Date: March 2015
If a line makes angles 90°, 60° and θ with x, y and z-axis respectively, where θ is acute, then find θ.
Chapter: [1.01] 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∣`
Chapter: [2.02] Matrices
Find the differential equation representing the family of curves v=A/r+ B, where A and B are arbitrary constants.
Chapter: [3.04] Differential Equations
Find the integrating factor of the differential equation.
`((e^(-2^sqrtx))/sqrtx-y/sqrtx)dy/dx=1`
Chapter: [3.04] Differential Equations
If `vec a=7hati+hatj-4hatk and vecb=2hati+6hatj+3hatk` , then find the projection of `vec a and vecb`
Chapter: [4.02] Vectors
Find λ, if the vectors `veca=hati+3hatj+hatk,vecb=2hati−hatj−hatk and vecc=λhatj+3hatk` are coplanar.
Chapter: [4.02] Vectors
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.
Chapter: [6.01] Probability
An unbiased coin is tossed 4 times. Find the mean and variance of the number of heads obtained.
Chapter: [6.01] Probability
If `vecr=xhati+yhatj+zhatk` ,find `(vecrxxhati).(vecrxxhatj)+xy`
Chapter: [4.02] Vectors
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
Chapter: [4.01] Three - Dimensional Geometry
If sin [cot^{−1} (x+1)] = cos(tan^{−}^{1}x), then find x.
Chapter: [1.01] Inverse Trigonometric Functions
If (tan^{−}^{1}x)^{2} + (cot^{−1}x)^{2} = 5π^{2}/8, then find x.
Chapter: [1.01] Inverse Trigonometric Functions
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.
Chapter: [3.01] Continuity and Differentiability
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`
Chapter: [3.01] Continuity and Differentiability
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 ?
Chapter: [3.02] Applications of Derivatives
Find : `int(x+3)sqrt(3-4x-x^2dx)`
Chapter: [3.05] Integrals
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.
Chapter: [2.02] 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
Chapter: [2.02] Matrices
If A = `[[1,-2,3],[0,-1,4],[-2,2,1]]` ,find (A')^{-1}
Chapter: [2.02] Matrices
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).
Chapter: [2.01] Determinants
Find: `I=intdx/(sinx+sin2x)`
Chapter: [3.05] Integrals
Integrate the following w.r.t. x `(x^3-3x+1)/sqrt(1-x^2)`
Chapter: [3.05] Integrals
Evaluate :
`∫_(-pi)^pi (cos ax−sin bx)^2 dx`
Chapter: [3.05] Integrals
Solve the differential equation : (tan^{−1}y−x)dy=(1+y^{2})dx.
Chapter: [3.04] Differential Equations
Find the particular solution of the differential equation `dy/dx=(xy)/(x^2+y^2)` given that y = 1, when x = 0.
Chapter: [3.04] Differential Equations
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.
Chapter: [4.01] Three - Dimensional Geometry
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).
Chapter: [6.01] Probability
Find the local maxima and local minima, of the function f(x) = sin x − cos x, 0 < x < 2π.
Chapter: [3.02] Applications of Derivatives
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
Chapter: [5.01] Linear Programming
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.
Chapter: [1.02] Relations and Functions
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)`
Chapter: [3.03] Applications of the Integrals
Evaluate `int_1^3(e^(2-3x)+x^2+1)dx` as a limit of sum.
Chapter: [3.05] Integrals
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