## 77th SSC CGL level Question Set, topic Trigonometry 7

This is the 77th question set for the 10 practice problem exercise for SSC CGL exam and 7th on topic Trigonometry. Answers and links to the corresponding solution set are given at the end.

Before taking the test, you may refer to the **tutorial**,

**Tutorial on Basic and rich concepts in Trigonometry and its applications.**

### 77th question set - 10 problems for SSC CGL exam: 7th on Trigonometry - testing time 12 mins

**Problem 1.**

If $7\sin^2 \theta+3\cos^2 \theta=4$, then the value of $\tan \theta$, where $\theta$ is acute, is,

- $1$
- $\displaystyle\frac{1}{\sqrt{3}}$
- $\sqrt{3}$
- $\displaystyle\frac{1}{\sqrt{2}}$

**Problem 2.**

If $\alpha + \beta=90^0$, then the expression $\displaystyle\frac{\tan \alpha}{\tan \beta}+ \sin^2 \alpha + \sin^2 \beta$ is equal to,

- $\sec^2 \beta$
- $\sec^2 \alpha$
- $\tan^2 \beta$
- $\tan^2 \alpha$

**Problem 3.**

If $0^0 \lt \theta \lt 90^0$ then the value of $\displaystyle\frac{\tan \theta-\sec \theta-1}{\tan \theta + \sec \theta +1}$ is,

- $\displaystyle\frac{1-\cos \theta}{\sin \theta}$
- $\displaystyle\frac{\sin \theta +1}{\cos \theta}$
- $\displaystyle\frac{1-\sin \theta}{\cos \theta}$
- $\displaystyle\frac{\sin \theta-1}{\cos \theta}$

**Problem 4.**

If $5\sin \theta=3$, then the numerical value of $\displaystyle\frac{\sec \theta - \tan \theta}{\sec \theta + \tan \theta}$ is,

- $\displaystyle\frac{1}{2}$
- $\displaystyle\frac{1}{4}$
- $\displaystyle\frac{1}{3}$
- $\displaystyle\frac{1}{5}$

**Problem 5.**

If $\displaystyle\frac{\sec \theta + \tan \theta}{\sec \theta - \tan \theta}=2\displaystyle\frac{51}{79}$, the value of $\sin \theta$ is,

- $\displaystyle\frac{39}{72}$
- $\displaystyle\frac{91}{144}$
- $\displaystyle\frac{65}{144}$
- $\displaystyle\frac{35}{72}$

**Problem 6.**

The value of $\theta$ ($0 \leq \theta \leq 90^0$) satisfying $2\sin^2 \theta=3\cos \theta$ is,

- $60^0$
- $45^0$
- $90^0$
- $30^0$

**Problem 7.**

If $a$, $b$, $c$ are the lengths of three sides of a $\triangle ABC$. If $a$, $b$ and $c$ are related by the relation, $a^2+b^2+c^2=ab+bc+ca$, then the value of $\sin^2 \text{A}+\sin^2 \text{B}+\sin^2 \text{C}$ is,

- $\displaystyle\frac{3}{4}$
- $\displaystyle\frac{9}{4}$
- $\displaystyle\frac{3}{2}$
- $\displaystyle\frac{3\sqrt{3}}{2}$

**Problem 8.**

If $x\cos^2 30^0.\sin 60^0=\displaystyle\frac{\tan^2 45^0.\sec 60^0}{\text{cosec } 60^0}$, then the value of $x$ is,

- $\displaystyle\frac{1}{\sqrt{3}}$
- $\displaystyle\frac{1}{2}$
- $2\displaystyle\frac{2}{3}$
- $\displaystyle\frac{1}{\sqrt{2}}$

**Problem 9.**

If $\sec \theta-\cos \theta = \displaystyle\frac{3}{2}$, where $\theta$ is a positive acute angle, the value of $\sec \theta$ is,

- $2$
- $0$
- $-\displaystyle\frac{1}{2}$
- $1$

**Problem 10.**

If $1+\cos^2 \theta=3\sin \theta.\cos \theta$, then the integral value of $\text{cot } \theta$ $\left(0 \lt \theta \lt \displaystyle\frac{\pi}{2}\right)$ is equal to,

- $0$
- $3$
- $1$
- $2$

The answers to the questions are given below, but you will find the * detailed conceptual solutions* to these questions in

*.*

**SSC CGL level Solution Set 77 on Trigonometry 7**### Answers to the questions

**Problem 1.** Answer: Option b: $\displaystyle\frac{1}{\sqrt{3}}$.

**Problem 2.** Answer: Option b: $\sec^2 \alpha$.

**Problem 3.** Answer: Option d: $\displaystyle\frac{\sin \theta-1}{\cos \theta}$.

**Problem 4.** Answer: Option b: $\displaystyle\frac{1}{4}$.

**Problem 5.** Answer: Option c: $\displaystyle\frac{65}{144}$.

**Problem 6.** Answer: Option a: $60^0$.

**Problem 7.** Answer: Option b: $\displaystyle\frac{9}{4}$.

**Problem 8.** Answer: Option c: $2\displaystyle\frac{2}{3}$.

**Problem 9.** Answer: Option a: $2$.

**Problem 10.** Answer: Option c: $1$.

### Resources on Trigonometry and related topics

You may refer to our useful resources on Trigonometry and other related topics especially algebra.

#### Tutorials on Trigonometry

**Basic and rich concepts in Trigonometry and its applications**

**Basic and Rich Concepts in Trigonometry part 2, proof of compound angle functions**

**Trigonometry concepts part 3, maxima (or minima) of Trigonometric expressions**

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**A note on usability:** The *Efficient math problem solving* sessions on **School maths** are **equally usable for SSC CGL aspirants**, as firstly, the "Prove the identity" problems can easily be converted to a MCQ type question, and secondly, the same set of problem solving reasoning and techniques have been used for any efficient Trigonometry problem solving.

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