Q. Find the integral ∫ (1/x) dx.
A.
ln
B.
x
C.
+ C
D.
x + C
.
1/x + C
.
e^x + C
Show solution
Solution
The integral of 1/x is ln|x| + C.
Correct Answer:
A
— ln
Learn More →
Q. Find the integral ∫ (2x + 1)/(x^2 + x) dx.
A.
ln
B.
x^2 + x
C.
+ C
D.
ln
.
x
.
+ C
.
ln
.
x^2 + x
.
+ 1
.
ln
.
x
.
+ 1
Show solution
Solution
Using partial fraction decomposition, we can integrate to find that ∫ (2x + 1)/(x^2 + x) dx = ln|x^2 + x| + C.
Correct Answer:
A
— ln
Learn More →
Q. Find the integral ∫ (tan(x))^2 dx.
A.
tan(x) - x + C
B.
tan(x) + x + C
C.
tan(x) + x
D.
tan(x) - x
Show solution
Solution
Using the identity tan^2(x) = sec^2(x) - 1, we find that ∫ (tan(x))^2 dx = tan(x) - x + C.
Correct Answer:
A
— tan(x) - x + C
Learn More →
Q. Find the integral ∫ (x^2 - 1)/(x - 1) dx.
A.
(1/3)x^3 - x + C
B.
(1/3)x^3 - x - 1 + C
C.
(1/3)x^3 - x + 1
D.
(1/3)x^3 - x - 1
Show solution
Solution
The integrand simplifies to x + 1. Therefore, ∫ (x + 1) dx = (1/2)x^2 + x + C.
Correct Answer:
A
— (1/3)x^3 - x + C
Learn More →
Q. Find the integral ∫ sin(2x) dx.
A.
-cos(2x)/2 + C
B.
cos(2x)/2 + C
C.
-sin(2x)/2 + C
D.
sin(2x)/2 + C
Show solution
Solution
The integral of sin(kx) is -1/k * cos(kx). Here, k = 2, so ∫ sin(2x) dx = -cos(2x)/2 + C.
Correct Answer:
A
— -cos(2x)/2 + C
Learn More →
Q. Find the intervals where the function f(x) = x^4 - 4x^3 has increasing behavior.
A.
(-∞, 0)
B.
(0, 2)
C.
(2, ∞)
D.
(0, 4)
Show solution
Solution
f'(x) = 4x^3 - 12x^2 = 4x^2(x - 3). Critical points are x = 0 and x = 3. Test intervals: f' is positive in (0, 3) and (3, ∞).
Correct Answer:
B
— (0, 2)
Learn More →
Q. Find the inverse of the matrix A = [[1, 2], [3, 4]].
A.
[[4, -2]; [-3, 1]]
B.
[[1, -2]; [-3, 4]]
C.
[[-2, 1]; [3, 4]]
D.
[[2, -1]; [-1.5, 0.5]]
Show solution
Solution
The inverse of A is (1/det(A)) * adj(A) = (1/-2) * [[4, -2], [-3, 1]] = [[-2, 1]; [1.5, -0.5]].
Correct Answer:
A
— [[4, -2]; [-3, 1]]
Learn More →
Q. Find the inverse of the matrix D = [[4, 7], [2, 6]]. (2023)
A.
[[3/2, -7/4], [-1/2, 2/4]]
B.
[[3/2, -7/4], [-1/4, 2/4]]
C.
[[6, -7], [-2, 4]]
D.
[[6, 7], [2, 4]]
Show solution
Solution
The inverse of D is (1/det(D)) * adj(D) = (1/(4*6 - 7*2)) * [[6, -7], [-2, 4]] = [[3/2, -7/4], [-1/2, 2/4]].
Correct Answer:
A
— [[3/2, -7/4], [-1/2, 2/4]]
Learn More →
Q. Find the inverse of the matrix F = [[4, 7], [2, 6]]. (2021)
A.
[[3, -7], [-1, 4]]
B.
[[6, -7], [-2, 4]]
C.
[[3, 7], [-1, 2]]
D.
[[2, -7], [-1, 4]]
Show solution
Solution
The inverse of F is given by (1/det(F)) * adj(F). Here, det(F) = 4*6 - 7*2 = 10, and adj(F) = [[6, -7], [-2, 4]]. Thus, F^(-1) = (1/10) * [[6, -7], [-2, 4]] = [[3/5, -7/10], [-1/5, 2/5]].
Correct Answer:
A
— [[3, -7], [-1, 4]]
Learn More →
Q. Find the item that does not belong to the group.
A.
Piano
B.
Guitar
C.
Violin
D.
Car
Show solution
Solution
Car is a vehicle, while the others are musical instruments.
Correct Answer:
D
— Car
Learn More →
Q. Find the item that does not belong:
A.
Pencil
B.
Pen
C.
Eraser
D.
Notebook
Show solution
Solution
Notebook is a collection of pages, while the others are writing instruments.
Correct Answer:
D
— Notebook
Learn More →
Q. Find the item that does not fit in the series:
A.
Circle
B.
Square
C.
Triangle
D.
Pyramid
Show solution
Solution
Pyramid is a 3D shape, while the others are 2D shapes.
Correct Answer:
D
— Pyramid
Learn More →
Q. Find the length of the diagonal of a cuboid with dimensions 3, 4, and 12 units. (2020)
A.
√169
B.
√145
C.
√153
D.
√157
Show solution
Solution
Diagonal = √(3² + 4² + 12²) = √(9 + 16 + 144) = √169 = 13.
Correct Answer:
C
— √153
Learn More →
Q. Find the length of the diagonal of a rectangular box with dimensions 2, 3, and 6. (2023)
A.
√49
B.
√36
C.
√45
D.
√50
Show solution
Solution
Diagonal = √(2² + 3² + 6²) = √(4 + 9 + 36) = √49 = 7.
Correct Answer:
A
— √49
Learn More →
Q. Find the length of the diagonal of a rectangular box with dimensions 2, 3, and 6 units. (2022)
A.
√49
B.
√45
C.
√36
D.
√50
Show solution
Solution
Diagonal = √(2² + 3² + 6²) = √(4 + 9 + 36) = √49 = 7 units.
Correct Answer:
A
— √49
Learn More →
Q. Find the length of the latus rectum of the parabola y^2 = 16x.
Show solution
Solution
The length of the latus rectum of a parabola y^2 = 4px is given by 4p. Here, 4p = 16, so p = 4. Therefore, the length of the latus rectum is 4 * 4 = 16.
Correct Answer:
B
— 8
Learn More →
Q. Find the length of the line segment joining the points (-1, -1) and (2, 3).
Show solution
Solution
Length = √[(2 - (-1))² + (3 - (-1))²] = √[3² + 4²] = √25 = 5.
Correct Answer:
A
— 5
Learn More →
Q. Find the length of the line segment joining the points (1, 1) and (4, 5).
Show solution
Solution
Length = √[(4-1)² + (5-1)²] = √[3² + 4²] = √[9 + 16] = √25 = 5.
Correct Answer:
C
— 5
Learn More →
Q. Find the length of the line segment joining the points (1, 2) and (1, 5).
Show solution
Solution
Length = |5 - 2| = 3.
Correct Answer:
A
— 3
Learn More →
Q. Find the limit lim x->0 (sin(3x)/x).
A.
0
B.
1
C.
3
D.
undefined
Show solution
Solution
Using the limit property, lim x->0 (sin(kx)/x) = k. Here, k = 3, so the limit is 3.
Correct Answer:
C
— 3
Learn More →
Q. Find the limit lim x->0 of (sin(3x)/x).
A.
0
B.
1
C.
3
D.
undefined
Show solution
Solution
Using L'Hôpital's rule, the limit evaluates to 3.
Correct Answer:
C
— 3
Learn More →
Q. Find the limit lim(x→0) (sin(5x)/x).
Show solution
Solution
Using L'Hôpital's rule, lim(x→0) (sin(5x)/x) = 5.
Correct Answer:
A
— 5
Learn More →
Q. Find the limit lim(x→∞) (1/x).
Show solution
Q. Find the limit: lim (x -> 0) (1 - cos(2x))/x^2
A.
0
B.
1
C.
2
D.
Undefined
Show solution
Solution
Using the identity 1 - cos(θ) = 2sin^2(θ/2), we have lim (x -> 0) (1 - cos(2x))/x^2 = lim (x -> 0) (2sin^2(x))/x^2 = 2.
Correct Answer:
C
— 2
Learn More →
Q. Find the limit: lim (x -> 0) (1 - cos(4x))/(x^2)
A.
0
B.
2
C.
4
D.
Undefined
Show solution
Solution
Using the identity 1 - cos(x) = 2sin^2(x/2), we have lim (x -> 0) (2sin^2(2x))/(x^2) = 8.
Correct Answer:
B
— 2
Learn More →
Q. Find the limit: lim (x -> 0) (1 - cos(x))/(x^2)
A.
0
B.
1/2
C.
1
D.
Infinity
Show solution
Solution
Using the identity 1 - cos(x) = 2sin^2(x/2), we have lim (x -> 0) (2sin^2(x/2))/(x^2) = 1.
Correct Answer:
B
— 1/2
Learn More →
Q. Find the limit: lim (x -> 0) (cos(x) - 1)/x^2
A.
0
B.
-1/2
C.
1
D.
Infinity
Show solution
Solution
Using the Taylor series expansion for cos(x), we find that lim (x -> 0) (cos(x) - 1)/x^2 = -1/2.
Correct Answer:
B
— -1/2
Learn More →
Q. Find the limit: lim (x -> 0) (e^x - 1)/x
A.
0
B.
1
C.
e
D.
Undefined
Show solution
Solution
Using the derivative of e^x at x = 0, we find lim (x -> 0) (e^x - 1)/x = 1.
Correct Answer:
B
— 1
Learn More →
Q. Find the limit: lim (x -> 0) (sin(5x)/x)
A.
0
B.
5
C.
1
D.
Infinity
Show solution
Solution
Using the limit property, lim (x -> 0) (sin(kx)/x) = k. Here, k = 5, so the limit is 5.
Correct Answer:
B
— 5
Learn More →
Q. Find the limit: lim (x -> 0) (x^2 * sin(1/x))
A.
0
B.
1
C.
Infinity
D.
Undefined
Show solution
Solution
Since |sin(1/x)| <= 1, we have |x^2 * sin(1/x)| <= |x^2|. As x approaches 0, |x^2| approaches 0, hence the limit is 0.
Correct Answer:
A
— 0
Learn More →
Showing 4351 to 4380 of 27896 (930 Pages)
