Q. The equation x^2 - 2x + 1 = 0 has:
A.
Two distinct roots
B.
One repeated root
C.
No real roots
D.
Infinitely many roots
Show solution
Solution
The discriminant is 0, indicating one repeated root.
Correct Answer:
B
— One repeated root
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Q. The equation x^2 - 2x + k = 0 has roots that are both positive. What is the range of k?
A.
k < 0
B.
k > 0
C.
k > 1
D.
k < 1
Show solution
Solution
For both roots to be positive, k must be greater than 1 (from Vieta's formulas).
Correct Answer:
C
— k > 1
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Q. The equation x^2 - 4x + k = 0 has equal roots when k is equal to:
Show solution
Solution
For equal roots, the discriminant must be zero: (-4)^2 - 4*1*k = 0 leads to k = 4.
Correct Answer:
A
— 4
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Q. The equation x^2 - 4x + k = 0 has no real roots if k is:
A.
< 4
B.
≥ 4
C.
≤ 4
D.
> 4
Show solution
Solution
The discriminant must be less than zero: (-4)^2 - 4*1*k < 0 leads to k > 4.
Correct Answer:
A
— < 4
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Q. The equation x^2 - 6x + 9 = 0 has how many distinct roots?
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Solution
The equation can be factored as (x - 3)(x - 3) = 0, indicating it has one distinct root (a double root).
Correct Answer:
B
— 1
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Q. The equation x^2 - 6x + k = 0 has roots that are both positive. What is the range of k?
A.
k < 0
B.
k > 0
C.
k > 9
D.
k < 9
Show solution
Solution
For both roots to be positive, k must be greater than the square of half the coefficient of x: k > (6/2)^2 = 9.
Correct Answer:
C
— k > 9
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Q. The equation x^2 - 7x + 10 = 0 has roots that are:
A.
1 and 10
B.
2 and 5
C.
3 and 4
D.
5 and 2
Show solution
Solution
Factoring the equation gives (x - 2)(x - 5) = 0, so the roots are 2 and 5.
Correct Answer:
C
— 3 and 4
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Q. The equation x^2 - 9 = 0 has roots that are: (2019)
A.
-3 and 3
B.
0 and 9
C.
3 and 9
D.
1 and -1
Show solution
Solution
Factoring gives (x-3)(x+3)=0, hence the roots are -3 and 3.
Correct Answer:
A
— -3 and 3
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Q. The equation x^3 - 3x^2 + 3x - 1 = 0 has a root at x = 1. What is the multiplicity of this root? (2023)
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Solution
The polynomial can be expressed as (x - 1)^3, indicating that the root x = 1 has a multiplicity of 3.
Correct Answer:
C
— 3
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Q. The equation x^3 - 3x^2 + 3x - 1 = 0 has how many distinct real roots? (2022)
Show solution
Solution
The given polynomial can be factored as (x - 1)^3 = 0, which has one distinct real root, x = 1.
Correct Answer:
A
— 1
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Q. The equilibrium constant for the reaction 2SO2(g) + O2(g) ⇌ 2SO3(g) is 4. What is the value of Kp? (2023)
A.
4
B.
16
C.
0.25
D.
0.0625
Show solution
Solution
Kp can be calculated from Kc using the relation Kp = Kc(RT)^(Δn), where Δn is the change in moles of gas. Here, Δn = 2 - 3 = -1, so Kp = Kc(0.0821T)^(-1). Assuming standard conditions, Kp = 4 * (1/RT) = 16.
Correct Answer:
B
— 16
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Q. The escape velocity from the surface of the Earth is approximately: (2019)
A.
7.9 km/s
B.
11.2 km/s
C.
9.8 km/s
D.
15 km/s
Show solution
Solution
The escape velocity from the surface of the Earth is approximately 11.2 km/s.
Correct Answer:
B
— 11.2 km/s
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Q. The establishment of national parks in the United States was significantly influenced by which of the following events?
A.
The Great Depression
B.
The Industrial Revolution
C.
The establishment of Yellowstone National Park
D.
The Civil Rights Movement
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Solution
The establishment of Yellowstone National Park in 1872 marked the beginning of the national park movement in the United States, influencing conservation efforts worldwide.
Correct Answer:
C
— The establishment of Yellowstone National Park
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Q. The establishment of the 'Aligarh Movement' was primarily aimed at:
A.
Promoting women's education
B.
Reviving traditional Indian education
C.
Encouraging modern scientific education among Muslims
D.
Establishing a new educational curriculum
Show solution
Solution
The Aligarh Movement, led by Sir Syed Ahmed Khan, aimed to promote modern scientific education among Muslims in India.
Correct Answer:
C
— Encouraging modern scientific education among Muslims
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Q. The expression 4^(x+1) can be rewritten as?
A.
2^(2x+2)
B.
2^(x+1)
C.
2^(x+2)
D.
4^x
Show solution
Solution
4^(x+1) = (2^2)^(x+1) = 2^(2(x+1)) = 2^(2x+2).
Correct Answer:
A
— 2^(2x+2)
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Q. The family of curves defined by the equation x^2 + y^2 = r^2 represents:
A.
Ellipses
B.
Hyperbolas
C.
Circles
D.
Parabolas
Show solution
Solution
The equation x^2 + y^2 = r^2 represents a circle with radius r.
Correct Answer:
C
— Circles
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Q. The family of curves defined by the equation y = a(x - h)^2 + k represents which type of function?
A.
Linear
B.
Quadratic
C.
Cubic
D.
Rational
Show solution
Solution
The equation y = a(x - h)^2 + k represents a quadratic function in vertex form.
Correct Answer:
B
— Quadratic
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Q. The family of curves defined by the equation y = a(x - h)^2 + k represents:
A.
Parabolas
B.
Circles
C.
Ellipses
D.
Hyperbolas
Show solution
Solution
The equation y = a(x - h)^2 + k represents a family of parabolas with vertex (h, k).
Correct Answer:
A
— Parabolas
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Q. The family of curves defined by the equation y = ax^2 + bx + c is known as:
A.
Linear functions
B.
Quadratic functions
C.
Polynomial functions
D.
Rational functions
Show solution
Solution
The equation y = ax^2 + bx + c represents a quadratic function.
Correct Answer:
B
— Quadratic functions
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Q. The family of curves defined by the equation y = e^(kx) is classified as:
A.
Linear
B.
Exponential
C.
Logarithmic
D.
Polynomial
Show solution
Solution
The equation y = e^(kx) represents a family of exponential curves.
Correct Answer:
B
— Exponential
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Q. The family of curves defined by the equation y = k/x represents which type of function?
A.
Linear
B.
Quadratic
C.
Rational
D.
Exponential
Show solution
Solution
The equation y = k/x represents a rational function.
Correct Answer:
C
— Rational
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Q. The family of curves defined by the equation y = k/x represents:
A.
Linear functions
B.
Hyperbolas
C.
Parabolas
D.
Circles
Show solution
Solution
The equation y = k/x represents a family of hyperbolas.
Correct Answer:
B
— Hyperbolas
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Q. The family of curves defined by y = kx^3 represents:
A.
Linear curves
B.
Cubic curves
C.
Quadratic curves
D.
Exponential curves
Show solution
Solution
The equation y = kx^3 represents a family of cubic curves.
Correct Answer:
B
— Cubic curves
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Q. The family of curves given by the equation y = a sin(bx + c) is known as:
A.
Linear functions
B.
Trigonometric functions
C.
Exponential functions
D.
Polynomial functions
Show solution
Solution
The equation y = a sin(bx + c) represents a family of trigonometric functions.
Correct Answer:
B
— Trigonometric functions
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Q. The family of curves given by the equation y = kx + b is characterized by:
A.
Different slopes
B.
Different intercepts
C.
Both a and b
D.
None of the above
Show solution
Solution
The equation y = kx + b represents a family of straight lines with different slopes (k) and intercepts (b).
Correct Answer:
C
— Both a and b
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Q. The family of curves given by y = a sin(bx) is characterized by:
A.
Linear behavior
B.
Periodic behavior
C.
Exponential growth
D.
Quadratic growth
Show solution
Solution
The equation y = a sin(bx) represents a family of periodic curves.
Correct Answer:
B
— Periodic behavior
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Q. The family of curves given by y = k(x - a)(x - b) is a representation of:
A.
Linear functions
B.
Quadratic functions
C.
Cubic functions
D.
Exponential functions
Show solution
Solution
The equation y = k(x - a)(x - b) represents a family of quadratic functions.
Correct Answer:
B
— Quadratic functions
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Q. The family of curves given by y^2 = 4ax represents which type of conic section?
A.
Circle
B.
Ellipse
C.
Parabola
D.
Hyperbola
Show solution
Solution
The equation y^2 = 4ax represents a parabola that opens to the right.
Correct Answer:
C
— Parabola
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Q. The family of curves represented by the equation x^2 + y^2 = r^2 describes which geometric shape?
A.
Ellipse
B.
Circle
C.
Hyperbola
D.
Parabola
Show solution
Solution
The equation x^2 + y^2 = r^2 represents a circle with radius r.
Correct Answer:
B
— Circle
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Q. The family of curves represented by the equation x^2 + y^2 = r^2 is known as:
A.
Ellipses
B.
Hyperbolas
C.
Circles
D.
Parabolas
Show solution
Solution
The equation x^2 + y^2 = r^2 represents a family of circles.
Correct Answer:
C
— Circles
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