(i) All questions are compulsory.

(ii) The question paper consists of 34 questions divided into four sections A, B, C and D.

(iii) Sections A contains 8 questions of one mark each, which are multiple choice type questions, section B contains 6 questions of two marks each, section C contains 10 questions of three marks each, and section D contains 10 questions of four marks each.

(iv) Use of calculators is not permitted.

Q1 :

If the height of a vertical pole is

(A) 30°

(B) 60°

(C) 45°

(D) 75°

**Answer :**

Let AO be the pole and OB be its shadow.

Let the length of the shadow be *x*.

Let* θ* be the angle of elevation of the
Sun at that time.

Given:

Height of pole (*h*) =

We have:

Thus, the angle of elevation of the Sun is 60°.

Hence, the correct option is B.

Q2 :

A bag contains cards numbered from 1 to 25. A card is drawn at
random from the bag. The probability that the number on this card
is divisible by both 2 and 3 is

(A)

(B)

(C)

(D)

**Answer :**

Total number of possible outcomes = 25

Let *E* be the event that the number on the card
drawn is divisible by both 2 and 3.

∴ *E* = {6, 12, 18, 24}

∴ Number of outcomes favourable to *E* =
4

Hence, the correct option is C.

Q3 :

Two different coins are tossed simultaneously. The probability of
getting at least one head is

(A)

(B)

(C)

(D)

**Answer :**

When two coins are tossed simultaneously, the possible outcomes are
{(H, H), (H, T), (T, H), (T, T)}.

∴ Total number of outcomes = 4

Let *E* be the event of getting at least one
head.

Then, the outcomes favourable to E are {(H, H), (H, T), (T,
H)}.

∴ Number of favourable outcomes = 3

P(*E*) =

Hence, the correct option is C.

Q4 :

Two concentric circles are of radii 5 cm and 3 cm. Length of the
chord of the larger circle, (in cm), which touches the smaller
circle is

(A) 4

(B) 5

(C) 8

(D) 10

**Answer :**

Q5 :

In Figure 1, a quadrilateral ABCD is drawn to circumscribe a
circle such that its sides AB, BC, CD and AD touch the circle at
P, Q, R and S respectively. If AB = *x* cm, BC =
7 cm, CR = 3 cm and AS = 5 cm, find *x*.

(A) 10

(B) 9

(C) 8

(D) 7

**Answer :**

Q6 :

The perimeter of a triangle with vertices (0, 4), (0, 0) and (3,
0) is

(A)

(B) 5

(C) 10

(D) 12

**Answer :**

Q7 :

A rectangular sheet of paper 40 cm × 22 cm, is rolled to
form a hollow cylinder of height 40 cm. The radius of the
cylinder (in cm) is

(A) 3·5

(B) 7

(C)

(D) 5

**Answer :**

Q8 :

The next term of the A.P.

(A)

(B)

(C)

(D)

**Answer :**

Q9 :

In Figure 2, XP and XQ are two tangents to the circle with centre
O, drawn from an external point X. ARB is another tangent,
touching the circle at R. Prove that XA + AR = XB + BR.

**Answer :**

Q10 :

Prove that the tangents drawn at the ends of any diameter of a circle are parallel.

**Answer :**

Q11 :

Two different dice are rolled simultaneously. Find the probability that the sum of numbers appearing on the two dice is 10.

**Answer :**

Q12 :

In Figure 3, OABC is a quadrant of a circle of radius 7 cm. If OD
= 4 cm, find the area of the shaded region.

**Answer :**

Q13 :

Solve for *x*:

**Answer :**

Q14 :

The sum of the first *n* terms of an A.P. is
5*n* *−* *n*^{2}. Find the *n*^{th} term of this A.P.

**Answer :**

Q15 :

Points P, Q, R and S divide the line segment joining the points A(1, 2) and B(6, 7) in 5 equal parts. Find the coordinates of the points P, Q and R.

**Answer :**

Q16 :

In Figure 4, from a rectangular region ABCD with AB = 20 cm, a
right triangle AED with AE = 9 cm and DE = 12 cm, is cut off. On
the other end, taking BC as diameter, a semicircle is added on
outside the region. Find the area of the shaded region.

**Answer :**

Q17 :

In Figure 5, ABCD is a quadrant of a circle of radius 28 cm and a
semi circle BEC is drawn with BC as diameter. Find the area of
the shaded region.

**Answer :**

Q18 :

A 5 m wide cloth is used to make a conical tent of base diameter
14 m and height 24 m. Find the cost of cloth used at the rate of
Rs 25 per metre.

**Answer :**

Q19 :

A girl empties a cylindrical bucket, full of sand, of base radius 18 cm and height 32 cm, on the floor to form a conical heap of sand. If the height of this conical heap is 24 cm, then find its slant height correct upto one place of decimal.

**Answer :**

Q20 :

The sum of the first 7 terms of an A.P. is 63 and the sum of its
next 7 terms is 161. Find the 28^{th} term of
this A.P.

**Answer :**

Q21 :

Two ships are approaching a light-house from opposite directions.
The angles of depression of the two ships from the top of the
light-house are 30° and 45°. If the distance between
the two ships is 100 m, find the height of the light-house.

**Answer :**

Q22 :

If 2 is a root of the quadratic equation 3*x*^{2} + *px* − 8 =
0 and the quadratic equation 4*x*^{2} − 2*px* + *k* =
0 has equal roots, find the value of *k*.

**Answer :**

Q23 :

Construct a triangle PQR, in which PQ = 6 cm, QR = 7 cm and PR =
8 cm. Then construct another triangle whose sides
are

**Answer :**

Q24 :

Find the value(s) of p for which the points (*p*
+ 1, 2*p* − 2), (*p*
−1, *p*) and (*p* −
3, 2*p* − 6) are collinear.

**Answer :**

Q25 :

The mid-point P of the line segment joining the points
A(−10, 4) and B(−2, 0) lies on the line segment
joining the points C(−9, −4) and D(−4,
*y*). Find the ratio in which P divides CD. Also
find the value of *y*.

**Answer :**

Q26 :

A quadrilateral is drawn to circumscribe a circle. Prove that the sums of opposite sides are equal.

**Answer :**

Q27 :

The angle of elevation of the top of a chimney from the foot of a tower is 60° and the angle of depression of the foot of the chimney from the top of the tower is 30°. If the height of the tower is 40 m, find the height of the chimney. According to pollution control norms, the minimum height of a smoke emitting chimney should be 100 m. State if the height of the above mentioned chimney meets the pollution norms. What value is discussed in this question?

**Answer :**

Q28 :

A hemispherical depression is cut out from one face of a cubical
block of side 7 cm, such that the diameter of the hemisphere is
equal to the edge of the cube. Find the surface area of the
remaining solid. [Use *π* =

**Answer :**

Q29 :

If S* _{n}* denotes the sum of
the first

**Answer :**

Q30 :

A metallic bucket, open at the top, of height 24 cm is in the
form of the frustum of a cone, the radii of whose lower and upper
circular ends are 7 cm and 14 cm respectively. Find :

(i) the volume of water which can completely fill the
bucket.

(ii) the area of the metal sheet used to make the
bucket.

[Use *π* =

**Answer :**

Q31 :

The sum of the squares of two consecutive even numbers is 340. Find the numbers.

**Answer :**

Q32 :

Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact.

**Answer :**

Q33 :

A dice is rolled twice. Find the probability that

(i) 5 will not come up either time.

(ii) 5 will come up exactly one time.

**Answer :**

Q34 :

Solve for *x*:

**Answer :**

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