PYQ Vault
EbooksTutor

Determinant with sin^2 x and cos^2 x entries whose columns add to a constant: zeros, maximum or minimum

This type of question, asked 6 times: 2021 Paper 6, 2021 Paper 9, 2021 Paper 16, 2021 Paper 18, 1 Feb 2023, 29 Jan 2025. All 6 questions below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    2021 Paper 6

    61
    The maximum value of f(x)=∣sin⁡2x1+cos⁡2xcos⁡2x1+sin⁡2xcos⁡2xcos⁡2xsin⁡2xcos⁡2xsin⁡2x∣,x∈Rf(x) =\left| \begin{matrix} \sin^{2}x & 1 +\cos^{2}x & \cos2x \\ 1 +\sin^{2}x & \cos^{2}x & \cos2x \\ \sin^{2}x & \cos^{2}x & \sin2x \end{matrix} \right|,x\in R is:

    Tap an option to check your answer.

  2. Question 2

    2021 Paper 9

    64
    The solutions of the equation ∣1+sin⁡2xsin⁡2xsin⁡2xcos⁡2x1+cos⁡2xcos⁡2x4sin⁡2x4sin⁡2x1+4sin⁡2x∣=0,(0<x<π)\left| \begin{matrix} 1 +\sin^{2}x & \sin^{2}x & \sin^{2}x \\ \cos^{2}x & 1 +\cos^{2}x & \cos^{2}x \\ 4\sin2x & 4\sin2x & 1 + 4\sin2x \end{matrix} \right|= 0,(0 <x<\pi), are

    Tap an option to check your answer.

  3. Question 3

    2021 Paper 16

    54
    Let f(x)=∣sin⁡2x−2+cos⁡2xcos⁡2x2+sin⁡2xcos⁡2xcos⁡2xsin⁡2xcos⁡2x1+cos⁡2x∣;x∈(0,π]f(x) =\left| \begin{matrix} \sin^{2}x & - 2 +\cos^{2}x & \cos2x \\ 2 +\sin^{2}x & \cos^{2}x & \cos2x \\ \sin^{2}x & \cos^{2}x & 1 + \cos2x \end{matrix} \right|;x\in (0,\pi\rbrack. Then the maximum value of f(x)f(x) is?

  4. Question 4

    2021 Paper 18

    66
    Let θ∈(0,π2)\theta\in\left( 0,\frac{\pi}{2} \right). If the system of linear equations
    (1+cos⁡2θ)x+sin⁡2θy+4sin⁡3θz=0\left( 1 +\cos^{2}\theta \right)x +\sin^{2}\theta y + 4\sin3\theta z = 0
    cos⁡2θx+(1+sin⁡2θ)y+4sin⁡3θz=0\cos^{2}\theta x +\left( 1 +\sin^{2}\theta \right)y + 4\sin3\theta z = 0
    cos⁡2θx+sin⁡2θy+(1+4sin⁡3θ)z=0\cos^{2}\theta x +\sin^{2}\theta y + (1 + 4\sin3\theta)z = 0
    has a non-trivial solution, then the value of θ\theta is:

    Tap an option to check your answer.

  5. Question 5

    1 Feb 2023

    79
    Let f(x)=∣1+sin⁡2xcos⁡2xsin⁡2xsin⁡2x1+cos⁡2xsin⁡2xsin⁡2xcos⁡2x1+sin⁡2x∣,x∈[π6,π3]f(x) =\left| \begin{matrix} 1 +\sin^{2}x & \cos^{2}x & \sin2x \\ \sin^{2}x & 1 +\cos^{2}x & \sin2x \\ \sin^{2}x & \cos^{2}x & 1 + \sin2x \end{matrix} \right|,x \in\left\lbrack \frac{\pi}{6},\frac{\pi}{3} \right\rbrack. If α\alpha and β\beta respectively are the maximum and the minimum values of ff, then

    Tap an option to check your answer.

  6. Question 6

    29 Jan 2025

    52
    Let M and m respectively be the maximum and the minimum values of
    f(x)=∣1+sin⁡2xcos⁡2x4sin⁡4xsin⁡2x1+cos⁡2x4sin⁡4xsin⁡2xcos⁡2x1+4sin⁡4x∣,x∈Rf(x) =\left| \begin{matrix} 1 +\sin^{2}x & \cos^{2}x & 4\sin4x \\ \sin^{2}x & 1 +\cos^{2}x & 4\sin4x \\ \sin^{2}x & \cos^{2}x & 1 + 4\sin4x \end{matrix} \right|,x \in R
    Then M4−m4M^{4}-m^{4} is equal to :

    Tap an option to check your answer.