__MATHEMATICS – QUESTION PAPER__

__TERM – II__

__Last 5 Year Questions__

**Whole Syllabus**

Q.1 Find the value of x for which 2x, (x + 10) and (3x + 2) are the three consecutive terms of an AP. ( Board exam 2020 set 1)

Ans – AP is in the form of

Here a1 = 2x , a2 = (x + 10) , a3 = (3x + 2)

So x+ 10 =

2(x + 10) = 5x + 2

2x + 20 = 5x + 2

2x – 5x = 2 – 20

-3x = -18

x= 6

Q- 2 The first term of an AP is p and the common difference is q, then What will be its10^{th} term?

Ans – We know that

An = a + (n -1)d a = p, d = q and n = 10

A10 = p + (10 – 1)q

A10 = p + 9d

Q- 3 In the figure, ∆ABC is circumscribing a circle, the length of BC is ______ cm.

Ans – PB = BQ = 3 cm (From the external point length of tangents are equal)

RC = QC

AC = AR + CR

11 = 4 + CR

CR = 11 – 4

= 7 cm

So QC = 7 cm

BC = BQ + QC = 3 + 7 = 10 cm

Q-4 The ratio of the length of a vertical rod and the length of its shadow is 1 : √3. Find the angle of elevation of the sun at that moment?

Ans – Let the length of rod be AB = x and shadow of rod BC = √3x In ∆ABC where angle B = 90º

Angle C = ?

TanC = AB/BC

= x/√3x

tanC = 1/√3

tanC = tan30

∠C = 30º

Q- 5 Two cones have their heights in the ratio 1:3 and radii in the ratio 3:1. What is the ratio of their volumes?

Ans – Let the height of 1^{st} cone and 2^{nd} cone be h1 = x and h2 = 3x respectively and radii of 1^{st} cone and 2^{nd} cone be r1 = 3y and r2 = y respectively

Volume of cone1 = 1/3**π**r_{1}^{2} h_{1}

Volume of cone2 = 1/3**π**r_{2}^{2} h_{2}

3 : 1 is the ratio.

Q- 6 If the mean of the first n natural numbers is 15, find n.

Ans – mean = Sum of observation numbers/total observation

Sum of n numbers Sn = n/2( 2a + (n-1)d) a = 1 and d = 1

So Sn = n/2(2 + n -1)

Sn = n(n + 1)/2

Mean = n(n + 1)/2n

15 = (n + 1)/2

30 = n + 1

n= 29

Q- 7 Show that (a – b)² , (a² + b²) and (a + b)² are in AP.

Ans – a1 = (a – b)² = a² – 2ab + b², a2 = a² + b², a3 = (a + b)² = a² + 2ab + b² d1 = a2 – a1 = 2ab and d2 = a3 – a2 = 2ab

Here d1 = d2 = 2ab, so it is an AP

Q- 8 In a flight of 600 km, an aircraft was slowed due to bad weather. Its average speed for the trip was reduced by 200km/hr and time of flight increased by 30 minutes. Find the original duration of the flight.

Q- 9 A cone of base radius 4 cm is divided into two parts by drawing a plane through the mid point of its height and parallel to its base. Compare the volume of the two parts.

Q- 10 The sum of four consecutive numbers in AP is 32 and the ratio of the product of the first and last terms to the product of two middle terms is 7 : 15. Find the numbers.

Q- 11 Solve 1 + 4 + 7 + 10 + ……. + x = 287

Q- 12 A vertical tower stands on a horizontal plane and is surmounted by a vertical flag staff of height 5 meters. At a point on the plane, the angles of elevation of the bottom and the top of the flag staff are respectively 30° and 60°. Find the height of the tower.

Q.1 What is the nth term of the AP a, 3a, 5a, … ?

Q- 2 The angle of depressions from the observing positions O1 and O2 respectively of the object A are __________ , _____________ .

Q- 3 Find the class marks of the classes 10-25 and 35-55.

Q- 4 Show that the sum of all terms of an AP whose first term is a, the second term is b and the last term is c is equal to (a + c)(b + c – 2a)/2(b – a)

Q- 5 If a circle touches the side BC of a triangle ABC at P and extended sides AB and AC at Q and R, respectively, prove that

AQ = 1/2 of (BC + CA + AB)

Q- 6 Draw a circle of radius 2 cm with center O and take a point P outside the circle such that OP = 6.5 cm. From P, draw two tangents to the circle.

Q- 7 From the point on the ground, the angles of elevation of the bottom and the top of a tower fixed at the top of a 20 m high building are 45º and 60º respectively. Find the height of the tower.

Q- 8 What is the common difference of an AP, whose nth term is an = (3n + 7) ?

Q- 9 Find the value of p for which (2p + 1), 10 and (5p + 5) are three consecutive terms of an AP ?

Q- 10 Find the number of terms of an AP 5,9,13,…. 185.

Q- 11 In the figure PA and PB are tangents to the circle with center O such that <APB = 50º, then find the measure of <OAB.

Q- 12 In figure, PQ is a chord of a circle and PT is tangent at P such that <QPT = 60º, then find the measure of <PRQ ?

Q- 13 Find the 11^{th} term from the last term(towards the first term) of the AP 12,8,4,…,-84.

Q- 14 In figure AB is a chord of circle with center O, AOC is diameter and AT is tangent at A. Prove that <BAT = < ACB

Q- 15 Case study –

Students of Class 12 presented a gift to their school in the form an electric lamp in the shape of a glass hemispherical base surmounted by a metallic cylindrical top of same radius 21 cm and height 3.5 cm. The top was silver coated and the glass surface was painted red.

- (a) What is the cost of silver coating the top at the rate of 5rs per 100cm²?
- (b) What is the surface area of glass to be painted red?

Q- 16 Find a, b and c if it is given that the numbers a, 7, b,23, c are in AP.

Q- 17 If m times the mth term of an AP is equal to n times its nth term, show that the (m +n)th term of the AP is zero.

Q- 18 Find the values of k, for which the quadratic equation (k + 4)x² + (k+1)x +1 = 0 has equal roots.

Q- 19 Prove that, a tangent to a circle is perpendicular to the radius through the point of contact.

Q- 20 A solid iron cuboidal block of dimensions 4.4 m x 2.6 m x 1m is cast into a hollow cylindrical pipe of internal radius 30cm and thickness 5 cm. Find the length of the pipe.