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If the roots of the given equation:
\( \Large \left(\cos p-1\right)x^{2}+ \left(\cos p\right)x+\sin p = 0 \) are real, then:

A) \( \Large P \epsilon \left(- \pi ,0\right) \)
B) \( \Large P \epsilon \left(- \frac{ \pi }{2}, \frac{ \pi }{2} \right) \)
C) \( \Large P \epsilon \left(0, \pi \right) \)
D) \( \Large P \epsilon \left(0, 2 \pi \right) \)

Correct Answer:



C) \( \Large P \epsilon \left(0, \pi \right) \)

Description for Correct answer:

Given equation

\( \Large \left(\cos P-1\right)x^{2}+ \left(\cos P\right)x+\sin P=0 \)

since, roots are real, its discriminant \( \Large \phi \ge 0 \)

=> \( \Large \cos^{2}P-4\cos P \sin P+4 \sin P \ge 0 \)

=> \( \Large \left(\cos P - 2 \sin P\right)^{2}-4 \sin^{2} P+4 \sin P \ge 0 \)...(i)

Now, \( \Large \left(1 - \sin P\right)\ge 0 \) for all real P and \( \Large \sin P > 0 \) for \( \Large 0 < P < \pi \). Therefore, \( \Large 4 \sin P \left(1-\sin P\right)\ge 0 \) when \( \Large 0 < P < \pi \) or \( \Large P \epsilon (0, \pi) \)

Part of solved Quadratic Equations questions and answers : >> Elementary Mathematics >> Quadratic Equations

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Similar Questions

1). The solution of the quadratic equation \( \Large \left(3|x|-3\right)^{2}=|x|+7 \) which belongs to the domain of definition of function \( \Large \gamma = \sqrt{x \left(x-3\right) } \) are given by:

A). \( \Large \pm \frac{1}{9}, \pm 2 \)

B). \( \Large -\frac{1}{9}, 2 \)

C). \( \Large \frac{1}{9}, -2 \)

D). \( \Large -\frac{1}{9}, -2 \)

2). The number of solution of \( \Large \frac{log 5 + log \left(x^{2}+1\right) }{log \left(x-2\right) }=2 \)

A). 2

B). 3

C). 1

D). none of these

3). If the expression \( \Large \left(mx-1+\frac{1}{x}\right) \) is always nonnegative, then the minimum value of m must be:

A). \( \Large -\frac{1}{2} \)

B). 0

C). \( \Large \frac{1}{4} \)

D). \( \Large \frac{1}{2} \)

4). The value of x in the given equation \( \Large 4^{x}-3^{x-\frac{1}{2}}=3^{x+\frac{1}{2}}-2^{2x-1} \) is:

A). \( \Large \frac{4}{3} \)

B). \( \Large \frac{3}{2} \)

C). \( \Large \frac{2}{1} \)

D). \( \Large \frac{5}{3} \)

5). The harmonic mean of the roots of equation \( \Large \left(5+\sqrt{2}x^{2}-14+\sqrt{5}\right)x+8+2\sqrt{5}=0 \) is:

A). 2

B). 4

C). 6

D). 8

6). For what value of \( \Large \lambda \) the sum of the squares of the roots of \( \Large x^{2}+ \left(2+\lambda\right)n-\frac{1}{2} \left(1+\lambda\right)=0 \) is minimum?

A). \( \Large \frac{3}{2} \)

B). 1

C). \( \Large \frac{1}{2} \)

D). \( \Large \frac{11}{4} \)

7). Given that \( \Large \tan \alpha \) and \( \Large \tan \beta \) are the roots of \( \Large x^{2}-px+q=0 \), then the value of \( \Large \sin^{2} \left(\alpha + \beta \right) \) is equal to:

A). \( \Large \frac{p^{2}}{p^{2}+ \left(1-q\right)^{2} } \)

B). \( \Large \frac{p^{2}}{p^{2}+q^{2}} \)

C). \( \Large \frac{q^{2}}{p^{2} \left(1-q\right)^{2} } \)

D). \( \Large \frac{p^{2}}{ \left(p+q\right)^{2} } \)

8). The number of values of K for which the system of equations \( \Large \left(K+1\right)x+84=4K \) and \( \Large Kx+ \left(K+3\right)y=3 K-1 \) has infinitely many solution, is:

A). 0

B). 1

C). 2

D). infinite

9). The set of all real numbers x for which \( \Large x^{2}-|x+2|+x>0 \) is:

A). \( \Large \left(-\infty, -2\right) \ \left(2, \infty\right) \)

B). \( \Large \left(-\infty, -\sqrt{2}\right) \ \left(\sqrt{2}, \infty\right) \)

C). \( \Large \left(-\infty, -1\right) \ \left(1, \infty\right) \)

D). \( \Large \left(\sqrt{2}, \infty\right) \)

10). Let a, b, c be positive numbers, the following systems of equations in x, y and z \( \Large \frac{x^{2}}{a^{2}} +\ \frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=1; \ \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}} +\ \frac{z^{2}}{c^{2}}=1 and\ \frac{-x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1 \) has;

A). no solutions

B). unique solution

C). infinitely many solution

D). finitely many solution.