Alternate Sets
If `vec a=7hati+hatj-4hatk and vecb=2hati+6hatj+3hatk` , then find the projection of `vec a and vecb`
Concept: Product of Two Vectors - Scalar (Or Dot) Product of Two Vectors
Find λ, if the vectors `veca=hati+3hatj+hatk,vecb=2hati−hatj−hatk and vecc=λhatj+3hatk` are coplanar.
Concept: Scalar Triple Product of Vectors
If a line makes angles 90°, 60° and θ with x, y and z-axis respectively, where θ is acute, then find θ.
Concept: Properties of Inverse Trigonometric Functions
Write the element a_{23} of a 3 ✕ 3 matrix A = (a_{ij}) whose elements a_{ij} are given by `a_(ij)=∣(i−j)/2∣`
Concept: Introduction of Operations on Matrices
Find the differential equation representing the family of curves v=A/r+ B, where A and B are arbitrary constants.
Concept: Formation of a Differential Equation Whose General Solution is Given
Find the integrating factor of the differential equation.
`((e^(-2^sqrtx))/sqrtx-y/sqrtx)dy/dx=1`
Concept: Solutions of Linear Differential Equation
If `A=([2,0,1],[2,1,3],[1,-1,0])` find A^{2}-5A+4I and hence find a matrix X such that A^{2}-5A+4I+X=O
Concept: Operations on Matrices - Addition of Matrices
If A = `[[1,-2,3],[0,-1,4],[-2,2,1]]` ,find (A')^{-1}
Concept: Inverse of a Matrix by Elementary Operations
If ` f(x)=|[a,-1,0],[ax,a,-1],[ax^2,ax,a]| ` , using properties of determinants find the value of f(2x) − f(x).
Concept: Properties of Determinants
Find: `I=intdx/(sinx+sin2x)`
Concept: Methods of Integration - Integration Using Partial Fractions
Integrate the following w.r.t. x `(x^3-3x+1)/sqrt(1-x^2)`
Concept: Evaluation of Simple Integrals of the Following Types and Problems
Evaluate :
`∫_(-pi)^pi (cos ax−sin bx)^2 dx`
Concept: Evaluation of Definite Integrals by Substitution
A bag A contains 4 black and 6 red balls and bag B contains 7 black and 3 red balls. A die is thrown. If 1 or 2 appears on it, then bag A is chosen, otherwise bag B, If two balls are drawn at random (without replacement) from the selected bag, find the probability of one of them being red and another black.
Concept: Probability Examples and Solutions
An unbiased coin is tossed 4 times. Find the mean and variance of the number of heads obtained.
Concept: Mean of a Random Variable
If `vecr=xhati+yhatj+zhatk` ,find `(vecrxxhati).(vecrxxhatj)+xy`
Concept: Vectors Examples and Solutions
Find the distance between the point (−1, −5, −10) and the point of intersection of the line `(x-2)/3=(y+1)/4=(z-2)/12` and the plane x-y+z=5
Concept: Three - Dimensional Geometry Examples and Solutions
If sin [cot^{−1} (x+1)] = cos(tan^{−}^{1}x), then find x.
Concept: Inverse Trigonometric Functions (Simplification and Examples)
If (tan^{−}^{1}x)^{2} + (cot^{−1}x)^{2} = 5π^{2}/8, then find x.
Concept: Inverse Trigonometric Functions (Simplification and Examples)
If `y=tan^(−1) ((sqrt(1+x^2)+sqrt(1−x^2))/(sqrt(1+x^2)−sqrt(1−x^2)))` , x^{2}≤1, then find dy/dx.
Concept: Derivatives of Inverse Trigonometric Functions
If x = a cos θ + b sin θ, y = a sin θ − b cos θ, show that `y^2 (d^2y)/(dx^2)-xdy/dx+y=0`
Concept: Second Order Derivative
The side of an equilateral triangle is increasing at the rate of 2 cm/s. At what rate is its area increasing when the side of the triangle is 20 cm ?
Concept: Increasing and Decreasing Functions
Find : `int(x+3)sqrt(3-4x-x^2dx)`
Concept: Methods of Integration - Integration by Substitution
Three schools A, B and C organized a mela for collecting funds for helping the rehabilitation of flood victims. They sold hand made fans, mats and plates from recycled material at a cost of Rs 25, Rs 100 and Rs 50 each. The number of articles sold are given below:
SchoolArticle | |||
A | B | C | |
Hand-fans | 40 | 25 | 35 |
Mats | 50 | 40 | 50 |
Plates | 20 | 30 | 40 |
Find the funds collected by each school separately by selling the above articles. Also find the total funds collected for the purpose.
Write one value generated by the above situation.
Concept: Multiplication of Two Matrices
Let N denote the set of all natural numbers and R be the relation on N × N defined by (a, b) R (c, d) if ad (b + c) = bc (a + d). Show that R is an equivalence relation.
Concept: Types of Relations
Using integration find the area of the triangle formed by positive x-axis and tangent and normal of the circle
`x^2+y^2=4 at (1, sqrt3)`
Concept: Area Under Simple Curves
Evaluate `int_1^3(e^(2-3x)+x^2+1)dx` as a limit of sum.
Concept: Definite Integral as the Limit of a Sum
Solve the differential equation : (tan^{−1}y−x)dy=(1+y^{2})dx.
Concept: General and Particular Solutions of a Differential Equation
Find the particular solution of the differential equation `dy/dx=(xy)/(x^2+y^2)` given that y = 1, when x = 0.
Concept: General and Particular Solutions of a Differential Equation
If lines `(x−1)/2=(y+1)/3=(z−1)/4 and (x−3)/1=(y−k)/2=z/1` intersect, then find the value of k and hence find the equation of the plane containing these lines.
Concept: Three - Dimensional Geometry Examples and Solutions
If A and B are two independent events such that `P(barA∩ B) =2/15 and P(A ∩ barB) = 1/6`, then find P(A) and P(B).
Concept: Independent Events
Find the local maxima and local minima, of the function f(x) = sin x − cos x, 0 < x < 2π.
Concept: Maximum and Minimum Values of a Function in a Closed Interval
Find graphically, the maximum value of z = 2x + 5y, subject to constraints given below :
2x + 4y ≤ 83
x + y ≤ 6
x + y ≤ 4
x ≥ 0, y≥ 0
Concept: Graphical Method of Solving Linear Programming Problems