Chapter 26

The equation \(2 \cos ^{2}\left(\frac{x}{2}\right) . \sin ^{2} x=x^{2}+\frac{1}{x^{2}} 0 \leq x \leq \frac{\pi}{2}\) has (A) one real solution (B) no solution (C) more than one real solution (D) none of these

The general solution of the equation \(\sin ^{50} x-\cos ^{30} x=1\) is (A) \(2 n \pi+\frac{\pi}{2}\) (B) \(2 n \pi+\frac{\pi}{3}\) (C) \(n \pi+\frac{\pi}{2}\) (D) \(n \pi+\frac{\pi}{3}\)

General solution of the equation \((\sqrt{3}-1) \sin \theta+(\sqrt{3}+1) \cos \theta=2\) is (A) \(2 n \pi \pm \frac{\pi}{4}+\frac{\pi}{12}\) (B) \(n \pi+(-1)^{n} \frac{\pi}{4}+\frac{\pi}{12}\) (C) \(2 n \pi \pm \frac{\pi}{4}-\frac{\pi}{12}\) (D) \(n \pi+(-1)^{n} \frac{\pi}{4}-\frac{\pi}{12}\)

The number of all possible triplets \(\left(a_{1}, a_{2}, a_{3}\right)\) such that \(a_{1}+a_{2} \cos 2 x+a_{3} \sin ^{2} x=0\) for all \(x\) is (A) 0 (B) 1 (C) 3 (D) infinite

The equation \(\sin ^{4} x-(k+2) \sin ^{2} x-(k+3)=0\) pos- sesses a solution if (A) \(k>-3\) (B) \(k<-2\) (C) \(-3 \leq k \leq-2\) (D) \(k\) is any positive integer

The least positive non-integral solution of the equation \(\sin \pi\left(x^{2}+x\right)=\sin \pi x^{2}\) is (A) rational (B) irrational of the form \(\sqrt{p}\) (C) irrational of the form \(\frac{\sqrt{p}-1}{4}\), where \(p\) is an odd integer (D) irrational of the form \(\frac{\sqrt{p}+1}{4}\), where \(p\) is an even integer

If \(\sin ^{2} x-2 \sin x-1=0\) has exactly four different solutions in \(x \in[0, n \pi]\), then minimum value of \(n\) can be \((n \in N)\) If \(\sin ^{2} x-2 \sin x-1=0\) has exactly four different solutions in \(x \in[0, n \pi]\), then minimum value of \(n\) can be \((n \in N)\)

A set of values of \(x\) satisfying the equation \(\cos ^{2}\left(\frac{1}{2} p x\right)+\cos ^{2}\left(\frac{1}{2} q x\right)=1\) form an arithmetic progression with common difference (A) \(\frac{2}{p+q}\) (B) \(\frac{2}{p-q}\) (C) \(\frac{\pi}{p+q}\) (D) none of these

If \(0 \leq x \leq 2 \pi\) and \(|\cos x| \leq \sin x\), then (A) \(x \in\left[0, \frac{\pi}{4}\right]\) (B) \(x \in\left[\frac{\pi}{4}, \frac{\pi}{2}\right]\) (C) \(x \in\left[\frac{\pi}{4}, \frac{3 \pi}{4}\right]\) (D) none of these

The general solution of the equation \(\frac{1-\sin x+\cdots+(-1)^{n} \sin ^{n} x+\cdots}{1+\sin x+\cdots+\sin ^{n} x+\cdots}=\frac{1-\cos 2 x}{1+\cos 2 x}\) \(x \neq(2 n+1) \frac{\pi}{2}, n \in Z\) is (A) \((-1)^{n}\left(\frac{\pi}{3}\right)+n \pi\) (B) \((-1)^{n}\left(\frac{\pi}{6}\right)+n \pi\) (C) \((-1)^{n+1}\left(\frac{\pi}{6}\right)+n \pi\) (D) \((-1)^{n-1}\left(\frac{\pi}{3}\right)+n \pi\)

The general solution of the equation \(\sum_{r=1}^{n} \cos r^{2} \theta \sin r \theta=\frac{1}{2}\) is (A) \(\frac{4 k-1}{n(n+1)} \frac{\pi}{2}, k \in Z\) (B) \(\frac{2 k+1}{n(n+1)} \frac{\pi}{2}, k \in Z\) (C) \(\frac{4 k+1}{n(n+1)} \frac{\pi}{2}, k \in Z\) (D) none of these

The solution of \(\sin ^{8} x+\cos ^{8} x=\frac{17}{32}\) is (A) \(\frac{n \pi}{2} \pm \frac{\pi}{8}\) (B) \(n \pi \pm \frac{\pi}{4}\) (C) \(n \pi \pm \frac{\pi}{8}\) (D) no solution

The general solution of the equation \(2^{\cos ^{2} \theta}+1=3.2^{-\sin ^{2} \theta}\) is (A) \(2 n \pi \pm \frac{\pi}{2}, n \pi, n \in Z\) (B) \(n \pi \pm \frac{\pi}{2}, 2 n \pi, n \in Z\) (C) \(n \pi \pm \frac{\pi}{2}, n \pi, n \in Z\) (D) none of these

The solution of the inequality \(\log _{1 / 2} \sin \theta>\log _{1 / 2} \cos \theta\) in \([0,2 p]\) is (A) \(\left(0, \frac{\pi}{2}\right)\) (B) \(\left(\frac{\pi}{4}, \frac{\pi}{2}\right)\) (C) \(\left(0, \frac{\pi}{4}\right)\) (D) none of these

If \(\cos 3 x+\sin \left(2 x-\frac{7 \pi}{6}\right)=-2\) then \(x=\) (A) \(\frac{\pi}{3}(6 k+1)\) (B) \(\frac{\pi}{3}(6 k-1)\) (C) \(\frac{\pi}{3}(2 k+1)\) (D) none of these where \(k \in Z\)

If \(\tan ^{2}[\pi(x+y)]+\cot ^{2}[\pi(x+y)]=1+\sqrt{\frac{2 x}{1+x^{2}}}\), where \(x, y \in R\), then least positive value of \(y\) is (A) \(\frac{5}{4}\) (B) \(\frac{1}{4}\) (C) \(\frac{3}{4}\) (D) 2

The general value of \(y\) satisfying the equation \(1-2 x\) \(-x^{2}=\tan ^{2}(x+y)+\cot ^{2}(x+y)\) is (A) \(2 n \pi \pm \frac{\pi}{4}\) (B) \(n \pi \pm \frac{\pi}{4}\) (C) \(\frac{n \pi}{2} \pm \frac{\pi}{4}\) (D) none of these

If \([\sin x]+[\sqrt{2} \cos x]=-3, x \in[0,2 \pi]([.]\) denotes the greatest integer function) then \(x\) belongs to (A) \(\left[\frac{5 \pi}{4}, 2 \pi\right]\) (B) \(\left(\frac{5 \pi}{4}, 2 \pi\right)\) (C) \(\left(\pi, \frac{5 \pi}{4}\right)\) (D) \(\left[\pi, \frac{5 \pi}{4}\right]\)

The number of solutions of the equation \(\sin \left(\frac{\pi x}{2 \sqrt{3}}\right)=x^{2}-2 \sqrt{3} x+4\) (A) forms an empty set (B) is only one (C) is only two (D) is more than 2

\(\sin x+2 \sin 2 x=3+\sin 3 x, 0 \leq x \leq 2 \pi\) has (A) 2 solutions in I quadrant (B) one solution in II quadrant (C) no solution in any quadrant (D) one solution in each quadrant

The solution of the equation \(1+\sin ^{2} a x=\cos x\), where \(a\) is irrational, is (A) \(x=0\) (B) \(x=\frac{n \pi}{a}\) (C) \(x=2 n \pi\) (D) none of these

The values of \(\alpha\) for which the equation \(\sin ^{4} x+\cos ^{4} x+\sin 2 x+\alpha=0\) may be valid, are (A) \(-\frac{3}{2} \leq \alpha \leq 1\) (B) \(0 \leq \alpha \leq \frac{1}{2}\) (C) \(-\frac{3}{2} \leq \alpha \leq \frac{1}{2}\) (D) none of these

If \(\alpha\) and \(\beta\) be two distinct values of \(\theta\) lying between 0 and \(2 \pi\), satisfying the equation \(3 \cos \theta+4 \sin \theta=2\), then the value of \(\sin (\alpha+\beta)\) is (A) \(\frac{12}{25}\) (B) \(\frac{24}{25}\) (C) \(\frac{13}{25}\) (D) none of these

\(|\tan x+\sec x|=|\tan x|+|\sec x|, x \in[0,2 p]\), if and only if \(x\) belongs to the interval (A) \((\pi, 2 \pi]\) (B) \([0, \pi]\) (C) \(\left[0, \frac{\pi}{2}\right) \cup\left(\frac{\pi}{2}, \pi\right]\) (D) \(\left[\pi, \frac{3 \pi}{2}\right) \cup\left(\frac{3 \pi}{2}, 2 \pi\right]\)

\(|\cos x|=\cos x-2 \sin x\) if (A) \(x=n \pi\) (B) \(x=2 n \pi\) or \((2 n+1) \pi+\frac{\pi}{4}\) (C) \(x=n \pi+\frac{\pi}{4}\) (D) \(x=n \pi\) or \(n \pi+\frac{\pi}{4}\)

A solution of the equation \((1-\tan \theta)(1+\tan \theta) \sec ^{2} \theta\) \(+2 \tan ^{2} \theta=0\), where \(\theta\) lies in the interval \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\) is given by (A) \(\theta=0\) (B) \(\theta=\frac{\pi}{3}\) or \(-\frac{\pi}{3}\) (C) \(\theta=\frac{\pi}{6}\) (D) \(\theta=-\frac{\pi}{6}\)

If \(\sqrt{p} \cos x-2 \sin x=\sqrt{2}+\sqrt{2-p}\) has a solution, then \(p \in\) (A) \([\sqrt{5}+1,4]\) (B) \([\sqrt{5}-1,2]\) (C) \([\sqrt{3}+1,3]\) (D) none of these

The value of ' \(b\) ' such that the equation \(\frac{b \cos x}{2 \cos 2 x-1}=\frac{b+\sin x}{\left(\cos ^{2} x-3 \sin ^{2} x\right) \tan x}\) possess solutions, belongs to the set (A) \(\left(-\infty, \frac{1}{2}\right)\) (B) \(\left(\frac{1}{2}, \infty\right)\) (C) \((-\infty, \infty)\) (D) \(\left(-\infty, \frac{1}{2}\right) \cup(1, \infty)\)

If \(\sin \theta=k\) for exactly one value of \(\theta, \theta \in\left[0, \frac{7 \pi}{3}\right]\), then the value of \(k\) is (A) 1 (B) \(-1\) (C) \(1 / \sqrt{2}\) (D) 0

The equation \(\sin ^{4} x+\cos ^{4} x=a\) has a solution for (A) all of values of \(a\) (B) \(a=1\) (C) \(a=\frac{1}{2}\) (D) \(\frac{1}{2}

If \(|\cos x|^{\sin ^{2} x-\frac{3}{2} \sin x+\frac{1}{2}}=1\), then possible values of \(x\) are (A) \(n \pi\) or \(n \pi+(-1)^{n} \frac{\pi}{6}, n \in I\) (B) \(n \pi\) or \(2 n \pi+\frac{\pi}{2}\) or \(n \pi+(-1)^{n} \frac{\pi}{6}, n \in I\) (C) \(n \pi+(-1)^{n} \frac{\pi}{6}, n \in I\) (D) \(n \pi, n \in I\)

The equation \(3^{\sin 2 x+2 \cos ^{2} x}+3^{1-\sin 2 x+2 \sin ^{1} x}=28\) is satis- fied for the values of \(x\) given by (A) \(\cos x=0\) (B) \(\tan x=-1\) (C) \(\tan x=1\) (D) none of these

The value of \(\theta\), lying between \(\theta=0\) and \(\theta=\frac{\pi}{2}\) and satisfying the equation \(\left|\begin{array}{ccc}1+\cos ^{2} \theta & \sin ^{2} \theta & 4 \sin 4 \theta \\ \cos ^{2} \theta & 1+\sin ^{2} \theta & 4 \sin 4 \theta \\ \cos ^{2} \theta & \sin ^{2} \theta & 1+4 \sin 4 \theta\end{array}\right|=0\), is (A) \(\frac{11 \pi}{24}\) (B) \(\frac{7 \pi}{24}\) (C) \(\frac{5 \pi}{24}\) (D) none of these

Solution of the system of equations \(2^{\sin x}+{ }^{\cos y}=1\), \(16^{\sin ^{2} x+\cos ^{2} y}=4\) is (A) \(x=n \pi+(-1)^{n} \frac{\pi}{6}, y=2 n \pi \pm \frac{2 \pi}{3}\) (B) \(x=n \pi+(-1)^{n} \frac{\pi}{6}, y=2 n \pi \pm \frac{\pi}{3}\) (C) \(x=n \pi-(-1)^{n} \frac{\pi}{6}, y=2 n \pi \pm \frac{2 \pi}{3}\) (D) \(x=n \pi-(-1)^{n} \frac{\pi}{2}, y=2 n \pi \pm \frac{\pi}{3}\)

Solution of the equation \(4 \sin ^{4} x+\cos ^{4} x=1\) is (A) \(x=n \pi\) (B) \(x=2 n \pi \pm \cos ^{-1}\left(\sqrt{\frac{3}{5}}\right)\) (C) \(x=(2 n+1) \frac{\pi}{2}\) (D) none of these

The solution of the equation \(\frac{\sqrt{3}}{2} \sin x-\cos x=\cos ^{2} x\) is (A) \(x=(2 n+1) \pi\) (B) \(x=2 n \pi \pm \frac{\pi}{3}\) (C) \(x=2 n \pi \pm \frac{\pi}{6}\) (D) none of these

Solution of the equation \(\sin ^{3} \theta+\sin \theta \cos \theta+\cos ^{3} \theta=1\) is (A) \(\theta=2 n \pi+\frac{\pi}{4}\) (B) \(\theta=2 n \pi-\frac{\pi}{4}\) (C) \(\theta=2 n \pi+\frac{\pi}{2}\) (D) \(\theta=2 n \pi\)

The values of \(x\) in \((-\pi, \pi)\) which satisfy the equation \(8^{1+\cos x\left|+\cos ^{2} x+\cos ^{3} x\right|+\ldots \text {.to infinity }}=4^{3}\) are (A) \(\pm \frac{\pi}{4}\) (B) \(\pm \frac{\pi}{3}\) (C) \(\pm \frac{2 \pi}{3}\) (D) none of these

Solution of the equation \(\sin 6 x+\cos 4 x+2=0 ; 0

\(\sqrt{\cos 2 x}+\sqrt{1+\sin 2 x}=\sqrt{\sin x+\cos x}\) if (A) \(x=2 n \pi\) (B) \(x=n \pi-\frac{\pi}{4}\) (C) \(\sin x+\cos x=0\) (D) \(x=n \pi\)

If \([x]\) denotes the greatest integer less than or equal to \(x\), then the equation \(\sin x=[1+\sin x]+[1-\cos x]\) has no solution in (A) \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) (B) \(\left[\frac{\pi}{2}, \pi\right]\) (C) \(\left[\pi, \frac{3 \pi}{2}\right]\) (D) \(R\)

Column-I I. If \(2 \sin ^{2} x+3 \sin x-2>0\) and \(x^{2}-x-2<0\) ( \(x\) is measured in radians), then \(x \in\) II. If \(\frac{\sin ^{3} \theta-\cos ^{3} \theta}{\sin \theta-\cos \theta}-\frac{\cos \theta}{\sqrt{1+\cos ^{2} \theta}}\) \(-2 \tan \theta \cot \theta=-1, \theta \in[0\) \(2 \pi]\), then \(\theta \in\) III. The set of all \(x\) in \((-\pi, \pi)\) satisfying \(|4 \sin x-1|<\sqrt{5}\) is given by Column-II (A) \((0, \pi)-\left\\{\frac{\pi}{4}, \frac{\pi}{2}\right\\}\) (B) \((0, \pi)\) (C) \(\left(-\frac{\pi}{10}, \frac{3 \pi}{10}\right)\) (D) \(\left(\frac{\pi}{6}, 2\right)\)

In the following questions an Assertion \((A)\) is given followed by a Reason \((R) .\) Mark your responses from the following options: (A) Assertion(A) is True and Reason(R) is True; Reason(R) is acorrect explanation for Assertion(A) (B) Assertion(A) is True, Reason(R) is True; Reason(R) is not a correct explanation for Assertion(A) (C) Assertion(A) is True, Reason(R) is False (D) Assertion(A) is False, Reason(R) is True Assertion: The general solution of the equation \(2^{\sin } \theta+\) \(2^{\cos } \theta=2^{1-\frac{1}{\sqrt{2}}}\) is \(\theta=n \pi+\frac{\pi}{4}\) Reason: For any two numbers \(a\) and \(b\), A.M. \(\geq\) G.M.

In the following questions an Assertion \((A)\) is given followed by a Reason \((R) .\) Mark your responses from the following options: (A) Assertion(A) is True and Reason(R) is True; Reason(R) is acorrect explanation for Assertion(A) (B) Assertion(A) is True, Reason(R) is True; Reason(R) is not a correct explanation for Assertion(A) (C) Assertion(A) is True, Reason(R) is False (D) Assertion(A) is False, Reason(R) is True Assertion: Solution of the equation \(4 \cot ^{2} \theta=\cot ^{2} \theta-\) \(\tan ^{2} \theta\) is \(\theta=n \pi \pm \frac{\pi}{4}\) Reason: \(\tan \theta=\tan \alpha \Rightarrow \theta=n \pi \pm \alpha\).

If \(0 \leq x<2 \pi\), then the number of real values of \(x\), which satisfy the equation \(\cos x+\cos 2 x+\cos 3 x+\) \(\cos 4 x=0\), is: (A) 9 (B) 3 (C) 5 (D) 7