SABHYA ACADEMY
DETERMINANTS
Class 12 - Mathematics
DPP-01
∘ ∘
∣ cos 15 sin 15 ∣
[1]
1. ∣ ∣ =?
Y
∘ ∘
∣ sin 15 cos 15 ∣
a) b)
√3 √3
M
4 2
c) d) 1
1
∣1 1 1 ∣
[1]
E
∣ ∣
2. The maximum value of ∣ 1 1 + sin θ 1
∣
is
∣1 1 + cos θ ∣
AD
1
a) −1
2
b) 4
c) 1
2
d) 0
∣ 2x 5 ∣ ∣6 −2 ∣ [1]
AC
3. If ∣ ∣ = ∣ ∣ , then value of x is
∣ 8 x∣ ∣7 3 ∣
a) 3 b) ± 3
c) 6 d) ± 6
YA
∣0 2 0 ∣
[1]
∣ ∣
4. There are two values of λ such that ∣ λ 3 λ
∣
= - 16 then the sum of two values of λ is
∣λ 5 6 ∣
a) 0 b) 5
BH
c) 6 d) 3
⎡
1 sin θ 1
⎤ [1]
5. The value of det A where A=⎢ − sin θ 1 sin θ ⎥ lies in the interval
SA
⎣ ⎦
−1 − sin θ 1
a) [ 0 ,2 ] b) (2,1)
c) [2,4] d) (1,2)
6. If A = ∣
∣1 2∣
∣ , then find the value of k if |2A| = k|A| [1]
∣4 2∣
a) - 4 b) 4
c) 0 d) 3
7. If A is a matrix of order 3 × 3, then |3A| is equal to [1]
a) 81 IAI b) 27 IAI
c) 9 IAI d) 3 IAI
[1]
1/6
∣α 3 4∣
∣ ∣
8. If 1 2 1 = 0, then the value of α is
∣ ∣
∣1 4 1∣
a) 4 b) 1
c) 3 d) 2
9. Let A be a 3 × 3 square matrix such that A (adj A) = 2I, where I is the identity matrix. The value of |adj A| is [1]
a) 2 b) 4
c) -4 d) 0
∣ 0 a −b ∣ [1]
∣ ∣
10. The value of the determinant −a 0 −c is
∣ ∣
Y
∣ b c 0 ∣
a) 0 b) a
M
c) - a d) b
11. If A is a 3-rowed square matrix and |3A| = k |A| then k = ? [1]
E
a) 9 b) 1
12.
c) 3
The value of ∣
∣ cos 20
∣ sin 70
∘
∘
sin 20
cos 70
∘
∘
∣
∣
∣
is
AD
d) 27
[1]
AC
a) 0 b) 1
c) -1 d) 1
∘ ∘
13.
∣ cos 70
∣
sin 20 ∣
∣ =? [1]
∘ ∘
∣ sin 70 cos 20 ∣
YA
a) cos 50o b) sin 50o
c) 1 d) 0
∣ 2 cos x 1 0 ∣
[1]
BH
∣ ∣
14. If f(x) = ∣ then, f ( ) =
π
1 2 cos x 1
∣ 3
∣ 0 1 2 cos x ∣
a) 0 b) 1
SA
c) –1 d) 2
∘ ∘
15.
∣ sin 23
∣
− sin 67 ∣
∣ =? [1]
∘ ∘
∣ cos 23 cos 67 ∣
a) √3
b) sin 16o
2
c) 1 d) cos 16o
16.
∣ a + ib
∣
c + id ∣
∣ =? [1]
∣ −c + id a − ib ∣
a) (a2 - b2 - c2 - d2) b) (a2 + b2 + c2 - d2)
c) (a2 - b2 + c2 - d2) d) (a2 + b2 + c2 + d2)
17. If ∣
∣2 4∣ ∣ 2x
∣ = ∣
4 ∣
∣ , then x is [1]
∣5 1∣ ∣ 6 x∣
2/6
– –
a) ± √2 b) ± √3
c) 1 d) 0
∣ cos θ − sin θ ∣ [1]
18. The value of ∣ ∣ is
∣ sin θ cos θ ∣
a) 0 b) 1
c) 5 d) 2
∣ 1 2 4∣ [1]
∣ ∣
19. The value of the determinant Δ = ∣
−1 3 0
∣
is
∣ 4 1 0∣
a) 44 b) -52
Y
c) 52 d) -44
∣ x sin θ cos θ ∣ [1]
M
∣ ∣
20. The determinant ∣ − sin θ −x 1
∣
is
∣ cos θ 1 x∣
E
a) independent of both θ and x b) independent of x only
c) independent of -x only d) independent of θ only
21. Evaluate Δ =
⎡
⎢ − sin α
⎣
0
cos α
sin α
− sin β
0
− cos α
sin β
0
⎤
⎥
⎦
AD [2]
22. A matrix A of order 3 × 3 is such that |A| = 4. Find the value of |2A|. [2]
AC
23. Evaluate [2]
∣ cos θ − sin θ ∣
i. ∣ ∣
∣ sin θ cos θ ∣
2
∣x − x+ 1 x− 1 ∣
ii. ∣ ∣
YA
∣ x+ 1 x+ 1 ∣
24. Find values of x, if ∣
∣2 3∣ ∣ x
∣ = ∣
3∣
∣
[2]
∣4 5∣ ∣ 2x 5∣
∣1 0 1∣ [2]
BH
∣ ∣
25. If A = 0 1 2 then show that |3A| = 27|A|
∣ ∣
∣0 0 4∣
∣x 3 7 ∣ [3]
∣ ∣
26. Solve the equation: ∣ 2 x 2
∣
= 0 .
SA
∣7 6 x∣
27. A matrix of order 3 × 3 has determinant 2. What is the value of |A(3I)|, where I is the identity matrix of order 3 [3]
× 3.
28. If the lines ax + y + 1 = 0, x + by + 1 = 0 and x + y + c = 0 are concurrent, prove that [3]
∣a 1 1∣
∣ ∣
1 b 1 = 0 .
∣ ∣
∣1 1 c∣
∣x 2 3 ∣ [3]
∣ ∣
29. If x= – 4 is a root of Δ = ∣
1 x 1
∣
= 0 , then find the other two roots.
∣ 3 2 x∣
∣ x −6 −1 ∣ [5]
∣ ∣
30. Show that x = 2 is a root of the equation 2 −3x x− 3 = 0 and solve it completely.
∣ ∣
∣ −3 2x x+ 2∣
[5]
3/6
∣ 1 cos C cos B ∣
∣ ∣
31. If A + B + C = 0, then prove that cos C 1 cos A = 0
∣ ∣
∣ cos B cos A 1 ∣
32. Read the following text carefully and answer the questions that follow: [4]
1 0
Let A = [ ] , and U1, U2 are first and second columns respectively of a 2 × 2 matrix U. Also, let the
2 1
1 2
column matrices U1 and U2 satisfying AU1 = [ ] and AU2 = [ ] .
0 3
i. What is the value of |X| if X is defined as X = [3 2]? (1)
ii. What is the minor of the element at position a22 in matrix U? (1)
iii. What is the result of adding matrices U1 and U2? (2)
OR
Y
What is the value of the determinant of matrix U? (2)
Section B
EM
33. Find the area of the triangle with vertices (0,0), (4,2), and (1,1). [1]
a) 1 sq.unit b) 2 sq.unit
c) 0 sq.unit d) 5 sq.unit
34.
a) 3
AD
If the area of a △ABD is 3 sq. units with vertices A(1, 3), B(0, 0) and D(k, 0), then k is equal to
b) 2
[1]
c) ±2 d) ±3
AC
35. If the points (a1, b1), (a2, b2) and (a1 + a2, b1 + b2) are collinear, then [1]
a) a1 - b1 = a2 - b1 b) a1a2 = b1b2
c) a1b2 = a2b1 d) a1b1 = a2b2
YA
36. The three points (a + 5, a - 4), (a - 2, a + 3) and (a, a) [1]
a) do not lie in a straight line b) coincide
BH
c) lie in a straight line d) are vertices of equilateral triangle
37. Let A be the area of a triangle having vertices (x1, y1), (x2, y2) and (x3, y3). Which of the following is correct? [1]
2
a) ∣ x 1 y 1∣ b) ∣ x1 y
1
1∣
SA
1
∣ ∣ ∣ ∣
∣ x2 y
2
1∣ = A2 ∣ x2 y
2
1 ∣ = ±A
∣ ∣ ∣ ∣
∣ x3 y3 1∣ ∣ x3 y3 1∣
c) ∣ x 1 y
1
1∣ d) ∣ x1 y
1
1∣
∣ ∣ ∣ ∣
A
∣ x2 y 1∣ = ± ∣ x2 y 1 ∣ = ±2A
2 2 2
∣ ∣ ∣ ∣
∣ x3 y3 1∣ ∣ x3 y3 1∣
38. The area of a triangle with vertices (–3, 0), (3, 0) and (0, k) is 9 sq. units. The value of k will be [1]
a) 6 b) 3
c) -9 d) 9
39. Find the area of triangle with vertices (1 ,1) , (2, 2) and (3, 3). [1]
a) 1 b) 3
c) 0 d) 2
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40. If area of triangle is 35 sq units with vertices (2, -6), (5, 4) and (k, 4). Then k is [1]
a) 12 b) 2, -2
c) 12, -2 d) -2
41. If the points representing the complex numbers - 4 +3i, 2 – 3i and 0 + pi are collinear, then the value of p is [1]
a) 1 b) - 1
c) 2 d) - 2
42. The equation of line joining (2, 3) and (-1, 2) is [1]
a) x - 3 y + 7 = 0 b) x + 3y - 7 = 0
c) x - 3y - 7 = 0 d) x + 3y + 7 = 0
Y
43. If the points (3, - 2), (x, 2) and (8, 8) are collinear, find x using determinant. [2]
44. Using determinants, find the area of △PQR with vertices P(3, 1), Q(9, 3) and R(5, 7). Also, find the equation of [2]
M
line PQ using determinants.
45. Use determinants to show that the following points are collinear. P (-2, 5),Q (-6 , -7) and P (-5, -4). [2]
E
46. Find value of k if area of triangle is 4 sq. units and vertices are :(k ,0), (4, 0), (0,2). [2]
47.
48.
49.
AD
Using determinant show that the (5, 5), (- 5, 1) and (10, 7) points are collinear.
Find the area of the triangle with vertices at the points given (3, 8), (-4, 2) and (5, -1)
Use determinants to show that the following points are collinear. A (3, 8), B (-4,2) and C(10,14).
[2]
[2]
[2]
50. Using determinant show that the points (3, - 2), (8, 8) and (5, 2) are collinear. [2]
AC
51. If the points (a1, b1), (a2, b2) and (a1 + a2, b1 + b2) are collinear, show that a1b2 = a2 b1. [2]
52. Show that points A( a, b + c), B(b, c + a), c(c, a + b) are collinear. [2]
53. Find the equation of line joining (1, 2) and (3, 6) using determinants. [3]
54. If the points A(a , 0), B(0, b) and C(1, 1) are collinear, prove that 1
+
1
= 1 . [3]
YA
a b
55. Find the area of Δ whose vertices are (3, 8) (-4, 2) and (5, 1). [3]
56. Find the equation of the line joining A (1, 3) and B (0, 0) using determinants and find the value of k if D (k, 0) is [3]
a point such that area of ΔABD is 3 square units.
BH
57. Using determinants prove that the points (a, b), (a', b') and ( a - a', b - b') are collinear if ab' = a'b. [3]
58. Area of a triangle whose vertices are (x1, y1), (x2, y2) and (x3, y3) is given by the determinant [4]
∣ x1 y1 1∣
1 ∣ ∣
SA
Δ = x2 y2 1
2 ∣ ∣
∣ x3 y3 1∣
Since, area is a positive quantity, so we always take the absolute value of the determinant A. Also, the area of the
triangle formed by three collinear points is zero.
i. Find the area of the triangle whose vertices are (-2, 6), (3, -6), and (1, 5). (1)
ii. If the points (2, -3), (k, -1) and (0, 4) are collinear, then find the value of 4k. (1)
iii. If the area of a triangle ABC, with vertices A(1, 3), B(0, 0) and C(k, 0) is 3 sq. units, then find the value of k.
(2)
OR
Using determinants, find the equation of the line joining the points A(1, 2) and B(3, 6). (2)
59. Read the following text carefully and answer the questions that follow: [4]
Each triangular face of the Pyramid of Peace in Kazakhstan is made up of 25 smaller equilateral triangles as
5/6
shown in the figure.
– –
i. If the vertices of one of the smaller equilateral triangle are (0, 0), (3, √3) and (3, - √3), then find the area of
such triangle. (1)
ii. Find the area of a face of the Pyramid. (1)
Y
iii. Find the length of a altitude of a smaller equilateral triangle. (2)
OR
EM
Let A(a, 0), B(0, b) and C(1, 1) be three points. If = 1, then find the relation between three points.(2)
1 1
+
a b
AD
AC
YA
BH
SA
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