### Mathematics

```SAMPLE QUESTION PAPER
Class- XI
Sub- MATHEMATICS
SAMPLE QUESTION PAPER
Class- XI
Sub- MATHEMATICS
Time : 3 Hrs.
Maximum Marks : 100
General Instructions :
1.
All the questions are compulsory.
2.
The question paper consists of 30 questions divided into four sections A, B, C, D. Section A
comprises of 6 questions of one mark each. Section B comprises of 6 questions of two marks
each. Section C comprises of 13 questions of four marks each and section D comprises of 5
questions of six marks each.
3.
This set of model questions is not unique in character. Several such set of model questions
can be framed. Teachers are requested to frame model questions of this kind for their
Students.
Section- A
Question numbers 1 to 6 carry 1 mark each. For each question four options are provided out of
which only one is correct. Write the correct option.
1.
1x6=6
Î!ò A = {a, b, c} ~ÓÇ B = {b, c} •Î˚ ï˛ˆÏÓ n (A x B) ~Ó˚ üyö •ˆÏÓ
(i) 3
(ii) 6
(iii) 9
(iv) 8
If A = {a, b, c} and B = {b, c} Then value of n (A x B) will be
(i) 3
(ii) 6 (iii) 9 (iv) 8
2.
n ˛~Ó˚ ˆÎ ˆÜ˛yö ôöydÜ˛ ˛õ)î≈üyˆÏö (- √ - 1)4n+3 ~Ó˚ üyö •ˆÏÓ
(i) 1
(ii) - 1 (iii) i
(iv) -i
For any positive integral value of n the value of (- √ - 1)4n+3 is
(i) 1
(ii) - 1 (iii) i (iv) -i
3.
| 4x - 5 | <
•ˆÏÓ (i)
if
, x C R •ˆÏ° x ~Ó˚ üyö
(ii) 76 < x <
< x < 43
| 4x - 5 | <
(i)
4.
7
6
1
3
7
6
< x<
1
3
4
3
(iii)
7
6
<x<
4
3
(iv)
7
6
<x<
4
3
, x C R then the value of x will be
4
3
7
6
(ii)
< x<
4
3
(iii)
7
6
<x<
4
3
(iv)
7
6
<x<
4
3
ˆÎ !Ó®%ˆÏï˛ S5, - 3) ~ÓÇ SÈ- 1, 3) !Ó®%mˆÏÎ˚Ó˚ §ÇˆÏÎyçÜ˛ ˆÓ˚áyÇ¢ 2 : 1 xö%˛õyˆÏï˛ xhsˇ!Ó≈û˛=˛ •Î˚ ï˛yÓ˚ ﬁÌyöyÇÜ˛ •ˆÏÓ
(i) (
4
3
, 1)
(ii) (3, 1)
(iii) (- 3, 0)
(iv) (1, 3)
The line segment joining the points (5, -3) and (-1, 3) is divided internally in the ratio 2 : 1 at the point
whose coordinates are
4
(i) ( 3 , 1)
(ii) (3, 1)
(iii) (- 3, 0)
(iv) (1, 3)
5.
ü)°!Ó®%àyü# ˆÎ Ó,ˆÏ_Ó˚ ˆÜ˛w S0, 2) !Ó®%ˆÏï˛ xÓ!ﬁÌï˛ ï˛yÓ˚ §ü#Ü˛Ó˚î •ˆÏÓ
(i) x2 + y2 - 4x = 0
(ii) x2 + y2 - 4y + 4 = 0
(iii) x2 + y2 - 4y = 0
(iv) x2 + y2 - 4x + 4 = 0
Page : 1
Equation of the circle with centre at (0, 2) which passes through the origin is
(i) x2 + y2 - 4x = 0
(ii) x2 + y2 - 4y + 4 = 0
(iii) x2 + y2 - 4y = 0
(iv) x2 + y2 - 4x + 4 = 0
6.
~Ü˛!ê˛ ï˛yˆÏ§Ó˚ ˛õƒyˆÏÜ˛ˆÏê˛ x!ÓöƒhﬂÏ û˛yˆÏÓ 52 !ê˛ ï˛y§ xyˆÏåÈ– ï˛yˆÏ§Ó˚ ˛õƒyˆÏÜ˛ê˛ ˆÌˆÏÜ˛ ÎˆÏÌFåÈ û˛yˆÏÓ ~Ü˛!ê˛ ï˛y§ ê˛yöy •ˆÏ°y–
ï˛y§!ê˛ ~Ü˛!ê˛ Ü˛yˆÏ°y Ó˚ˆÏäÓ˚ Ó˚yçy •ÄÎ˚yÓ˚ §Ω˛yÓöy •ˆÏÓ–
2
1
1
(i) 13 (ii) 13 , (iii) 26 (iv) ˆÜ˛yö!ê˛•z öˆÏ•–
From a well-shuffled pack of 52 cards, a card is drawn at random. The probability of it
being a king of black colour is
2
1
1
(i) 13 (ii) 13 , (iii) 26 (iv) none of these
Section- B
Question numbers 7 to 12 carry two marks each.
7.
f (x) =
1
1 - x2
2 x 6 = 12
xˆÏ˛õ«˛Ü˛!ê˛Ó˚ §ÇK˛yÓ˚ ˆ«˛e ~ÓÇ ≤Ã§yÓ˚ !öî≈Î˚ Ü˛ˆÏÓ˚y–
Find the domain and range of the function f (x) =
1
1 - x2
8.
– x - 3–, x C R xˆÏ˛õ«˛Ü˛!ê˛Ó˚ ˆ°á!ã˛e xAÜ˛ö Ü˛ˆÏÓ˚y–
Draw a graph of the function – x - 3–, x C R
9.
Cos 3300 Sin (- 2400) + Sin (- 2100) Cos (- 4200) ~Ó˚ üyö !öî≈Î˚ Ü˛Ó˚–
Find the value of
Cos 3300 Sin (- 2400) + Sin (- 2100) Cos (- 4200)
10.
(- 1 - i) ˆÜ˛ ˆüÓ˚% xyÜ˛yˆÏÓ˚ ≤ÃÜ˛y¢ Ü˛Ó˚–
Represent (- 1 - i) in polar form.
11.
lim
x_0
1- Cosx
x2
Simplify : lim
x_0
1- Cosx
x2
§Ó˚° Ü˛Ó˚ É
lim
x_0
Sin 7x - Sin 5x
x
Simplify : lim
x_0
Sin 7x - Sin 5x
x
§Ó˚° Ü˛Ó˚ É
12.
!ö¡¨!°!áï˛ ï˛Ìƒà%!°Ó˚ §üÜ˛ !Óã%˛ƒ!ï˛ !öî≈Î˚ Ü˛Ó˚ É
3, 6, 7, 9, 13, 15, 17
Find the standard deviation for the following data :
3, 6, 7, 9, 13, 15, 17
Section- C
Question numbers 13 to 25 carry 4 marks each.
!ï˛ö!ê˛ ˆ§ê˛ A, B ~ÓÇ C ~Ó˚)˛õ ˆÎ AUB = AUC
~ÓÇ A B = A C, ˆòáyÄ ˆÎñ B = C
U
13.
U
Let A, B and C be three sets such that AUB = AUC
and A B = A C, Show that B = C
U
U
14.
ABC ˛!eû%˛ˆÏç A =
∧
3
•ˆÏ° ˆòáyÄ ˆÎñ
Page : 2
4 x 13 = 52
b+c = 2a Cos B - C
2
In triangle ABC, if A=
b+c = 2a Cos
B-C
2
∧
3
then show that,
OR
∧
2
≤Ãüyî Ü˛Ó˚ Cos
Cos 4 ∧
Cos 8 ∧ Cos 14 ∧ = 1
15
15
15
16
15
Prove that Cos 2 ∧ Cos 4 ∧
Cos 8 ∧ Cos 14 ∧ = 1
15
15
15
16
15
15.
x = a + b, y = aw + bw2 ˛~ÓÇ z = aw3 + bw ˛•ˆÏ° ˆòáyÄ ˆÎñ
xyz = a3+b3, ˆÎáyˆÏö w ˛•ˆÏ° 1 ~Ó˚ âöü)°–
If w is the cube root of unity and x = a + b, y = aw + bw2, z = aw2 + bw, then
show that xyz = a3 + b3
OR
z
˛Î!ò z1 = 3i ˛~ÓÇ z2 = -i, •Î˚ ï˛ˆÏÓ ˛arg ( z1 ) ~Ó˚ üyö !öî≈Î˚ Ü˛Ó˚–
2
z
If z1 = 3i and z2 = -i then find the value of arg ( z1 )
2
16.
ày!î!ï˛Ü˛ xyˆÏÓ˚y•Ó˚ ï˛ˆÏ_¥Ó˚ §y•yˆÏÎƒ ≤Ãüyî Ü˛ˆÏÓ˚y ˆÎ 4n + 15n - 1, n C N. ˛§Ó≈òy 9 ˛myÓ˚y !Óû˛yçƒ–
Using the principle of mathematical induction prove that 4n + 15n - 1 is divisible by 9 for all n C N.
17.
n ˛§ÇáƒÜ˛ ˛õò ˛õÎ≈hsˇ !öˆÏ¡¨Ó˚ ˆ◊î#!ê˛Ó˚ ˆÎyàú˛° !öî≈Î˚ Ü˛Ó˚–
1
1
1
+
+...............
+
2.5
8.11
5.8
Find the sum of n terms of the series
1
1
1
+
+...............
+
2.5
8.11
5.8
18.
(9x2 - 1 )12 ˛~Ó˚ !Óhﬂ,Ï!ï˛ˆÏï˛ x ˛Ó!ç≈ï˛ ˛õò!ê˛ !öî≈Î˚ Ü˛ˆÏÓ˚y–
3x
Find the term independent of x in the expanssion of (9x2 - 1 )12
3x
OR
(1 + x)15 ~Ó˚ !Óhﬂ,Ï!ï˛Ó˚ (r + 1) ï˛ü ˛õò ~ÓÇ (r + 2) ï˛ü ˛õˆÏòÓ˚ §•àmˆÏÎ˚Ó˚ xö%˛õyï˛ 3 : 5 •ˆÏ° r ˛~Ó˚ üyö !öî≈Î˚ Ü˛Ó˚–
If the ratio of the co-efficients of (r + 1)th and (r + 2)th term of the expanssion (1 + x)15
are in the ratio 3 : 5, find r.
19.
˛!ö¡¨!°!áï˛ x§ü#Ü˛Ó˚îà%!°Ó˚ §üyôyö xM˛° !öö≈Î˚ Ü˛ˆÏÓ˚y É
x - 2y < 3, 3x + 4y > 12, x > 0, y > 1.
Find the solution region of the following in equations
x - 2y < 3, 3x + 4y > 12, x > 0, y > 1.
20.
y2 = 4 ax ˛x!ôÓ,ˆÏ_Ó˚ öy!û˛àyü# çƒy ~Ó˚ xˆÏ«˛Ó˚ §ˆÏAà Ø ˆÜ˛yî í˛zÍ˛õß¨ Ü˛Ó˚ˆÏ°ñ ˆòáyÄ ˆÎñ
çƒy!ê˛Ó˚ ˜òâ≈ƒ •ˆÏÓ 4a cosec2Ø
Focal chord of a parabola y2 = 4 ax makes an angle Ø with its axis. Show that the length of the
focal chord is 4a Cosec2 Ø.
21.
˛~Ü˛!ê˛ xô≈ÈÙÈí˛z˛õÓ,_yÜ˛yÓ˚ !á°yˆÏöÓ˚ !Óhﬂ,Ï!ï˛ 8 !üê˛yÓ˚ ~ÓÇ ˆÜ˛w ˆÌˆÏÜ˛ í˛zFã˛ï˛y 2 !üê˛yÓ˚– !Óhﬂ,Ï!ï˛Ó˚ ˆÜ˛w !Ó®% ˆÌˆÏÜ˛ 2.5 ˛!üê˛yÓ˚ ò)ˆÏÓ˚
!á°yö!ê˛Ó˚ í˛zFã˛ï˛y !öî≈Î˚ Ü˛ˆÏÓ˚y–
An arch is of the form of a semi-ellipse. It is 8m wide and 2m high from the centre. Find the height of the arch
Page : 3
at
22.
a point 2.5 m from the centre.
~Ó˚)˛õ ~Ü˛!ê˛ ˛õÓ˚yÓ,ˆÏ_Ó˚ §ü#Ü˛Ó˚Ïî !öî≈Î˚ Ü˛ˆÏÓ˚y ÎyÓ˚ öy!û˛ S0 , ± √10 ) ~ÓÇ Îy S2, 3) !Ó®%àyü#–
Find the equation of a hyperbola which passes through the point (2, 3) and has foci at (0 – ± √10 )
23.
˛≤ÃÌü §)ˆÏeÓ˚ §y•yˆÏÎƒ x ~Ó˚ §yˆÏ˛õˆÏ«˛ Cosec (2x + 1) ~Ó˚ xÓÜ˛° §•à !öî≈Î˚ Ü˛Ó˚
Differentiate Cosec (2x + 1) from the first principle.
24.
˛§ï˛ƒ§yÓ˚î# ÓƒÓ•yÓ˚ Ü˛ˆÏÓ˚ ≤Ãüyî Ü˛ˆÏÓ˚y ˆÎ ~ q V ~ p = ~ (p ∧ q)
Use truth table to verify that, ~ q V ~ p = ~ (p ∧ q)
25.
ò%•z!ê˛ ˆéÑ˛yÜ˛¢)öƒ åÈE˛yˆÏÜ˛ ÎˆÏÌFåÈ û˛yˆÏÓ ~Ü˛ÓyÓ˚ ã˛y°yöy Ü˛Ó˚ˆÏ°ñ ò%•z!ê˛ åÈE˛yˆÏï˛•z ~Ü˛•z §Çáƒy xÌÓy ˆÎyàú˛° 6 öy •ÄÎ˚yÓ˚
§Ω˛yÓöy !öî≈Î˚ Ü˛ˆÏÓ˚y–
In a single throw of a pair of dice, find the probability that neilher a doublet nor a total of 6 will appear.
Section - D
Question numbers 26 to 30 carry 6 marks each.
6 x 5 = 30
26.
˛§yôyÓ˚î §üyôyö !öî≈Î˚ Ü˛ˆÏÓ˚y tanx + tan2x + tan3x = 0
27.
Find the general solution of tanx + tan2x + tan3x = 0
˛~Ü˛!ê˛ à%ˆÏîy_Ó˚ ≤Ãà!ï˛Ó˚ ≤ÃÌü n, 2n ~ÓÇ 3n §ÇáƒÜ˛ ˛õˆÏòÓ˚ §ü!T˛
ÎÌyÜ ˛ˆÏü S1, S2 ~ÓÇ S3 ˛•ˆÏ° ≤Ãüyî Ü˛ˆÏÓ˚y ˆÎ S1(S3 - S2) = (S2 - S1)2
If the sum of first n, 2n and 3n terms of a geometric progression are respectively S1, S2 and S3 then
prove that S1 (S3 - S2) = (S2 - S1)2
28.
˛˛õ%öÓ˚yÓ,!_ öy Ü˛ˆÏÓ˚ 1, 2, 3, 4, 5 ˛xAÜ˛à%!° myÓ˚y à!ë˛ï˛ §ÇáƒyÓ˚ §Çáƒy !öî≈Î˚ Ü˛ˆÏÓ˚y– à!ë˛ï˛ ~•z çyï˛#Î˚ §Çáƒyà%!°Ó˚
ˆÎyàú˛° Ä !öî≈Î˚ Ü˛ˆÏÓ˚y–
Find the number of numbers that can be formed with the digits 1, 2, 3, 4, 5 without repetition.
Find also the sum of the numbers thus formed.
˛xÌÓy
ˆÜ˛yö §üï˛ˆÏ° xÓ!ﬁÌï˛ 10˛!ê˛ !Ó®%Ó˚ üˆÏôƒ 4!ê˛ §üˆÏÓ˚áñ x˛õÓ˚ !Ó®%à%!°Ó˚ ˆÎ ˆÜ˛yö !ï˛ö!ê˛ §üˆÏÓ˚á öÎ˚–
~•z !Ó®%à%!°Ó˚myÓ˚y à!ë˛ï˛ (i) ˛§Ó˚° ˆÓ˚áyÓ˚ §Çáƒy ~ÓÇ (ii) !eû%˛ˆÏçÓ˚ §Çáƒy !öî≈Î˚ Ü˛ˆÏÓ˚y–
There are 10 points in a plane no three of which are on the same straight line, except 4 points which
are collinear. Find
(i) The number of lines obtained from these points.
(ii) The number of triangles that can be formed with these points.
29.
(2, 3) ˛!Ó®%àyü# ò%•z!ê˛ §Ó˚° ˆÓ˚áy ˛õÓ˚ﬂõˆÏÓ˚Ó˚ §!•ï˛ 600 ˛ˆÜ˛yî í˛zÍ˛õß¨ Ü˛ˆÏÓ˚– Î!ò ~Ü˛!ê˛ §Ó˚° ˆÓ˚áyÓ˚ ≤ÃÓîï˛y 2 ˛•Î˚ ï˛ˆÏÓ
x˛õÓ˚ §Ó˚° ˆÓ˚áy!ê˛Ó˚ §ü#Ü˛Ó˚î !öî≈Î˚ Ü˛ˆÏÓ˚y–
Two straight lines passing through the point (2, 3) make an angle 600 with each other. It the slope of
one straight line is 2 find the equation of the other straight line.
30.
!ö¡¨!°!áï˛ ˛õ!Ó˚§Çáƒy !Óû˛yçö ˆÌˆÏÜ˛ ˆû˛òyAÜ˛ !öî≈Î˚ Ü˛ˆÏÓ˚y
ˆ◊î#
10 - 29
30 - 49
50 - 69
70 - 89
90 - 109
110 - 129
˛˛õ!Ó˚§Çáƒy
2
5
13
18
7
6
Calculate the coefficient of variation from the following frequency distribution table.
Class
10 - 29
30 - 49
50 - 69
70 - 89
90 - 109
110 - 129
frequency
2
5
13
18
7
6
Page : 4
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