DPP-53-58

Subject : Maths DPP No. ____ Course Name : Batch : CC
53
Time: 30 Min.
DPP No. – 1
1. Set of all values of a for which the inequality (a – 4) x2
– 2ax + (2a – 6) < 0 is satisfying for all real values
of x, is
(A*) (– , 2) (B) (– , 2)  (C) (2,12) (D) (– , 4)
2. Set of all values of x satisfying the inequation 2 sin2
x – 3 sin x – 3 > 0, is
(A) 




 






3
)
1
n
2
(
,
3
2
n
2 (B) 




 





6
11
n
2
,
6
n
2
(C*) 




 





3
5
n
2
,
3
4
n
2 (D) No value of x
3. If ai
> 0  i  N such that 


n
1
i
i 1
a , minimum value (1 + a1
) (1 + a2
) (1 + a3
) ........ (1 + an
) is
(A) 1 (B*) 2n
(C) 2n (D) does not exist
4. The system of equations
2x + py + 6z = 8
x + 2y + qz = 5
& x + y + 3z = 4 has
(i) No solution if p  2
(ii) a unique solution if p  2, q  3
(iii) Infinitely many solutions if p = 2, then
(A) (i) and (ii) are correct (B*) (ii) and (iii) are correct
(C) (i) and (iii) are correct (D) none is correct
5. Value of x satisfying the equation sin–1
6x + sin–1
6 3 x = –
2

is
(A) –
6
1
(B)
12
1
(C*) –
12
1
(D)
6
1
6. Domain of definition of f(x) =
2
x
1
x
3
1







 is
(A) R – {2} (B) (–, –3)  (2, )
(C) R – {0, 2} (D*) (–, –3)  (0, ) – {2}
7.
2
x
lim











 x
2
x
3
x
)
3
x
2
(
2
2
x
1
2
3 =
(A) 1 (B) – 1 (C*) –
2
1
(D) does not exist

8. f(x) =









4
x
1
,
x
4
1
x
1
,
3x
is
(A) not continuous at x = 1 (B*) continuous and but not differentiable at x = 1
(C) continuous at x = 1 (D) continuous and differentiable at x = 1
9. y
x
a
In
y
x
x



, then
dx
dy
=
(A) 2 +
y
x
(B*) 2 –
y
x
(C) 2 +
x
y
(D) 2 –
x
y
10. STATEMENT-1 : 






2
1
x = [x] if {x} <
2
1
STATEMENT-2 : [nx] = n[x] if {x} <
n
1
where [.], {.} stands for greatest integer and fraction part functions respectively and n is a natural
number
(A*) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.
(B) Statement-1 isTrue, Statement-2 isTrue; Statement-2 is NOT a correct explanation for Statement-1
(C) Statement-1 is True, Statement-2 is False
(D) Statement-1 is False, Statement-2 is True
11. STATEMENT-1:Perpendicular from origin O to the line joining the points A (c cos , c sin ) and
B (c cos , c sin ) divides it in the ratio 1 : 1
STATEMENT-2:Perpendicular from opposite vertex to the base of an isosceles triangle bisects it.
12. STATEMENT-1:The number of terms in the expansion of (x2 + x + 1)10 is 21.
STATEMENT-2:The number of terms in the expansion of (x + y + z)n is
2
1
(n + 1) (n + 2)
(A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.
(B*) Statement-1 isTrue, Statement-2 isTrue; Statement-2 is NOT a correct explanation for Statement-1
13. STATEMENT-1:If one A.M. ‘A’ and two G.M.'s p and q be inserted between any two numbers,
then p3 + q3 = 2Apq
STATEMENT-2:If x, y, z are in G.P., then y2 = xz

Time: 30 Min.
DPP No. – 2
Comprehension # 1
The number of distinct real roots of 2x3
+ 3x2
– 12x + d = 0 is n.
1. Set of all values of d if n = 1, is
(A) (–, –7)  (20, ) (B) (–20, 7)
(C*) (–, –20)  (7, ) (D) {–20, 7}
(A) (–, –7)  (20, ) (B*) (–20, 7)
(C) (–, –20)  (7, ) (D) {–20, 7}
(A) (–, –7)  (20, ) (B) (–20, 7)
(C) (–, –20)  (7, ) (D*) {–20, 7}
Comprehension # 2
If f(x) =

































x
2
if
|)
x
cos
|
1
(
2
x
if
2
b
2
x
0
if
5
6
b
|
x
|tan
a
x
5
tan
x
6
tan
is continuous at x =
2

consider an infinite geometric
progression whose first term is a + 1 and common ratio is
3
b
1

. Let it's sum be S. Further A is square
matrix of order 3 × 3 such that aij
= 





j
i
,
S
j
i
,
2
S
4. The value of a + b is
(A*) – 1 (B) 0 (C) 1 (D) 2
5. trace of A is equal to
(A) 8 (B) – 4 (C*) 6 (D) – 6
6. If A (adj A) = 3
where 3
is a unit matrix of order 3, then  is equal to
(A*) 8 (B) 64 (C) 16 (D) 6
MATCH THE COLUMN
7. Column -  Column - 
(A) The minimum value of (p) 1
f(x) = |x – 4| + |x – 6| + |x – 2| is
(B) The number of solution/solutions (q) 4
of |x| = |cosx| is/are
(C) The value of x satisfying the equation (r) 0
|2x + 3| – |x – 1| = 5 is
(D) The product of all the solutions (s) 2
of the equation (x – 2)2
– 3 |x – 2| + 2 = 0 is
Ans. (A) (q), (B)  (s), (C)  (p), (D)  (r)
54

8. Column –  Column – 
(A) Number of solutions of tan–1
x = sin–1
(x + k) (k  R) is (p) 0
(B) Number of solutions of sin-1
x = sgn(x) is (q) 1
(C) Number of solutions of |1 – |1 – x2
|| = 5 is (r) 2
(D) Number of solutions of the equation x . 2x
= x + 1 is (s) 3
Ans. (A)  (q), (B)  (s), (C)  (r), (D)  (r)
9. If y = f(x) has following graph
then match the column.
Column –  Column – 
(A) y = |f(x)| (p)
(B) y = f(|x|) (q)
(C) y = f(– |x|) (r)
(D) y  | f ( |x| ) | (s)
Ans. (A) (r), (B)  (p), (C)  (q), (D)  (s)

55
Time: 30 Min.
DPP No. – 3
1. The function f(x) = x2
e–2x
, x > 0. If  is maximum value of f(x) then ]
[ 1
2


is equal to
{where [.] represents greater integer function}
(A) 64 (B*) 128 (C) 256 (D) none of these
2. For the function f(x) = n (cos–1
log3
x)
(A*) domain is [1/3, 3) (B) range is (– , n /2)
(C) domain [1, 3) (D) range (– , n )
3. If x  2, y  2, z  2 and
2
y
x
z
2
x
z
y
2
= 0, then the value of
z
2
2
y
2
2
x
2
2





=
(A) 1 (B*) 2 (C) 3 (D) 4
4. y = et
sin t and x = et
cos t and y =
dt
dy
, y = 2
2
dt
y
d
, x =
dt
dx
, x = 2
2
dt
x
d
, then
(A) y = x (B) y = – 2x (C*) x = – 2y (D) x = – y
5. A straight line passing through the point P(2, 5) intersects the circle x2
+ y2
= 4 is two different points
Q and R. Then the value of PQ . PR is
(A) 27 (B) 29 (C*) 25 (D) 5
6. Equation 3x2
– 8xy – 3y2
= 0 and x – 2y = 3 represents the sides of a triangle which is
(A*) right angled but not isosceles (B) isosceles
(C) equilateral (D) right angled and isosceles
7. There are two equal circles with centres at C1
and C2
, each is of radius 2 and both are touching the
lines x = 2 and y = 3. x-coordinate of C1
is less than 2 and that of C2
is greater than 2. y-coordinate of
C1
is greater than 3 and that of C2
is less than 3. Then length of a direct common tangent to these
circles is
(A) 4 (B*) 2
4 (C) 2
2 (D) 8
8. If r is an odd natural number greater than 1 and n is an even natural number, then the value of the
expression n
C2
. n
Cr
– n
C3
. n
Cr + 1
+ .......... + (–1)n – r
. n
Cn – r
. n
Cn – 2
, considering all those terms which are
defined, is
(A*) 0 (B) n (C) 2n
Cr – 2
(D) n
Cr + 2
9. There are 10 speakers S1
, S2
, ........ S10
. Speaker S4
wants to speak before S9
but S4
can not be the first and
S9
can not be the last speaker. If at least 2 speaksers are to speak after S4
and before S9
, then the number
of ways in which the speaches can be arranged is
(A) 12 (B*) 15 (C) 28 (D) 32
10. STATEMENT-1 : e
is bigger then e
.
STATEMENT-2 : f(x) = x1/x
is a decreasing function when x  [e, )

11. STATEMENT-1 : The number of solutions of the equation sin–1
x + 2x
= 3 is one.
STATEMENT-2 : If f(x) is strictly increasing real function defined on R and c is a real constant, then number
of solutions of f(x) = c is always equal to one.
(C*) Statement-1 is True, Statement-2 is False
12. In both the statements [.] represents greatest integer function.
STATEMENT-1 : The greatest value of sin 





 ]
x
[
2
3
x
2
3
is sin
2
3
.
STATEMENT-2 : The greatest value of [sin x] is 1, where x  R.
(D*) Statement-1 is False, Statement-2 is True
13. In both the statements [.] represents greatest integer function.
STATEMENT-1 :
0
x
lim








x
sin
x
= –1.
STATEMENT-2 :
0
x
lim







x
x
sin
= 0.
(D*) Statement-1 is False, Statement-2 is True

Time: 30 Min.
DPP No. – 4
Comprehension
Let L1
and L2
be the lines whose equation are
1
3
z
1
8
y
3
3
x 





and
3
3
x


=
2
7
y 
=
4
6
z 
respectively.
.
A and B are two points on L1
and L2
respectively such that AB is perpendicular both the lines L1
and L2
.
1. Shartest distence between the lines L1
and L2
is
(A) 30 (B) 2 30 (C*) 3 30 (D) none of these
2. Co-ordinates of the point Aare
(A) (1, 8, 2) (B*) (3, 8, 3) (C) (–3, 8, 3) (D) none of these
3. Co-ordinates of the point B are
(A*) (–3, –7, 6) (B) (2, 7, 6) (C) (1, 6, 3) (D) none of these
Comprehension
Let f(x) =







0
x
,
x
ax
x
0
x
,
)
x
(
g
x
3
2 , where g(t) =
0
x
lim

(1 + a tan x)t/x
, a is positive constant.
4. If a is even prime number, then g(2) =
(A) e2
(B) e3
(C*) e4
(D) none of these
5. Set of all values of a for which function f(x) is continuous at x = 0
(A) (–1, 10) (B) (–, ) (C*) (0, ) (D) none of these
6. If f(x) differentiable at x = 0, then a belongs to
(A) (–5, –1) (B) (–10, 3) (C*) (0, ) (D) none of these
MATCH THE COLUMN
(A) Minimum value of , for which 61+x
+ 61–x
, , 4x
+ 4–x
are in (p) 1
A.P., is
(B) Sn
= 3
1
1
+ 3
3
2
1
2
1


+ ........... + 3
3
3
3
n
.......
3
2
1
n
.....
3
2
1








, (q) 3
then the greatest integer less than or equal to S100
is
(C) If x = 2 + 3 , then 





 2
2
x
1
x + 4 






x
1
x = (r) 7
(D) In a cyclic quadrilateral ABCD, tan A =
12
5
, cos B = –
5
3
, (s) 30
then
D
tan
C
cos
D
tan
C
cos

=
Ans. (A)  (r), (B)  (p), (C)  (s), (D)  (q)
56

(A) If f(x) =
a
ax
ax
1
a
ax
0
1
a
2


, then f(x) – f(–x) is a polynomial (p) 1
of degree
(B) Number of points of discontinuity of f(x) = tan2
x – sec2
x in (q) 0
(0, 2) is
(C) Fundamental period of f(x) = sin 















2
x
x
2 is (r) 3
{.} represents fractional part function.
(D) The integral part of a for which the point (1, a) lies in (s) 2
between the lines x + y = 1 and 2(x + y) = 3
Ans. (A)  (p), (B)  (s), (C)  (s), (D)  (q)
(A) The reflection of the point (t – 1, 2t + 2) in a line is (p) 0
(2t + 1, t), then the equation of the line has slope
equals to
(B) Let u

= i

+ j

, v = i

– j

, w = k
3
j
2
i




 . If n

is a (q) 3
unit vector such that u

. n

= 0, n
.
v


= 0, then the
value of |
n
.
w
|


is
(C) The number of solutions of the equation (r) 1
9
log
2
x
log x
2
3
|
1
x
| 
 = (x – 1)7
, is
(D) Ladder of length 5m leaning against a wall is being (s) 2
pulled along the ground at 2cm/s. When the foot of
the ladder is 4m away from the wall, if the top of the
ladder slides down on the wall at

8
cm/s, then  =
Ans. (A)  (r), (B)  (q), (C)  (s), (D)  (q)

Time: 30 Min.
DPP No. – 5
1. If x2
– 2x + 4 = 0 and x2
– bx + c = 0 have atleast one common root then [b + c] =
{where [.] represents greastest integer function and b & c are real}
2. If 14a2
+ 11b2
+ 6c2
+ 14 = 4ab + 12bc + 6ac + 2a + 4b + 6c, then a, b, c are in
(A) G.P. (B) H.P. (C*)A.P. (D) none of these
3. If m and n are number of points (not end points of the domain) where y = sin–1
(3x – 4x3
) is non-differentiable
and discontinuous respectively, then [2m
+ 3n
] is equal to. {where [.] represents greastest integer function}
4. If  =
0
x
lim
 x
6
sin
x
1
x
2
tan 2
1








and m =
0
x
lim







x
x
tan
, then 8
+ 3m
=
{where [.] represents greastest integer function}
(A) 2 (B) 3 (C*) 5 (D) none of these
5. A fair coin is tossed 50 times. The probability of getting tails an odd number of times is
(A)
3
1
(B*)
2
1
(C)
3
2
(D)
4
1
6. The equation of the plane, passing through the intersection of the planes x + 2y + 3z + 4 = 0 and
4x + 3y + 2z + 8 = 0 and whose x-intercept is 4, is
(A*) x –3y – 7z – 4 = 0 (B) 2x + 3y + 4z = 8 (C) x + y + z – 4 = 0 (D) none of these
7. If |
a
|

= 2, |
b
|

= 3 and |
c
|

= 4, then the maximum value of ]
a
c
c
b
b
a
[








 is
(A) 625 (B*) 576 (C) 596 (D) 288
8. If A is the area of equalateral triangle inscribed in the circle x2
+ y2
– 4x + 6y – 3 = 0, then [A] =
{where [.] represents greastest integer function}
(A*) 15 (B) 16 (C) 17 (D) 5
9. The function f(x) = 2
x
1
x

, where x > 0, decrease in the interval.
(A) [0, ) (B*) [1, ) (C) [1/2, ) (D) none of these
10. STATEMENT-1 : Range of the function f(x) = 6 cos2
x + 8 sin x cos x is [–2, 8]
STATEMENT-2 : – 2
2
b
a   a cos  + b sin  2
2
b
a  ,  , a, b R.
57

11. STATEMENT-1 : Lines x(cos  + sin ) + y(cos  – 2 sin ) – 3cos  = 0 passes through fixed point (2, 1)
STATEMENT-2 :  L1
+ L2
= 0 represents family of all straight lines passing through the point of intersection
of L1
= 0 & L2
= 0
12. STATEMENT-1 : The largest integer n for which 35! is divisible by 3n
is 15.
STATEMENT-2 : n
C2
=
2
)
1
n
(
n 
13. STATEMENT-1 : If Aand B are two symmetric matrices of the same order, thenAB – BA is skew symmetric.
STATEMENT-2 : AB = –BA always, where A and B are matrices of same order.
(C*) Statement-1 is True, Statement-2 is False

Time: 30 Min.
DPP No. – 6
Comprehension
Let M be the set of all members of the family of circles x2
+ y2
– 2x – 2by – 8 = 0 where b is a variable and
{.} represents fraction part function.
1. If every member of set M passes through two fixed points (x1
, y1
) and (x2
, y2
), then x1
+ x2
+ y1
+ y2
=
(A) 0 (B) 1 (C*) 2 (D) none of these
2. If a member of set M cuts the circle x2
+ y2
– 4x – 2y – 7 = 0 orthogenally, then {b} =
(A) 0 (B*) 1/2 (C) 3/4 (D) none of these
3. If n is the number of those circles belonging to set M to whom y = x is tangent, then n =
(A*) 0 (B) 1 (C) 2 (D) none of these
Comprehension
For points P(x1
, y1
) and Q(x2
, y2
), a new distance formula d(P, Q) is defined by d(P, Q) = |x1
– x2
| + |y1
– y2
|
Let O(0, 0), A(1, 2), B(2, 3), C(4, 3) are four fixed points on x – y plane.
4. If 0  x < 1 and 0  y < 2, then locus of point (x, y), which is equi-distant from point O and point A, with
respect to the defined distance formula, is a part of
(A) x + y = 3 (B) x + 2y = 3 (C) 2x + y = 3 (D*) 2x + 2y = 3
5. Let S(x, y), such that S is equidistant from O and B, with respect to the defined distance formula and if
x  2, 0  y < 3, then locus of S is
(A) a line segment of finite length (B) y = 1/2
(C*) a ray of infinite length (D) none of these
6. Let T (x, y), a moving point is such that it is equidistant from O and C, with respect to the defined distance
formula. If T lies in first quadrant, then graph of moving point T is
(A*) (B)
(C) (D)
58

7. Column – I Column – II
(A) If y =
100
x
12
x
100
x
12
x
2
2




, x  R, then greatest value of y is (p) 6
(B) If
1
4
.
2
6
.
3
8
.
4
10
...........
30
62
.
31
64
= (64)x
, then x is (q) 4
(C) If there are n integers between 1 and 1000 whose number (r) 3
of proper divisors is odd, then the value of 





5
n
,
where [.] represents greatest integer function, is
(D) If V1
and V2
are the maximum and minimum value of (s) 2
parallelopiped formed by the vectors k̂
ĵ
a
î 
 , k̂
a
ĵ 
and k̂
î
a  respectively, where –1 < a < 1, then the value of (V1
+ V2
)2
is equal to
Ans. (A)  (q), (B)  (p), (C)  (p), (D)  (q)
(A) If m is absolute value of the difference of all integral (p) 8
roots of (3 – x)5
+ (x – 5)5
+ 32 = 0, then 3m
is
(B) If m & n are the largest natural numbers for which 100! (q) 7
is divisible by 2m
and 5n
, then
n
1
n
3
m 

is
(C) If the domain at y = cos–1 




 
4
x
2
1
is [a, b], then twice (r) 9
of b – a is equal to
(D) If a chord of the circle x2
+ y2
– 4x – 2y + k = 0 (s) 1
is trisected at the points 





3
1
,
3
1
and 





3
8
,
3
8
and
 is the length of the chord then
2

is equal to
Ans. (A)  (r), (B)  (s), (C)  (p), (D)  (q)
(A) Let D(20, 25), E(8, 16) and F(8, 9) be the feet of perpendicular (p) 3
from the verticesA, B and C of a triangle ABC to the respective
opposite sides. If (a, b) is the orthocentre of the triangle ABC,
then b – a is equal to
(B) Let f(x + y) = f(x) . f(y) for all x, y  R and suppose that f is (q) 4
differentiable for all x and f(0) = 2. It f(a) = 3, then f(a) is equal
to
(C) If x2
+ y2
= 2 and 2y + ay–3
= 0, then a is equal to (r) 5
(D) If 




 

5
,
2
is the reflection of (1, 4) about the line y = x, (s) 6
then  –  is equal to
Ans. (A)  (r), (B)  (s), (C)  (q), (D)  (p)

DPP-53-58

DPP-53-58

DPP-53-58

Recomendados

Más contenido relacionado

Similar a DPP-53-58

Similar a DPP-53-58(20)

Más de STUDY INNOVATIONS

Más de STUDY INNOVATIONS(20)

Último

Último(20)

DPP-53-58