Question : If $ \frac{k-k \cot ^2 30^{\circ}}{1+\cot ^2 30^{\circ}}=\sin ^2 60^{\circ}+4 \tan ^2 45^{\circ}-\operatorname{cosec}^2 60^{\circ}$, then the value of k (correct to two decimal places) is:

Option 1: 5.55

Option 2: – 6.83

Option 3: – 5.58

Option 4: 6.83

Question : If $x\sin ^{2}60^{\circ}-\frac{3}{2}\sec 60^{\circ}\tan^{2}30^{\circ}+\frac{4}{5}\sin ^{2}45^{\circ}\tan ^{2}60^{\circ}=0$, then $x$ is:

Option 1: $-\frac{1}{15}$

Option 2: $–4$

Option 3: $-\frac{4}{15}$

Option 4: $–2$

Question : The value of $\frac{4 \tan ^2 30^{\circ}+\sin ^2 30^{\circ} \cos ^2 45^{\circ}+\sec ^2 48^{\circ}-\cot ^2 42^{\circ}}{\cos 37^{\circ} \sin 53^{\circ}+\sin 37^{\circ} \cos 53^{\circ}+\tan 18^{\circ} \tan 72^{\circ}}$ is:

Option 1: $\frac{35}{48}$

Option 2: $\frac{59}{48}$

Option 3: $\frac{49}{24}$

Option 4: $\frac{35}{24}$

Question : The value of $\frac{3\left(\operatorname{cosec}^2 26^{\circ}-\tan ^2 64^{\circ}\right)+\left(\cot ^2 42^{\circ}-\sec ^2 48^{\circ}\right)}{\cot \left(22^{\circ}-\theta\right)-\operatorname{cosec}^2\left(62^{\circ}+\theta\right)-\tan \left(\theta+68^{\circ}\right)+\tan ^2\left(28^{\circ}-\theta\right)}$ is:

Option 1: 3

Option 2: 4

Option 3: –1

Option 4: –2

Question : If ${\operatorname{cosec} 39^{\circ}} = x$, then the value of $ \frac{1}{\operatorname{cosec}^2 51^{\circ}} +\sin^239^{\circ} +\tan ^251^{\circ} -\frac{1}{\sin ^2 51^{\circ} \sec ^2 39^{\circ}}$ is:

Option 1: $x^2-1$

Option 2: $\sqrt{x^2-1}$

Option 3: $ \sqrt{1-x^2}$

Option 4: $1-x^2$

Question : The value of $\frac{5 \cos ^2 60^{\circ}+4 \sec ^2 30^{\circ}-\tan ^2 45^{\circ}}{\tan ^2 60^{\circ}-\sin ^2 30^{\circ}-\cos ^2 45^{\circ}}$ is:

Option 1: $\frac{67}{27}$

Option 2: $\frac{22}{9}$

Option 3: $\frac{67}{24}$

Option 4: $\frac{19}{9}$