CBSE BOARD EXAM PAPER ANSWER - 2016 Class 10 - Mathematics Set – 1, Code - 30/1

 

 

 

Max. Marks: 90                                                                                  Duration: 3 hrs.

 

 

General Instructions:

 

a) All questions are compulsory

b) The question paper consists of 30 questions divided into four sections A, B, C & D.

c) Section A comprises of 4 questions of 1 mark each.

d) Section B comprises of 6 questions of 2 marks each.

e) Section C comprises of 10 questions of 3 marks each.

f) Section D comprises 11 questions of 4 marks each.

g) Use of calculators is not permitted.

 

SECTION - A

 

Question 1:

In the figure, PQ is a tangent at a point C to a circle with centre O. If AB is a diameter and

∠CAB = 30⁰, find ∠PCA.

 

Question 2:

For what value of k, will k + 9, 2k – 1 and 2k + 7 are the consecutive terms of an A.P.?

 

Question 3:

A ladder leaning against a wall makes an angle of 60⁰ with the horizontal. If the foot of the ladder is 2.5 m away from the wall, find the length of the ladder.

 

Question 4:

A card is drawn at random from a well shuffled pack of 52 playing cards. Find the probability of getting neither a red card nor a queen.

 

SECTION - B

 

Question 5:

If -5 is a root of the quadratic equation 2x² + px – 15 = 0 and quadratic equation

p(x² + x) + k = 0 has equal roots, find the value of k.

 

Question 6:

Let P and Q be the points on trisection of the line segment joining the points A(2, -2) and

B(-7, 4) such that P is nearer to A. Find the coordinates of P and Q.

 

Question 7:

In the figure, a quadrilateral ABCD is drawn to circumscribe a circle with centre O in such a way that the sides AB, BC, CD and DA touch the circle at the points P, Q, R and S respectively. Prove that AB + CD = BC + DA.

 

Question 8:

Prove that the points (3, 0), (6, 4) and (-1, 3) are the vertices of a right angled isosceles triangle.

 

Question 9:

The 4^th term of an AP is zero. Prove that 25^th term of the AP is three times its 11^th term.

 

Question 10:

In figure, from an external point P, two tangents PT and PS are drawn to a circle with centre O and radius r. If OP = 2r, show that ∠OTS = ∠OST = 30⁰.

 

SECTION - C

 

Question 11:

In the figure, O is the centre of a circle such that diameter AB = 13 cm and AC = 12 cm. BC is joined. Find the area of the shaded region. (Take π = 3.14)

 

Question 12:

In the 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 Rs 500/sq. Meter. [Use π = ]

                                                                            

 

 

Question 13:

If the point P(x, y) is equidistant from the points A(a + b, b - a) and B(a – b, a + b).

Prove that bx = ay.

 

Question 14:

In the figure, find the area of the shaded region, enclosed between two concentric radii 7 cm and 14 cm, where ∠AOC = 40°.      [Use π = ]

 

Question 15:

If the ratio of the sum of the first n terms of two APs is (7n + 1) : (4n + 27), find the ratio of their m^th terms.

 

Question 16:

Solve for x:     

 

Question 17:

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.      (Use π = )

 

Question 18:

A sphere of diameter 12 cm, is dropped in a right circular cylindrical vessel, partly filled with water. If the sphere is completely submerged in water, the water level in the cylindrical vessel rises be 3 cm. Find the diameter of the cylindrical vessel.

 

Question 19:

A man standing on the deck of a ship, which is 10 m above water level, observes the angle of elevation of the top of a hill as 60⁰ and the angle of depression of the base of hill as 30⁰. Find the distance of the hill from the ship and the height of the hill.

 

Question 20:

Three different coins are tossed together. Find the probability of getting

(i) exactly two heads        (ii) at least two heads        (iii) at least two tails

 

SECTION - D

 

Question 21:

Due to heavy floods in a state, thousands were rendered homeless, 50 schools collectively offered to the state government to provide place and the canvas for 1500 tents to be fixed by the government and decided to share the whole expenditure equally. The lower part of each tent is cylindrical of base radius 2.8 m and height 3.5 m, with conical upper part of same base radius but of height 2.1 m. If the canvas used to make the tents costs Rs 120 per sq.m, find the amount shared by each school to set up the tents. What value is generated by the above problem?     (Use π = )

 

Question 22:

Prove that the lengths of the tangents drawn from an external point to a circle are equal.

 

Question 23:

Draw a circle of radius 4 cm. Draw two tangents to the circle inclined at an angle of 60⁰ to each other.

 

Question 24:

In the figure, two equal circles, with centres O and O’, touch each other at X. OO’ produced meets the circle with centre O’ at A. AC is tangent to the circle with centre O, at the point C. O’D is perpendicular to AC. Find the value of .

 

Question 25:

Solve for x:     

 

Question 26:

The angle of elevation of the top Q of a vertical tower PQ from a point X on the ground is 60⁰. From a point Y, 40 m vertically above x, the angle of elevation of the top Q of tower is 45⁰. Find the height of the tower PQ and the distance PX. (Use √3 = 1.73)

 

Question 27:

The houses in the row are numbered consecutively from 1 to 49. Show that there exists a value X such that sum of numbers of houses proceeding the house numbered X is equal to sum of numbers of houses following X.

 

Question 28:

In the figure, the vertices of ΔABC are A(4, 6), B(1, 5) and C(7, 2). A line-segment DE is drawn to intersect the sides AB and AC at D and E, respectively, such that . Calculate the area of ΔADE and compare it with area of ΔABC.

 

Question 29:

A number x is selected at random from the numbers 1, 2, 3 and 4. Another number y is selected at random from the numbers 1, 4, 9 and 16. Find the probability that product of x and y is less than 16.

 

Question 30:

In the figure, is shown a sector OAP of a circle with centre O, containing ∠θ. AB is perpendicular the radius OA and meets OP produced at B. Prove that the perimeter of shaded region is r [tan θ + sec θ +  - 1].

 

Question 31:

A motor boat whose speed is 24 km/h in still water takes 1 hour more to go 32 km/h upstream than to return downstream to the same spot. Find the speed of the stream.

 

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