Maths Class X Question Bank

Question 1 of 39

1. Question
1 point(s)
Which of the following is a root of the equation $2x^2-5x-3=0 $?
- $x=3 $
- $x=4 $
- $x=1 $
- $x=-3 $
Correct

Incorrect

Hint

$2x^2-5x-3\\ \\ 2x^2-6x+x-3\\ \\ 2x(x-3)+1(x-3)\\ \\ x=3,\ \dfrac{-1}{2}$
Question 2 of 39

2. Question
1 point(s)
Which of the following is a root of the equation $3x^2-2x-1=0 $?
- $x=-1 $
- $x=-2 $
- $x=1 $
- $x=2 $
Correct

Incorrect

Hint

$3x^2-2x-1\\ \\ 3x^2-3x+x-1\\ \\ 3x(x-1)+1(x-1)\\ \\ (3x+1)\ (x-1)=0\\ \\ x=\dfrac{-1}{3},1$
Question 3 of 39

3. Question
1 point(s)
$x=\sqrt2 $ is a solution of the equation :
- $x^2+\sqrt2x-4=0 $
- $x^2-\sqrt2x-4=0 $
- $3x^2+5x+2=0 $
- (A) and (B) both
Correct

Incorrect

Hint

$x^2-\sqrt2x-4=0\\ \\ x^2+\sqrt2x-4=(\sqrt2)^2+\sqrt2\times\sqrt2-4=2+2-4=0$
Question 4 of 39

4. Question
1 point(s)
Which of the following is a root of the equation $2x^2-x+\dfrac{1}{8} $?
- $\dfrac{1}{4} $
- $\dfrac{1}{3} $
- $\dfrac{1}{2} $
- $4 $
Correct

Incorrect

Hint

$2x^2-x+\dfrac{1}{8}\Rightarrow 16x^2-8x+1=0\\ \\ \Rightarrow 16x^2-4x-4x+1=0\\ \\ \Rightarrow 4x(4x-1)-1(4x-1)=0\\ \\ \Rightarrow x=\dfrac{1}{4}$
Question 5 of 39

5. Question
1 point(s)
Which of the following is a quadratic equation in x?
- $\sqrt x+\dfrac{1}{\sqrt x}=4 $
- $x^2+\dfrac{2}{x^2}=3 $
- $x+\dfrac{1}{x^2}=3 $
- $\dfrac{4}{3}x+x^2=0 $
Correct

Incorrect

Hint

(a) $x+1=4\sqrt x$
(b) $x^4+2=3x^2$
(c) $x^3+1=3x^2$
(d) $4x+3x^2=0\Rightarrow 3x^2+4x+0=0$
Question 6 of 39

6. Question
1 point(s)
Which of the following is not a quadratic equation?
- $2(x-1)^2=4x^2-2x+1 $
- $2x-x^2=x^2+5 $
- $(\sqrt2x+\sqrt3)^2+x^2=3x^2-5x $
- $(x^2+2x)^2=x^4+3+4x^3 $
Correct

Incorrect

Hint

$(\sqrt2x+\sqrt3)^2+x^2=3x^2-5x\\ \\ \Rightarrow 2x^2+3+2\sqrt6x+x^2=3x^2-5x\\ \\ \Rightarrow (5+2\sqrt6)x+3=0$
Question 7 of 39

7. Question
1 point(s)
Which of the following equations has 2 as a root ?
- $x^2-4x+5=0 $
- $x^2+3x-12=0 $
- $2x^2-7x+6=0 $
- $3x^2-6x-2=0 $
Correct

Incorrect

Hint

$2x^2-7x+6\\ \\ 2(2)^2-7(2)+6\\ \\ =8-14+6=14-14=0$
Question 8 of 39

8. Question
1 point(s)
The roots of $4x^2+4\sqrt3x+3=0 $ are:
- real and equal
- real and unequal
- not real
- none of these
Correct

Incorrect

Hint

$4x^2+4\sqrt3x+3=0\\ \\ D=b^2-4ac=(4\sqrt3)^2-4\times4\times3\\ \\ =48-48=0$
Question 9 of 39

9. Question
1 point(s)
Discriminant of $x^2+px+2q=0 $ is:
- $p-8q $
- $p^2+8q $
- $p^2-8q $
- $q^2-8p $
Correct

Incorrect

Hint

$x^2+px+2q=0\\ \\ D=b^2-4ac=p^2-4\times2q\times1\\ \\ =p^2-8q$
Question 10 of 39

10. Question
1 point(s)
If $ax^2+bx+c=0 $ has equal roots, then $c $ is equal to:
- $-\dfrac{b}{2a} $
- $\dfrac{b}{2a} $
- $\dfrac{-b^2}{4a} $
- $\dfrac{b^2}{4a} $
Correct

Incorrect

Hint

$D=0\\ \\ b^2-4ac=0\\ \\ \Rightarrow b^2=4ac\\ \\ \Rightarrow c=\dfrac{b^2}{4a}$
Question 11 of 39

11. Question
1 point(s)
If the equation $9x^2+6kx+4=0 $ has equal roots, then the roots are:
- $\pm\dfrac{2}{3} $
- $\pm\dfrac{3}{2} $
- $0 $
- $\pm3 $
Correct

Incorrect

Hint

$b^2-4ac=0\\ \\ (6k)^2-4\times9\times4=0\\ \\ 36k^2=36\times4\\ \\ k^2=4\Rightarrow k=\pm2$
Roots $=\dfrac{-b}{2a}=\dfrac{-12}{18}=\pm\dfrac{2}{3}$

Question 12 of 39

12. Question

1 point(s)

If the equation $(a^2+b^2)x^2-2(ac+bd)x+c^2+d^2=0 $ has equal roots, then:

$ad=cd $
$ad=bc $
$ad=\sqrt{bc} $
$ab=\sqrt{cd} $

Correct

Incorrect

Hint

*** QuickLaTeX cannot compile formula:
(a^2+b^2)x^2-2(ac+bd)x+c^2+d^2=0\\ \\ b^2-4ac=0\\ \\ [-2(ac+bd)^2]-4(a^2+b^2)\ (c^2+d^2)=0\\ \\ \Rightarrow 4(a^2c^2+b^2d^2+2abcd-a^2c^2-a^2d^2-b^2c^2-b^2d^2)=0\\ \\ \Rightarrow (ad-bc)^2=0\Rightarrow ad=bc 

*** Error message:
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leading text: ...^2+d^2=0\\ \\ b^2-4ac=0\\ \\ [-2(ac+bd)^2]
Missing $ inserted.
leading text: ...^2+d^2=0\\ \\ b^2-4ac=0\\ \\ [-2(ac+bd)^2]
Extra }, or forgotten $.
leading text: ...^2+d^2=0\\ \\ b^2-4ac=0\\ \\ [-2(ac+bd)^2]
You can't use `\end' in internal vertical mode.
leading text: \end{document}
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leading text: \end{document}
Missing $ inserted.
leading text: \end{document}
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leading text: \end{document}
Emergency stop.

Question 13 of 39

13. Question
1 point(s)
If the roots of the equation $(a^2+b^2)x^2-2b\ (a+c)x+(b^2+c^2)=0 $ are equal, then:
- $2b=a+c $
- $b^2=ac $
- $b=\dfrac{2ac}{a+c} $
- $b=ac $
Correct

Incorrect

Hint

$(a^2+b^2)x^2-2b(a+c)x+b^2+c^2=0\\ \\ b^2-4ac=0\\ \\ [-2b(a+c)]^2-4(a^2+b^2)\ (b^2+c^2)=0\\ \\ \Rightarrow 4[b^2(a^2+c^2+2ac)-a^2b^2-a^2c^2-b^4-b^2c^2]=0\\ \\ \Rightarrow 2acb^2-a^2c^2-b^4=0\\ \\ \Rightarrow (b^2-ac)^2=0\Rightarrow b^2=ac$
Question 14 of 39

14. Question
1 point(s)
If $x=1 $ is a common root of the equation $ax^2+ax+3=0 $ and $x^2+x+b=0 $, then $ab $ is equal to:
- 3
- 3.5
- 6
- -3
Correct

Incorrect

Hint

$ax^2+ax+3=0$ …(1)
$x^2+x+b=0$ …(2)
$1^2+1+b=0\Rightarrow b=-2\\ \\ a(1)^2+a(1)+3=0\ \Rightarrow 2a=-3\\ \\ ab=\dfrac{-3}{2}\times(-2)=3$
Question 15 of 39

15. Question
1 point(s)
Which constant must be added and subtracted to solve the quadratic equation $9x^2+\dfrac{3}{4}x-\sqrt2=0 $ by the method of completing the square ?
- $\dfrac{1}{8} $
- $\dfrac{1}{64} $
- $\dfrac{1}{4} $
- $\dfrac{9}{64} $
Correct

Incorrect

Hint

$ax^2+\dfrac{3}{4}x-\sqrt2=0\\ \\ (3x)^2+\dfrac{1}{4}(3x)-\sqrt2=0$
Let $3x=y$
$y^2+\dfrac{y}{4}-\sqrt2=0\Rightarrow y^2+\dfrac{1}{4}y+\left(\dfrac{1}{8}\right)^2-\left(\dfrac{1}{8}\right)^2-\sqrt2=0\\ \\ \Rightarrow \left(y+\dfrac{1}{8}\right)^2=\dfrac{1+64\sqrt2}{64}$
Question 16 of 39

16. Question
1 point(s)
Which of the following equations has two distinct real roots ?
- $2x^2-3\sqrt2x+\dfrac{9}{4}=0 $
- $x^2+x-5=0 $
- $x^2+3x+2\sqrt2=0 $
- $5x^2-3x+1=0 $
Correct

Incorrect

Hint

$x^2+x-5\\ \\ D=b^2-4ac=1^2-4(1)(-5)=21>0$
Question 17 of 39

17. Question
1 point(s)
Which of the following equations has no real roots?
- $x^2-4x+3\sqrt2=0 $
- $x^2+4x-3\sqrt2=0 $
- $x^2-4x-3\sqrt2=0 $
- $3x^2+4\sqrt3x+4=0 $
Correct

Incorrect

Hint

$D=b^2-4ac=(-4)^2-4(1)(3\sqrt2)\\ \\ =16-12\sqrt2=-0.92<0$
Question 18 of 39

18. Question
1 point(s)
$(x^2+1)^2-x^2=0 $ has:
- four real roots
- two real roots
- no real roots
- one real root
Correct

Incorrect

Hint

$(x^2+1)^2-x^2=0\\ \\ \Rightarrow x^4+1+2x^2-x^2=0\\ \\ \Rightarrow x^4+x^2+1=0$
Let $x^2=y$
$y^2+y+1=0\\ \\ D=b^2-4ac=1^2-4(1)(1)=-3<0$
Question 19 of 39

19. Question
1 point(s)
The quadratic equation $2(p+q)^2x^2+2(p+q)x+1=0 $ has:
- equal roots
- no real roots
- real but not equal roots
- none of these
Correct

Incorrect

Hint

$2(p+q)^2x^2+2(p+q)x+1=0\\ \\ b^2-4ac\\ \\ [2(p+q)]^2-4(2(p^2+q^2)\times1)\\ \\ =4(p+q)^2-8(p^2+q^2)\\ \\ 4(p^2+q^2+2pq)-8p^2-8q^2\\ \\ =-4(p^2+q^2-2pq)=-4(p-q)^2<0$
Question 20 of 39

20. Question
1 point(s)
If $p^2x^2+(p^2+q^2)x+q^2=0 $ has equal roots, then $p^2-q^2 $ is equal to:
- $-2q^2 $
- $2q^2 $
- $0 $
- $-1 $
Correct

Incorrect

Hint

$p^2x^2+(p^2+q^2)x+q^2=0\\ \\ b^2-4ac=0\\ \\ (p^2+q^2)^2-4\times p^2\times q^2=0\\ \\ \Rightarrow p^4+q^4+2p^2q^2-4p^2q^2=0\\ \\ \Rightarrow (p^2-q^2)=0\Rightarrow p^2=q^2\Rightarrow p^2-q^2=0$
Question 21 of 39

21. Question
1 point(s)
If the price of a pen is increased by Rs 2, a person can buy 1 pen less for Rs 40, then the original price of one pen is :
- Rs 10
- Rs 8
- Rs 6
- Rs 4
Correct

Incorrect

Hint

The original price is ₹8 so he buys 5 pens. If price is increased to $8+2=10$ he buy 4 pens
Question 22 of 39

22. Question
1 point(s)
If the sum of the square of three consecutive integers is 29, then one of the integer is :
- 1
- 2
- 5
- 6
Correct

Incorrect

Hint

$x^2+(x+1)^2+(x+2)^2=29\\ \\ \Rightarrow x^2+x^2+1+2x+x^2+4+4x=29\\ \\ \Rightarrow 3x^2+6x+5=29\\ \\ \Rightarrow 3(x^2+2x-8)=0\\ \\ \Rightarrow (x-2)\ (x+4)=0\Rightarrow x=2$
There consecutive integers are 2, 3, 4
Question 23 of 39

23. Question
1 point(s)
The roots of the quadratic equation $\sqrt3x^2-2x-\sqrt3=0 $ are:
- $-\sqrt3,\sqrt{\dfrac{1}{3}} $
- $2,3 $
- $\dfrac{\sqrt3}{2},-\dfrac{2}{\sqrt3} $
- $\sqrt3,\dfrac{-1}{\sqrt3} $
Correct

Incorrect

Hint

$\sqrt3x^2-2x-\sqrt3=0\\ \\ \sqrt3x^2-3x+x-\sqrt3=0\\ \\ \sqrt3(x-\sqrt3)+1(x-\sqrt3)=0\\ \\ x=\sqrt3,\dfrac{-1}{\sqrt3}$
Question 24 of 39

24. Question
1 point(s)
The roots of the quadratic equations $x^2+7x+12=0 $ are
- –4, –3
- 4, –3
- 4, 3
- –4, 3
Correct

Incorrect

Hint

$x^2+7x+12=0\\ \\ x^2+4x+3x+12=0\\ \\ x(x+4)+3(x+4)=0\\ \\ x=-3,\ -4$
Question 25 of 39

25. Question
1 point(s)
If the discriminant of $3x^2+2x+a=0 $, is double the discriminant of $x^2-4x+2=0 $, then the value of $a $ is:
- 2
- -2
- 1
- -1
Correct

Incorrect

Hint

A/Q $2^2-4\times3\times a=2\times[(-4)^2-4\times1\times2]\\ \\ 4-12a=16\Rightarrow 12a=-12\Rightarrow a=-1$
Question 26 of 39

26. Question
1 point(s)
One of the roots of the quadratic equation $6x^2-x-2=0 $ is:
- $\dfrac{1}{2} $
- $-\dfrac{1}{2} $
- $-\dfrac{2}{3} $
- $-1 $
Correct

Incorrect

Hint

$6x^2-x-2=0\\ \\ 6x^2-4x+3x-2=0\\ \\ 2x(3x-2)+1(3x-2)=0\\ \\ x=\dfrac{-1}{2},\dfrac{2}{3}$
Question 27 of 39

27. Question
1 point(s)
For what value of $k $ will $\dfrac{7}{3} $ be a root of $3x^2-13x-k=0 $?
- $14 $
- $\dfrac{3}{7} $
- $\dfrac{-7}{2} $
- $-14 $
Correct

Incorrect

Hint

$3\left(\dfrac{7}{3}\right)^2-13\times\dfrac{7}{3}-K=0\\ \\ 3\times\dfrac{49}{9}-\dfrac{91}{3}-K=0\\ \\ \Rightarrow \dfrac{49}{3}-\dfrac{91}{3}-K=0\Rightarrow \dfrac{49-91-3K}{3}=0\\ \\ \Rightarrow -42=3K\Rightarrow K=-14$
Question 28 of 39

28. Question
1 point(s)
The value of $ k$ for which the equation $2x^2-(k-1)\ x+8=0 $ will have real and equal roots are :
- 9 and –7
- only 9
- only –7
- –9 and –7
Correct

Incorrect

Hint

$2x^2-(K-1)x+8=0\\ \\ b^2-4ac=0\\ \\ (K-1)^2-4\times2\times8=0\\ \\ K^2+1-2K-64=0\\ \\ K^2-2K-63=0\\ \\ K^2-9K+7K-63=0\\ \\ K(K-9)+7(K-9)=0\Rightarrow K=9,-7$
Question 29 of 39

29. Question
1 point(s)
Which of the following is a solutions of the quadratic equation $x^2-b^2=a(2x-a) $?
- $a+b $
- $2b-a $
- $ab $
- $\dfrac{a}{b} $
Correct

Incorrect

Hint

$x^2-b^2=a\ (2x-a)\\ \\ x^2-2ax+a^2-b^2=0\\ \\ x=\dfrac{2a\pm\sqrt{4a^2-4(a^2-b^2)}}{2}=\dfrac{2a\pm\sqrt{4b^2}}{2}\\ \\ =\dfrac{2a\pm2b}{2}\\ \\ =\dfrac{2a+2b}{2},\dfrac{2a-2b}{2}\\ \\ =a+b,a-b$
Question 30 of 39

30. Question
1 point(s)
If $\dfrac{1}{2} $ is the root of the equation $x^2+kx-\dfrac{5}{4}=0 $, then the value of $k $ is:
- $2 $
- $-2 $
- $\dfrac{1}{4} $
- $\dfrac{1}{2} $
Correct

Incorrect

Hint

$x^2+kx-\dfrac{5}{4}=0\\ \\ \Rightarrow \left(\dfrac{1}{2}\right)^2+\dfrac{k}{2}-\dfrac{5}{4}=0\Rightarrow \dfrac{1}{4}-\dfrac{5}{4}+\dfrac{k}{2}=0\\ \\ \Rightarrow \dfrac{-4}{4}+\dfrac{k}{2}=0\\ \\ \Rightarrow -1\times2=k\Rightarrow k=-2$
Question 31 of 39

31. Question
1 point(s)
The roots of the equation $3x^2-4x+3=0 $ are:
- real and unequal
- real and equal
- imaginary
- none of these
Correct

Incorrect

Hint

$3x^2-4x+3=0\\ \\ D=b^2-4ac=(-4)^2-4\times3\times3\\ \\ =-20<0$
Question 32 of 39

32. Question
1 point(s)
If the equation $kx^2+4x+1=0 $ has real and distinct root, then :
- k<4 [/latex]
- $k>4 $
- $k\le4 $
- $k\ge4 $
Correct

Incorrect

Hint

$b^2-4ac\ge0\\ \\ 4^2-4\times k\times1\ge0\\ \\ 16-4k\ge0\\ \\ 16=4k\Rightarrow k\ge4$
Question 33 of 39

33. Question
1 point(s)
Value of $k $ for which quadratic equation $2x^2-kx+k=0 $ has equal roots is :
- -4
- 4
- 8
- -4
Correct

Incorrect

Hint

$b^2-4ac=0\\ \\ (-k)^2-4\times2\times k=0\\ \\ k^2-8k=0\ \Rightarrow k^2=8k\ \Rightarrow k=8$
Question 34 of 39

34. Question
1 point(s)
If $r=3 $ is a root of quadratic equation $kr^2-kr-3=0 $, value of $k $ is:
- $\dfrac{1}{2} $
- $2 $
- $-2 $
- $-\dfrac{1}{2} $
Correct

Incorrect

Hint

$kr^2-kr-3=0\\ \\ k(3)^2-k(3)-3=0\\ \\ 9k-3k-3=0\\ \\ 6k=3\Rightarrow k=\dfrac{1}{2}$
Question 35 of 39

35. Question
1 point(s)
The value of $k $, for which the quadratic equation $4x^2+4\sqrt3x+k=0 $ has equal roots is :
- $k=2 $
- $k=-2 $
- $k=-3 $
- $k=3 $
Correct

Incorrect

Hint

$b^2-4ac=0\\ \\ (4\sqrt3)^2-4\times4\times k=0\\ \\ 48-16k=0\Rightarrow 48=16k\\ \\ \Rightarrow k=3$
Question 36 of 39

36. Question
1 point(s)
For what value of $k $ the equation $kx^2-6x-2=0 $ has equal roots ?
- $\dfrac{7}{2} $
- $\dfrac{-9}{2} $
- $-2 $
- $\dfrac{-7}{2} $
Correct

Incorrect

Hint

$b^2-4ac=0\\ \\ (-6)^2-4\times k\times (-2)=0\\ \\ 36+8k=0\\ \\ 36=-8k\Rightarrow k=\dfrac{-36}{8}=\dfrac{-9}{2}$
Question 37 of 39

37. Question
1 point(s)
The value of $p $ for which the quadratic equation $x(x – 4) + p = 0$ has real roots, is :
- p<4 [/latex]
- $p\ge4 $
- $p=4 $
- None of these
Correct

Incorrect

Hint

$x(x-4)+P=0\\ \\ x^2-4x+P=0\\ \\ b^2-4ac=0\\ \\ (-4)^2-4\times1\times P=0\\ \\ 16-4P=0\Rightarrow P=4$
Question 38 of 39

38. Question
1 point(s)
The value of $k $ for which $3x^2+2x+k=0 $ has real roots is:
- $k=\dfrac{1}{3} $
- $k\le\dfrac{1}{3} $
- $k\ge\dfrac{1}{3} $
- k<\dfrac{1}{3} [/latex]
Correct

Incorrect

Hint

$b^2-4ac=0\\ \\ (2)^2-4\times3\times k=0\\ \\ 4-12k=0\ \Rightarrow 4=12k\\ \\ \Rightarrow k=\dfrac{1}{3}$
Question 39 of 39

39. Question
1 point(s)
For the quadratic equation $x^2-2x+1=0 $ the value of $x+\dfrac{1}{x} $ is:
- -1
- 1
- 2
- -2
Correct

Incorrect

Hint

$x^2-2x+1=0\\ \\ \Rightarrow (x-1)^2=0\\ \\ \Rightarrow x-1=0\Rightarrow x=1\\ \\ x+\dfrac{1}{x}=1+\dfrac{1}{1}=2$