Relations between roots and coefficients; nature of roots — practice questions

Question 1 · difficulty L2 · understanding

If $\alpha, \beta$, where $\alpha<\beta$, are the roots of the equation $\lambda x^2-(\lambda+3) x+3=0$ such that $\frac{1}{\alpha}-\frac{1}{\beta}=\frac{1}{3}$, then the sum of all possible values of $\lambda$ is

• A. 2
• B. 6
• C. 8
• D. 4

Question 2 · difficulty L2 · understanding

cosec18$$^\circ$$ is a root of the equation :

• A. x 2 + 2x $$-$$ 4 = 0
• B. 4x 2 + 2x $$-$$ 1 = 0
• C. x 2 $$-$$ 2x + 4 = 0
• D. x 2 $$-$$ 2x $$-$$ 4 = 0

Question 3 · difficulty L3 · understanding

If the set of all $\mathrm{a} \in \mathbf{R}-\{1\}$, for which the roots of the equation $(1-\mathrm{a}) x^2+2(\mathrm{a}-3) x+9=0$ are positive is $(-\infty,-\alpha] \cup[\beta, \gamma)$, then $2 \alpha+\beta+\gamma$ is equal to $\qquad$ .

Question 4 · difficulty L3 · understanding

If the equation $\mathrm{a}(\mathrm{b}-\mathrm{c}) \mathrm{x}^2+\mathrm{b}(\mathrm{c}-\mathrm{a}) \mathrm{x}+\mathrm{c}(\mathrm{a}-\mathrm{b})=0$ has equal roots, where $\mathrm{a}+\mathrm{c}=15$ and $\mathrm{b}=\frac{36}{5}$, then $a^2+c^2$ is equal to _________

Question 5 · difficulty L3 · understanding

Let $$\alpha, \beta$$ be roots of $$x^2+\sqrt{2} x-8=0$$. If $$\mathrm{U}_{\mathrm{n}}=\alpha^{\mathrm{n}}+\beta^{\mathrm{n}}$$, then $$\frac{\mathrm{U}_{10}+\sqrt{2} \mathrm{U}_9}{2 \mathrm{U}_8}$$ is equal to ________.

Question 6 · difficulty L3 · understanding

The number of points, where the curve $$f(x)=\mathrm{e}^{8 x}-\mathrm{e}^{6 x}-3 \mathrm{e}^{4 x}-\mathrm{e}^{2 x}+1, x \in \mathbb{R}$$ cuts $$x$$-axis, is equal to _________.

Question 7 · difficulty L3 · understanding

If $$a$$ and $$b$$ are the roots of the equation $$x^{2}-7 x-1=0$$, then the value of $$\frac{a^{21}+b^{21}+a^{17}+b^{17}}{a^{19}+b^{19}}$$ is equal to _____________.

Question 8 · difficulty L3 · understanding

Let m and $$\mathrm{n}$$ be the numbers of real roots of the quadratic equations $$x^{2}-12 x+[x]+31=0$$ and $$x^{2}-5|x+2|-4=0$$ respectively, where $$[x]$$ denotes the greatest integer $$\leq x$$. Then $$\mathrm{m}^{2}+\mathrm{mn}+\mathrm{n}^{2}$$ is equal to __________.

Question 9 · difficulty L3 · understanding

Let $$\alpha_1,\alpha_2,....,\alpha_7$$ be the roots of the equation $${x^7} + 3{x^5} - 13{x^3} - 15x = 0$$ and $$|{\alpha _1}| \ge |{\alpha _2}| \ge \,...\, \ge \,|{\alpha _7}|$$. Then $$\alpha_1\alpha_2-\alpha_3\alpha_4+\alpha_5\alpha_6$$ is equal to _________.

Question 10 · difficulty L3 · understanding

Let $$\alpha \in\mathbb{R}$$ and let $$\alpha,\beta$$ be the roots of the equation $${x^2} + {60^{{1 \over 4}}}x + a = 0$$. If $${\alpha ^4} + {\beta ^4} = - 30$$, then the product of all possible values of $$a$$ is ____________.