Major Competitive Exams
Q. The equation x^2 - 6x + k = 0 has roots that are both positive. What is the range of k?
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A.
k < 0
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B.
k > 0
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C.
k > 9
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D.
k < 9
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 family of curves defined by the equation x^2 + y^2 = r^2 represents:
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A.
Ellipses
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B.
Hyperbolas
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C.
Circles
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D.
Parabolas
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:
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A.
Parabolas
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B.
Circles
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C.
Ellipses
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D.
Hyperbolas
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 = e^(kx) is classified as:
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A.
Linear
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B.
Exponential
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C.
Logarithmic
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D.
Polynomial
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:
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A.
Linear functions
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B.
Hyperbolas
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C.
Parabolas
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D.
Circles
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:
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A.
Linear curves
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B.
Cubic curves
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C.
Quadratic curves
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D.
Exponential curves
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 y = a sin(bx) is characterized by:
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A.
Linear behavior
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B.
Periodic behavior
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C.
Exponential growth
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D.
Quadratic growth
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:
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A.
Linear functions
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B.
Quadratic functions
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C.
Cubic functions
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D.
Exponential functions
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 represented by the equation x^2 + y^2 = r^2 is known as:
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A.
Ellipses
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B.
Hyperbolas
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C.
Circles
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D.
Parabolas
Solution
The equation x^2 + y^2 = r^2 represents a family of circles.
Correct Answer: C — Circles
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Q. The family of curves represented by the equation y = e^(kx) is characterized by:
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A.
Linear growth
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B.
Exponential growth
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C.
Quadratic growth
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D.
Logarithmic growth
Solution
The equation y = e^(kx) represents exponential growth for different values of k.
Correct Answer: B — Exponential growth
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Q. The family of curves represented by the equation y = kx^2, where k is a constant, is known as:
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A.
Linear curves
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B.
Parabolic curves
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C.
Circular curves
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D.
Exponential curves
Solution
The equation y = kx^2 represents a parabola for different values of k.
Correct Answer: B — Parabolic curves
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Q. The family of curves represented by the equation y = kx^n, where n is a constant, is known as:
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A.
Polynomial curves
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B.
Rational curves
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C.
Trigonometric curves
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D.
Exponential curves
Solution
The equation y = kx^n represents a family of polynomial curves.
Correct Answer: A — Polynomial curves
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Q. The family of curves represented by y = a sin(bx + c) is known as:
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A.
Linear functions
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B.
Trigonometric functions
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C.
Polynomial functions
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D.
Exponential functions
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 represented by y = mx + c can be described as:
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A.
Quadratic functions
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B.
Linear functions
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C.
Cubic functions
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D.
Exponential functions
Solution
The equation y = mx + c describes linear functions for varying m and c.
Correct Answer: B — Linear functions
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Q. The family of curves represented by y^2 = 4ax is known as:
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A.
Parabolas
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B.
Ellipses
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C.
Hyperbolas
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D.
Circles
Solution
The equation y^2 = 4ax represents a family of parabolas.
Correct Answer: A — Parabolas
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Q. The family of curves y = ax^3 + bx^2 + cx + d is classified as:
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A.
Linear
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B.
Quadratic
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C.
Cubic
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D.
Quartic
Solution
The equation y = ax^3 + bx^2 + cx + d represents a family of cubic curves.
Correct Answer: C — Cubic
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Q. The family of curves y = kx^3 is known for having:
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A.
One turning point
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B.
Two turning points
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C.
No turning points
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D.
Three turning points
Solution
The cubic function y = kx^3 has one turning point at x = 0.
Correct Answer: A — One turning point
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Q. The family of curves y = kx^n, where n is a constant, represents:
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A.
Linear functions
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B.
Polynomial functions
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C.
Rational functions
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D.
Trigonometric functions
Solution
The equation y = kx^n represents a family of polynomial functions.
Correct Answer: B — Polynomial functions
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Q. The foci of the ellipse x^2/25 + y^2/16 = 1 are located at which points?
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A.
(±3, 0)
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B.
(±4, 0)
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C.
(±5, 0)
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D.
(±6, 0)
Solution
For the ellipse, c = √(a^2 - b^2) = √(25 - 16) = √9 = 3. The foci are at (±3, 0).
Correct Answer: B — (±4, 0)
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Q. The Freundlich adsorption isotherm is applicable to which type of adsorption?
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A.
Physisorption only
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B.
Chemisorption only
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C.
Both physisorption and chemisorption
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D.
None of the above
Solution
The Freundlich isotherm can describe both physisorption and chemisorption processes.
Correct Answer: C — Both physisorption and chemisorption
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Q. The function f(x) = ln(x) + x has a minimum at:
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A.
x = 1
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B.
x = 0
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C.
x = e
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D.
x = 2
Solution
Finding f'(x) = 1/x + 1. Setting f'(x) = 0 gives x = 1 as the minimum point.
Correct Answer: A — x = 1
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Q. The function f(x) = ln(x) is differentiable at x = 1?
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A.
Yes
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B.
No
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C.
Only for x > 1
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D.
Only for x < 1
Solution
f'(x) = 1/x; f'(1) = 1/1 = 1, hence it is differentiable at x = 1.
Correct Answer: A — Yes
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Q. The function f(x) = x^2 + 2x + 1 is differentiable everywhere?
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A.
True
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B.
False
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C.
Only at x = 0
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D.
Only for x > 0
Solution
f(x) is a polynomial function, which is differentiable everywhere.
Correct Answer: A — True
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Q. The function f(x) = x^2 - 2x + 1 is differentiable at all points?
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A.
True
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B.
False
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C.
Only at x = 0
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D.
Only for x > 0
Solution
f(x) is a polynomial function, which is differentiable everywhere.
Correct Answer: A — True
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Q. The function f(x) = x^2 - 4 is:
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A.
Always increasing
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B.
Always decreasing
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C.
Neither increasing nor decreasing
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D.
Both increasing and decreasing
Solution
The function has a minimum at x = 0, hence it is neither always increasing nor decreasing.
Correct Answer: C — Neither increasing nor decreasing
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Q. The function f(x) = x^2 - 4x + 4 can be expressed in which form?
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A.
(x - 2)^2
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B.
(x + 2)^2
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C.
(x - 4)^2
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D.
(x + 4)^2
Solution
f(x) = (x - 2)^2 is the completed square form.
Correct Answer: A — (x - 2)^2
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Q. The function f(x) = x^2 - 4x + 4 is differentiable at x = 2?
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A.
Yes
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B.
No
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C.
Only left
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D.
Only right
Solution
f(x) is a polynomial function, hence differentiable everywhere including at x = 2.
Correct Answer: A — Yes
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Q. The function f(x) = x^2 - 4x + 4 is differentiable everywhere?
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A.
True
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B.
False
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C.
Only at x = 0
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D.
Only at x = 2
Solution
f(x) is a polynomial function, hence it is differentiable everywhere.
Correct Answer: A — True
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Q. The function f(x) = x^2 for x < 1 and f(x) = 2x - 1 for x ≥ 1 is differentiable at x = 1?
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A.
Yes
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B.
No
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C.
Only continuous
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D.
Only from the left
Solution
f'(1) from left = 2 and from right = 2; hence, f is continuous but not differentiable at x = 1.
Correct Answer: B — No
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Q. The function f(x) = x^2 sin(1/x) for x ≠ 0 and f(0) = 0 is differentiable at x = 0. True or False?
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A.
True
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B.
False
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C.
Depends on x
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D.
Not enough information
Solution
True, as the limit of f'(x) as x approaches 0 exists and equals 0.
Correct Answer: A — True
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