If $\sin \alpha$ and $\cos \alpha$ are the roots of the equations $a x^2+b x+c=0$, then $b^2=$
- A$a^2-2 a c$
- ✓$a^2+2 a c$
- C$a^2-a c$
- D$a 2+a c$
Answer: B.
View full solution →Question types
Quadratic Equations question types
465 questions across 8 question groups — pick any mix to generate a Maths paper with step-by-step answer keys.
465
Questions
8
Question groups
5
Question types
01
M.C.Q (1 Marks)
100 Q→02Assertion (A) & Reason (B) MCQ
11 Q→03Fill In The Blanks[1 Marks ]
14 Q→041 Marks Question
20 Q→052 Marks Questions
40 Q→063 Marks Question
129 Q→075 Marks Questions
144 Q→08Case study (4 Marks)
7 Q→Sample Questions
Quadratic Equations questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
If $2$ is a root of the equation $x^2+b x+12=0$ and the equation $x^2+b x+q=0$ has equal roots, then $q=$
- A$8$
- B$-8$
- ✓$16$
- D$-16$
Answer: C.
View full solution →If the equation $ax^2 + 2x + a = 0$ has two distinct roots, if:
- ✓$\text{a}=\pm1$
- B$a = 0$
- C$a = 0, 1$
- D$a = -1, 0$
Answer: A.
View full solution →If the equation $x^2+4 x+k=0$ has real and distinct roots, then:
- ✓$\text{k}<4$
- B$\text{k}>4$
- C$\text{k}\geq4$
- D$\text{k}\leq4$
Answer: A.
View full solution →The value of $c$ for which the equation $ax^2 + 2bx + c = 0$ has equal roots is:
- ✓$\frac{\text{b}^2}{\text{a}}$
- B$\frac{\text{b}^2}{4\text{a}}$
- C$\frac{\text{a}^2}{\text{b}}$
- D$\frac{\text{a}^2}{4\text{b}}$
Answer: A.
View full solution →Statement-1 (A) : If the difference of roots of the equation $x^2-2 p x+q=0$ is same as the difference of the roots of the equation $x^2-2 r x+s=0$, then $s-q=r^2-p^2$.
Statement-2 (R): The roots of the quadratic equation $a x^2+b x+c=0$ are given by $x=\frac{-b \pm \sqrt{D}}{2 a}$, where $D$ is the discriminant.
Statement-2 (R): The roots of the quadratic equation $a x^2+b x+c=0$ are given by $x=\frac{-b \pm \sqrt{D}}{2 a}$, where $D$ is the discriminant.
- AStatement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
- ✓Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1
- CStatement-1 is True, Statement-2 is False.
- DStatement-1 is False, Statement-2 is True.
Answer: B.
View full solution →Statement-1 (A): If $5+\sqrt{7}$ is a root of a quadratic equation with rational coefficients, then its other root is $5-\sqrt{7}$.
Statement-2 (R) : Surd roots of a quadratic equation with rational coefficients occur a conjugate pairs.
Statement-2 (R) : Surd roots of a quadratic equation with rational coefficients occur a conjugate pairs.
- ✓Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
- BStatement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1
- CStatement-1 is True, Statement-2 is False.
- DStatement-1 is False, Statement-2 is True.
Answer: A.
View full solution →Statement-1 (A): If $a c \neq 0$, then at least one of the two equations $a x^2+b x+c=0$ and $a x^2+b x-c=0$ has real and distinct roots.
Statement-2 (R): A quadratic equation has real and distinct roots if its discriminant is positive.
Statement-2 (R): A quadratic equation has real and distinct roots if its discriminant is positive.
- ✓Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
- BStatement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1
- CStatement-1 is True, Statement-2 is False.
- DStatement-1 is False, Statement-2 is True.
Answer: A.
View full solution →Statement-1 (A): If $a$ and $c$ are of opposite signs, then the quadratic equation $a x^2+b x+c=0$ has real and distinct roots.
Statement-2(R): If discrimmant Dof a quadratic equationt is not equal to zero, it has roal and distinct rooks.
Statement-2(R): If discrimmant Dof a quadratic equationt is not equal to zero, it has roal and distinct rooks.
- AStatement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
- BStatement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1
- ✓Statement-1 is True, Statement-2 is False.
- DStatement-1 is False, Statement-2 is True.
Answer: C.
View full solution →Statement-1 (A) : If roots of the equation $(2 k-1) x^2+4 x-3=0$ are reciprocal of each other, then $k=-1$.
Statement- 2(R) : If $a=c$, then roots of $a x^2+b x+c=0$ are reciprocal of each other.
Statement- 2(R) : If $a=c$, then roots of $a x^2+b x+c=0$ are reciprocal of each other.
- ✓Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
- BStatement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1
- CStatement-1 is True, Statement-2 is False.
- DStatement-1 is False, Statement-2 is True.
Answer: A.
View full solution →If the coefficient of $x^2$ and the constant term of a quadratic equation have the same sign and if the coefficient of x term is zero, then the quadratic equation has __________ roots.
View full solution →If the coefficient of $x^2$ and the constant term of a quadratic equation have opposite signs, then the roots of the quadratic equation are __________ and __________.
View full solution →The number of real roots of the equation $x^2+3|x|+2=0$ is __________.
View full solution →The total number of values of x satisfying $x^2-3|x|+2=0$ is __________.
View full solution →The quadratic equation with rational coefficients having $\frac{\sqrt{3}}{2}$ as a root is __________.
View full solution →(a) Find the nature of the roots of the quadratic equation $x^2-5 x+9=0$.
(b) Write a quadratic equation with roots -3 and 5.
View full solution →(b) Write a quadratic equation with roots -3 and 5.
Solve the quadratic equation $2 x^2-5 x-1=0$ for $x$.
View full solution →Find the value of k for which the roots of the equation $3 x^2-10 x+k=0$ are reciprocal of each other.
View full solution →For what value of k, the roots of the equation $x^2+4 x+k=0$ are real?
View full solution →If x = 3 is one root of the quadratic equation $x^2-2 k x-6=0$, then find the value of k.
View full solution →Which of the following are quadratic equations?
$(x + 2)^3= x^3 - 4$
View full solution →$(x + 2)^3= x^3 - 4$
In the following, find the value of k for which the given value is a solution of the given equation:
$x^2 + 3ax + k = 0, x = -a$
View full solution →$x^2 + 3ax + k = 0, x = -a$
Write the discriminant of the following quadratic equation.
$2x^2 - 5x + 3 = 0$
View full solution →$2x^2 - 5x + 3 = 0$
Which of the following are quadratic equations?
$\text{x}+\frac{1}{\text{x}}=1$
View full solution →$\text{x}+\frac{1}{\text{x}}=1$
Write the discriminant of the following quadratic equation.
$(x - 1)(2x - 1) = 0$
View full solution →$(x - 1)(2x - 1) = 0$
Find the values of k for which the given quadratic equation has real and distinct root:
$kx^2 + 2x + 1 = 0$
View full solution →$kx^2 + 2x + 1 = 0$
Find the value of k for which the following equation have real root:
$4x^2 + kx + 3 = 0$
View full solution →$4x^2 + kx + 3 = 0$
Find the least positive value of k for which the equation $x^2 + kx + 4 = 0$ has real roots.
View full solution →Find the value of k for which the root are real and equal in the following equations:
$4x^2 + kx + 9 = 0$
View full solution →$4x^2 + kx + 9 = 0$
Find the value of k for which the following equation have real root:
$kx(x - 3) + 9 = 0$
View full solution →$kx(x - 3) + 9 = 0$
Find a natural number whose square diminished by 84 is equal to thrice of 8 more than the given number.
If the roots the equations $ax^2 + 2bx + c = 0$ and $\text{bx}^2-2\sqrt{\text{ac}}\text{x}+\text{b}=0$ are simultaneously real, then prove that $b^2 = ac.$
View full solution →A pole has to be erected at a point on the boundary of a circular park of diameter 13 meters in such a way that the difference of its distances from two diametrically opposite fixed gates A and B on the boundary is 7 meters. Is it the possible to do so? If yes, at what distances from the two gates should the pole be erected?
A two-digit number is such that the product of digit is 12. When 36 is added to the number the digits interchange their places. Determine the number.
Find the value of k for which root are real and equal in the following equations:
$x^2 - 2(5 + 2k)x + 3(7 + 10k) = 0$
View full solution →$x^2 - 2(5 + 2k)x + 3(7 + 10k) = 0$
A rectangular floor area can be completely tiled with 200 square tiles. If the side leng th of each tile is increased by 1 unit, it would take only 128 tiles to cover the floor
(i) Assuming the original length of each side of a tile be $x$ units, make a quadratic equation from the above information.
(ii) Write the corresponding quadratic equation in standard form.
(iii) (a) Find the value of $x$, the length of side of tile by factorisation.
OR
(b) Solve the quadratic equation for $x$, using quadratic formula.
View full solution →(i) Assuming the original length of each side of a tile be $x$ units, make a quadratic equation from the above information.
(ii) Write the corresponding quadratic equation in standard form.
(iii) (a) Find the value of $x$, the length of side of tile by factorisation.
OR
(b) Solve the quadratic equation for $x$, using quadratic formula.
Two schools 'P' and 'Q' decided to award prizes to their students for two games of Hockey ₹$ x$ per student and Cricket ₹ y per student. School 'P' decided to award a total of ₹ 9,500 for the two games to 5 and 4 students respectively; while school 'Q' decided to award ₹ 7,370 for the two games to 4 and 3 students respectively.
(i) Represent the following information algebraically (in terms of $x$ and $y$ ).
(ii) What is the prize amount for hockey?
(iii) Prize amount of which game is more and by how much?
(iv) What will be the total prize amount if there are 2 students each from two games?
View full solution →(i) Represent the following information algebraically (in terms of $x$ and $y$ ).
(ii) What is the prize amount for hockey?
(iii) Prize amount of which game is more and by how much?
(iv) What will be the total prize amount if there are 2 students each from two games?
While designing the school year book, a teacher asked the student that the length and width of a particular photo is increased by $x$ units each to double the area of the photo. The original photo is 18 cm long and 12 cm wide.
Based on the above information, answer the following questions:
(i) Write an algebraic equation depicting the above information.
(ii) Write the corresponding quadratic equation in standard form.
(iii) What should be the new dimensions of the enlarged photo?
(iv) Can any rational value of $x$ make the new area equal to $220 cm^2$ ?
View full solution →Based on the above information, answer the following questions:
(i) Write an algebraic equation depicting the above information.
(ii) Write the corresponding quadratic equation in standard form.
(iii) What should be the new dimensions of the enlarged photo?
(iv) Can any rational value of $x$ make the new area equal to $220 cm^2$ ?
Johan and Jayant are very close friends. They decided to go to Ranikhet with their families in Separate cars. Johan's car travels at a speed of $x km / hr$ while Jayant's car travels $5 km / hr$ faster than Johan's car. Johan took 4 hours more than Jayant to complete the Journey of 400 km .
(i) The distance covered by Jayant's car in two hours is
(a) $2(x+5) km$ $\qquad$ (b) $(x-5) km$
(c) $2(x+10) km$ $\qquad$ (d) $(2 x+5) km$
(ii) The quadratic equation describing the speed of Johan's car is
(a) $x^2-5 x-500=0$ $\qquad$ (b) $x^2+4 x-400=0$
(c) $x^2+5 x-500=0$ $\qquad$ (d) $x^2-4 x+400=0$
(iii) The speed of Johan's car is (in km/hour)
(a) 20 $\qquad$ (b) 15 $\qquad$ (c) 25 $\qquad$ (d) 10
(iv) The speed of Jayant's car in $km / hr$ is
(a) 25 $\qquad$ (b) 20 $\qquad$ (c) 30 $\qquad$ (d) 15
View full solution →(i) The distance covered by Jayant's car in two hours is
(a) $2(x+5) km$ $\qquad$ (b) $(x-5) km$
(c) $2(x+10) km$ $\qquad$ (d) $(2 x+5) km$
(ii) The quadratic equation describing the speed of Johan's car is
(a) $x^2-5 x-500=0$ $\qquad$ (b) $x^2+4 x-400=0$
(c) $x^2+5 x-500=0$ $\qquad$ (d) $x^2-4 x+400=0$
(iii) The speed of Johan's car is (in km/hour)
(a) 20 $\qquad$ (b) 15 $\qquad$ (c) 25 $\qquad$ (d) 10
(iv) The speed of Jayant's car in $km / hr$ is
(a) 25 $\qquad$ (b) 20 $\qquad$ (c) 30 $\qquad$ (d) 15
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