Class 11 Mathematics Trigonometric Equations MCQs Set 05

SINGLE CORRECT OPTION

Question. \( A = \{x \in \mathbb{R} | \cos 2x + 3\cos x + 2 = 0\} \) and \( B = \{x \in \mathbb{R} | \sqrt{2x - 5} < 1\} \). Then the number of elements in the set \( A \cap B \) is
(a) 0
(b) 1
(c) 2
(d) 4
Answer: (c) 2

Question. The number of distinct real values of the parameter 'a' for which the equation \( \int_{0}^{x} \sin^{2} \frac{t}{2} dt = a^{2}x^{2} - \left( \frac{3x - 1}{2} \right) + \frac{1}{a^{2}} \) possesses real roots belonging to the interval \( [0, 4\pi] \)
(a) 0
(b) 2
(c) 4
(d) infinite
Answer: (c) 4

Question. The number of distinct real roots of the equation \( \cos \left( \frac{\sqrt{2} + 1}{2} \right) x \cos \left( \frac{\sqrt{2} - 1}{2} \right) x = 1 \) is
(a) Infinite
(b) 1
(c) 0
(d) 4
Answer: (b) 1

Question. The variable 'x' stisfying the equation \( |\sin x \cos x| + \sqrt{2 + \tan^{2} x + \cot^{2} x} = \sqrt{3} \), belongs to the interval
(a) \( \left[0, \frac{\pi}{3} \right] \)
(b) \( \left( \frac{\pi}{3}, \frac{\pi}{2} \right) \)
(c) \( \left[ \frac{3\pi}{4}, \pi \right) \)
(d) non existent
Answer: (d) non existent

Question. The number of points P(x, y) lying inside or on the circle \( x^2 + y^2 = 9 \) and satisfying the equation \( \tan^4 x + \cot^4 x + 2 = 4 \sin^2 y \), is
(a) 2
(b) 4
(c) 8
(d) Infinite
Answer: (c) 8

Question. The equation \( 2x = (2n + 1) \pi (1 - \cos x) \), (where n is a positive integer)
(a) has infinitely many real roots
(b) has exactly one real root
(c) has exactly 2n + 2 real roots
(d) has exactly 2n + 3 real roots
Answer: (c) has exactly 2n + 2 real roots

Question. The number of solutions of the equation \( 16 (\sin^5 x + \cos^5 x) = 11 (\sin x + \cos x) \) in the interval \( [0, 2\pi] \) is
(a) 6
(b) 7
(c) 8
(d) 9
Answer: (a) 6

Question. The number of non similar isosceles possible triangle's suchthat \( \tan A + \tan B + \tan C = 100 \) is
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (b) 2

Question. The number of soultions of \( [\sin x + \cos x] = 3 + [-\sin x] + [-\cos x] \) (where [.] denotes the greatest integer function), \( x \in [0, 2\pi] \) is
(a) 0
(b) 4
(c) infinite
(d) 1
Answer: (a) 0

Question. The equation \( \cos^8 x + b \cos^4 x + 1 = 0 \) will have a solution if b belongs to
(a) \( (-\infty, 2] \)
(b) \( [2, \infty) \)
(c) \( (-\infty, -2] \)
(d) \( [-2, 0] \)
Answer: (c) \( (-\infty, -2] \)

Question. The number of values of y in \( [-2\pi, 2\pi] \) satisfying the equation \( |\sin 2x| + |\cos 2x| = |\sin y| \) is
(a) 3
(b) 4
(c) 5
(d) 6
Answer: (b) 4

Question. If both the distinct roots of the equation \( |\sin x|^2 + |\sin x| + b = 0 \) in \( [0, \pi] \) are real, then the values of b are
(a) \( [-2, 0] \)
(b) \( (-2, 0) \)
(c) \( [-2, 0) \)
(d) \( (0, 0) \)
Answer: (b) \( (-2, 0) \)

Question. The total number of solutions of \( \sin \{x\} = \cos \{x\} \) (where {.} denotes the fractional part) in \( [0, 2\pi] \) is equal to
(a) 5
(b) 6
(c) 8
(d) 9
Answer: (b) 6

Question. The number of integral pairs (x,y) that satisfy the following two equations
\( \begin{cases} \cos(xy) = x \\ \tan(xy) = y \end{cases} \) is
(a) 1
(b) 2
(c) 4
(d) 6
Answer: (a) 1

Question. Number of solutions of the equation \( \cos(\theta) \cos(\pi\theta) = 1 \) is
(a) 0
(b) 2
(c) 1
(d) infinite
Answer: (c) 1

Question. Number of solutions of the equations \( y = \frac{1}{3} [\sin x + [\sin x + [\sin x]]] \) and \( [y + [y]] = 2\cos x \), (where [.] denotes the greatest integer function) is
(a) 0
(b) 1
(c) 2
(d) infinite
Answer: (a) 0

Question. The value of a for which the equation \( 4\text{cosec}^{2} \{\pi (a + x)\} + a^{2} - 4a = 0 \) has a real solution, is
(a) 0
(b) 1
(c) 2
(d) 3
Answer: (c) 2

Question. The equation \( \tan |x| = \tan x, x \in [-2\pi, 2\pi] \)
(a) has exactly four solutions
(b) has exactly eight solutions
(c) infinite solutions
(d) has the set \( \left\{ -\frac{5\pi}{2}, -2\pi \right\} \) as solution
Answer: (c) infinite solutions

Question. The greatest possible value of the expression \( \tan \left( x + \frac{2\pi}{3} \right) - \tan \left( x + \frac{\pi}{6} \right) + \cos \left( x + \frac{\pi}{6} \right) \) on the interval \( [-5\pi/12, -\pi/3] \) is
(a) \( \frac{12}{5}\sqrt{2} \)
(b) \( \frac{11}{6}\sqrt{2} \)
(c) \( \frac{12}{5}\sqrt{3} \)
(d) \( \frac{11}{6}\sqrt{3} \)
Answer: (d) \( \frac{11}{6}\sqrt{3} \)

ONE OR MORE CORRECT ANSWERS

Question. Consider the equation \( \sqrt{\sin x} - \sqrt{\cos x} = a \) where \( a \in \mathbb{R} \) is a parameter. Then the true statements among the following are
(a) The equation has solution iff \( |a| \le 1 \).
(b) If \( |a| \le 1 \) then the equation has no solution
(c) \( |a| \le \frac{1}{\sqrt{3}} \) then the equation possess infinitely many solutions.
(d) The equation has no solutions if \( a \le 0 \).
Answer: (a) The equation has solution iff \( |a| \le 1 \)., (c) \( |a| \le \frac{1}{\sqrt{3}} \) then the equation possess infinitely many solutions.

Question. Consider the equation \( \cot x + \log_{\frac{\pi}{4}} x = 2, 0 < x < \pi \) then the true statements among the following are
(a) The equation has no real root
(b) The equation has two distinct real roots.
(c) The equation has a unique root.
(d) A root of the equation lies between 0 and 1
Answer: (c) The equation has a unique root., (d) A root of the equation lies between 0 and 1

Question. Consider the equation \( \int_{0}^{x} \cos (t + x^2) dt = \sin x, 2 \le x \le 3 \) then
(a) The equation has no real root.
(b) The equation has two distinct real roots.
(c) The equation has exactly one real root.
(d) \( \frac{\sqrt{1 + 8\pi} - 1}{2} \) is a root of the equation.
Answer: (b) The equation has two distinct real roots., (d) \( \frac{\sqrt{1 + 8\pi} - 1}{2} \) is a root of the equation.

Question. Consider the equation \( -2\sqrt{3}\pi \sin x = |x + \pi| + |x - 2\pi| \) then the true Statements among the following are.
(a) The equation has no real root.
(b) The equation has exactly four distinct real roots.
(c) All the roots of the equation are irrational.
(d) All the roots of the equation lie between \( -\pi \) and \( 2\pi \).
Answer: (b) The equation has exactly four distinct real roots., (c) All the roots of the equation are irrational., (d) All the roots of the equation lie between \( -\pi \) and \( 2\pi \).

Question. Let \( \theta, \phi \in [0, 2\pi] \) be such that
\( 2\cos \theta (1 - \sin \phi) = \sin^2 \theta \left( \tan \frac{\theta}{2} + \cot \frac{\theta}{2} \right) \cos \phi - 1 \),
\( \tan (2\pi - \theta) > 0 \) and \( -1 < \sin \theta < -\frac{\sqrt{3}}{2} \). Then \( \phi \) cannot satisfy [IIT 2012]
(a) \( 0 < \phi < \frac{\pi}{2} \)
(b) \( \frac{\pi}{2} < \phi < \frac{4\pi}{3} \)
(c) \( \frac{4\pi}{3} < \phi < \frac{3\pi}{2} \)
(d) \( \frac{3\pi}{2} < \phi < 2\pi \)
Answer: (a) \( 0 < \phi < \frac{\pi}{2} \), (c) \( \frac{4\pi}{3} < \phi < \frac{3\pi}{2} \), (d) \( \frac{3\pi}{2} < \phi < 2\pi \)

Question. The trigonometric equation
\( \sin 2x + \sin 3x + \sin 4x + ........ + \sin nx = n - i \) (n is a natural number greater than 2)
(a) has unique solution for any n
(b) has no solution for any n
(c) has infinite solutions
(d) has no solutrion in \( [2, n\pi] \)
Answer: (b) has no solution for any n, (d) has no solutrion in \( [2, n\pi] \)

Question. If \( f(x) = [1 + \sin x] + \left[ 2 + \sin \frac{x}{2} \right] + \left[ 3 + \sin \frac{x}{3} \right] + .... + \left[ n + \sin \frac{x}{n} \right] \) (\( 0 < x < \pi \))
(a) min.value is \( \frac{n(n + 1)}{2} \)
(b) max. value is \( \frac{n(n + 1)}{2} + 1 \)
(c) Range is \( \left[ \frac{n(n + 1)}{2}, \frac{n(n + 1)}{2} + 1 \right] \)
(d) minimum value is \( \frac{n(n + 1)}{2} + 1 \)
Answer: (a) min.value is \( \frac{n(n + 1)}{2} \), (b) max. value is \( \frac{n(n + 1)}{2} + 1 \)

Question. \( \sqrt{(\cos 2x)} + \sqrt{(1 + \sin 2x)} = 2\sqrt{(\sin x + \cos x)} \), If
(a) \( \sin x + \cos x = 0 \)
(b) \( x = 2n\pi, n \in I \)
(c) \( x = n\pi - \frac{\pi}{4}, n \in I \)
(d) \( x = 2n\pi \pm \cos^{-1}\left(-\frac{1}{5}\right), n \in I \)
Answer: (a) \( \sin x + \cos x = 0 \), (b) \( x = 2n\pi, n \in I \), (c) \( x = n\pi - \frac{\pi}{4}, n \in I \)

Question. The inequation \( 2\sin^{2}\left(x - \frac{\pi}{3}\right) - 5\sin\left(x - \frac{\pi}{3}\right) + 2 > 0 \)
(a) is satisfied in \( \left( -\frac{5\pi}{6}, \frac{5\pi}{6} \right) \)
(b) is satisfied in \( \left( -\frac{5\pi}{6}, \frac{\pi}{2} \right) \)
(c) is satisfied in \( \left( \frac{(12n - 5)\pi}{6}, \frac{(4n + 1)\pi}{2} \right) \)
(d) is satisfied for all x for which \( \sin \left( x - \frac{\pi}{2} \right) < \frac{1}{2} \)
Answer: (b) is satisfied in \( \left( -\frac{5\pi}{6}, \frac{\pi}{2} \right) \), (c) is satisfied in \( \left( \frac{(12n - 5)\pi}{6}, \frac{(4n + 1)\pi}{2} \right) \), (d) is satisfied for all x for which \( \sin \left( x - \frac{\pi}{2} \right) < \frac{1}{2} \)

Question. The equation \( 1 + \cos^2 ax = \sin x \), where a is a parameter
(a) has a solution if a is not rational
(b) does not have a solution if a is not rational
(c) can have \( a < \frac{\pi^2}{2} \) as a solution for some 'a'
(d) can not have \( (2009)\pi / 2 \) as a solution for any 'a'
Answer: (b) does not have a solution if a is not rational, (c) can have \( a < \frac{\pi^2}{2} \) as a solution for some 'a'

Question. The equation \( \sin^2 x + \sin x - a = 0 (0 \le x < 2\pi) \)
(a) has solutions for every \( a \ge -\frac{1}{4} \)
(b) has two solutions for \( a = -\frac{1}{4} \)
(c) has four solutions for \( -\frac{1}{4} < a < 0 \)
(d) has two solutions for \( -\frac{1}{4} < a < 0 \)
Answer: (b) has two solutions for \( a = -\frac{1}{4} \), (c) has four solutions for \( -\frac{1}{4} < a < 0 \)

COMPREHENSION PASSAGES

(P) Consider the equation
\( \tan^2 \left(\cos \sqrt{4\pi^2 - x^2}\right) - 4a \tan \left(\cos \sqrt{4\pi^2 - x^2}\right) + 2 + 2a = 0 \) , 'a' being a parameter. (Given \(\tan 1 = 1.56\))

Question. If a = 1 then the number of distinct real roots of the equation is
(a) 0
(b) 1
(c) 2
(d) Infinite
Answer: (a) 0

Question. If \( a = -\frac{1}{2} \) then the number of distinct real roots of the equation is
(a) 0
(b) 2
(c) 3
(d) 4
Answer: (d) 4

Passage-2:
Consider the equation
\( \sec \theta + \text{cosec } \theta = a, \theta \in (0, 2\pi) - \left\{ \frac{\pi}{2}, \pi, \frac{3\pi}{2} \right\} \)

Question. If the equation has four real roots, then
(a) \( |a| \ge 2\sqrt{2} \)
(b) \( |a| < 2\sqrt{2} \)
(c) \( |a| \ge -2\sqrt{2} \)
(d) \( |a| > 2\sqrt{2} \)
Answer: (d) \( |a| > 2\sqrt{2} \)

Question. If the equation has exactly two real roots, then
(a) \( |a| \ge 2\sqrt{2} \)
(b) \( a < 2\sqrt{2} \)
(c) \( |a| < 2\sqrt{2} \)
(d) None of the options
Answer: (c) \( |a| < 2\sqrt{2} \)

Question. If the equation has exactly three real solutions then
(a) \( |a| \ge 4\sqrt{2} \)
(b) \( a < 2\sqrt{2} \)
(c) \( |a| < 2\sqrt{2} \)
(d) \( a = 2\sqrt{2} \) or \( a = -2\sqrt{2} \)
Answer: (d) \( a = 2\sqrt{2} \) or \( a = -2\sqrt{2} \)

Passage-3:
Suppose equation is \( f(x) - g(x) = 0 \) or \( f(x) = g(x) = y \) say, then draw the graphs of \( y = f(x) \) and \( y = g(x) \). If graphs of \( y = f(x) \) and \( y = g(x) \) cuts at one, two, three, ...., no points, then number of solutions are one, two, three, ......, zero respectively.
On the basis of above information, answer the following questions:

Question. The number of solutions of \( \sin x = \frac{|x|}{10} \) is
(a) 4
(b) 6
(c) 8
(d) none of the options
Answer: (b) 6

Question. Total number of solutions of the equation \( 3x + 2\tan x = \frac{5\pi}{2} \) in \( x \in [0, 2\pi] \) is equal to
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (c) 3

Question. If \( 1 - \sin x = \frac{\sqrt{3}}{2} \left| x - \frac{\pi}{2} \right| + a \) has no solution when \( a \in \mathbb{R}^+ \), then
(a) \( a \in \mathbb{R}^+ \)
(b) \( a > \frac{3}{2} - \frac{\pi}{\sqrt{3}} \)
(c) \( a \in \left( 0, \frac{3}{2} + \frac{\pi}{\sqrt{3}} \right) \)
(d) \( a \in \left( \frac{3}{2}, \frac{3}{2} + \frac{\pi}{\sqrt{3}} \right) \)
Answer: (b) \( a > \frac{3}{2} - \frac{\pi}{\sqrt{3}} \)