Quadratic Equations · NDA previous year questions

If 5 and 7 are the roots of the equation $\left|\begin{array}{ccc}x& 4& 5\\ 7& x& 7\\ 5& 8& x\end{array}\right|=0$ then what is the third root?

1. A.$-12$
2. B.9
3. C.13
4. D.14

What is the sum of the roots of the equation $(2-\sqrt{3})x^2-(7-4\sqrt{3})x+(2+\sqrt{3})=0$?

1. A.$2-\sqrt{3}$
2. B.$2+\sqrt{3}$
3. C.$7-4\sqrt{3}$
4. D.4

One of the roots of the quadratic equation $a{x}^2+bx+c=0,a\ne 0$ is positive and the other root is negative. The condition for this to happen is

1. A.$a>0,b>0,c>0$
2. B.$a>0,b<0,c>0$
3. C.$a<0,b>0,c<0$
4. D.$a<0,c>0$

What is the condition that one root of the equation $a{x}^2+bx+c=0,a\ne 0$ should be double the other?

1. A.$2a^2=9bc$
2. B.$2b^2=9ac$
3. C.$2c^2=9ab$
4. D.None of the above

If $\alpha$ and $\beta$ are the roots of the equation $4x^2+3x+7=0$, then what is the value of $({\alpha }^{-2}+{\beta }^{-2})$?

1. A.$47/49$
2. B.$49/47$
3. C.$-47/49$
4. D.$-49/47$

What is the equation of the line passing through $(2,-3)$ and parallel to $y$-axis?

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

If the roots of the equation $(a^2+b^2)x^2-2b(a+c)x+(b^2+c^2)=0$ are equal, then which one of the following is correct?

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

If $\alpha$ and $\beta$ are the roots of the equation $x^2-2x+4=0$, then what is the value of ${\alpha }^3+{\beta }^3$?

1. A.16
2. B.-16
3. C.8
4. D.-8

What is the value of $(-1+i\sqrt{3}{)}^{48}$?

1. A.1
2. B.2
3. C.$2^{24}$
4. D.$2^{48}$

Which of the following are the two roots of the equation $(x^2+2{)}^2+8x^2=6x(x^2+2)$?

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

If $\alpha$ and $\beta$ are the roots of the equation $x^2+x+1=0$, then which of the following are the roots of the equation $x^2-x+1=0$?

1. A.${\alpha }^7$ and ${\beta }^{13}$
2. B.${\alpha }^{13}$ and ${\beta }^7$
3. C.${\alpha }^{20}$ and ${\beta }^{20}$
4. D.None of the above

If the product of the roots of the equation $x^2-5x+k=15$ is $-3$, then what is the value of k?

1. A.12
2. B.15
3. C.16
4. D.18

If p and q are the roots of the equation $x^2-px+q=0$, then what are the values of p and q respectively?

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

If $\alpha ,\beta$ are the roots of the quadratic equation $x^2-x+1=0$, then which one of the following is correct?

1. A.$({\alpha }^4-{\beta }^4)$ is real
2. B.$2({\alpha }^5+{\beta }^5)=(\alpha \beta {)}^5$
3. C.$({\alpha }^6-{\beta }^6)=0$
4. D.$({\alpha }^8+{\beta }^8)=(\alpha \beta {)}^8$

What is the value of $\sqrt{3}\mathrm{cosec}20^{\circ }-\sec 20^{\circ }$?

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

Which one of the following is one of the roots of the equation $(b-c)x^2+(c-a)x+(a-b)=0$?

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

What is the value of $x$ satisfying the equation $16{\left(\frac{a-x}{a+x}\right)}^3=\frac{a+x}{a-x}$?

1. A.$a/2$
2. B.$a/3$
3. C.$a/4$
4. D.$0$

If $\alpha ,\beta$ are the roots of the equation $2x^2-2(1+n^2)x+(1+n^2+n^4)=0$, then what is the value of ${\alpha }^2+{\beta }^2$?

1. A.$2n^2$
2. B.$2n^4$
3. C.$2$
4. D.$n^2$

The roots of $A{x}^2+Bx+C=0$ are $r$ and $s$. For the roots of $x^2+px+q=0$ to be $r^2$ and $s^2$, what must be the value of $p$?

1. A.$(B^2-4AC)/A^2$
2. B.$(B^2-2AC)/A^2$
3. C.$(2AC-B^2)/A^2$
4. D.$B^2-2C$

What is the value of $r$ if $P(5,r)=P(6,r-1)$?

1. A.$9$
2. B.$5$
3. C.$4$
4. D.$2$

What is the number of words formed from the letters of the word 'JOKE' so that the vowels and consonants alternate?

1. A.4
2. B.8
3. C.12
4. D.None of the above

If $(x+a)$ is a factor of both the quadratic polynomials $x^2+px+q$ and $x^2+lx+m$, where $p,q,l$ and $m$ are constants, then which one of the following is correct?

1. A.$a=(m-q)/(l-p),(l\ne p)$
2. B.$a=(m+q)/(l+p),(l\ne -p)$
3. C.$l=(m-q)/(a-p),(a\ne p)$
4. D.$p=(m-q)/(a-l),(a\ne l)$