If the sum of the roots of the quadratic equation ax2 + bx + c = 0 is equal to the sum of the squares of their reciprocals, then \(\frac{a}{c},\frac{b}{a}\) and \(\frac{c}{b}\) are in

arithmetic-geometric progression

If the graph of a quadratic polynomial lies entirely above x-axis, then which one of the following is correct?

One root is real and the other is complex

All ‘x’ satisfying the inequality (cot-1 x)2 - 7(cot-1 x) + 10 > 0, lie in the interval:

(-∞, cot 5) ∪ (cot 4, cot 2)

Let α and β be the roots of the quadratic equation x2 sin θ - x(sin θ cos θ + 1) + cos θ = 0(0 < θ < 45°), and α < β. Then \(\mathop \sum \limits_{n = 0}^\infty \left( {{\alpha ^n} + \frac{{{{( - 1)}^n}}}{{{\beta ^n}}}} \right)\) is equal to:

\(\frac{1}{{1 + {\rm{cos\;}}\theta }} - \frac{1}{{1 - {\rm{sin\;}}\theta }}\)

\(\frac{1}{{1 - {\rm{cos\;}}\theta }} - \frac{1}{{1 + {\rm{sin\;}}\theta }}\)

\(\frac{1}{{1 + {\rm{cos\;}}\theta }} + \frac{1}{{1 - {\rm{sin\;}}\theta }}\)

\(\frac{1}{{1 - {\rm{cos\;}}\theta }} + \frac{1}{{1 + {\rm{sin\;}}\theta }}\)

\(\frac{1}{{1 + {\rm{cos\;}}\theta }} - \frac{1}{{1 - {\rm{sin\;}}\theta }}\)

Let S be the set of all real values of λ such that a plane passing through the points (-λ2, 1, 1),(1, -λ2, 1)and (1 , 1, -λ2) also passes through the point (-1, -1, 1). Then, S is equal to

\(\left\{ {\sqrt 3 , - \sqrt 3 } \right\}\)

\(\left\{ {\sqrt 3 } \right\}\)

78 + Mcqs in Quadratic Equations in Mathematics Page 1 McqOptions

1.	If the roots of the equation x2 + px + q = 0 are in the same ratio as those of the equation x2 + lx + m = 0, then which one of the following is correct?
A.	p2m = l2q
B.	m2p = l2q
C.	m2p = q2l
D.	m2p2 = l2q
Answer» B. m2p = l2q

Discussion

2.	If 5, 5r, 5r2 are the lengths of the sides of a triangle, then r cannot be equal to:
A.	3/4
B.	5/4
C.	7/4
D.	3/2
Answer» D. 3/2

Discussion

3.	It is given that the roots of the equation x2 – 4x – log3 P = 0 are real. For this, the minimum value of P is
A.	\(\frac{1}{{27}}\)
B.	\(\frac{1}{{64}}\)
C.	\(\frac{1}{{81}}\)
D.	1
Answer» D. 1

Discussion

4.	If c > 0 and 4a + c < 2b, then ax2 – bx + c = 0 has a root in which one of the following intervals?
A.	(0, 2)
B.	(2, 3)
C.	(3, 4))
D.	(-2, 0)
Answer» B. (2, 3)

Discussion

5.	If the roots of the equation x2 - nx + m = 0, Differ by 1, then
A.	n2 - 4m - 1 = 0
B.	n2 + 4m - 1 = 0
C.	m2 + 4n + 1 = 0
D.	m2 - 4n - 1 = 0
Answer» B. n2 + 4m - 1 = 0

Discussion

6.	If α and β are the roots of x2 + x + 1 = 0, then what is \(\mathop \sum \limits_{j = 0}^3 \left( {{\alpha ^j} + {\beta ^j}} \right)\) equal to?
A.	8
B.	6
C.	4
D.	2
Answer» E.

Discussion

7.	Consider the quadratic equation (c – 5) x2 – 2cx + (c – 4) = 0, c ≠ 5. Let S be the set of all integral values of c for which one root of the equation lies in the interval (0, 2) and its other root lies in the interval (2, 3). Then the number of elements in S is:
A.	18
B.	12
C.	10
D.	11
Answer» E.

Discussion

8.	Let [x] denote the greatest integer function. What is the number of solutions of the equation x2 - 4x + [x] = 0 in the interval [0, 2]?
A.	Zero (No solution)
B.	One
C.	Two
D.	Three
Answer» C. Two

Discussion

9.	If cos α and cos β (0 < α < β < π) are the roots of the quadratic 4x2 – 3 = 0, then what is the value of sec α × sec β?
A.	\( - \frac{4}{3}\)
B.	\(\frac{4}{3}\)
C.	\(\frac{3}{4}\)
D.	\(- \frac{3}{4}\)
Answer» B. \(\frac{4}{3}\)

Discussion

10.	If (a2 + b2) x2 + 2(ab + bd) x + c2 + d2 = 0 has no real roots then
A.	ad = bc
B.	ab = cd
C.	ac = bd
D.	ad ≠ bc
Answer» E.

Discussion

11.	If 2 + i is a root of the equation x2 - ax + 1 = 0, then the value of a is:
A.	2
B.	4
C.	1
D.	8
Answer» C. 1

Discussion

12.	Let P(x) be a quadratic polynomial such that p(0) = 1. If p(x) leaves remainder 4 when divided by x - 1 and it leaves remainder 6 when divided by x + 1, then
A.	p(-2) = 11
B.	p(2) = 11
C.	p(2) = 19
D.	p(-2) = 19
Answer» E.

Discussion

13.	If \(\frac{{{{\rm{x}}^2} + 1}}{{\rm{x}}} = 4\frac{1}{4}\), then what is the value of \({{\rm{x}}^3} + \frac{1}{{{{\rm{x}}^3}}}?\)
A.	529/16
B.	527/64
C.	4913/64
D.	4097/64
Answer» E.

Discussion

14.	If α, β are the roots of the equation (x - a) (x - b) + c = 0 (c ≠ 0), then the roots of the equation (x - c - α) (x - c - β) = c are
A.	(a + c) and (b + c)
B.	a and b
C.	a and (b + c)
D.	(a + b) and c
Answer» B. a and b

Discussion

15.	If α and β (≠ 0) are the roots of the quadratic equation x2 + αx - β = 0, then the quadratic expression - x2 + αx + β where x ϵ R has
A.	Least value \(- \frac{1}{4}\)
B.	Least value \(- \frac{9}{4}\)
C.	Greatest Value \(\frac{1}{4}\)
D.	Greatest value \(\frac{9}{4}\)
Answer» E.

Discussion

16.	If α and β are the roots of the equation x2 - 12x + 32 = 0 then the value of \(\dfrac{\alpha^2 + \beta^2}{\alpha + \beta}\) is:
A.	\(\dfrac{8}{3}\)
B.	\(-\dfrac{8}{3}\)
C.	\(-\dfrac{20}{3}\)
D.	\(\dfrac{20}{3}\)
Answer» E.

Discussion

17.	If the sum of the roots of the quadratic equation ax2 + bx + c = 0 is equal to the sum of the squares of their reciprocals, then \(\frac{a}{c},\frac{b}{a}\) and \(\frac{c}{b}\) are in
A.	arithmetic progression
B.	geometric progression
C.	arithmetic-geometric progression
D.	harmonic progression
Answer» E.

Discussion

18.	Find the smallest positive integer value of k for which quadratic equation \(\rm \sqrt{3}x^2-\sqrt{2}kx +2 \sqrt{3}=0\), will have distinct real roots ?
A.	3
B.	4
C.	1
D.	2
Answer» C. 1

Discussion

19.	If x2 + 2ax + 10 - 3a > 0 for all x ∈ R, then
A.	-5 < a < 2
B.	a < -5
C.	a > 5
D.	2 < a < 5
Answer» B. a < -5

Discussion

20.	Let α and β be two roots of the equation x2 + 2x + 2 = 0, then α15 + β15 equal to:
A.	-256
B.	512
C.	-512
D.	256
Answer» B. 512

Discussion

21.	If x = 2 + 22/3 + 21/3, then what is the value of x3 – 6x2 + 6x?
A.	3
B.	2
C.	1
D.	0
Answer» C. 1

Discussion

22.	If α and β are the roots of the equation 3x2 + 2x + 1 = 0, then the equation whose roots are α + β-1 and β + α-1 is
A.	3x2 + 8x + 16 = 0
B.	3x2 – 8x – 16 = 0
C.	3x2 + 8x – 16 = 0
D.	x2 + 8x + 16 = 0
Answer» B. 3x2 – 8x – 16 = 0

Discussion

23.	If α and β are the roots of the equation 2x2 + 2px + p2 = 0, where p is a non-zero real number, and α4 and β4 are the roots of x2 - rx + s = 0, then the roots of 2x2 - 4p2x + 4p4 - 2r = 0 are:
A.	Real and unequal.
B.	Equal and zero.
C.	Imaginary.
D.	Equal and non-zero.
Answer» D. Equal and non-zero.

Discussion

24.	If \({\left( {{\rm{x}} + \frac{1}{{\rm{x}}}} \right)^2} = 5\) and x > 0, then what is the value of \({{\rm{x}}^3} + \frac{1}{{{{\rm{x}}^3}}}\) ?
A.	2√5
B.	3√5
C.	4√5
D.	5√5
Answer» B. 3√5

Discussion

25.	If a + b + c = 0 then the roots of the equation 4ax2 + 3bx + 2c = 0 are (a, b, c ϵ R)
A.	Real
B.	Equal
C.	Imaginary
D.	None of these
Answer» B. Equal

Discussion

26.	If λ be the ratio of the roots of the quadratic equation in x, 3m2x2 + m(m – 4)x + 2 = 0, then the least value of m for which \(\lambda + \frac{1}{\lambda } = 1\), is
A.	\(- 2 + \sqrt 2\)
B.	\(4 - 2\sqrt 3\)
C.	\(4 - 3\sqrt 2\)
D.	\(2 + \sqrt 3\)
Answer» D. \(2 + \sqrt 3\)

Discussion

27.	If the graph of a quadratic polynomial lies entirely above x-axis, then which one of the following is correct?
A.	Both the roots are real
B.	One root is real and the other is complex
C.	Both the roots are complex
D.	Cannot say
Answer» D. Cannot say

Discussion

28.	If (x-1)2 + (y -2)2 = (x – 1) (y -2), where x and y are integers, then value fo 2x + 3y is:
A.	5
B.	8
C.	11
D.	7
Answer» C. 11

Discussion

29.	If the equations x2 - px + q = 0 and x2 + qx - p = 0 have a common root, then which one of the following is correct?
A.	p - q = 0
B.	p + q - 2 = 0
C.	p + q - 1 = 0
D.	p - q - 1 = 0
Answer» E.

Discussion

30.	If m and n are the roots of the equation (x + p) (x + q) – k = 0, then the roots of the equation(x – m) (x - n) + k = 0 are
A.	P and q
B.	1/p and 1/q
C.	-p and -q
D.	P + q and p - q
Answer» D. P + q and p - q

Discussion

31.	All ‘x’ satisfying the inequality (cot-1 x)2 - 7(cot-1 x) + 10 > 0, lie in the interval:
A.	(-∞, cot 5) ∪ (cot 4, cot 2)
B.	(cot 2, ∞)
C.	(-∞, cot 5) ∪ (cot 2, ∞)
D.	(cot 5, cot 4)
Answer» D. (cot 5, cot 4)

Discussion

32.	If \({{\rm{x}}^2}{\rm{}}-{\rm{\;px}} + {\rm{\;}}4{\rm{}} > {\rm{}}0\) for all real values of x, then which one of the following is correct?
A.	\|p\| < 4
B.	\|p\| ≤ 4
C.	\|p\| > 4
D.	\|p\| ≥ 4
Answer» B. \|p\| ≤ 4

Discussion

33.	If both p and q belong to the set {1, 2, 3, 4}, then how many equations of the form px2 + qx + 1 = 0 will have real roots?
A.	12
B.	10
C.	7
D.	6
Answer» D. 6

Discussion

34.	If x + y + z = 0, then what is the value of \(\frac{{{\rm{xy}} + {\rm{yz}} + {\rm{zx}}}}{{{{\rm{x}}^2} + {{\rm{y}}^2} + {{\rm{z}}^2}}}?\)
A.	1
B.	-1
C.	1/2
D.	-1/2
Answer» E.

Discussion

35.	If the roots of the equation ax2 - 2bx + c = 0 are n and m, then the value of \(\rm \frac{b}{an^2+c}+\frac{b}{am^2+c}\) is:
A.	\(\rm \frac{c}{a}\)
B.	\(\rm \frac{b}{a}\)
C.	\(\rm \frac{a}{c}\)
D.	\(\rm \frac{b}{c}\)
Answer» E.

Discussion

36.	If \(\sqrt {\frac{{\rm{x}}}{{\rm{y}}}} = \frac{{24}}{5} + \sqrt {\frac{{\rm{y}}}{{\rm{x}}}} \) and x + y = 26, then what is the value of xy?
A.	5
B.	15
C.	25
D.	30
Answer» D. 30

Discussion

37.	If the roots of the equation ax2 + bx + c = 0 are reciprocal to each other, then
A.	a + c = 0
B.	b = 0
C.	a - c = 0
D.	None of these
Answer» D. None of these

Discussion

38.	Let α and β be the roots of the quadratic equation x2 sin θ - x(sin θ cos θ + 1) + cos θ = 0(0 < θ < 45°), and α < β. Then \(\mathop \sum \limits_{n = 0}^\infty \left( {{\alpha ^n} + \frac{{{{( - 1)}^n}}}{{{\beta ^n}}}} \right)\) is equal to:
A.	\(\frac{1}{{1 - {\rm{cos\;}}\theta }} - \frac{1}{{1 + {\rm{sin\;}}\theta }}\)
B.	\(\frac{1}{{1 + {\rm{cos\;}}\theta }} + \frac{1}{{1 - {\rm{sin\;}}\theta }}\)
C.	\(\frac{1}{{1 - {\rm{cos\;}}\theta }} + \frac{1}{{1 + {\rm{sin\;}}\theta }}\)
D.	\(\frac{1}{{1 + {\rm{cos\;}}\theta }} - \frac{1}{{1 - {\rm{sin\;}}\theta }}\)
Answer» D. \(\frac{1}{{1 + {\rm{cos\;}}\theta }} - \frac{1}{{1 - {\rm{sin\;}}\theta }}\)

Discussion

39.	If (a + 4)3 = a3 + 12a2 + ka + 64, then what is the value of k?
A.	12
B.	24
C.	36
D.	48
Answer» E.

Discussion

40.	If p and q are the roots of x2 + px + q = 0, then which of the following is correct?
A.	p = 0 or 1
B.	p = 1 only
C.	p = -2 or 0
D.	p = -2 only
Answer» C. p = -2 or 0

Discussion

41.	If α, β are the roots of an equation x2 - 2x cos θ + 1 = 0 then the equation having αn and βn is ?
A.	x2 - (2cos nθ )x + 1 = 0
B.	2x2 - (2 cos nθ) x - 1 = 0
C.	x2 + (2 cos nθ) x + 1 = 0
D.	x2 + (2 cos nθ) x - 1 = 0
Answer» B. 2x2 - (2 cos nθ) x - 1 = 0

Discussion

42.	Let S be the set of all real values of λ such that a plane passing through the points (-λ2, 1, 1),(1, -λ2, 1)and (1 , 1, -λ2) also passes through the point (-1, -1, 1). Then, S is equal to
A.	\(\left\{ {\sqrt 3 , - \sqrt 3 } \right\}\)
B.	{3, -3}
C.	{1, -1}
D.	\(\left\{ {\sqrt 3 } \right\}\)
Answer» B. {3, -3}

Discussion

43.	For a quadratic equation, ax2 + bx + c = 0, if b2 – 4ac = 0, then the roots are,
A.	Real and equal
B.	Real and distinct
C.	Imaginary and equal
D.	Imaginary and distinct
Answer» B. Real and distinct

Discussion

44.	Consider the following statements in respect of the quadratic equation 4(x - p)(x - q) - r2 = 0, where p, q and r are real numbers:1. The roots are real2. The roots are equal if p = q and r = 0Which of the above statements is/are correct?
A.	1 only
B.	2 only
C.	Both 1 and 2
D.	Neither 1 nor 2
Answer» D. Neither 1 nor 2

Discussion

45.	If α and β are the roots of the equation 375x2 – 25x – 2 = 0, then \(\underset{n\to \infty }{\mathop{lim}}\,\underset{r=1}{\overset{n}{\mathop \sum }}\,{{\alpha }^{r}}+\underset{n\to \infty }{\mathop{lim}}\,\underset{r=1}{\overset{n}{\mathop \sum }}\,{{\beta }^{r}}\) is equal to:
A.	21/346
B.	29/358
C.	1/12
D.	7/116
Answer» D. 7/116

Discussion

46.	If m is chosen in the quadratic equation (m2 + 1) x2 – 3x + (m2 + 1)2 = 0 such that the sum of its roots is greatest, then the absolute difference of the cubes of its roots is:
A.	\(10\sqrt 5 \)
B.	\({\rm{\;}}8\sqrt 3 \)
C.	\(8\sqrt 5 \)
D.	\(4\sqrt 3 \)
Answer» D. \(4\sqrt 3 \)

Discussion

47.	Let the sum of the nth terms of a non-constant A.P., a1, a2, a3,……… be, \(50n + \frac{{n\left( {n - 7} \right)}}{2}A\), where A is a constant. If d is the common difference of this A.P., then the ordered pair (d, a50)is equal to:
A.	(50, 50 + 46A)
B.	(50, 50 + 45A)
C.	(A, 50 + 45A)
D.	(A, 50 + 46A)
Answer» E.

Discussion

48.	a, b, c are positive integers such that a2 + 2b2 - 2bc = 100 and 2ab - c2 = 100. Then the value of \(\dfrac{a+b}{c}\) is
A.	10
B.	100
C.	2
D.	20
Answer» D. 20

Discussion

49.	In a quadratic equation, x2 + (√5 - √3)x - √15 = 0, sum and product of roots are,
A.	(√3 - √5) and -√15
B.	(√5 - √3) and -√15
C.	(√5 + √3) and √15
D.	(√3 + √5) and -√15
Answer» B. (√5 - √3) and -√15

Discussion

50.	If \({\rm{\;\alpha }}\) and \({\rm{\;\beta }}\) be the roots of the equation x2 – 2x + 2 = 0 then the least value of n for which \({\left( {\frac{{\rm{\alpha }}}{{\rm{\beta }}}} \right)^{\rm{n}}} = 1\) is:
A.	2
B.	5
C.	4
D.	3
Answer» D. 3

Discussion

Explore topic-wise MCQs in Mathematics.

If the roots of the equation x2 + px + q = 0 are in the same ratio as those of the equation x2 + lx + m = 0, then which one of the following is correct?

If 5, 5r, 5r2 are the lengths of the sides of a triangle, then r cannot be equal to:

It is given that the roots of the equation x2 – 4x – log3 P = 0 are real. For this, the minimum value of P is

If c > 0 and 4a + c < 2b, then ax2 – bx + c = 0 has a root in which one of the following intervals?

If the roots of the equation x2 - nx + m = 0, Differ by 1, then

If α and β are the roots of x2 + x + 1 = 0, then what is \(\mathop \sum \limits_{j = 0}^3 \left( {{\alpha ^j} + {\beta ^j}} \right)\) equal to?

Consider the quadratic equation (c – 5) x2 – 2cx + (c – 4) = 0, c ≠ 5. Let S be the set of all integral values of c for which one root of the equation lies in the interval (0, 2) and its other root lies in the interval (2, 3). Then the number of elements in S is:

Let [x] denote the greatest integer function. What is the number of solutions of the equation x2 - 4x + [x] = 0 in the interval [0, 2]?

If cos α and cos β (0 < α < β < π) are the roots of the quadratic 4x2 – 3 = 0, then what is the value of sec α × sec β?

If (a2 + b2) x2 + 2(ab + bd) x + c2 + d2 = 0 has no real roots then

If 2 + i is a root of the equation x2 - ax + 1 = 0, then the value of a is:

Let P(x) be a quadratic polynomial such that p(0) = 1. If p(x) leaves remainder 4 when divided by x - 1 and it leaves remainder 6 when divided by x + 1, then

If \(\frac{{{{\rm{x}}^2} + 1}}{{\rm{x}}} = 4\frac{1}{4}\), then what is the value of \({{\rm{x}}^3} + \frac{1}{{{{\rm{x}}^3}}}?\)

If α, β are the roots of the equation (x - a) (x - b) + c = 0 (c ≠ 0), then the roots of the equation (x - c - α) (x - c - β) = c are

If α and β (≠ 0) are the roots of the quadratic equation x2 + αx - β = 0, then the quadratic expression - x2 + αx + β where x ϵ R has

If α and β are the roots of the equation x2 - 12x + 32 = 0 then the value of \(\dfrac{\alpha^2 + \beta^2}{\alpha + \beta}\) is:

If the sum of the roots of the quadratic equation ax2 + bx + c = 0 is equal to the sum of the squares of their reciprocals, then \(\frac{a}{c},\frac{b}{a}\) and \(\frac{c}{b}\) are in

Find the smallest positive integer value of k for which quadratic equation \(\rm \sqrt{3}x^2-\sqrt{2}kx +2 \sqrt{3}=0\), will have distinct real roots ?

If x2 + 2ax + 10 - 3a > 0 for all x ∈ R, then

Let α and β be two roots of the equation x2 + 2x + 2 = 0, then α15 + β15 equal to:

If x = 2 + 22/3 + 21/3, then what is the value of x3 – 6x2 + 6x?

If α and β are the roots of the equation 3x2 + 2x + 1 = 0, then the equation whose roots are α + β-1 and β + α-1 is

If α and β are the roots of the equation 2x2 + 2px + p2 = 0, where p is a non-zero real number, and α4 and β4 are the roots of x2 - rx + s = 0, then the roots of 2x2 - 4p2x + 4p4 - 2r = 0 are:

If \({\left( {{\rm{x}} + \frac{1}{{\rm{x}}}} \right)^2} = 5\) and x > 0, then what is the value of \({{\rm{x}}^3} + \frac{1}{{{{\rm{x}}^3}}}\) ?

If a + b + c = 0 then the roots of the equation 4ax2 + 3bx + 2c = 0 are (a, b, c ϵ R)

If λ be the ratio of the roots of the quadratic equation in x, 3m2x2 + m(m – 4)x + 2 = 0, then the least value of m for which \(\lambda + \frac{1}{\lambda } = 1\), is

If the graph of a quadratic polynomial lies entirely above x-axis, then which one of the following is correct?

If (x-1)2 + (y -2)2 = (x – 1) (y -2), where x and y are integers, then value fo 2x + 3y is:

If the equations x2 - px + q = 0 and x2 + qx - p = 0 have a common root, then which one of the following is correct?

If m and n are the roots of the equation (x + p) (x + q) – k = 0, then the roots of the equation(x – m) (x - n) + k = 0 are

All ‘x’ satisfying the inequality (cot-1 x)2 - 7(cot-1 x) + 10 > 0, lie in the interval:

If \({{\rm{x}}^2}{\rm{}}-{\rm{\;px}} + {\rm{\;}}4{\rm{}} > {\rm{}}0\) for all real values of x, then which one of the following is correct?

If both p and q belong to the set {1, 2, 3, 4}, then how many equations of the form px2 + qx + 1 = 0 will have real roots?

If x + y + z = 0, then what is the value of \(\frac{{{\rm{xy}} + {\rm{yz}} + {\rm{zx}}}}{{{{\rm{x}}^2} + {{\rm{y}}^2} + {{\rm{z}}^2}}}?\)

If the roots of the equation ax2 - 2bx + c = 0 are n and m, then the value of \(\rm \frac{b}{an^2+c}+\frac{b}{am^2+c}\) is:

If \(\sqrt {\frac{{\rm{x}}}{{\rm{y}}}} = \frac{{24}}{5} + \sqrt {\frac{{\rm{y}}}{{\rm{x}}}} \) and x + y = 26, then what is the value of xy?

If the roots of the equation ax2 + bx + c = 0 are reciprocal to each other, then

Let α and β be the roots of the quadratic equation x2 sin θ - x(sin θ cos θ + 1) + cos θ = 0(0 < θ < 45°), and α < β. Then \(\mathop \sum \limits_{n = 0}^\infty \left( {{\alpha ^n} + \frac{{{{( - 1)}^n}}}{{{\beta ^n}}}} \right)\) is equal to:

If (a + 4)3 = a3 + 12a2 + ka + 64, then what is the value of k?

If p and q are the roots of x2 + px + q = 0, then which of the following is correct?

If α, β are the roots of an equation x2 - 2x cos θ + 1 = 0 then the equation having αn and βn is ?

Let S be the set of all real values of λ such that a plane passing through the points (-λ2, 1, 1),(1, -λ2, 1)and (1 , 1, -λ2) also passes through the point (-1, -1, 1). Then, S is equal to

For a quadratic equation, ax2 + bx + c = 0, if b2 – 4ac = 0, then the roots are,

Consider the following statements in respect of the quadratic equation 4(x - p)(x - q) - r2 = 0, where p, q and r are real numbers:1. The roots are real2. The roots are equal if p = q and r = 0Which of the above statements is/are correct?

If α and β are the roots of the equation 375x2 – 25x – 2 = 0, then \(\underset{n\to \infty }{\mathop{lim}}\,\underset{r=1}{\overset{n}{\mathop \sum }}\,{{\alpha }^{r}}+\underset{n\to \infty }{\mathop{lim}}\,\underset{r=1}{\overset{n}{\mathop \sum }}\,{{\beta }^{r}}\) is equal to:

If m is chosen in the quadratic equation (m2 + 1) x2 – 3x + (m2 + 1)2 = 0 such that the sum of its roots is greatest, then the absolute difference of the cubes of its roots is:

Let the sum of the nth terms of a non-constant A.P., a1, a2, a3,……… be, \(50n + \frac{{n\left( {n - 7} \right)}}{2}A\), where A is a constant. If d is the common difference of this A.P., then the ordered pair (d, a50)is equal to:

a, b, c are positive integers such that a2 + 2b2 - 2bc = 100 and 2ab - c2 = 100. Then the value of \(\dfrac{a+b}{c}\) is

In a quadratic equation, x2 + (√5 - √3)x - √15 = 0, sum and product of roots are,

If \({\rm{\;\alpha }}\) and \({\rm{\;\beta }}\) be the roots of the equation x2 – 2x + 2 = 0 then the least value of n for which \({\left( {\frac{{\rm{\alpha }}}{{\rm{\beta }}}} \right)^{\rm{n}}} = 1\) is: