Download mathematics - Kendriya Vidyalaya No.1 Salt Lake

SAMPLE QP FOR XI HALF-YEARLY EXAMINATION
SUB: MATHEMATICS
CLASS - XI
TIME: 3 HOURS
M.M: 100
General Instructions:
1. All questions are compulsory.
2. The question paper consists of 26 questions divided into three sections viz. A, B
and C. Section A comprises of 6 questions of one mark each, Section B comprises
of 13 questions of four marks each and Section C comprises of 7 questions of six
marks each.
3. There is no overall choice .However, internal choice has been provided in 4
questions of four marks each and 2 questions of six marks each. You have to
attempt only one of the alternatives in all such questions.
4. Use of calculator is not permitted.
SECTION-A
1. What is the number of proper subsets of a set containing n elements?
(1)
2. If R is a relation from a finite set A having 4 elements to a finite set B having 3
elements, then what is the number of relations from A to B?
(1)
3. What is the value of (1+i)(1+ )(1+ )(1+ ) ?
(1)
4. Write the solution of 4x+3 6x+7 in interval form.
(1)
5. Find the coefficient of
(1)
in
.
6. Find the sum to infinity of the G.P.
.
(1)
SECTION B
Page 1 of 8
7. If
U={1,2,3,4,5,6,7,8,9,10} ,A={2,4,6,8,10}, B={1,3,5,7,9} and C={3,4,7,8,!0},
(i) ( B  C )
Find :
(ii) ( A  C )
8. If A,B and C are any three sets, then prove that : A - (B
(4)
) = (A-B)
(A-C)
(4)
9. Let R be the relation on the set N of natural numbers defined by
R  (a, b) : a  3b  12, a, b  N  Find:
(a)R
(b)
Domain of R
(c) Range of R
(4)
OR
Find the domain and range of the function f(x) given by f(x) =
.
10. Let f,g : R→R be defined , respectively by f(x) = x+1, g(x) = 2x-3 . Find f+g, f-g,
fg and
.
(4)
11. Prove that:
=
(4)
12. Solve for x : 4 sin 2 x  4 cos x  1
(4)
OR
Solve:
=√
13. In a triangle ABC , if a=18, b=24 and c=30, find cosA, cosB and cosC
(4)
14. Find the square root of the complex number 5  12i
(4)
15. If
= a  ib, x, y, a, b  R , then prove that :
4(
)
(4)
OR
Convert the complex number
√
into polar form.
Page 2 of 8
16. a). How many different words can be formed using the letters of the word
HARYANA?
(4)
b).How many of these begin with H and end with N ?
c).In how many of these H and N are together?
17. Find the term independent of x in the expansion of
.
(4)
18. Find the image of the point (3,8) with respect to the line x+3y=7 assuming the
line to be a plane mirror.
(4)
19. Find the equation of the line passing through the intersection of the lines
(4)
3x-4y+1=0 and 5x+y-1 =0 and cutting equal intercepts on the coordinate axes.
OR
Verify that the area of the triangle with vertices (4,6),(7,10) and (1,-2) remains
invariant under the translation of axes when origin is shifted to the point (-2,1).
SECTION - C
20. In a survey of 25 students, it was found that 15 had taken Mathematics,12 had
taken Physics and 11 had taken Chemistry, 5 had taken Mathematics and
Chemistry,9 had taken Mathematics and Physics,4 had taken Physics and
Chemistry and 3 had taken all the three subjects. Find the number of students
that had taken
(a) at least one of the three subjects,
(b) only one of the subjects.
(6)
Page 3 of 8
21. . Prove that :
=
(6)
OR
Prove that:
A+
=
22. A committee of 8 persons is to be constituted from a group of 5 women and 7
men. In how many ways can this be done? In how many ways can the
committee be formed if it consists of atleast 3 women and 3 men? What value is
depicted by taking men and women in forming the committee?
(6)
23. Solve the following system of inequalities graphically:
(6)
x+2y 10, x+y 1, x-y 0, x 0 , y 0
24. Prove the following by principle of mathematical induction for all n N :
1.3+3.5+5.7+……………………….+(2n-1)(2n+1) =
(6)
25. Find the equations of the medians of a triangle formed by the lines x+y-6=0,
x-3y-2=0 and 5x-3y+2=0.
26. If a and b are the roots of
(6)
-3x+p =0 and c,d are the roots of
-12x+q
=0,where a,b,c,d form a G.P.. Prove that (q+p):(q-p) = 17:15.
(6)
OR
Find the sum of the first n terms of the series: 3+7+13+21+31+………………
Page 4 of 8
Marking Scheme
Answers:
1.
SECTION-A
-1
4. (-2, )
2.
3.0
5.1512
6. 1
For each correct answer, 1mark
SECTION-B
7. (B
) ={1,3,4,5,7,8,9,10}
(A-C) = {2,6}
)’
= (A
(1)
=A
(1)
= (A-B)
9. R= {(9,1),(6,2),(3,3)}
For domain 3-x
(1)
(1)
(A-C)
(1)
0
x=
(1)
Range of R={1,2,3}
(1)
–x – 3
(1)
domain of R = R- {3]
(1)
10. f+g = 3x-2
Fg = 2
(1)
(2)
Domain of R={9,6,3}
OR
)’ ={2,6}
(B
(A-C)’ = {1,3,4,5,7,8,9,10}
(1)
’
8. LHS=A
(1)
(1)
Range of R= R-{-1}
(1)
f-g = -x+4
(1)
11.
=
(1)
(1)
,x
(1)
( )
= 2sin4x .cos2x +2sin4x
(1)
= 2sin4x(cos2x+1)
(1)
=4
(
x sin4x
)
12. 4 sin 2 x  4 cos x  1  4 cos 2 x  4 cos x  3  0 (1)
3
cos x   ( Not Acceptable)
2

 1
x  2n  , n  N
1 
3
 2
cos x 
OR
1
1
, ( )
2
2
cosθ -√ sinθ = -1
(1)
1
√
=
(1)
Page 5 of 8
Cos( +θ) = cos
(1)
13. cosA =
=
cosB =
=
cosC =
=0
14. Let √
( +1)
(
= x+iy
=5+12i
(1)
OR
( )
2
√
16. a.
and y= ±2
17. Let it be
12-3r =0
y=
-
(1)
(1)
(1 )
r= | | = 8
(1)
(1)
(1)
polar form= 8(cos
b.
term
r= 4
(1)
x+iy=
= -4+4i√
= 840
(1)
( )
= arg(z) =
18.
(2)
Solving x=
= 3+2i or -3-2i
15. (x+iy) = (a+ib)3
=
( )
=5 and 2xy=12
= 13
x=
(1)
( +1)
+2xyi =5+12i
√
θ = 2nπ +
(1)
(1)
)
(1)
c.
(1)
X2
(2)
term = C(n,r)
(1)
term (independent of x)=C(6,4) .
(2)
x+3y=7
Equation of vertical line 3x-y =1
(3,8)
Foot of the perpendicular =(1,2)
(1)
19. Finding point of intersection (3/23,8/23)
Considering the equation in intercept form
image = (-1,-4)
(1)
(1)
(1)
Page 6 of 8
To find the intercept 11/23
(1)
(1)
Finding the equation of line 23x+23y=11
OR
Let P(4,6), Q(7,10) and R(-1,2) be the given points.
Area of triangle PQR= 6 sq. units
When origin is (-2,1)
(1 )
New coordinates are (6,5), (9,9) and (1,1)
(1)
Now the area of the triangle = 6 sq. units
(1 )
SECTION-C
20.
n(P
n(U) = 25
n(M) =15 n(P)=12
n(C)=11
)=4
n(M
) =3
(2)
(a). n(M
n(M
)=15+12+11-5-9-4+3=23
) =5
n(M
(2)
(b). Taken only one of the subjects= 23-5-9-4+6=11
21.
= (cos20
√
=
cos20 sin80 -
√
√
OR
sin80 +
LHS=
√
√
(1 )
cos60 .sin80
)-
-
()
√
).sin80
= (sin100
=
(2)
) sin 80 sin 60
(2sin40
√
(1)
sin80
sin80
+
(
(1)
( )
+
(
= (3+cos2A+2cos2A( ))
(2)
)
(1)
(3+cos2A +2cos2A.cos240 )
22. C(12,8) =495
(3)
)
=
= (3+cos2A+cos(2A+240 )+cos(2A-240 ))
=
) =9
(1 )
(1) =
(1)
.C(5,3).C(7,5)+ C(5,4)C(7,4) +C(5,5)C(7,3) =420
For any one value one mark.
23. For each correct line 1 mark (1X3)
For correct region 3 marks
Page 7 of 8
24. To prove for n=1
1mark
To assume that the statement is true for n=k
1mark
To prove that the statement is true for n=k+1
4marks
25. Finding the vertices A(2,4),B(5,1) and C(-1,-1)
(
)
Finding the mid points of AB,BC and AC i.e. F(7/2,5/2),D(2,0) and E(1/2,3/2)
(
)
Equation of the median AD x=2
(1)
Equation of the median BE x+9y -14 =0
(1)
Equation of the median CF 7x-9y-2=0
(1)
26. a+b= 3 ,ab= p
b =ar , c = a
r= 2
OR
(1)
,d =a
c+d =12 , cd = q
(1)
( )
a(1+r) = 3 , a (1+r) = 12
=
=
=
(1)
(1)
(
)
S =3+7+13+21+…………………………………….+.
S=
3+7+13+21+………………………………………….+
0 = 3+4+6+8+……………………………………………………….(1)
= 3+4+6+8+……………………………………….(n-1)terms
=
=∑
+n+1
∑
=
=
∑
+
(
(1)
(1)
+n
)
(
)
(
)
Page 8 of 8