Q. The equation of the line passing through the points (1, 2) and (3, 6) is:
A.
y = 2x
B.
y = 3x - 1
C.
y = 4x - 2
D.
y = x + 1
Show solution
Solution
Slope = (6-2)/(3-1) = 2. Using point-slope form: y - 2 = 2(x - 1) => y = 2x.
Correct Answer: A — y = 2x
Learn More →
Q. The equation of the pair of lines through the origin is given by y = mx. If m1 and m2 are the slopes, what is the condition for them to be perpendicular?
A.
m1 + m2 = 0
B.
m1 * m2 = 1
C.
m1 - m2 = 0
D.
m1 * m2 = -1
Show solution
Solution
For two lines to be perpendicular, the product of their slopes must equal -1.
Correct Answer: D — m1 * m2 = -1
Learn More →
Q. The equation of the pair of lines through the origin with slopes m1 and m2 is given by:
A.
y = mx
B.
y^2 = mx
C.
x^2 + y^2 = 0
D.
x^2 - 2mxy + y^2 = 0
Show solution
Solution
The correct form of the equation representing the lines through the origin is x^2 - 2mxy + y^2 = 0.
Correct Answer: D — x^2 - 2mxy + y^2 = 0
Learn More →
Q. The equation of the pair of lines through the origin with slopes m1 and m2 is:
A.
y = m1x + m2x
B.
y = (m1 + m2)x
C.
y = m1x - m2x
D.
y = m1x * m2x
Show solution
Solution
The equation of the lines can be expressed as y = (m1 + m2)x, representing the sum of the slopes.
Correct Answer: B — y = (m1 + m2)x
Learn More →
Q. The equation of the tangent line to the curve y = x^2 at the point (2, 4) is:
A.
y = 2x
B.
y = 4x - 4
C.
y = 4x - 8
D.
y = x + 2
Show solution
Solution
The slope of the tangent at x = 2 is f'(x) = 2x, so f'(2) = 4. The equation of the tangent line is y - 4 = 4(x - 2), which simplifies to y = 4x - 8.
Correct Answer: C — y = 4x - 8
Learn More →
Q. The equation of the tangent to the curve y = x^2 at the point (2, 4) is:
A.
y = 2x - 4
B.
y = 2x
C.
y = x + 2
D.
y = x^2 - 2
Show solution
Solution
The derivative f'(x) = 2x. At x = 2, f'(2) = 4. The equation of the tangent line is y - 4 = 4(x - 2), which simplifies to y = 2x - 4.
Correct Answer: A — y = 2x - 4
Learn More →
Q. The equation x^2 + 2x + 1 = 0 can be factored as:
A.
(x + 1)(x + 1)
B.
(x - 1)(x - 1)
C.
(x + 2)(x + 1)
D.
(x - 2)(x - 1)
Show solution
Solution
This is a perfect square: (x + 1)^2 = 0.
Correct Answer: A — (x + 1)(x + 1)
Learn More →
Q. The equation x^2 + 4x + 4 = 0 has:
A.
Two distinct roots
B.
One repeated root
C.
No real roots
D.
None of these
Show solution
Solution
The discriminant is 0, indicating one repeated root.
Correct Answer: B — One repeated root
Learn More →
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
Learn More →
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
Learn More →
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
Learn More →
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
Learn More →
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
Learn More →
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
Learn More →
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
Show solution
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
Learn More →
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
Learn More →
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)
Learn More →
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
Learn More →
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
Learn More →
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
Learn More →
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
Learn More →
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
Learn More →
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
Learn More →
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
Learn More →
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
Learn More →
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
Learn More →
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
Learn More →
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
Learn More →
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
Learn More →
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
Learn More →
Showing 9121 to 9150 of 19340 (645 Pages)
