Mathematics Sample paper, Part- B for class 10th

 Mathematics Sample paper

Part-B

19. Case study III

Elevation of Helicopter

From a point 100 m above a lake,the angle of elevation of a stationary helicopter is 30° and the angle of depression of reflection of the helicopter in the lake is 60°.

a) If any point above the lake is in height h and the reflection of point in the lake is of depth

i. h

ii. h/2

iii. 2h

iv. 3h

b) Find the height of the helicopter.

i. 200m

ii. 150m

iii. 145 m

iv. 160 m

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c) Find the distance between the helicopter and the given position of point.

i. 150m

ii. 250 m

iii. 200 m

iv. 300 m

d) If the observer moves away the perpendicular line (tower/building), the angle of elevation

i. Decreases

ii. Increases

iii. Increases or Decreases

iv. None of the above

e) The angle of elevation of a point P as seen from a point O is always equal to the angle of O as seen from P.

i. Depression

ii. Elevation

iii. Don't say anything

iv. None of the above

20. Case Study IV

Tossing of Pair of Coins

Neha, Lata and Sonia were fighting to get first chance in a game. Neha says, " Let us toss two coins. If both heads appears,Lata will take first chance. If both tails appear, Sonia will get it and is one head and one tail appears, I will get the chance."

a) What is the probability of Neha getting the first chance?

i. 1/4

ii. 1/2

iii. 1/3

iv. 2/3

b) What is the probability of Lata getting the first chance?

i. 1/4

ii. 1/3

iii. 1/2

iv. 2/3

c) What is the probability of Sonia getting the first chance?

i. 1/4

ii. 1/3

iii. 1/2

iv. 3/4

d) Is her decision fair?

i. No

ii. Yes

iii. Don't say anything

iv. None of the above

e) The probability of impossible event is

i. 0

ii. 1

iii. 1/2

iv. 1/4

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Part- B

Directions (Q.nos. 21-26 are of 2 marks each)

21. Prove that the points (-2,-1), (1,0), (4,3) and (1,2) are the vertices of a parallelogram. Is it a rectangle?

22. If tan∅=1/√3, evaluate cosec²∅ - sec²∅/ cosec²∅ + sec²∅

Or

Prove that

sin∅/1-cos∅= cosec∅ + cot∅

23. If a, b and c are the sides of a right angled triangle, where c is hypotenuse, then prove that the radius r of the circle which touches the sides of the triangle is given by

r= a+b-c/2

24. Calculate the average age of workers in a factory whose ages are as follows:

Or

Given below is a cumulative frequency distribution showing the marks secured by 50 students of a class.

Form the frequency distribution table for is the above data.

25. If -4 is a root of the equation x²+px-4 =0 and the equation x²+px+q =0 has equal rootz, then find the values of p and q.

26. Prove that √5-√7 is irrational.

Directions ( Q.nos. 27-33 are of 3 marks each)

27. A cylinder whose height is equal to it's diameter has the same volume as a sphere of radius 4 cm. Calculate the cube of radius to the base of the cylinder correct to one decimal place.

Or

A hollow sphere of internal and external radii 6cm and 8cm, respectively, is melted and recast into small cones of base radius 2cm and height 8cm. Find the number of cones.

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28. Prove that the points (-4,-1), (-2,-4), (4,0) and (2,3) are the vertices of a rectangle. Also,find the area of the rectangle.

Or

Find the equation of the perpendicular bisector of AB, where A and B the points (3,6) and (-3,4) respectively. Also,find it's point of intersection with X- axis.

29. In the given figure, PA, QB and RC are each perpendicular to AC and AP=x, QB=z,RC=y. Prove that 1/x + 1/y= 1/z.

30. If the zeroes of the polynomial f(x)= x³- 3x² +x+1 are a-b, a, a+b, find the value of a²+b².

31. The following frequency distribution gives the monthly consumption of electricity of 68 consumers of a locality. Find the median of the data.

32. Draw a circle of radius 3cm. Take two points P and Q on one of it's extended diameter each at a distance of 7cm from it's centre. Draw tangents to the circle from these two points P and Q.

33. In the given figure, angle P= ∅ and R=ø.

Find

i. √x+√1 cot of

ii. √x³ + √x² tan∅hi

Directions ( Q.nos. 34-36 are of 5 marks each)

34. If the equation 

(1+m²)c + (2mc)x + (c² - a²) = 0 has equal roots, then prove that c²=a² (1 + m²).

Or

Find the solution of the equation

1/1+x + 2/x+2 =4/x+4, x≠1, -2, -4.

35. A railway half ticket cost half the full fare but the reservation charges are the same on a half ticket as on a full ticket. One reserved first class ticket from the stations A to B costs Rs 2530. Also, one reserved first class ticket and one reserved first class half ticket from stations A to B costs Rs 3810. Find the full first class fare from stations A to B and also the reservation charges for a ticket.

36. From each corner of a square of side 4cm a quadrant of a circle of radius 1 cm is cut and also a circle of diameter 2 cm is cut as shown in figure. Find the area of remaining portion of the square.

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Posted By:- Anuranjan Gadekar

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