Quadratic Equations · NDA previous year questions

If the highest degree coefficient is equal to 1, then what is the total number of quadratic equations which are unchanged on squaring their roots ?

1. A.6
2. B.4
3. C.2
4. D.None

For the following two (02) items: Let $\alpha$ and $\beta$ be the roots of the quadratic equation $x^2+({\log }_{0.5}({\alpha }^2))x+({\log }_{0.5}({\alpha }^2){)}^4=0$ where ${\alpha }^2\ne 1$ and ${\log }_{0.5}({\alpha }^2)>0$ Further, ${\beta }^2=\alpha ({\log }_{{\alpha }^2}(0.5))$ What is $\beta$ equal to?

1. A.${\log }_{{\alpha }^2}(0.5)$
2. B.${\log }_{0.5}({\alpha }^2)$
3. C.$2({\log }_{{\alpha }^2}(0.5))$
4. D.$2{\log }_{0.5}({\alpha }^2)$

For the following two (02) items: Let $\alpha$ and $\beta$ be the roots of the quadratic equation $x^2+({\log }_{0.5}({\alpha }^2))x+({\log }_{0.5}({\alpha }^2){)}^4=0$ where ${\alpha }^2\ne 1$ and ${\log }_{0.5}({\alpha }^2)>0$ Further, ${\beta }^2=\alpha ({\log }_{{\alpha }^2}(0.5))$ What is the relation between $\alpha$ and $\beta$?

1. A.$\alpha =2\beta$
2. B.$2\alpha =\beta$
3. C.$\alpha =-2\beta$
4. D.$2\alpha =-\beta$

For the following two (02) items: Let $p={\sum }_{j=1}^n{\log }_{10}{2}^j\text{ and }q={\sum }_{j=1}^n{\log }_{10}{5}^j$ If p + q = 66 then which one of the following is correct?

1. A.$n<7$
2. B.$7<n<9$
3. C.$9<n<12$
4. D.$n>12$

For the following two (02) items: Let $p={\sum }_{j=1}^n{\log }_{10}{2}^j\text{ and }q={\sum }_{j=1}^n{\log }_{10}{5}^j$ If p + q = 15 then what is equal to? q - p

1. A.${\log }_{10}2\cdot 5$
2. B.$5{\log }_{10}2\cdot 5$
3. C.$10{\log }_{10}2\cdot 5$
4. D.$15{\log }_{10}2\cdot 5$

If the roots of the equation are equal, then which one of the following is correct?

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

For the following two (02) items: Let $f(x)=a{x}^2+bx+c$ be a quadratic polynomial such that f(1) = f(4) = 2 Further, 2 is a root of f(x) = 0 What is the other root of f(x) = 0

1. A.1
2. B.2
3. C.3
4. D.Cannot be determined

For the following two (02) items: Let $f(x)=a{x}^2+bx+c$ be a quadratic polynomial such that f(1) = f(4) = 2 Further, 2 is a root of f(x) = 0 What is (a + b + c) equal to?

1. A.0
2. B.1
3. C.2
4. D.Cannot be determined

If one root of the equation $x^2-kx+k=0$ exceeds the other by $2\sqrt{3}$ then which one of the following is a value of k?

1. A.3
2. B.6
3. C.9
4. D.12

If $x+\frac{5}{y}=4$, and $y+\frac{5}{x}=-4$, then what is (x + y) equal to?

1. A.0
2. B.1
3. C.4
4. D.5

If $n$ is a root of the equation $x^2+px+m=0$ and $m$ is a root of the equation $x^2+px+n=0$, where $m\ne n$, then what is the value of $p+m+n$?

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

What is the number of real roots of the equation $(x-1{)}^2+(x-3{)}^2+(x-5{)}^2=0$?

1. A.None
2. B.Only one
3. C.Only two
4. D.Three

In a class of 240 students, 180 passed in English, 130 passed in Hindi and 150 passed in Sanskrit. Further, 60 passed in only one subject, 110 passed in only two subjects and 10 passed in none of the subjects. How many passed in all three subjects?

1. A.60
2. B.55
3. C.40
4. D.35

Which one of the following is a root of the equation?

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

Direction : Consider the following for the items that follow : $$\text{Let A}=\left[\begin{array}{ccc}3& -3& 4\\ 2& -3& 4\\ 0& -1& 1\end{array}\right]$$ What is A(adj A) equal to?

1. A.$\left[\begin{array}{ccc}5& 0& 0\\ 0& 5& 0\\ 0& 0& 5\end{array}\right]$
2. B.$\left[\begin{array}{ccc}2& 0& 0\\ 0& 2& 0\\ 0& 0& 2\end{array}\right]$
3. C.$\left[\begin{array}{ccc}1/2& 0& 0\\ 0& 1/2& 0\\ 0& 0& 1/2\end{array}\right]$
4. D.$\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

1. A.$\left[\begin{array}{ccc}1& -1& 0\\ -2& 3& -4\\ -2& 3& -3\end{array}\right]$
2. B.$\left[\begin{array}{ccc}1/2& -1/2& 0\\ -1& 3/2& -2\\ -1& 3/2& -3/2\end{array}\right]$
3. C.$\left[\begin{array}{ccc}2& -2& 0\\ -4& 6& -8\\ -4& 6& -6\end{array}\right]$
4. D.$\left[\begin{array}{ccc}1/5& -1/5& 0\\ -2/5& 3/5& -4/5\\ -2/5& 3/5& -3/5\end{array}\right]$

If $\sqrt{2}$ and $\sqrt{3}$ are roots of the equation $a_0+a_{1}x+a_2{x}^2+a_3{x}^3+x^4=0$ where $a_0,a_1,a_2,a_3$ are integers, then which one of the following is correct?

1. A.$a_2=a_3=0$
2. B.$a_2=0\text{ and }{a}_3=-5$
3. C.$a_0=6,a_3=0$
4. D.$a_1=0\text{ and }{a}_2=5$

If $\tan \alpha$ and $\tan \beta$ are the roots of the equation $x^2-6x+8=0$, then what is the value of $\cos (2\alpha +2\beta )$?

1. A.$\frac{13}{75}$
2. B.$\frac{13}{85}$
3. C.$\frac{17}{85}$
4. D.$\frac{19}{85}$

What is the value of $\tan 65^{\circ }+2\tan 45^{\circ }-2\tan 40^{\circ }-\tan 25^{\circ }$?

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

Consider the following statements: 1. In a triangle $ABC$, if $\cot A\cdot \cot B\cdot \cot C>0$, then the triangle is an acute-angled triangle. 2. In a triangle $ABC$, if $\tan A\cdot \tan B\cdot \tan C>0$, then the triangle is an obtuse-angled triangle. Which of the statements given above is/are correct?

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 items that follow: A quadratic equation is given by $(a+b)x^2-(a+b+c)x+k=0$, where a, b, c are real. If $k=\frac{c}{2}$, $(c\ne 0)$, then the roots of the equation are : If $k=c$, then the roots of the equation are:

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

Consider the following for the next items that follow: Let $(1+x{)}^n=1+T_{1}x+T_2{x}^2+T_3{x}^3+\cdots +T_n{x}^n$. What is $T_1+2T_2+3T_3+\cdots +n{T}_n$ equal to ?

1. A.0
2. B.1
3. C.$2^n$
4. D.$n{2}^{n-1}$

What is $1-T_1+2T_2-3T_3+\cdots +(-1{)}^{n}n{T}_n$ equal to ?

1. A.0
2. B.$-2^{n-1}$
3. C.$n{2}^{n-1}$
4. D.1

Consider the following for the next items that follow: Let $(1+x{)}^n=1+T_{1}x+T_2{x}^2+T_3{x}^3+\cdots +T_n{x}^n$. What is $T_1+T_2+T_3+\cdots +T_n$ equal to ?

1. A.$2^n$
2. B.$2^n-1$
3. C.$2^{n-1}$
4. D.$2^n+1$

Consider the following for the next items that follow: Let $f(x)=x^2-1$ and $gof(x)=x-\sqrt{x}+1$. Which one of the following is a possible expression for g(x)?

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