. Prove that ∆ABC is an isosceles triangle. What values can be inculcated through celebration of national festivals?

**Question 25.**

A straight highway leads to the foot of a tower. A man standing at the top of the tower observes a car at an angle of depression of 30°, which is approaching the foot of the tower with a uniform speed. Six seconds later the angle of depression of the car is found to be 60°. Find the time taken by the car to reach the foot of the tower from this point.

**Question 26.**

From a solid cylinder whose height is 8 cm and radius 6 cm, a conical cavity of height 8 cm and of base radius 6 cm, is hollowed out. Find the volume of the remaining solid correct to two places of decimals. Also find the total surface area of the remaining solid. (Take π = 3.1416)

In the figure, ABC is a right triangle at A. Find the area of shaded region if AB = 6 cm, BC – 10 cm and O is the centre of the incircle of ∆ABC. (Take π = 3.14)

**Question 27.**

Prove that: .

**Question 28.**

Construct a ∆ABC in which BC = 6.5 cm, AB = 4.5 cm and ∠ABC = 60°. Construct a triangle similar to this triangle whose sides are of the corresponding sides of the triangle ABC.

**Question 29.**

The sum of n, 2n, 3n terms of an AP are S_{1} S_{2} and S_{3} respectively. Prove that S_{3} = 3(S_{2} – S_{1}).

** OR**

If S_{n} denotes the sum of the first n terms of an AP, prove that S_{30} = 3 (S_{20} – S_{10}).

**Question 30.**

Draw a ‘less than’ ogive and a ‘more than’ ogive for the following frequency distribution:

Marks obtained |
0-10 | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 | 60-70 | 70-80 | 80-90 | 90-100 |

No. of students |
2 | 5 | 8 | 12 | 15 | 20 | 18 | 17 | 16 | 2 |

Hence, find the median from the curves.

**Answers**

**SECTION-A**

**Answer 1.**

We have,

**Answer 2.**

**Answer 3.**

**Answer 4.**

We have,

**Answer 5.**

∴ The point (3, a) lies on the line 2x – 3y = 5

∴ 2 x 3 – 3 x a = 5 ⇒ 6 – 3a = 5 ⇒ 1 = 3a

⇒

**Answer 6.**

**SECTION-B**

**Answer 7.**

We have,

**Answer 8.**

We have,

**Answer 9.**

3 x 5 x7 + 7= 7 (3 x 5 +1) = 7 x 16, which has more than two factors.

**Answer 10.**

Let A (5, – 2), B (6,4) and C (7, – 2) be the vertices of a triangle.

Then we have,

**Answer 11.**

As we removed three cards king, queen and jack of clubs

So, the total number of remaining cards = 49

Total number of hearts = 13, Total number of clubs = 10

Total number of king = 3; Total number of queen = 3;

Total number of jacks = 3

Therefore,

- P(a heart) =
- P(a queen) =

**Answer 12.**

Let S and R denote the events that Sangeeta and Reshma wins the match, respectively.

The probability of Sangeeta’s winning = P(S) = 0.62

As the events R and S are complementary

∴ The probability of Reshma’s winning = P(R) = 1 – P(S)

= 1-0.62 = 0.32

**SECTION-C**

**Answer 13.**

Let us assume, to the contrary, that is a rational number.

Let,

So,

Squaring on both sides, we have

a^{2} = 5b^{2} …(i)

Therefore 5 divides a2, it follows that 5 divides a

So we can write a = 5c, (where c is any integer)

Putting the value of a = 5c in (i), we have

25c^{2} = 5b^{2} => 5c^{2} = b^{2}

It means 5 divides b2 and so 5 divides b.

So, 5 is a common factor of both a and b which contradicts that a and b are co-prime. Our assumption is wrong. Hence, is an irrational number.

**Answer 14.**

We have

**Answer 15.**

In ∆ABC and ∆ADE, we have

**Answer 16.**

Consider an equilateral triangle PQR with each side of length 2a. Each angle of an equilateral triangle PQR is of 60°.

Let PM be the perpendicular from P on QR. Since the triangle is equilateral.

Therefore, PM is the bisector of P and M is the mid-point of QR.

Therefore, QM = MR – a and ∠QPM = 30°

Now, In ∆PQM,

∠M is a right angle, hypotenuse PQ=2a and QM=a

So, by hypotenuse theorem, we have

**Answer 17.**

Let the required point on y-axis is (0, y).

Then, according to question

**Answer 18.**

We have

**Answer 19.**

**Answer 20.**

As the lengths of tangents drawn from an external point to a circle are equal

**Answer 21.**

**Answer 22.**

Here, assumed mean, a = 150 and class-size, h = 20

**SECTION-D**

**Answer 23.**

Let two consecutive odd numbers be x, x + 2.

Now, according to question

**Answer 24.**

Given: A triangle ABC in which a line parallel to side BC intersects other two sides AB and AC at D and E respectively.

**Answer 25.**

Let OA be the tower of height h and P be the initial position of the car when the angle of depression is 30°.

After 6 seconds the car reaches to Q such that the angle of depression at Q is 60°. Let the speed of the car be v metre per second. Then,

PQ = 6v (v Distance = speed x time)

and let the car takes t seconds to reach the tower OA from Q. Then OQ = vt metres

Now, in ∆AQO, we have

**Answer 26.**

We have,

Height of cylinder = height of cone = 8 cm

Radius of cylinder = radius of cone = 6 cm

Thus, volume of remaining solid

= Volume of cylinder – Volume of cone

**Answer 27.**

**Answer 28.**

**Steps of Construction:**

**Step I:** Construct AABC in which BC = 6.5 cm,

AB = 4.5 cm and A ∠BC = 60°.

**Step II:** Below BC, make an acute ∠CBX

**Step III:** Along BX, mark off four points: B_{1}, B_{2}, B_{3} and

B_{4}, such that BB_{1} = B_{1}B_{2} = B_{2}B_{3} = B_{3}B_{4}.

**Step IV: **Join B_{4}C. Draw B_{3}D || B_{4} C meeting BC at D.

**Step V:** From D, draw ED || AC, meeting BA at E. Now, we have AEBD which is the required triangle whose sides are th of the corresponding sides of ∆ABC.

**Justification:** Here, DE||CA

**Answer 29.**

Let a be the first term and d be the common difference of the AP

**Answer 30.**

Table for ‘less than’ ogive:

Median of the given data is the x-coordinate of the point of intersection of the two types of ogives.

Hence, median = 57.5 (approx.)

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