Quadratic Equations · NDA previous year questions

Consider the following for the next items that follow: Let $f(x)=x^2-1$ and $gof(x)=x-\sqrt{x}+1$. What is g(15) equal to ?

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

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

If $2-i\sqrt{3}$ where $i=\sqrt{-1}$ is a root of the equation $x^2+ax+b=0$, then what is the value of $(a+b)$?

1. A.-11
2. B.-3
3. C.0
4. D.3

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

Consider the following for the next two (02) items that follow : A quadratic equation is given by $(3+2\sqrt{2})x^2-(4+2\sqrt{3})x+(8+4\sqrt{3})=0$ What is the HM of the roots of the equation ?

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

1. A.√2(√6 - √3 + √2 - 1)
2. B.√2(√6 + √3 - √2 - 1)
3. C.(√6 - √3 + √2 - 1)
4. D.(√6 + √3 + √2 - 1)

For how many quadratic equations, the sum of roots is equal to the product of roots?

1. A.0
2. B.1
3. C.2
4. D.Infinitely many

Let p, q($p>q$) be the roots of the quadratic equation $x^2+bx+c=0$ where $c>0$. If $p^2+q^2-11pq=0$, then what is p - q equal to ?

1. A.$3\sqrt{c}$
2. B.$3c$
3. C.$9\sqrt{c}$
4. D.$9c$

Let $\alpha$ and $\beta$ be the roots of the equation $x^2+px+q=0$. If ${\alpha }^3$ and ${\beta }^3$ are the roots of the equation $x^2+mx+n=0$, then what is the value of $m+n$ ?

1. A.$p^3+q^3+pq$
2. B.$p^3+q^3-pq$
3. C.$p^3+q^3+3pq$
4. D.$p^3+q^3-3pq$

Let $\alpha$ and $\beta$ be the roots of the equation $x^2-ax-bx+ab-c=0$. What is the quadratic equation whose roots are a and b?

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

If the roots of the equation $x^2-ax-bx-cx+bc+ca=0$ are equal, then which one of the following is correct?

1. A.$a+b+c=0$
2. B.$a-b+c=0$
3. C.$a+b-c=0$
4. D.$-a+b+c=0$

Let $\alpha$ and $\beta$ ($\alpha >\beta$) be the roots of the equation $x^2-8x+q=0$. If ${\alpha }^2-{\beta }^2=16$, then what is the value of q?

1. A.$-15$
2. B.$-10$
3. C.10
4. D.15

The quadratic equation $3x^2-(k^2+5k)x+3k^2-5k=0$ has real roots of equal magnitude and opposite sign. Which one of the following is correct?

1. A.$0<k<\frac{5}{3}$
2. B.$0<k<\frac{3}{5}$ only
3. C.$\frac{3}{5}<k<\frac{5}{3}$
4. D.No such value of k exists

If $a_n=n(n!)$, then what is $a_1+a_2+a_3+\cdots +a_{10}$ equal to?

1. A.$10!-1$
2. B.$11!+1$
3. C.$10!+1$
4. D.$11!-1$

If p and q are the non-zero roots of the equation $x^2+px+q=0$, then how many possible values can q have?

1. A.Nil
2. B.One
3. C.Two
4. D.Three

1. A.$\Delta$
2. B.$7\Delta$
3. C.$72\Delta$
4. D.$-72\Delta$

If $\sin \theta$ and $\cos \theta$ are the roots of the equation $a{x}^2+bx+c=0$, then which one of the following is correct?

1. A.$a^2+b^2-2ac=0$
2. B.$-a^2+b^2+2ac=0$
3. C.$a^2-b^2+2ac=0$
4. D.$a^2+b^2+2ac=0$

Consider all the real roots of the equation $x^4-10x^2+9=0$. What is the sum of the absolute values of the roots?

1. A.4
2. B.6
3. C.8
4. D.10

If the roots of the equation $4x^2-(5k+1)x+5k=0$ differ by unity, then which one of the following is a possible value of k?

1. A.-3
2. B.-1
3. C.$-\frac{1}{5}$
4. D.$-\frac{3}{5}$

Consider the digits 3, 5, 7, 9. What is the number of 5-digit numbers formed by these digits in which each of these four digits appears?

1. A.240
2. B.180
3. C.120
4. D.60

If $\alpha$ and $\beta$ are the roots of the equation $4x^2+2x-1=0$, then which one of the following is correct?

1. A.$\beta =-2{\alpha }^2-2\alpha$
2. B.$\beta =4{\alpha }^3-3\alpha$
3. C.$\beta ={\alpha }^2-3\alpha$
4. D.$\beta =-2{\alpha }^2+2\alpha$

If one root of $5x^2+26x+k=0$ is reciprocal of the other, then what is the value of $k$?

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

The equation ${\sin }^{-1}x-{\cos }^{-1}x=\frac{\pi }{6}$ has

1. A.no solution
2. B.unique solution
3. C.two solutions
4. D.infinite number of solutions

If the roots of the quadratic equation $x^2+2x+k=0$ are real, then

1. A.$k<0$
2. B.$k\le 0$
3. C.$k<1$
4. D.$k\le 1$

If n = 100!, then what is the value of the following?

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