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Number of solutions in an interval of a quadratic in one trig ratio

This type of question, asked 7 times: 2021 Paper 25, 14 Jun 2022, 25 Jun 2022, 29 Jun 2022, 2 Apr 2025, 3 Apr 2025, 23 Jan 2026 S1. All 9 questions below, oldest first.

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  1. Question 1

    2021 Paper 25

    86
    If 3(cos⁡2x)=(3−1)cos⁡x+1\sqrt{3}\left( \cos^{2}x \right)= (\sqrt{3}- 1)\cos x+ 1, the number of solutions of the given equation when x∈[0,π2]x\in\left\lbrack 0,\frac{\pi}{2} \right\rbrack is

  2. Question 2

    14 Jun 2022

    163
    The number of solutions of the equation cos⁡(x+π3)cos⁡(π3−x)=14cos⁡22x,x∈[−3π\cos\left( x +\frac{\pi}{3} \right)\cos\left( \frac{\pi}{3}- x \right)=\frac{1}{4}\cos^{2}2x,x \in \lbrack - 3\pi, 3π]3\pi\rbrack is:

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  3. Question 3

    25 Jun 2022

    85
    The number of values of xx in the interval (π4,7π4)\left( \frac{\pi}{4},\frac{7\pi}{4} \right) for which 14csc⁡2x−2sin⁡2x=21−4cos⁡2x14\csc^{2}x- 2\sin^{2}x= 21 - 4\cos^{2}x holds, is

  4. Question 4

    29 Jun 2022

    84
    The number of elements in the set S=S= {θ∈[−4π,4π]:3cos⁡22θ+6cos⁡2θ−\left\{ \theta\in \lbrack - 4\pi,4\pi\rbrack:3\cos^{2}2\theta+ 6\cos2\theta- \right.  10cos⁡2θ+5=0}\left. \ 10\cos^{2}\theta+ 5 = 0 \right\} is

  5. Question 5

    29 Jun 2022

    174
    The number of solutions of the equation sin⁡x=\sin x= cos⁡2x\cos^{2}x in the interval (0,10)(0,10) is

  6. Question 6

    2 Apr 2025

    134
    If θ∈[−7π6,4π3]\theta\in\left\lbrack -\frac{7\pi}{6},\frac{4\pi}{3} \right\rbrack, then the number of solutions of 3csc⁡2θ−2(3−1)csc⁡θ−4=0\sqrt{3}\csc^{2}\theta- 2(\sqrt{3}- 1)\csc\theta- 4 = 0, is equal to.

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  7. Question 7

    2 Apr 2025

    57
    If θ∈[−2π,2π]\theta\in \lbrack - 2\pi,2\pi\rbrack, then the number of solutions of 22cos⁡2θ+(2−6)cos⁡θ−3=02\sqrt{2}\cos^{2}\theta+ (2 -\sqrt{6})\cos\theta-\sqrt{3}= 0 is equal to:

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  8. Question 8

    3 Apr 2025

    141
    The number of solutions of equation (4−3)sin⁡x\left( 4 -\sqrt{3} \right)\sin x −23cos⁡2x=−41+3,x∈- 2\sqrt{3}\cos^{2}x= -\frac{4}{1 +\sqrt{3}},x\in [−2π,5π2]\left\lbrack - 2\pi,\frac{5\pi}{2} \right\rbrack is.

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  9. Question 9

    23 Jan 2026 S1

    66
    Number of solutions of 3cos⁡2θ+8cos⁡θ+33=0,θ∈[−3π,2π]\sqrt{3}\cos2\theta + 8\cos\theta + 3\sqrt{3}= 0,\theta \in \lbrack - 3\pi,2\pi\rbrack is:

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