## 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**

#### General guidelines for success in SSC CGL

**7 steps for sure success in SSC CGL tier 1 and tier 2 competitive tests**

#### Efficient problem solving in Trigonometry

**How to solve a difficult SSC CGL level problem in a few reasoned steps, Trigonometry 10**

**How to solve not so difficult SSC CGL level problem in a few light steps, Trigonometry 9**

**How to solve a difficult SSC CGL level problem in a few conceptual steps, Trigonometry 8 **

**How to solve not so difficult SSC CGL level problem in a few light steps, Trigonometry 7**

**How to solve a difficult SSC CGL level problem in few quick steps, Trigonometry 6**

**How to solve a School Math problem in a few direct steps, Trigonometry 5**

**How to solve difficult SSC CGL level School math problems in a few quick steps, Trigonometry 5**

**How to solve School Math problem in a few steps and in Many Ways, Trigonometry 4**

**How to solve a School Math problem in a few simple steps, Trigonometry 3**

**How to solve difficult SSC CGL level School math problems in a few simple steps, Trigonometry 4**

**How to solve difficult SSC CGL level School math problems in a few simple steps, Trigonometry 3**

**How to solve School math problems in a few simple steps, Trigonometry 2**

**How to solve School math problems in a few simple steps, Trigonometry 1**

**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.

#### SSC CGL Tier II level question and solution sets on Trigonometry

**SSC CGL Tier II level Solution Set 18 Trigonometry 4, questions with solutions**

**SSC CGL Tier II level Question Set 18 Trigonometry 4, questions with answers**

**SSC CGL Tier II level Solution set 12 Trigonometry 3, questions with solutions**

**SSC CGL Tier II level Question set 12 Trigonometry 3, questions with answers**

**SSC CGL Tier II level Solution set 11 Trigonometry 2**

**SSC CGL Tier II level Question set 11 Trigonometry 2**

**SSC CGL Tier II level Solution set 7 Trigonometry 1**

**SSC CGL Tier II level Question set 7 Trigonometry 1**

#### SSC CGL level question and solution sets on Trigonometry

**SSC CGL level Solution set 82 on Trigonometry 8**

**SSC CGL level Question set 82 on Trigonometry 8**

**SSC CGL level solution set 77 on Trigonometry 7**

**SSC CGL level question set 77 on Trigonometry 7**

**SSC CGL level Solution Set 65 on Trigonometry 6**

**SSC CGL level Question Set 65 on Trigonometry 6**

**SSC CGL level Solution Set 56 on Trigonometry 5**

**SSC CGL level Question Set 56 on Trigonometry 5**

**SSC CGL level Solution Set 40 on Trigonometry 4**

**SSC CGL level Question Set 40 on Trigonometry 4**

**SSC CGL level Solution Set 19 on Trigonometry**

**SSC CGL level Question set 19 on Trigonometry**

**SSC CGL level Solution Set 16 on Trigonometry**

**SSC CGL level Question Set 16 on Trigonometry**

**SSC CGL level Question Set 2 on Trigonometry**

**SSC CGL level Solution Set 2 on Trigonometry**

#### Algebraic concepts

**Basic and rich Algebraic concepts for elegant solutions of SSC CGL problems**

**More rich algebraic concepts and techniques for elega****n****t solutions of SSC CGL problems**