18 Questions
Which of the following points lies on the curve E1?
(3, 1)
What is the value of the definite integral $\int_{0}^{4\pi} (11 + 4 \cos^2 x + \cos^4 x) dx$?
12 - 2
In the arithmetic progression of natural numbers a1, a2, ..., an, what is the value of a6 if the volume of the parallelepiped formed by vectors a and c is 54?
5
If P = [1 0 0; 4 1 0] and Q is such that q12 + q32 = P50 - Q, what is the sum of digits in q21?
6
If b = ai + aj + ak and c = ai + aj + ak, what is the value of a such that volume of the parallelepiped formed by a, b, and c is 54?
7
What is the point that does not lie on E2?
(3, 1)
What is the value of $\lambda + \mu$ if the equation $y^2 + \lambda xy - \mu x^2 = 0$ represents lines through the origin parallel to lines dividing triangle ABC into three equal areas?
8
What is the value of $p - q$ if the area enclosed by the curves $y = x$ and $x + y - 2 = 0$ is expressed as $\frac{p}{q}$?
3
The curve satisfying the differential equation and passing through (2, 1) with eccentricity e is most likely a:
Hyperbola
If $\lim_{n \to \infty} 2m[(n+1)(n+2)...(n+n)]^n = \frac{1}{8}$, then the value of 'a' in the expression is:
$\frac{1}{4}$
If the distance of a plane from the origin that passes through (1,1,1) and is perpendicular to the line $\frac{x-1}{3} = \frac{y-1}{0} = \frac{z-1}{4}$ is $rac{p}{q}$, what is the value of $p-q$?
-1
If the interval [0, 4] is divided into n equal sub-intervals by points $x_0, x_1, x_2, ..., x_{n-1} = 4$, what is the sum of all these sub-intervals?
$4(n-1)$
What is the value of the limit lim Δx → ∞ ∑(x_i Δx) for i from 1 to n?
n
For the function f(x) = xf(x) + (1 - x)f(-x) = x^2 + x + 1, what is the greatest real number M such that f(x) ≥ M for all real numbers x?
7
In a trapezium ABCD with AB parallel to CD, AD perpendicular to AB, and AB = 3CD, what is the area of the circle tangent to all sides if its radius is r?
$\frac{8}{r}$
How many values of x in the form 1/n, where n is a natural number, lie in the interval [1/15, 10] and satisfy the equation {x} + {2x} + ... + {12x} = 78x?
15
In the expansion of (2x^2 + 3x + 4)^10 = ∑a_xr, what is the value of a_9?
$\binom{10}{9} \times 2^3 \times 3^7 \times 4^0$
If a derivable function f: R+ → R satisfies f(xy) = f(x) + f(y) for all positive real numbers x, y, what is true about f?
$f(x) = kln(x)$ for some constant k.
Test your knowledge on definite integrals and volume of parallelepipeds. Solve questions on calculating definite integrals and finding the volume of parallelepipeds formed by vectors. Challenge yourself with these calculus problems.
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