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Show that four points A, B, C and D whose position vectors are
`4hati+5hatj+hatk,-hatj-hatk-hatk, 3hati+9hatj+4hatk and 4(-hati+hatj+hatk)` respectively are coplanar.
Concept: Coplanarity of Two Lines
Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is `(4r)/3`. Also find maximum volume in terms of volume of the sphere
Concept: Maxima and Minima
If u and v are two functions of x then prove that
`intuvdx=uintvdx-int[du/dxintvdx]dx`
Hence evaluate, `int xe^xdx`
Concept: Methods of Integration> Integration by Parts
Find: `int(x+3)sqrt(3-4x-x^2dx)`
Concept: Methods of Integration> Integration by Substitution
Find: `I=intdx/(sinx+sin2x)`
Concept: Methods of Integration> Integration Using Partial Fraction
Given is X ~ B(n, p). If E(X) = 6, and Var(X) = 4.2, find the value of n.
Concept: Mean and Variance of Binomial Distribution
Find the symbolic form of the given switching circuit. Construct its switching table and interpret your result.
Concept: Application of Logic to Switching Circuits
Write the following compound statement symbolically.
Nagpur is in Maharashtra and Chennai is in Tamil Nadu.
Concept: Logical Connectives
Find k, if the sum of the slopes of the lines represented by x2 + kxy – 3y2 = 0 is twice their product.
Concept: Homogeneous Equation of Degree Two
If the lines `(x-1)/2=(y+1)/3=(z-1)/4 ` and `(x-3)/1=(y-k)/2=z/1` intersect each other then find value of k
Concept: Distance in Lines (Point & Parallel Lines)
Find the co-ordinates of the foot of the perpendicular drawn from the point (0, 2, 3) to the line `(x + 3)/(5) = (y - 1)/(2) = (z + 4)/(3)`.
Concept: Equation of a Plane
A manufacturer produces two products A and B. Both the products are processed on two different machines. The available capacity of first machine is 12 hours and that of second machine is 9 hours per day. Each unit of product A requires 3 hours on both machines and each unit of product B requires 2 hours on first machine and 1 hour on second machine. Each unit of product A is sold at Rs 7 profit and B at a profit of Rs 4. Find the production level per day for maximum profit graphically.
Concept: Methods to Solve LPP (Graphical / Corner Point Method)
If y = f(x) is a differentiable function of x such that inverse function x = f–1 (y) exists, then prove that x is a differentiable function of y and `dx/dy=1/((dy/dx)) " where " dy/dx≠0`
Concept: Derivative of Inverse Function
If y = cos(m cos–1x), then show that `(1 - x^2) ("d"^2y)/("d"x^2) - x("d"y)/("d"x) + "m"^2y` = 0
Concept: Higher Order Derivatives
The displacement of a particle at time t is given by s = 2t3 – 5t2 + 4t – 3. Find the velocity when 𝑡 = 2 sec
Concept: Derivatives as a Rate Measure
Find `intsqrtx/sqrt(a^3-x^3)dx`
Concept: Methods of Integration> Integration by Substitution
Choose the correct alternative:
`int (x + 2)/(2x^2 + 6x + 5) "d"x = "p"int (4x + 6)/(2x^2 + 6x + 5) "d"x + 1/2 int 1/(2x^2 + 6x + 5)"d"x`, then p = ?
Concept: Methods of Integration> Integration Using Partial Fraction
Find the symbolic form of the following switching circuit, construct its switching table and interpret it.
Concept: Application of Logic to Switching Circuits
Simplify the following circuit so that the new circuit has minimum number of switches. Also, draw the simplified circuit.
Concept: Application of Logic to Switching Circuits
Use quantifiers to convert the following open sentences defined on N, into a true statement.
n2 ≥ 1
Concept: Quantifier, Quantified and Duality Statements in Logic
