Quadratic Equations · NDA previous year questions

Consider the following values of x: 1. $8$ 2. $-4$ 3. $\frac{1}{6}$ 4. $-\frac{1}{4}$ Which of the above values of x is/are the solution(s) of the equation ${\tan }^{-1}(2x)+{\tan }^{-1}(3x)=\frac{\pi }{4}$?

1. A.3 only
2. B.2 and 3 only
3. C.1 and 4 only
4. D.4 only

Suppose $f(x)$ is such a quadratic expression that it is positive for all real x. If $g(x)=f(x)+f^{\prime }(x)+f^{\prime \prime }(x)$, then for any real x

1. A.$g(x)<0$
2. B.$g(x)>0$
3. C.$g(x)=0$
4. D.$g(x)\ge 0$

The ratio of roots of the equations $a{x}^2+bx+c=0$ and $p{x}^2+qx+r=0$ are equal. If $D_1$ and $D_2$ are respective discriminants, then what is $\frac{D_1}{D_2}$ equal to?

1. A.$\frac{a^2}{p^2}$
2. B.$\frac{b^2}{q^2}$
3. C.$\frac{c^2}{r^2}$
4. D.None of the above

The equation $|1-x|+x^2=5$ has

1. A.a rational root and an irrational root
2. B.two rational roots
3. C.two irrational roots
4. D.no real roots

The roots of the equation $(q-r)x^2+(r-p)x+(p-q)=0$ are

1. A.$(r-p)/(q-r),1/2$
2. B.$(p-q)/(q-r),1$
3. C.$(q-r)/(p-q),1$
4. D.$(r-p)/(p-q),1/2$

The sum of all real roots of the equation $|x-3{|}^2+|x-3|-2=0$ is

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

If $\alpha$ and $\beta$ are the roots of the equation $3x^2+2x+1=0$, then the equation whose roots are $\alpha +{\beta }^{-1}$ and $\beta +{\alpha }^{-1}$ is

1. A.$3x^2+8x+16=0$
2. B.$3x^2-8x-16=0$
3. C.$3x^2+8x-16=0$
4. D.$x^2+8x+16=0$

In $△PQR$, $\angle R=\frac{\pi }{2}$. If $\tan \left(\frac{P}{2}\right)$ and $\tan \left(\frac{Q}{2}\right)$ are the roots of the equation $a{x}^2+bx+c=0$, then which one of the following is correct?

1. A.$a=b+c$
2. B.$b=c+a$
3. C.$c=a+b$
4. D.$b=c$

The number of roots of the equation $z^2=2\bar{z}$ is

1. A.2
2. B.3
3. C.4
4. D.zero

If the difference between the roots of the equation $x^2+kx+1=0$ is strictly less than $\sqrt{5}$, where $|k|\ge 2$, then $k$ can be any element of the interval

1. A.$(-3,-2]\cup [2,3)$
2. B.$(-3,3)$
3. C.$[-3,-2]\cup [2,3]$
4. D.None of the above

If $\cot \alpha$ and $\cot \beta$ are the roots of the equation $x^2+bx+c=0$ with $b\ne 0$, then the value of $\cot (\alpha +\beta )$ is

1. A.$\frac{c-1}{b}$
2. B.$\frac{1-c}{b}$
3. C.$\frac{b}{c-1}$
4. D.$\frac{b}{1-c}$

If the roots of the equation $x^2+px+q=0$ are in the same ratio as those of the equation $x^2+lx+m=0$, then which one of the following is correct?

1. A.$p^{2}m=l^{2}q$
2. B.$m^{2}p=l^{2}q$
3. C.$m^{2}p=q^{2}l$
4. D.$m^2{p}^2=l^{2}q$

Consider the following for the next two (02) items that follow : Let $\alpha$ and $\beta$ be the roots of the equation $x^2-(1-2a^2)x+(1-2a^2)=0$. Under what condition does the above equation have real roots ?

1. A.$a^2<\frac{1}{2}$
2. B.$a^2>\frac{1}{2}$
3. C.$a^2\le \frac{1}{2}$
4. D.$a^2\ge \frac{1}{2}$

Consider the following for the next two (02) items that follow : Let $\alpha$ and $\beta$ be the roots of the equation $x^2-(1-2a^2)x+(1-2a^2)=0$. Under what condition is $\frac{1}{{\alpha }^2}+\frac{1}{{\beta }^2}<1$ ?

1. A.$a^2<\frac{1}{2}$
2. B.$a^2>\frac{1}{2}$
3. C.$a^2>1$
4. D.$a^2\in \left(\frac{1}{3},\frac{1}{2}\right)$ only

Consider the following for the next two (02) items that follow : $2x^2+3x-\alpha =0$ has roots $-2$ and $\beta$ while the equation $x^2-3mx+2m^2=0$ has both roots positive, where $\alpha >0$ and $\beta >0$. What is the value of $\alpha$?

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

If both the roots of the equation $x^2-2kx+k^2-4=0$ lie between $-3$ and 5, then which one of the following is correct ?

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

For the next two (2) items that follow : Let $\alpha$ and $\beta$ ($\alpha <\beta$) be the roots of the equation $x^2+bx+c=0$, where $b>0$ and $c<0$. Consider the following : 1. $\beta <-\alpha$ 2. $\beta <|\alpha |$ Which of the above is/are correct?

1. A.1 only
2. B.2 only
3. C.Both 1 and 2
4. D.Neither 1 nor 2

For the next two (2) items that follow : Let $\alpha$ and $\beta$ ($\alpha <\beta$) be the roots of the equation $x^2+bx+c=0$, where $b>0$ and $c<0$. Consider the following : 1. $\alpha +\beta +\alpha \beta >0$ 2. ${\alpha }^2\beta +{\beta }^2\alpha >0$ Which of the above is/are correct?

1. A.1 only
2. B.2 only
3. C.Both 1 and 2
4. D.Neither 1 nor 2

If $x^2-px+4>0$ for all real values of $x$, then which one of the following is correct?

1. A.$|p|<4$
2. B.$|p|\le 4$
3. C.$|p|>4$
4. D.$|p|\ge 4$

If one root of the equation $$(l-m)x^2+lx+1=0$$ is double the other and $l$ is real, then what is the greatest value of $m$?

1. A.$-\frac{9}{8}$
2. B.$\frac{9}{8}$
3. C.$-\frac{8}{9}$
4. D.$\frac{8}{9}$

For the next two (2) items that follow : Given that $\tan \alpha$ and $\tan \beta$ are the roots of the equation $x^2+bx+c=0$ with $b\ne 0$. What is $\tan (\alpha +\beta )$ equal to?

1. A.$b(c-1)$
2. B.$c(b-1)$
3. C.$c(b-1{)}^{-1}$
4. D.$b(c-1{)}^{-1}$

For the next two (2) items that follow : Given that $\tan \alpha$ and $\tan \beta$ are the roots of the equation $x^2+bx+c=0$ with $b\ne 0$. What is $\sin (\alpha +\beta )\sec \alpha \sec \beta$ equal to?

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

The number of real roots of the equation $x^2-3|x|+2=0$ is

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

If the sum of the roots of the equation $a{x}^2+bx+c=0$ is equal to the sum of their squares, then

1. A.$a^2+b^2=c^2$
2. B.$a^2+b^2=a+b$
3. C.$ab+b^2=2ac$
4. D.$ab-b^2=2ac$

If the roots of the equation $x^2-nx+m=0$ differ by 1, then

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