Quadratic Equations · NDA previous year questions

If $m$ and $n$ are the roots of the equation $(x+p)(x+q)-k=0$, then the roots of the equation $$(x-m)(x-n)+k=0$$ are

1. A.$p\text{ and }q$
2. B.$1/p\text{ and }1/q$
3. C.$-p\text{ and }-q$
4. D.$p+q\text{ and }p-q$

For the next two (2) items that follow : Let $\alpha$ be the root of the equation $25{\cos }^2\theta +5\cos \theta -12=0$, where $\frac{\pi }{2}<\alpha <\pi$. What is $\tan \alpha$ equal to?

1. A.$-3/4$
2. B.3/4
3. C.$-4/3$
4. D.$-4/5$

For the next two (2) items that follow: Let $\alpha$ be the root of the equation $25{\cos }^2\theta +5\cos \theta -12=0$, where $\frac{\pi }{2}<\alpha <\pi$. What is $\sin 2\alpha$ equal to?

1. A.24/25
2. B.$-24/25$
3. C.$-5/12$
4. D.$-21/25$

$(1-\sin A+\cos A{)}^2$ is equal to

1. A.$2(1-\cos A)(1+\sin A)$
2. B.$2(1-\sin A)(1+\cos A)$
3. C.$2(1-\cos A)(1-\sin A)$
4. D.None of the above

What is $$\frac{\cos \theta }{1-\tan \theta }+\frac{\sin \theta }{1-\cot \theta }$$ equal to?

1. A.$\sin \theta -\cos \theta$
2. B.$\sin \theta +\cos \theta$
3. C.$2\sin \theta$
4. D.$2\cos \theta$

The equation ${\tan }^{-1}(1+x)+{\tan }^{-1}(1-x)=\frac{\pi }{2}$ is satisfied by

1. A.$x=1$
2. B.$x=-1$
3. C.$x=0$
4. D.$x=1/2$

For the next three (3) items that follow: Consider $x=4{\tan }^{-1}\left(\frac{1}{5}\right),y={\tan }^{-1}\left(\frac{1}{70}\right)\text{ and }z={\tan }^{-1}\left(\frac{1}{99}\right)$. What is $x$ equal to?

1. A.${\tan }^{-1}\left(\frac{60}{119}\right)$
2. B.${\tan }^{-1}\left(\frac{120}{119}\right)$
3. C.${\tan }^{-1}\left(\frac{90}{169}\right)$
4. D.${\tan }^{-1}\left(\frac{170}{169}\right)$

For the next three (3) items that follow: Consider $x=4{\tan }^{-1}\left(\frac{1}{5}\right),y={\tan }^{-1}\left(\frac{1}{70}\right)\text{ and }z={\tan }^{-1}\left(\frac{1}{99}\right)$. What is $x-y$ equal to?

1. A.${\tan }^{-1}\left(\frac{828}{845}\right)$
2. B.${\tan }^{-1}\left(\frac{8287}{8450}\right)$
3. C.${\tan }^{-1}\left(\frac{8281}{8450}\right)$
4. D.${\tan }^{-1}\left(\frac{8287}{8471}\right)$

For the next three (3) items that follow: Consider $x=4{\tan }^{-1}\left(\frac{1}{5}\right),y={\tan }^{-1}\left(\frac{1}{70}\right)\text{ and }z={\tan }^{-1}\left(\frac{1}{99}\right)$. What is $x-y+z$ equal to?

1. A.$\frac{\pi }{2}$
2. B.$\frac{\pi }{3}$
3. C.$\frac{\pi }{6}$
4. D.$\frac{\pi }{4}$

Every quadratic equation $a{x}^2+bx+c=0$ where $a,b,c\in \mathbf{R},a\ne 0$ has

1. A.exactly one real root.
2. B.at least one real root.
3. C.at least two real roots.
4. D.at most two real roots.

If $\alpha ,\beta$ are the roots of $a{x}^2+bx+c=0$ and $\alpha +h,\beta +h$ are the roots of $p{x}^2+qx+r=0$, then what is h equal to ?

1. A.$\frac{1}{2}\left(\frac{b}{a}-\frac{q}{p}\right)$
2. B.$\frac{1}{2}\left(-\frac{b}{a}+\frac{q}{p}\right)$
3. C.$\frac{1}{2}\left(\frac{b}{p}+\frac{q}{a}\right)$
4. D.$\frac{1}{2}\left(-\frac{b}{p}+\frac{q}{a}\right)$

If $\alpha$ and $\beta$ are the roots of the equation $a{x}^2+bx+c=0$, where $a\ne 0$, then $(a\alpha +b)(a\beta +b)$ is equal to:

1. A.$ab$
2. B.$bc$
3. C.$ca$
4. D.$abc$

What is the positive square root of $7+4\sqrt{3}$?

1. A.$\sqrt{3}-1$
2. B.$\sqrt{3}+1$
3. C.$\sqrt{3}-2$
4. D.$\sqrt{3}+2$

If $A=\{1,2\},B=\{2,3\}$ and $C=\{3,4\}$, then what is the cardinality of $(A\times B)\cap (A\times C)$?

1. A.8
2. B.6
3. C.2
4. D.1

If $\alpha ,\beta$ are the roots of the equation $x^2+x+2=0$, then what is $$\frac{{\alpha }^{10}+{\beta }^{10}}{{\alpha }^{-10}+{\beta }^{-10}}$$ equal to?

1. A.4096
2. B.2048
3. C.1024
4. D.512

If $a$ and $b$ are rational and $b$ is not perfect square, then the quadratic equation with rational coefficients whose one root is $3a+\sqrt{b}$ is

1. A.$x^2-6ax+9a^2-b=0$
2. B.$3a{x}^2+x-\sqrt{b}=0$
3. C.$x^2+3ax+\sqrt{b}=0$
4. D.$\sqrt{b}{x}^2+x-3a=0$

How many real roots does the quadratic equation have?

1. A.One
2. B.Two
3. C.Four
4. D.No real root

If $\alpha ,\beta$ are the roots of the equation $a{x}^2+bx+b=0$, then what is the value of $\sqrt{\frac{\alpha }{\beta }}+\sqrt{\frac{\beta }{\alpha }}+\sqrt{\frac{b}{a}}$?

1. A.$-1$
2. B.0
3. C.1
4. D.2

What is the difference in the roots of the equation $x^2-10x+9=0$?

1. A.2
2. B.3
3. C.5
4. D.8

If $8x-9y=20$ and $7x-10y=9$, then what is $2x-y$ equal to?

1. A.10
2. B.11
3. C.12
4. D.13

The quadratic equation $x^2+bx+4=0$ will have real roots if

1. A.$b\le -4$ only
2. B.$b\ge 4$ only
3. C.$-4<b<4$
4. D.$b\le -4,b\ge 4$

If the sum of the roots of a quadratic equation is 3 and the product is 2, then the equation is

1. A.$2x^2-x+3=0$
2. B.$x^2-3x+2=0$
3. C.$x^2+3x+2=0$
4. D.$x^2+3x-2=0$

If $\alpha$ and $\beta$ are the roots of the equation $x^2+bx+c=0$, then what is the value of ${\alpha }^{-1}+{\beta }^{-1}$ ?

1. A.$-b/c$
2. B.$b/c$
3. C.$c/b$
4. D.$-c/b$

The area of a rectangle whose length is five more than twice its width is 75 square unit. The length is

1. A.5 unit
2. B.10 unit
3. C.15 unit
4. D.20 unit

What is $\sqrt{-i}$ where $i=\sqrt{-1}$ equal to ?

1. A.$\pm \frac{1-i}{\sqrt{2}}$
2. B.$\pm \frac{1+i}{\sqrt{2}}$
3. C.$\pm \frac{1-i}{2}$
4. D.$\pm \frac{1+i}{2}$