Sign in
12thPass logo
New chatPYQ LibraryDoubtsRank report
Sign in to see Recents

Your guest activity stays on this device

Sign in to save progress →
Sign in

Matrices and Determinants question

2020 · 4 Sep · Shift 1 · Q31
Guest · filters and generic practice availableBrowsing as a guest · PYQ filters and generic practice are available. Sign in only for personalised features and saved progress.
  1. PYQ Library
  2. /JEE Main
  3. /Mathematics
  4. /Matrices and Determinants
  5. /2020 · 4 Sep · Shift 1 · Q31

Matrices and Determinants question

2020 · 4 Sep · Shift 1 · Q31

JEE MainMathematicsMatrices and DeterminantsMCQ+4 / −1
If A=[cos⁡θisin⁡θisin⁡θcos⁡θ]A = \left[ {\begin{matrix} {\cos \theta } & {i\sin \theta } \\ {i\sin \theta } & {\cos \theta } \\ \end{matrix} } \right]A=[cosθisinθ​isinθcosθ​], (θ=π24)\left( {\theta = {\pi \over {24}}} \right)(θ=24π​) and A5=[abcd]{A^5} = \left[ {\begin{matrix} a & b \\ c & d \\ \end{matrix} } \right]A5=[ac​bd​], where i=−1i = \sqrt { - 1}i=−1​ then which one of the following is not true?
  1. A
    aaa 2 - ccc 2 = 1
  2. B
    0≤a2+b2≤10 \le {a^2} + {b^2} \le 10≤a2+b2≤1
  3. C
    aaa 2 - ddd 2 = 0
  4. D
    a2−b2=12{a^2} - {b^2} = {1 \over 2}a2−b2=21​
View written solutionFree

Correct answer: D

  1. Identify the matrix form

Given

A=[cos⁡θisin⁡θisin⁡θcos⁡θ]=cos⁡θ I+isin⁡θ [0110].A=\begin{bmatrix} \cos\theta & i\sin\theta\\ i\sin\theta & \cos\theta \end{bmatrix} = \cos\theta\,I + i\sin\theta\,\begin{bmatrix}0&1\\1&0\end{bmatrix}.A=[cosθisinθ​isinθcosθ​]=cosθI+isinθ[01​10​].

Let

J=[0110].J=\begin{bmatrix}0&1\\1&0\end{bmatrix}.J=[01​10​].

Then J2=IJ^2=IJ2=I, so

A=cos⁡θ I+isin⁡θ J.A=\cos\theta\,I+i\sin\theta\,J.A=cosθI+isinθJ.

This behaves like De Moivre's form, hence

An=cos⁡(nθ)I+isin⁡(nθ)J.A^n=\cos(n\theta)I+i\sin(n\theta)J.An=cos(nθ)I+isin(nθ)J.
  1. Compute A5A^5A5

Since θ=π24\theta=\dfrac{\pi}{24}θ=24π​,

5θ=5π24.5\theta=\frac{5\pi}{24}.5θ=245π​.

Therefore,

A5=[cos⁡5π24isin⁡5π24isin⁡5π24cos⁡5π24].A^5=\begin{bmatrix} \cos\frac{5\pi}{24} & i\sin\frac{5\pi}{24}\\ i\sin\frac{5\pi}{24} & \cos\frac{5\pi}{24} \end{bmatrix}.A5=[cos245π​isin245π​​isin245π​cos245π​​].

So,

a=d=cos⁡5π24,b=c=isin⁡5π24.a=d=\cos\frac{5\pi}{24},\qquad b=c=i\sin\frac{5\pi}{24}.a=d=cos245π​,b=c=isin245π​.
  1. Check each option

Option A: a2−c2=1a^2-c^2=1a2−c2=1

Here

a2=cos⁡25π24,a^2=\cos^2\frac{5\pi}{24},a2=cos2245π​,

and since c=isin⁡5π24c=i\sin\frac{5\pi}{24}c=isin245π​,

c2=i2sin⁡25π24=−sin⁡25π24.c^2=i^2\sin^2\frac{5\pi}{24}=-\sin^2\frac{5\pi}{24}.c2=i2sin2245π​=−sin2245π​.

Thus,

a2−c2=cos⁡25π24−(−sin⁡25π24)=cos⁡25π24+sin⁡25π24=1.a^2-c^2=\cos^2\frac{5\pi}{24}-\left(-\sin^2\frac{5\pi}{24}\right) =\cos^2\frac{5\pi}{24}+\sin^2\frac{5\pi}{24}=1.a2−c2=cos2245π​−(−sin2245π​)=cos2245π​+sin2245π​=1.

So A is true.

Option B: 0≤a2+b2≤10\le a^2+b^2\le 10≤a2+b2≤1

Now

b=isin⁡5π24  ⟹  b2=−sin⁡25π24.b=i\sin\frac{5\pi}{24}\implies b^2=-\sin^2\frac{5\pi}{24}.b=isin245π​⟹b2=−sin2245π​.

Hence

a2+b2=cos⁡25π24−sin⁡25π24=cos⁡5π12.a^2+b^2=\cos^2\frac{5\pi}{24}-\sin^2\frac{5\pi}{24} =\cos\frac{5\pi}{12}.a2+b2=cos2245π​−sin2245π​=cos125π​.

Since

5π12=75∘,\frac{5\pi}{12}=75^\circ,125π​=75∘,

we get

a2+b2=cos⁡75∘>0,a^2+b^2=\cos 75^\circ>0,a2+b2=cos75∘>0,

and certainly less than 111. Therefore

0≤a2+b2≤10\le a^2+b^2\le 10≤a2+b2≤1

is true.

Option C: a2−d2=0a^2-d^2=0a2−d2=0

Since a=da=da=d,

a2−d2=0.a^2-d^2=0.a2−d2=0.

So C is true.

Option D: a2−b2=12a^2-b^2=\dfrac12a2−b2=21​

We have

a2−b2=cos⁡25π24−(−sin⁡25π24)=1.a^2-b^2=\cos^2\frac{5\pi}{24}-\left(-\sin^2\frac{5\pi}{24}\right)=1.a2−b2=cos2245π​−(−sin2245π​)=1.

So

a2−b2≠12.a^2-b^2\ne \frac12.a2−b2=21​.

Thus D is not true.

  1. Conclusion

The statement which is not true is:

D\boxed{\text{D}}D​
PreviousNext

More from Matrices and Determinants

  • If the system of equations x+y+z=2 2x+4y–z=6 3x+2y+λ z=μ has infinitely many solutions, then2020 · MCQ
  • Suppose the vectors x1, x2 and x3 are the solutions of the system of linear equations, Ax = b when the vector b on the right side is equal to b1, b2 and b3 respectively. if x1​=​111​​…2020 · MCQ
  • Let λ∈ R . The system of linear equations 2x1 - 4x2 +λ x3 = 1 x1 - 6x2 + x3 = 2 λ x1 - 10x2 + 4x3 = 3 is inconsistent for:2020 · MCQ
  • If the minimum and the maximum values of the function f:[4π​,2π​]→R, defined by f(θ)=​−sin2θ−cos2θ12​−1−sin2θ−1−cos2θ10​11−2​​…2020 · MCQ
  • If a + x = b + y = c + z + 1, where a, b, c, x, y, z are non-zero distinct real numbers, then ​xyz​a+yb+yc+y​x+ay+bz+c​​…2020 · MCQ
  • If the system of linear equations x + y + 3z = 0 x + 3y + k2z = 0 3x + y + 3z = 0 has a non-zero solution (x, y, z) for some k ∈ R, then x + (zy​) is equal to :2020 · MCQ
  • The values of λ and μ for which the system of linear equations x + y + z = 2 x + 2y + 3z = 5 x + 3y +λ z = μ has infinitely many solutions are, respectively:2020 · MCQ
  • Let m and M be respectively the minimum and maximum values of ​cos2x1+cos2xcos2x​1+sin2xsin2xsin2x​sin2xsin2x1+sin2x​​…2020 · MCQ