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 6. According to new CBSE Exam Pattern, MCQ Questions for Class 10 Maths Carries 20 Marks.
CBSE Sample Papers for Class 10 Maths Paper 6
| Board | CBSE |
| Class | X |
| Subject | Maths |
| Sample Paper Set | Paper 6 |
| 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 6 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 the HCF of the smallest composite number and the smallest prime number.
Question 2.
For what value of k, are the roots of the quadratic equation 3x2 + 2kx + 27 real and equal?
Question 3.
Find the next term of the AP \(\sqrt { 2 } \), \(\sqrt { 8 } \), \(\sqrt { 18 } \),…
Question 4.
Given that tan θ = \(\frac { 1 }{ \sqrt { 3 } } \), what is the value of \(\frac { { cosec }^{ 2 }\theta -{ sec }^{ 2 }\theta }{ { cosec }^{ 2 }\theta +{ sec }^{ 2 }\theta } \)?
Question 5.
The perimeter of two similar triangles ABC and LMN are 60 cm and 48 cm respectively. If LM = 8 cm, then what is
the length of AB?
Question 6.
In the figure, if A (-1, 3), B (1, -1) and C (5, 1) are the vertices of a triangle ABC, what is the length of the median through vertex A?
SECTION-B
Question 7.
For what value of p, are the points (- 3, 9), (2, p) and (4, – 5) collinear?
Question 8.
A bag contains 5 red, 4 blue and 3 green balls. A ball is taken out of the bag at random. Find the probability that the selected ball is (i) of red colour (ii) not of green colour.
Question 9.
A card is drawn at random from a well-shuffled deck of playing cards. Find the probability of drawing a (i) face card (ii) card which is neither a king nor a red card.
Question 10.
Find the sum of \(0.\overline { 68 } +0.\overline { 73 } \).
Question 11.
On comparing the ratios \(\frac { { a }_{ 1 } }{ { a }_{ 2 } } ,\frac { { b }_{ 1 } }{ { b }_{ 2 } } \) and \(\frac { { c }_{ 1 } }{ { c }_{ 2 } } \), find out whether the lines representing the following pair of linear equations intersect at a point, are parallel or coincident:
5x – 4y + 8 = 0 and 7x + 6y – 9 = 0
Question 12.
If the ratio of sum of the first m and n terms of an AP is m2 : n2, show that the ratio of its mth and nth terms is (2m – 1): (2n – 1).
SECTION-C
Question 13.
Show that \(2+\sqrt { 3 } \) is an irrational number.
OR
Show that only one of the numbers n,n + 2 and n + 4 is divisible by 3.
Question 14.
Find all the zeros of 2x4 – 9x3 + 5x2 + 3x -1, if two of its zeros are \(2+\sqrt { 3 } \) and \(2-\sqrt { 3 } \).
Question 15.
Solve the following system of equations for x and y :
\(\frac { 5 }{ x-1 } +\frac { 1 }{ y-2 } =2\quad and\quad \frac { 6 }{ x-1 } =\frac { y }{ y-2 } =1\)
OR
For what values of a and b does the following pair of linear equations have an infinite number of solutions?
2x+3y=7
a(x+y)-b(x-y)=3a+b-2
Question 16.
Prove that: \(\frac { tan\theta }{ 1-cot\theta } +\frac { cot\theta }{ 1-tan\theta } \quad =1+sec\theta cosec\theta \)
Question 17.
Show that the points (3, 2), (0, 5), (- 3, 2) and (0,-1) are the vertices of a square.
OR
Find the area of the quadrilateral whose vertices are A (0, 0), B (6, 0), C (4, 3) and D (0, 3).
Question 18.
Prove that the angle between the two tangents to a circle drawn from an external point, is supplementary to the angle subtended by the line segment joining the points of contact at the centre.
Question 19.
In the figure, ABCD is a quadrant of a circle of radius 14cm and a semicircle BED is drawn with BD as diameter. Find the area of the shaded region.
OR
Find the area of the shaded region in figure, if PR = 24 cm, PQ = 7 cm and O is the centre of the circle.
Question 20.
If the diagonals of a quadrilateral divide each other proportionally, prove that it is a trapezium.
Question 21.
A conical vessel, with base radius 5 cm and height 24 cm, is full of water. This water is emptied into a cylindrical vessel of base radius 10 cm. Find the height to which the water will rise in the cylindrical vessel. \(\left( Use\quad \Pi =\frac { 22 }{ 7 } \right) \)
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.
An aeroplane left 30 minutes later than its scheduled time; and in order to reach its destination 1500 km away in time, it has to increase its speed by 250 km/hour from its usual speed. Determine its usual speed.
What value is reflected in this question?
Question 24.
A person standing on the bank of a river observes that the angle of elevation of the top of a tower standing on the opposite bank is 60°. When he moves 40 m away from the bank, he finds the angle of elevation to be 30°. Find the height of the tower and the width of the river. \(\left( Use\quad \sqrt { 3 } ={ 1.732 } \right) \)
Question 25.
Prove that the ratio of the areas of two similar triangles is equal to the ratio of squares of their corresponding sides.
Using the above result, do the following:
If D is a point on the side AB of ∆ABC such that AD : DB = 3:2 and E is a point on BC such that DE || AC; find the ratio of the areas of ∆ABC and ∆BDE.
OR
Prove that the lengths of the two tangents drawn from an external point to a circle are equal.
Using the above, do the following:
Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which is tangent to the smaller circle.
Question 26.
A wooden article was made by scooping out a hemisphere from each end of a solid cylinder. If the height of the cylinder is 20 cm and radius of the base is 3.5 cm, find the total surface area of the article.
OR
A bucket of height 16 cm is made up of metal sheet in the form of frustum of a right circular cone with radii of its lower and upper ends as 8 cm and 20 cm respectively. Find the volume of milk which can be filled in the bucket. Also find the cost of making the bucket when the metal sheet costs ₹ 15 per 100 cm2. (Use π = 3.14)
Question 27.
Find the mean, mode and median for the following data :
| Class | Frequency |
| 0-10 | 8 |
| 10-20 | 16 |
| 20-30 | 36 |
| 30-40 | 34 |
| 40-50 | 6 |
| Total | 100 |
Question 28.
The first and the last term of an AP are 4 and 81 respectively. If the common difference is 7, how many terms are there in the AP and what is their sum ?
Question 29.
Construct a ∆ABC in which BC = 5 cm, CA = 6 cm and AB = 7 cm. Construct a ∆A’BC’ similar to ∆ABC, each of the whose sides are times the corresponding sides of ∆ABC.
Question 30.
If 3 cot A = 4, check whether \(\frac { 1-{ tan }^{ 2 }A }{ 1+{ tan }^{ 2 }A } ={ cos }^{ 2 }A-{ sin }^{ 2 }A\) or not.
OR
If tan θ + sin θ = m and tan θ – sin θ = n, show that (m2 – n2) = \(4\sqrt { mn } \).
Answers
SECTION-A
Answer 1.
Smallest composite number = 4
Smallest prime number = 2
Hence, HCF of 4 and 2 is 2.
Answer 2.
3x2 + 2kx + 27 = 0
For real and equal roots, we have
Answer 3.
Given AP is \(\sqrt { 2 } \), \(\sqrt { 8 } \), \(\sqrt { 18 } \),…
Answer 4.
Answer 5.
Here ∆ABC ~ ∆LMN
Answer 6.
Answer 7.
As points are collinear.
Hence, Area of triangle = 0
Answer 8.
Total number of outcomes = 5 + 4 + 3 = 12
Answer 9.
Answer 10.
Answer 11.
Answer 12.
Answer 13.
Answer 14.
Answer 15.
Answer 16.
Answer 17.
Answer 18.
Answer 19.
Answer 20.
Answer 21.
Volume of water in conical vessel \(=\frac { 1 }{ 3 } \times \frac { 22 }{ 7 } \times 25\times 24{ cm }^{ 3 }\)
Let h be the height to which the water will rise in the cylindrical vessel
Volume of conical vessel = Volume of cylindrical vessel
\(\therefore \quad \quad \quad \frac { 1 }{ 3 } \times \frac { 22 }{ 7 } \times 25\times 24=\frac { 22 }{ 7 } \times 10\times 10\times h\) => h=2cm
Answer 22.
Calculation of median
SECTION-D
Answer 23.
Let the usual speed of the aeroplane be x km/h.
Time taken to cover 1500 km with usual speed = \(\frac { 1500 }{ x } \) hours
Time taken to cover 1500 km with the speed of (x + 250) km/h =\(\frac { 1500 }{ x+250 } \) hours
According to given condition,
Value: Importance of time.
Answer 24.
Let AB be the tower of height h.
Let C is the point lying on the opposite bank and person moves 40m away from C to D.
Answer 25.
Answer 26.
Answer 27.
Answer 28.
Answer 29.
Steps of Construction:
Step I: Draw BC = 5 cm.
Step II: With B as centre and radius BA = 7 cm, draw an arc.
Step III: With C as centre and radius CA = 6 cm, draw another arc, intersecting the arc drawn in step 2 at A.
Step IV: Join AB and AC to obtain ∆ABC.
Step V: Below BC, make an acute angle ∠CBX.
Step VI: Mark seven points as B1 B2, B3, B4, B5, B6 and B7 on BX, such that BB1 = B1B2 and so on.
Step VII: Join B5 to C and draw line through B7 parallel to B5C intersecting the extended line segment BC at C.
Draw a line through C’ parallel to CA intersecting the extended line segment BA at A’.
Thus, ∆A’BC’ is the required triangle.
Answer 30.
Let us consider a right triangle ABC in which ∠B = 90°
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