Find the distance of the point (−1, −5, −10) from the point of intersection of the line `vecr=2hati-hatj+2hatk+lambda(3hati+4hatj+2hatk) ` and the plane `vec r (hati-hatj+hatk)=5`
Concept: Three - Dimensional Geometry Examples and Solutions
Find the area of the region in the first quadrant enclosed by the x-axis, the line y = x and the circle x^{2} + y^{2} = 32.
Concept: Area Under Simple Curves
If y = x^{x}, prove that `(d^2y)/(dx^2)−1/y(dy/dx)^2−y/x=0.`
Concept: Simple Problems on Applications of Derivatives
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
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 Fractions
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
Let A = {1, 2, 3,......, 9} and R be the relation in A × A defined by (a, b) R (c, d) if a + d = b + c for (a, b), (c, d) in A × A. Prove that R is an equivalence relation. Also, obtain the equivalence class [(2, 5)].
Concept: Types of Relations
Let f : N→N be a function defined as f(x)=`9x^2`+6x−5. Show that f : N→S, where S is the range of f, is invertible. Find the inverse of f and hence find `f^-1`(43) and` f^−1`(163).
Concept: Inverse of a Function
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
if `2[[3,4],[5,x]]+[[1,y],[0,1]]=[[7,0],[10,5]]` , find (x−y).
Concept: Equality of Matrices
Solve the following matrix equation for x: `[x 1] [[1,0],[−2,0]]=0`
Concept: Operations on Matrices > Addition of Matrices
Prove that `|(yz-x^2,zx-y^2,xy-z^2),(zx-y^2,xy-z^2,yz-x^2),(xy-z^2,yz-x^2,zx-y^2)|`is divisible by (x + y + z) and hence find the quotient.
Concept: Elementary Transformations
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
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 = 0
Concept: Operations on Matrices > Addition of Matrices
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
If `[[3x,7],[-2,4]]=[[8,7],[6,4]]`, find the value of x
Concept: Introduction of Operations on Matrices
The monthly incomes of Aryan and Babban are in the ratio 3 : 4 and their monthly expenditures are in the ratio 5 : 7. If each saves Rs 15,000 per month, find their monthly incomes using matrix method. This problem reflects which value?
Concept: Inverse of Matrix > Inverse of a Square Matrix by the Adjoint Method
Prove that `|(yz-x^2,zx-y^2,xy-z^2),(zx-y^2,xy-z^2,yz-x^2),(xy-z^2,yz-x^2,zx-y^2)|`is divisible by (x + y + z) and hence find the quotient.
Concept: Elementary Transformations
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