Determinants+[Practice+Question].pdf
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Determinants+[Practice+Question].pdf...
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Practice Question Question based on
LEVEL –1 Q.7
Expansion of Determinants
Determinants The minors of the elements of the first row in the determinant
Q.1
a 1 1 1 If 1 1 1 = 4, then the value a is 1
(A) 1 Q.2
y
4 2
= 7 and
(A) x = – 3, y = – (C) x = 3, y =
Q.3
The value of (A) 12
Q.4
(C) –2 2 3 y x
5 2
(C) x =
5i
3i
4i
5i
(B) 17
sec x sin x 0 1
= 4, then -
5 ,y=3 2
is -
(C) 14
Minors & Cofactor and their properties
1
2
3
4
c1 c2
a3
b3
c3
(B) –4, 3, 2, –1 (D) –4, –3, –2, –1
and A2, B2, C2 are
(B) 0 (D) None of these
Q.10
If cofactor x 1 x 1
x
(A) 0 (C) 1 Question based on
Q.11
of 2x
in the
determinant
1 2 2x x 1 is zero, then x equals to0
(B) 2 (D) –1
Some basic properties
a1 The value of the determinant a 2
ma1 ma 2
b1 b2
a3
ma 3
b3
–3 and 4 in
are-
(A) 4, 3, 2, 1 (C) 4, –3, –2, 1
b1 b2
If A = (aij) is a 4 × 4 matrix and cij is the cofactor of the element aij in Det (A), then the expression a11c11+ a12c12+ a13c13 + a14c14 equals(A) 0 (B) – 1 (C) 1 (D) Det. (A)
(D) 24
(B) x2 – xy + y2 (D) x3 – y3
a1 If = a 2
Q.9
1 0 0 1 The value of 3 x 3 1 is xy 5 y3 1
–2,
(B) 7, 11, 2 (D) 7, 2, 11
(A) – (C)
(B) – 1 (D) None of these
The cofactors of 1,
2
respectively cofactors of a2, b2, c2 then a1A2 + b1B2 + c1C2 is equal to-
tan x 0 is equal to -
(A) x + y (C) x2 + xy + y2
Q.6
Q.8
5 5 (B) x = – , y = – 3 2 2
(A) 0 (C) 1
Question based on
(A) 2, 7, 11 (C) 11, 2, 7
1
(D) 0
tan x cot x sec x
Q.5
1
1
(B) –1
x
If
1
2 1 4 4 2 3 are-
is (A) 0 (C) ma1b2a2
(B) ma1a2a3 (D) mb1b2b3
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Q.12
p 2a 0 0 a 0 0 If = b c a , then pb c a pc
c a b
r
is equal
Q.19
x
If Dr = 2r 1 y 3r 2 z
a b
n (n 1) / 2 n2 , then
(B) p2 (D) 2p
(A)
1 1 n(n + 1)(2n + 1) (B) n2(n + 1)2 6 4
(C) 0 1 / a 1 bc The value of the determinant 1 / b 1 ca is 1 / c 1 ab
Q.15
Q.21
The sum of infinite series 6 4
+
1/ 2 2 2
+
4
(A) –10 (C) 10
1/ 4
2
2/3 4
+ ........ is -
a ma nx x The value of b mb ny y ismc nz
Q.22
3a
equal to (A) a3 (C) c3 Q.18
The ka
value 1
2
2
1 is -
kb k b kc
of
2
k a
a bc 4a 3b 2c is
c c 2 is -
bc
ca
ab
a2 If (a 1) 2
b2 ( b 1) 2
c2 a2 2 = k (c 1) a
b2 b
c2 c ,
(a 1) 2
(b 1) 2
(c 1) 2
1
1
a
Q.23
the
(A) k (a + b) (b + c) (c + a) (B) k abc (a2 + b2 + c2) (C) k (a – b) (b – c) ( c – a) (D) k (a + b – c) (b + c – a) (c + a – b)
(B) 2 (D) 0 b c a3
c a b c3
(A) (a – b) (b – c) (c – a) (B) abc (a – b) (b – c) (c – a) (C) – (a + b + c)2 (a – b) (b – c) (c – a) (D) None of these
determinant
k 2 c2 1
1
The value of b c a b 3 is-
6a 3b 10a 6b 3c
(B) b3 (D) a3 + b3 + c3
2
b b2
then k is equal to(A) 1 (C) 4
(A) a + b + c (B) x + y + z (C) m(a + b + c) + n(x + y + z) (D) 0 The value of
a The value of the determinant a 2
z
a ab 2a 3a 2b
(B) 0, – a (D) 0, 3a
(A) abc (a – b) (b – c) (c – a) (B) (a – b) (b – c) (c – a) (a + b + c) (C) (a – b) (b – c) (c – a) (ab + bc + ca) (D) None of these
(B) 0 (D)
c
Q.17
If
ax ax ax a x a x a x = 0, then value of x
are(A) 0, a (C) a, – a
(B) 1/abc (D) None of these
If each row of a determinant of third order of value is multiplied by 3, then the value of new determinant is (A) (B) 27 (C) 21 (D) 54 1 2
Q.16
Q.20
(D) None of these
ax ax ax
equal to (A) abc (C) 0 Q.14
r 1
n(3n 1) / 2
is equal to -
to(A) p (C) p3 Q.13
n
Dr
Q.24
If
x
is
real
number
such
that
x 1 x 2 x x 2 x 3 x = 0 then are in x3 x4
(A) A.P. (C) H.P.
x
(B) G.P. (D) None of these
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Q.25
1 a
The determinant
1 b
1 c
is
Q.32
a 2 bc b 2 ca c 2 ab
equal to (A) 0 (C) – 1
Q.26
(B) 1 (D) None of these
1 m C1
1 m 1 C1
m2
C1 =
m
m 1
m2
C2
C2
(A) m(m + 1) (C) 1 Q.27
Question based on
(B) m(m – 1) (D) 0
Q.34
(A) –1, 2 (C) 2, 0
(B) –1, 0 (D) 1, 2
x 2 x 3 x 2a x 3 x 4 x 2b equals -
Q.29
i
(C) 2a
(D) 2x Q.37
a b
abc 5a 4b 3c
= 3a 4a 3b
value
( x 2) 2 ( x 1) x
2
(A) 0
2
of
(x 1) 2 x
2
(x 1)
If A + B + C = , then sin (A B C) sin B
sin B 0
cos (A B)
tan A
the
where a = i,
2
(B) 8x2
determinant
x2 (x 1) 2 ( x 2)
is-
2
(C) 8
cos C tan A equals0
(B) 2sinB tanA cosC (D) None of these
The value of
0 a b a c ba 0 b c iscb
0
(A) 0 (B) abc (C) (a – b) (b – c) (c – a) (D) None of these
b = c = then is equal to(A) i (B) – 2 (C) (D) – i The
Symmetric and skew symmetric Determinants
ca
(B) 7 – 4i (D) 4 – 7i
6a 9a 6b 11a 9b 6c
Q.31
(B) 3 (D) None of these
The value of an odd order skew symmetric determinant is(A) perfect square (B) negative (C) ±1 (D) 0
1 i 1 i
a
Q.30
6 x
Q.36
(where i = 1 ) equals -
(A) 7 + 4i (C) 4 + 7i
= 0 then x =
The value of an even order skew symmetric determinant is(A) 0 (B) perfect square (C) ±1 (D) None of these
x 4 x 5 x 2c
(B) 0
(D) 4
Q.35
If a, b, c are in A.P., then the value of
1 i 1 i i 1 i i 1 i
3
(C) – 20 3 3
(A) 0 (C) 1
5x 2
(A) 1
3 x 6 6 3 x
(A) 6 (C) 0
20 5 =0
1 2x
If
(B) – 2
3
Find the value of x in the equation 1 4 1 2
Q.28
Q.33
=
7581 7591
(A) 20
1
C2
7579 7589
(D) –8
Question based on
Crammer's Rule
Q.38
The equations x + 2y + 3z = 1, 2x + y + 3z = 2 and 5x + 5y + 9z = 4 have(A) unique solution (B) many solutions (C) inconsistent (D) None of these
Q.39
The existence of unique solution of the system x + y + z = b, 2x + 3y – z = 6, 5x – y + az = 10 depends on(A) b only (B) a only (C) a and b (D) neither a nor b
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Q.40
Given the system of equations px + y + z = 1, x + py + z = p, x + y + pz = p2, then for what value of p does this system have no solution (A) –2 (B) –1 (C) 1 (D) 0
Q.41
The value of k for which the set of equations 3x + ky – 2z = 0, x + ky + 3z = 0 and 2x + 3y – 4z = 0 has a non – trivial solution is(A) 15 (B) 16 (C) 31/2 (D) 33/2
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LEVEL – 2 a 2 b2 c
c
a
b2 c2 a
b
b
Q.1
Q.2
c
(C) 4abc
(D) 0
33
3
3
4
3
3.3 3.3 1 is equal to-
4
3
5
3
3.4 2 3.4 1
a is expressible as m a
ab bc
value of m is(A) – 1 (B) 0
(D) 4
a b c If = x y z and 2 = r
y b q x a p then z
c
b b
c c , then the
(C) 1
(D) 2
In a third order determinant each element of the first column consists of sum of two terms, each element of the second column consists of sum of three terms and each element of third column consists of sum of four terms, then it can be decomposed into n determinants, where n has the value(A) 1 (B) 9 (C) 16 (D) 24
Q.9
For any ABC, the value of determinant
r
sin 2 A cot A 1
1 is equal to-
sin 2 B cot B 1 issin 2 C
(A) 22
(B) 2
(C) –2
(D) None of these
ax The determinant x 2 1 1 a
1 b
1 c
2
2
2
z
1 x
1 y
1 z
2
2
2
b
a b a b
Q.8 then
c a b
(C) 3
y
ca c a
a b c
(B) 1 (D) None of these
(B) 2
a
If the determinant b c
b c c a a b
2
is equal to(A) 1
(C)
(B) 25 (D) None of these
3.2 2 3.2 1
a b bc ca a b c If b c c a a b = b c a
x
25 10
bc
Q.7
23
(A)
, then
(A) 0 (C) 625
(B) 2abc
p q
Q.5
3
8 9
D1 + D2 + D3 + D4 + D5 is equal to-
c2 a 2 b
(A) abc
ca
Q.4
p
is equal to-
a
(A) 0 (C) 92
Q.3
Q.6
p 15 If Dp = p 2 35
c
(A) 0 (B) 1 (C) sin A sin B sin C (D) sin A + sin B + sin C
by cz y 2 z 2 is equal to1
1
a (B) x
b y
c z
yz zx xy
cot C 1
Q.10
If Sr =
2r 6r 2 1
x n (n 1) y n 2 (2n 3)
3
4r 2nr z
then
3
n (n 1)
n
S
r
does not depends on
r 1
(D) None of these
(A) x (C) n
(B) y (D) all of these
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Q.11
If a, b, c are non-zero real numbers, then b 2c 2
bc b c
c2a 2
ca
2 2
a b
ab a b
(B) a2b2c2 (D) None of these
x p q p x q is equal to p q
Q.17
[x]
[ z] 1
5
value 5
C3 14
C1
5
C4
1
C2
5
C5
1
b c
If = b c a , then 2 is equal to-
bc a 2
ca b 2
ab c 2
2
2
bc a 2
ab c 2
bc a 2
ca b 2
bc a 2
ca b 2
ab c 2
(B) ca b 2
ab c 2
bc a 2
ab c 2
bc a 2
ca b 2
a 2 bc b 2 ca
c 2 ab
(A) ca b
(C) (2x) !. (x + 3) ! (D) None of these The determinant is-
ab c
(C) b 2 ca c 2 ab a 2 bc
cos
c 2 ab a 2 bc b 2 ca
(A) 0
(D) None of these
(B) independent of (C) independent of
Q.19
If ax + by + cz = 1, bx + cy + az = 0 = cx + ay +
(D) independent of both and bz, then Q.15
determinant
c a b
(A) (2x) !. (x + 1) !. (x + 2) !. (x + 3) ! (B) 2 (x) !. (x + 1) !. (x + 2) !
sin
the
(B) – (6!) (D) None of these a
Q.18
is equal to-
is-
(A) 0 (C) 80
( x 2)! ( x 3)! ( x 4)!
cos
of
C0
If x is a positive integer then the value of
cos ( ) sin ( ) cos 2 sin cos sin
[ y]
(B) [y] (D) None of these
The 5
x! (x 1)! ( x 2)! determinant ( x 1)! (x 2)! (x 3)! is-
[z ] [z ]
5
x
(D) (x + p) (x + q) (x + p + q)
Q.14
[ x ] 1 [ y] 2, then [ x ] [ y] 1
(A) [x] (C) [z]
(A) (x + p) (x + q) (x – p – q) (B) (x – p) (x – q) (x + p + q) (C) (x – p) (x – q) (x – p – q)
Q.13
If [a] denotes the greatest integer less than or equal to a and – 1 x < 0, 0 y < 1, 1 z <
c a is equal to
(A) abc (C) ab + bc + ca
Q.12
Q.16
13 3
2 5
5
15 26
5
10 equals-
65 3
15
5
(A) 0 (C) – 1
x y z x
z y
a b c c a b is equal to-
y
x
b c a
z
(B) 1 (D) 2
(A) 0 (B) 5 3 ( 6 – 5) (C) 5 3 (5 – 6 ) (D) None of these
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LEVEL – 3 Q.1
xn
x n2
x n 3
If y n
y n2
y n3
n
n 2
z n 3
z
z
1 1 1 = (x–y) (y–z) (z–x) , then n = x y z (A) 1 (B) –1 (C) 2 (D) –2
Q.2
Q.6
The values and for which the system of equations x + y + z = 6, x + 2y + 3z = 10 and x + 2y + z = have unique solution are(A) 3, R (B) 3, 10 (C) 3, 10 (D) 3, 10
Q.7
The system of linear equations x + y + z = 2, 2x + y – z = 3, 3x + 2y + kz = 4 has a unique solution if(A) k (B) –1 < k > 1 (C) –2 < k < 2 (D) k = 0
If are the roots of x3 + ax2 + b = 0, then the value of (A) – (C) a3
is equals to-
a3
x2 x
(B) a3 –3b (D) a2 – 3b
Q.8
If
x 1
2
2 x 3x 1 2
x 2x 3
Q.3
If A, B and C are the angles of a triangle and 1 1 sin A
1 1 sin B
1 1 sin C
2
2
2
3x
x2 3x 3 = Ax – 12,
2x 1 2x 1
then the value of A is(A) 12 (B) 24 (C) –12 (D) –24
0 ,
sin A sin A sin B sin B sin C sin C
then the triangle ABC is(A) isosceles (B) equilateral (C) right angled isosceles (D) none of these Q.4
Q.9
sin cos
(A) 441 × 446 × 451 (B) 0 (C) –1 (D) 1
1 1
x 3 2x 2 18
Q.10
1
Let
r
Q.11
216 1
D r b 3(4 r ) 2(416 1) ,
2
3
f(1).f(3) + f(3).f(5) + f(5).f(1) is equal to(A) f(1) (B) f(3) (C) f(1) + f(3) (D) f(1) + f(5)
(A) independent of (B) independent of (C) independent of and (D) none of these 2r
3x 3 81
If f(x) = x 5 2x 2 50 4x 3 500 then
is-
cos( ) sin( ) 1
a
441 442 443 445 446 447 is449 450 451
The value of the determinant cos sin
Q.5
The value of
then the
If the system of equations, x + 2y – 3z =1, (k + 3)z = 3,(2k + 1)x + z = 0 is inconsistent, then the value of k is(A) –3 (B) 1/2 (C) 0 (D) 2
16
c 7(8 ) 4(8 1) 16
value of Σ D r is equals tor 1
(A) 0 (B) a + b + c (C) ab + bc + ca (D) none of these
Q.12
1 a b In a ABC, if 1 c a 0 then 1 b c
sin2 A
+ (A) 9/4 (C) 1
sin2 B
+ sin2 C is equal to(B) 4/9 (D) 3 3
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Q.13
Q.14
The equation x + 2y + 3z = 1, 2x + y + 3z = 2, 5x + 5y + 9z = 4 have(A) unique solution (B) infinitely many solutions (C) inconsistent (D) None of these 1 1 1 cos (nx) cos (n 1) x cos (n 2) x sin (nx)
sin (n 1) x
dependent(A) on x (C) both on x and n
is not
sin (n 2) x
(B) on n (D) None of these
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ANSWER KEY LEVEL- 1 Q.No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
Ans.
D
B
C
C
C
A
B
B
D
C
A
B
C
B
A
D
A
C
C
D
C
Q.No.
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
Ans.
C
C
A
A
C
A
B
C
A
D
C
C
A
B
D
A
A
B
A
D
LEVEL- 2 Q.No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
Ans.
C
A
B
B
B
D
B
D
A
D
D
B
B
B
B
C
D
A
B
LEVEL- 3 Q.No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Ans.
B
C
A
A
A
A
A
B
B
B
A
A
A
B
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