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## Thursday, 28 January 2021

### CBSE Class 12 Maths - MCQ and Online Tests - Unit 4 - Determinants

#### CBSE Class 12 Maths – MCQ and Online Tests – Unit 4 – Determinants

Every year CBSE conducts board exams for 12th standard. These exams are very competitive to all the students. So our website provides online tests for all the 12th subjects. These tests are also very effective and useful for those who preparing for competitive exams like NEET, JEE, CA etc. It can boost their preparation level and confidence level by attempting these chapter wise online tests.

These online tests are based on latest CBSE Class 12 syllabus. While attempting these our students can identify the weak lessons and continuously practice those lessons for attaining high marks. It also helps to revise the NCERT textbooks thoroughly.

#### CBSE Class 12 Maths – MCQ and Online Tests – Unit 4 – Determinants

Question 1.
If 7 and 2 are two roots of the equation $$\left[\begin{array}{ccc} x & 3 & 7 \\ 2 & x & 2 \\ 7 & 6 & x \end{array}\right]$$ then the third root is
(a) -9
(b) 14
(c) $$\frac{1}{2}$$
(d) None of these

Question 2.
$$\left[\begin{array}{ccc} 1 & a & a^{2}-bc \\ 1 & b & b^{2}-ca \\ 1 & c & c^{2}-ab \end{array}\right]$$ is equal to
(a) abc
(b) ab + bc + ca
(c) 0
(d) (a – b)(b – c)(c – a)

Question 3.
$$\left|\begin{array}{cc} x & -7 \\ x & 5 x+1 \end{array}\right|$$
(a) 3x² + 4
(b) x(5x + 8)
(c) 3x + 4x²
(d) x(3x + 4)

Question 4.
A = $$\left[\begin{array}{ll} \alpha & q \\ q & \alpha \end{array}\right]$$ |A³| = 125 then α =
(a) ±3
(b) ±2
(c) ±5
(d) 0

Question 5.
$$\left[\begin{array}{ccc} 1 & x & x^{2} \\ 1 & y & y^{2} \\ 1 & z & z^{2} \end{array}\right]$$
(a) (x – y) (y + z)(z + x)
(b) (x + y) (y – z)(z – x)
(c) (x – y) (y – z)(z + x)
(d) (x – y) (y – z) (z – x)

Answer: (d) (x – y) (y – z) (z – x)

Question 6.
If a ≠ 0 and $$\left[\begin{array}{ccc} 1+a & 1 & 1 \\ 1 & 1+a & 1 \\ 1 & 1 & 1+a \end{array}\right]$$ = 0 then a =
(a) a = -3
(b) a = 0
(c) a = 1
(d) a = 3

Question 7.
If x, y, z are all different from zero and
$$\left[\begin{array}{ccc} 1+x & 1 & 1 \\ 1 & 1+y & 1 \\ 1 & 1 & 1+z \end{array}\right]$$ = 0, then value of x-1 + y-1 + z-1 is
(a) xyz
(b) x-1y-1z-1
(c) -x – y – z
(d) -1

Question 8.
If x > 0 and x ≠ 1. y > 0. and y ≠ 1, z > 0 and z ≠ 1 then
$$\left[\begin{array}{ccc} 1 & log_{x}y & log_{x}z \\ log_{y}x & 1 & log_{y}z \\ log_{z}x & log_{z}y & 1 \end{array}\right]$$ is equal to
(a) 1
(b) -1
(c) 0
(d) None of these

Question 9.
The value of the determinant
$$\left[\begin{array}{ccc} 3 & 1 & 7 \\ 5 & 0 & 2 \\ 2 & 5 & 3 \end{array}\right]$$
(a) 124
(b) 125
(c) 134
(d) 144

Question 10.
$$\left[\begin{array}{ccc} y+z & z & x \\ y & z+x & y \\ z & z & x+y \end{array}\right]$$ is equal to
(a) 6xyz
(b) xyz
(c) 4xyz
(d) xy + yz + zx

Question 11.
If $$\left[\begin{array}{cc} 2 & 4 \\ 5 & 1 \end{array}\right]$$ = $$\left[\begin{array}{cc} 2x & 4 \\ 6 & x \end{array}\right]$$ then the value of x is
(a) ±2
(b) ±$$\frac{1}{3}$$
(c) ±√3
(d) ± (0.5)

Question 12.
If f(x) = $$\left[\begin{array}{ccc} 0 & x-a & x-b \\ x+a & 0 & x-c \\ x+b & x+c & 0 \end{array}\right]$$ then
(a) f(a) = 0
(b) f(b) = 0
(c) f(0) = 0
(d) f(1) = 0

Question 13.
If a, b, c are in A.P. then the determinant
$$\left[\begin{array}{ccc} x+2 & x+3 & x+2a \\ x+3 & x+4 & x+2b \\ x+4 & x+5 & x+2c \end{array}\right]$$
(a) 1
(b) x
(c) 0
(d) 2x

Question 14.
If $$\left[\begin{array}{cc} 2x & 5 \\ 8 & x \end{array}\right]$$ = $$\left[\begin{array}{cc} 6 & -2 \\ 7 & 3 \end{array}\right]$$ then the value of x is
(a) 3
(b) ±3
(c) ±6
(d) 6

Question 15.
Let f(t) = $$\left[\begin{array}{ccc} cot t & t & 1 \\ 2 sin t & t & 2t \\ sin t & t & t \end{array}\right]$$ then $$_{t→0}^{lim}$$ $$\frac{f(t)}{t^2}$$ is equal to
(a) 0
(b) -1
(c) 2
(d) 3

Question 16.
If w is a non-real root of the equation x² – 1 = 0. then
$$\left[\begin{array}{ccc} 1 & ω & ω^{2} \\ ω & ω^{2} & 1 \\ ω^{2} & 1 & ω \end{array}\right]$$ =
(a) 0
(b) 1
(c) ω
(d) ω²

Question 17.
If Δ = $$\left[\begin{array}{cc} 10 & 2 \\ 30 & 6 \end{array}\right]$$ then A =
(a) 0
(b) 10
(c) 12
(d) 60

Question 18.
The number of distinct real roots of $$\left[\begin{array}{ccc} sin x & cos x & cos x \\ cos x & sin x & cos x \\ cos x & cos x & sin x \end{array}\right]$$ = 0 in the interval –$$\frac{π}{4}$$ ≤ x ≤ $$\frac{π}{4}$$ is
(a) 0
(b) 2
(c) 1
(d) 3

Question 19.
The value of determinant $$\left[\begin{array}{ccc} a-b & b+c & a \\ b-c & c+a & b \\ c-a & a+b & c \end{array}\right]$$
(a) a³ + b³ + c ³
(b) 3bc
(c) a³ + b³ + c³ – 3abc
(d) None of these

Answer: (c) a³ + b³ + c³ – 3abc

Question 20.
The area of a triangle with vertices (-3, 0) (3, 0) and (0, k) is 9 sq. units. The value of k will be
(a) 9
(b) 3
(c) -9
(d) 6

Question 21.
The determinant $$\left[\begin{array}{ccc} b^{2}-ab & b-c & bc-ac \\ ab-a^{2} & a-b & b^{2}-ab \\ bc-ac & c-a & ab-a^{2} \end{array}\right]$$ equals
(a) abc(b – c)(c -a)(a – b)
(b) (b – c)(c – a)(a – b)
(c) (a + b + c)(b – c)(c – a)(a – b)
(d) None of these

Question 22.
If A, B and C are angles of a triangle, then the determinant
$$\left[\begin{array}{ccc} -1 & cos C & cos B \\ cos C & -1 & cos A \\ cos B & cos A & -1 \end{array}\right]$$
(a) 0
(b) -1
(c) 1
(d) None of these

Question 23.
There are two values of a which makes determinant
Δ = $$\left[\begin{array}{ccc} 1 & -2 & 5 \\ 2 & a & -1 \\ 0 & 4 & 2a \end{array}\right]$$ = 86, then sum of these number is
(a) 4
(b) 5
(c) -4
(d) 9

Question 24.
The maximum value of $$\left[\begin{array}{ccc} 1 & 1 & 1 \\ 1 & 1+sin θ & 1 \\ 1+cos θ & 1 & 1 \end{array}\right]$$ is (θ is real number)
(a) $$\frac{1}{2}$$
(b) $$\frac{√3}{2}$$
(c) $$\frac{2√3}{4}$$
(d) √2

Answer: (a) $$\frac{1}{2}$$

Question 25.
If A = $$\left[\begin{array}{ccc} 2 & \lambda & -3 \\ 0 & 2 & 5 \\ 1 & 1 & 3 \end{array}\right]$$ then A-1 exists if
(a) λ = 2
(b) λ ≠ 2
(c) λ ≠ -2
(d) None of these

Question 26.
$$\left|\begin{array}{cc} \cos \theta & -\sin \theta \\ \sin \theta & \cos \alpha \end{array}\right|$$
(a) 0
(b) 1
(c) 2
(d) 3

Question 27.
If A and B are invertible matrices, then which of the following is not correct?
(b) det (a)-1 = [det (a)]-1
(c) (AB)-1 = B-1A-1
(d) (A + B)-1 = B-1 + A-1

Answer: (d) (A + B)-1 = B-1 + A-1

Question 28.
The value of the determinant $$\left[\begin{array}{ccc} x & x+y & x+2y \\ x+2y & x & x+y \\ x+y & x+2y & x \end{array}\right]$$ is
(a) 9x² (x + y)
(b) 9y² (x + y)
(c) 3y² (x + y)
(d) 7x² (x + y)

Answer: (b) 9y² (x + y)

Question 29.
Evaluate the determinant Δ = $$\left|\begin{array}{cc} log_{3}512 & log_{4}3 \\ log_{3}8 & log_{4}9 \end{array}\right|$$
(a) $$\frac{15}{2}$$
(b) 12
(c) $$\frac{14}{3}$$
(d) 6

Answer: (a) $$\frac{15}{2}$$

Question 30.
If $$\left[\begin{array}{cc} x & 2 \\ 18 & x \end{array}\right]$$ = $$\left[\begin{array}{cc} 6 & 2 \\ 18 & 6 \end{array}\right]$$ x is equal to
(a) 6
(b) ±6
(c) -1
(d) -6