Advanced Calculus Drill

11 Modules•Curated Set

differential equations

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All questions in Advanced Calculus Drill

Two vectors of magnitudes 5 and 12 act at a point with an included angle of 90 degrees; evaluate the
If the scalar triple product [a b c] equals 7 for three given vectors, what is the value of the rear
For two vectors with magnitudes 4 and 3 enclosing a 30 degree angle, evaluate the sum of squares |a
Three unit vectors a, b and c satisfy a + b + c = 0; using this constraint, evaluate the sum a . b +
The shortest distance between two skew lines is computed using which expression involving their dire
Determine the value of the scalar t for which the vectors i + 2j, 2i + tj and the resulting set beco
A point P divides the line segment joining A(1,2,3) and B(4,8,9) internally in the ratio 2:1; locate
If two nonzero vectors a and b satisfy |a + b| = |a - b|, what can be concluded about the angle betw
What is the unit vector directed along the vector obtained by adding a = 2i - j + 2k and b = i + j -
A vector makes equal acute angles with all three coordinate axes; determine the value of each of its
Three coterminous edges of a parallelepiped are a = i + j, b = j + k and c = k + i; calculate the vo
For what value of the scalar lambda are the three vectors i + j + k, i + lambda j + 2k and 2i + j +
Three vectors a = i + 2j + 3k, b = 2i + j + k and c = 3i + j + 2k are given; compute their scalar tr
Two vectors a and b have magnitudes 3 and 5 respectively while their dot product equals 15/2; find t
Using the standard expansion identity, simplify the vector triple product a x (b x c) where a, b and
The position vectors of three points are A(1,1,1), B(2,3,5) and C(3,5,1); find the area of triangle
Find the scalar value m so that the vector 2i + m j + k becomes perpendicular to the vector i - 3j +
Consider the vectors a = 3i + 4j and b = i + 2j + 2k; find the scalar projection of vector a onto th
Given that $\tan^{-1} x + \tan^{-1} y = \frac{\pi}{4}$ holds for positive reals with $xy < 1$, the e
If $\cos\theta = \frac{3}{5}$ with $\theta$ in the first quadrant, then applying the half-angle iden
The number of integer values of $k$ for which the equation $\cos x = \frac{k}{3}$ admits at least on
The fundamental period of the composite function $g(x) = \sin^4 x + \cos^4 x$, after simplification
Given two vectors a = 2i + 3j - k and b = i - 2j + 4k, what is the value of their scalar product a .
In a triangle with sides $a$, $b$, $c$ and semiperimeter $s$, the area computed by Heron's formula f
Applying the sine rule in a triangle where angle $A = 30^\circ$, angle $B = 45^\circ$, and the side
Statement based: assertion that $\sin^{-1}(\sin\frac{2\pi}{3}) = \frac{2\pi}{3}$ and reason that arc
By repeatedly using the triple-angle identity, the exact value of $\sin 3\theta$ when $\sin\theta =
For the vectors a = i + j + k and b = 2i - j + 3k, determine the magnitude of their vector product a
How many distinct values of $x$ in the open interval $(0, \pi)$ satisfy the equation $\sin x + \sqrt
Match the column conversion: the product $2\sin 75^\circ \cos 15^\circ$, when transformed by the pro
A point divides the segment joining (2, -1, 3) and (4, 3, -1) internally in the ratio 1:3, so find t
In a triangle $ABC$ the sides satisfy $a = 13$, $b = 14$, and $c = 15$ units; using the cosine rule,
Evaluate the principal value of the expression $\tan^{-1}(1) + \cos^{-1}\left(-\frac{1}{2}\right) +
Consider the general solution of $\tan 3x = \tan x$; the set of all real values of $x$ satisfying th
Decide whether the three points (1, 2, 3), (2, 4, 6), and (4, 8, 12) are collinear in three-dimensio
The total number of solutions of the equation $2\sin^2 x - 3\sin x + 1 = 0$ that lie within the inte
If $\sin\theta + \cos\theta = \frac{1}{5}$ for an angle $\theta$ lying in the second quadrant, then
Using exact angle-sum reasoning, the value of the product $\cos 20^\circ \cdot \cos 40^\circ \cdot \
Within a triangle, the identity relating angles states that $\tan A + \tan B + \tan C$ equals which
From a point on level ground, the angle of elevation of a tower top is $30^\circ$, and moving $40$ m
When the expression $\frac{1 - \cos 2\theta + \sin 2\theta}{1 + \cos 2\theta + \sin 2\theta}$ is sim
For all real $x$, the maximum possible value of the function $f(x) = 3\sin x + 4\cos x + 7$ is deter
Find the radius of the sphere whose equation is x^2 + y^2 + z^2 - 2x + 4y - 6z + 5 = 0 in three-dime
Find the acute angle between the two planes represented by the equations x + y + z = 5 and x - y + 2
Examine whether the lines (x-1)/2 = (y-2)/3 = (z-3)/4 and (x-2)/3 = (y-3)/4 = (z-4)/5 are coplanar.
A family of planes passes through the line of intersection of x + y + z = 1 and 2x + 3y - z = 4 in s
Two skew lines pass through (1, 2, 3) and (2, 4, 5) with direction vectors (1, 0, -1) and (0, 1, 1)
Determine the perpendicular distance from the point (1, 2, -1) to the plane whose equation is given
Given the point (1, 0, 0) and its reflection across the plane x + y + z = 3, identify the image poin
Suppose a plane passes through the three non-collinear points (1, 0, 0), (0, 1, 0), and (0, 0, 1) in
A directed line segment joins the point (3, 4, 5) to the point (5, 6, 7), so what are its direction
Locate the foot of the perpendicular drawn from the point (1, 2, 3) onto the line given by x/1 = y/2
Evaluate the perpendicular distance between the two parallel planes 2x - y + 2z = 5 and 2x - y + 2z
Verify whether the line (x-1)/2 = (y-2)/(-1) = (z-3)/4 lies entirely within the plane x + 2y - z = 2
Calculate the sine of the angle that the line (x-1)/2 = (y+1)/(-1) = (z)/2 makes with the plane x +
Determine the vector equation of the plane that is at a distance of 3 units from the origin along th
Where does the line (x-1)/2 = (y+1)/3 = (z-2)/1 intersect the plane defined by x + 2y + 3z = 14 in s
A group of ten numbers has a recorded mean of fifteen, but later one entry originally read as eight
A binomial random variable representing successes in n trials has mean equal to 4 and variance equal
From a committee containing seven men and five women, a subcommittee of three persons is selected at
An urn holds four red and six blue balls, and one ball drawn at the first stage is set aside without
Two friends agree to meet between five and six o'clock, each arriving at a uniformly random time and
When every observation in a dataset is first multiplied by three and then increased by five, how doe
Suppose events A and B satisfy P(A) = 1/2, P(B) = 1/3, and the conditional probability P(A given B)
A diagnostic test detects a disease present in one percent of a population with sensitivity ninety-n
Two lines have direction ratios (1, 2, 2) and (2, -2, 1) respectively, so what is the acute angle be
Consider the line passing through the point (2, -1, 3) and parallel to the vector 4i + 0j - 3k in th
In a binomial setting where a marksman hits a target with probability 1/4 on each independent shot,
A line makes equal acute angles with the three coordinate axes, so what is the value of each of its
Two distinct groups contain twenty and thirty students with mean marks of sixty and seventy respecti
Two players alternately roll a fair die starting with the first player, and whoever first rolls a si
A fair six-faced die is rolled once and the random variable X equals the number shown on the top fac
The number of real solutions of the equation given by the absolute value |x - 1| + |x - 3| = 4 over
If set A has 3 elements and set B has 4 elements, then the number of subsets of the Cartesian produc
A function f from positive reals to reals satisfies f(xy) = f(x) + f(y) for all positive x, y, and f
Let f from real numbers to real numbers be defined by f(x) = x^3 + x + 1; which statement about the
The total number of functions that can be defined from a set with 5 elements to a set with 3 element
Given two events with P(A) = 0.6 and P(B) = 0.5 inside a sample space, while their union has probabi
For finite sets the symmetric difference A Δ B is defined as the set of elements in exactly one of A
A single card is drawn at random from a standard well-shuffled deck and is observed to be a face car
A biased coin showing heads with probability 1/3 is tossed five independent times in succession; wha
Two fair dice are rolled together once, and the recorded sum of the two top faces turns out to be a
A discrete random variable X takes values 0, 1, and 2 with respective probabilities 0.2, 0.5, and 0.
Two factories supply identical bolts in the ratio 60 to 40, with defect rates 2 percent and 5 percen
For the signum function sgn(x), which equals -1, 0, or 1 according to the sign of x, the value of th
Consider the observations 2, 4, 6, 8, and 10 taken from an experiment, and from these five data poin
Three independent switches in a circuit close successfully with probabilities 0.9, 0.8, and 0.7 resp
For the real function g(x) = (x^2 + x + 1)/(x^2 + x + 2) defined for every real x, the complete rang
The real function defined by the rule f(x) = the square root of (x - 3) added to the square root of
Which of the following real functions defined for all real x is an odd function according to the sym
If the function f from real numbers to real numbers is defined by f(x) = (3x - 2)/(x + 4) for x not
Let [x] denote the greatest integer not exceeding x; the value of the sum [2.7] + [-2.7] + [0.4] + [
If a real function f satisfies f(x + 2) = 1 + the square root of (f(x) - f(x)^2) for all real x, the
Let f(x) = 2x + 3 and g(x) = x^2 - 1 be real functions; the value of the composite (f ∘ g)(2) minus
If the numbers a, b, c are simultaneously in arithmetic progression and in geometric progression wit
The general term of a series is given by T_k = k(k+2); evaluate the sum of the first n terms of this
If positive numbers x, y, z are in geometric progression, then the logarithms log x, log y, log z to
The sum of the first n terms of a certain sequence equals S_n = 3n^2 + 5n for every positive integer
Assertion: the series with terms 1/n diverges; Reason: every series whose terms tend to zero must co
Determine the exact value of the convergent infinite sum whose kth term is k/2^k, taken over all pos
If A and B are two finite sets such that n(A)=18, n(B)=23 and n(A ∪ B)=33, then the number of elemen
The number of onto functions that can be defined from a set of 4 distinct elements onto a set of 3 d
Let A be a set with exactly 3 elements; the total number of relations that can be defined from A to
On the set of all integers, define a relation R by aRb if and only if the difference a-b is divisibl
How many one-to-one functions can be defined from a set containing 4 distinct elements into a set co
Consider a set S having exactly n elements; the number of subsets of S that contain at least two ele
Determine the sum to \\infty of the arithmetico-geometric series 1 + 2x + 3x^2 + 4x^3 + ... when the
Three numbers are in geometric progression, their product equals 216, and the sum of the outer two n
For two distinct positive numbers, their arithmetic mean is 10 and their geometric mean is 8; comput
Find the sum of the telescoping series whose general term is 1/(k(k+1)) summed from k equal to 1 up
A worker is paid 200 rupees in the first month and receives an increment of 25 rupees each subsequen
Express the recurring decimal 0.overline{36}, meaning 0.363636 repeating endlessly, as a fully reduc
Consider the infinite geometric series 9 - 6 + 4 - 8/3 + ... that converges; what is its exact sum t
For positive reals a, b, c whose product abc equals 27, what is the minimum possible value of the su
If three arithmetic means are inserted between the numbers 4 and 28 to form an arithmetic progressio
What is the value of the sum 1^3 + 2^3 + 3^3 + ... + n^3 expressed as a closed-form function of the
Evaluate the value of the finite sum 1^2 + 2^2 + 3^2 + ... + 10^2 using the standard formula for the
Given that the fourth term of a harmonic progression is 1/12 and its ninth term is 1/27, find the tw
The number of ways to arrange 4 boys and 4 girls in a row so that no two girls are adjacent to each
The number of ways to place 5 distinct letters into 3 distinct mailboxes, where any mailbox may rece
Given 10 points in a plane of which no three are collinear, the number of distinct triangles that ca
If the combinatorial equation C(n, 2) equals 28 holds for a positive integer n, then the value of n
Using the standard combinatorial identity, the value of the sum C(8,0) + C(8,1) + C(8,2) + ... + C(8
From 20 points placed on a straight line, the number of line segments that can be formed by joining
Using all five digits 1, 2, 3, 4 and 5 exactly once to form five-digit numbers, the sum of all such
If all the distinct arrangements of the letters of the word RANK are listed in dictionary order, the
At a party every pair of guests shakes hands exactly once, and the total number of handshakes record
In an arithmetic progression whose seventh term is 34 and whose fifteenth term is 74, what is the su
Suppose a geometric progression of positive terms has third term equal to 12 and sixth term equal to
The number of ways to arrange 5 boys and 3 girls in a row so that all 3 girls always stand together
A student must travel from city A to city C passing through city B, with 4 roads from A to B and 5 r
The number of ways to arrange the letters of the word MONDAY so that all the vowels occupy only the
The area of the triangle with vertices at the points (1, 2), (4, 6) and (7, 2) in the coordinate pla
The number of distinct ways in which 8 different people can be seated around a circular table, where
Let A be the two by two matrix with rows (2, 1) and (1, 2); using the Cayley-Hamilton theorem, the m
From a group of 7 men and 6 women, the number of ways to form a committee consisting of exactly 3 me
The total number of diagonals that can be drawn in a convex polygon having 12 sides, counted using c
A box contains 5 red and 4 blue balls; the number of ways to select 4 balls so that at least 2 of th
The number of ways to distribute 10 identical chocolates among 3 distinct children such that each ch
The number of ways to divide 12 distinct books into three groups containing 4, 4 and 4 books respect
The number of distinct arrangements that can be formed using all the letters of the word BANANA, acc
The number of distinct four-digit numbers greater than 3000 that can be formed using the digits 1, 2
For any skew-symmetric matrix of odd order, the determinant always takes a fixed value; the determin
The homogeneous system of equations with coefficient matrix rows (1, 2, 3), (2, k, 6) and (1, 1, 1)
The value of the real parameter k for which the matrix with rows (k, 2, 3), (2, k, 1) and (3, 1, k)
If A and B are square matrices of order 3 with det(A) = 3 and det(B) = -2, then the determinant of t
A square matrix A is called idempotent when it satisfies A squared equal to A; for such a matrix the
If A is a square matrix of order 2 with trace equal to 5 and determinant equal to 6, then the sum of
A square matrix A is nilpotent if some positive integer power of A equals the zero matrix; the deter
Applying the property that adding a multiple of one row to another leaves a determinant unchanged, t
For two invertible square matrices A and B of the same order, the inverse of the product AB is corre
A real square matrix A is orthogonal when it satisfies the relation A transpose times A equal to the
Using Cramer's rule for the system 2x + 3y = 8 and 5x + 4y = 13, the value of the variable x obtaine
A function defined as f(x) = \frac{1 - \cos 4x}{x^2} for x \neq 0 and f(0) = m is continuous; find m
For the two by two matrix A with rows (4, 7) and (2, 6), the inverse matrix A inverse, computed usin
Using L'Hopital's rule or expansion, the limit \lim_{x \to 0} \frac{e^x - 1 - x}{x^2} evaluates to w
Evaluate \lim_{x \to 0} \frac{x - \sin x}{x - \tan x}, a ratio of two third-order vanishing trigonom
The value of the determinant of the three by three matrix with rows (2, -1, 3), (1, 0, 4) and (3, 2,
For a non-singular square matrix A of order 3 with determinant equal to 4, the determinant of the ad
Define f(x) = x^2 for x \le 1 and f(x) = ax + b for x > 1; find a, b making f differentiable everywh
If A and B are two square matrices of the same order such that AB is defined, which statement about
If A is a square matrix of order 3 and the determinant of A equals 5, then the value of the determin
Every square matrix can be written uniquely as the sum of a symmetric matrix and a skew-symmetric ma
The system of equations x + y + z = 6, x + 2y + 3z = 14 and x + 4y + 9z = 36 is analyzed using deter
For the greatest integer function, at how many points in the open interval (0,3) is g(x) = [x] + [-x
Consider h(x) = |x - 1| + |x + 1| on the real line; at how many points does h fail to be differentia
Let q(x) = \max\{x, x^3\} for real x; determine the set of points where q is not differentiable.
For f(x) = x^3 - 3x on the closed interval [-\sqrt{3}, \sqrt{3}], how many values of c satisfy Rolle
Let p(x) = x^2 \sin(1/x) for x \neq 0 and p(0) = 0; which statement about p at the origin is correct
If y = x^x for x > 0, then the derivative dy/dx evaluated through logarithmic differentiation equals
Applying the Lagrange mean value theorem to f(x) = \ln x on [1, e], the point c guaranteed by the th
Assertion: every differentiable function is continuous. Reason: differentiability requires the diffe
The value of \lim_{x \to 0} \frac{\ln(1 + 2x)}{\sin 3x} is required; choose the correct evaluation u
Determine \lim_{x \to \infty} \left(\frac{x+3}{x-1}\right)^{x+2}, a classic exponential indeterminat
Suppose f(x) = \frac{\sqrt{1+x} - \sqrt{1-x}}{x} for x \neq 0 and f(0) = k; which k makes f continuo
Considering the greatest integer function within the integrand, evaluate the definite integral from
The figure shows the curve y equals four minus x-squared sitting above the x-axis between its two ro
Evaluate the value of the limit \lim_{x \to 0} \frac{\tan x - \sin x}{x^3}, which arises frequently
Find the constants for which \lim_{x \to 0} \frac{a e^x - b \cos x + c e^{-x}}{x \sin x} equals 2, t
Exploiting the periodicity of the absolute sine function, evaluate the definite integral of the abso
Compute the right-hand limit \lim_{x \to 0^+} x^x, a standard 0^0 indeterminate form often tested th
What is the value of \lim_{n \to \infty} \left( \frac{1}{n^2} + \frac{2}{n^2} + \cdots + \frac{n}{n^
Given the piecewise function shown, with f(x)=ax+1 for x \le 2 and f(x)=bx-3 for x>2, find a relatio
Find the indefinite integral of one over the square root of the quantity a-squared minus x-squared a
Evaluate the definite integral from zero to \pi/2 of cosine raised to the fourth power, applying the
The region under one full arch of the sine curve from zero to \pi is shaded in the figure; calculate
Assertion: the integral from zero to two of the function defined as x for x below one and as two min
The shaded region lies between the parabola y equals x-squared and the line y equals x as drawn; det
Determine the indefinite integral of the natural logarithm of x treated as a single function, using
Using the Weierstrass half-angle substitution, evaluate the indefinite integral of one divided by th
By treating the expression as a Riemann sum, evaluate the limit as n tends to \\infty of the sum of
If a function F is defined as the integral from one to x of the reciprocal of t, then differentiate
Evaluate the definite integral from zero to \\infty of e raised to negative x multiplied by x, recog
Given an integrand combining an odd cube term with an even cosine term over a symmetric interval, ev
Apply the Leibniz differentiation rule to find the derivative of the accumulation function whose low
Using the king property of definite integrals, evaluate the integral from 0 to \pi of x times \sin x
Compute the indefinite integral of one divided by the quantity x-squared minus one, expressing the a
Evaluate the definite integral of \sin^2 x from 0 to \pi/2 and identify which standard symmetry resu
A cup of coffee cools according to Newton's law so that \frac{dT}{dt} = -k(T - T_s); identify the co
Solve the repeated-root equation \frac{d^2y}{dx^2} - 6\frac{dy}{dx} + 9y = 0 and state the general s