Quadratic Equations · NDA previous year questions

2018 8 questions

1. Q122018 · 9 Sept 2018

   If $\alpha$ and $\beta (\ne 0)$ are the roots of the quadratic equation $x^2+\alpha x-\beta =0$, then the quadratic expression $-x^2+\alpha x+\beta$ where $x\in \mathbf{R}$ has

   1. A.Least value $-\frac{1}{4}$
   2. B.Least value $-\frac{9}{4}$
   3. C.Greatest value $\frac{1}{4}$
   4. D.Greatest value $\frac{9}{4}$
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   Answer: D

2. Q132018 · 9 Sept 2018

   Let x be the number of integers lying between 2999 and 8001 which have at least two digits equal. Then x is equal to

   1. A.2480
   2. B.2481
   3. C.2482
   4. D.2483
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   Answer: B

3. Q412018 · 9 Sept 2018

   If $\cos \alpha$ and $\cos \beta$ $(0<\alpha <\beta <\pi )$ are the roots of the quadratic equation $4x^2-3=0$, then what is the value of $\sec \alpha \times \sec \beta$?

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

4. Q422018 · 9 Sept 2018

   If $A={\sin }^2\theta +{\cos }^4\theta$, then for all real $\theta$, which one of the following is correct?

   1. A.$1\le A\le 2$
   2. B.$\frac{3}{4}\le A\le 1$
   3. C.$\frac{13}{16}\le A\le 1$
   4. D.$\frac{3}{4}\le A\le \frac{13}{16}$
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   Answer: B

5. Q432018 · 9 Sept 2018

   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
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   Answer: A

6. Q1122018 · 9 Sept 2018

   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.$g(x)\ge 0$
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   Answer: B

7. Q1152018 · 9 Sept 2018

   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
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   Answer: B

8. Q172018 · 22 Apr 2018

   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
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   Answer: A