Write the principal value of `tan^(-1)+cos^(-1)(-1/2)`

Concept: Inverse Trigonometric Functions - Inverse Trigonometric Functions - Principal Value Branch

Write the value of `tan(2tan^(-1)(1/5))`

Concept: Inverse Trigonometric Functions (Simplification and Examples)

Find the value of *a* if `[[a-b,2a+c],[2a-b,3c+d]]=[[-1,5],[0,13]]`

Concept: Applications of Determinants and Matrices

If `|[x+1,x-1],[x-3,x+2]|=|[4,-1],[1,3]|`, then write the value of *x*.

Concept: Applications of Determinants and Matrices

if `[[9,-1,4],[-2,1,3]]=A+[[1,2,-1],[0,4,9]]`, then find the matrix A.

Concept: Subtraction of Matrices

Write the degree of the differential equation `x^3((d^2y)/(dx^2))^2+x(dy/dx)^4=0`

Concept: Order and Degree of a Differential Equation

If `veca=xhati+2hatj-zhatk and vecb=3hati-yhatj+hatk` are two equal vectors ,then write the value of x+y+z

Concept: Types of Vectors

If a unit vector `veca` makes angles `pi/3` with `hati,pi/4` with `hatj` and acute angles θ with ` hatk,` then find the value of θ.

Concept: Vectors Examples and Solutions

Find the Cartesian equation of the line which passes through the point (−2, 4, −5) and is parallel to the line `(x+3)/3=(4-y)/5=(z+8)/6`

Concept: Equation of a Line in Space

The amount of pollution content added in air in a city due to *x*-diesel vehicles is given by P(*x*) = 0.005*x*^{3} + 0.02*x*^{2} + 30*x*. Find the marginal increase in pollution content when 3 diesel vehicles are added and write which value is indicated in the above question.

Concept: Increasing and Decreasing Functions

Show that the function f in `A=R-{2/3} ` defined as `f(x)=(4x+3)/(6x-4)` is one-one and onto hence find f^{-1}

Concept: Types of Functions

Find the value of the following: `tan(1/2)[sin^(-1)((2x)/(1+x^2))+cos^(-1)((1-y^2)/(1+y^2))],|x| <1,y>0 and xy <1`

Concept: Inverse Trigonometric Functions (Simplification and Examples)

Prove that: `tan^(-1)(1/2)+tan^(-1)(1/5)+tan^(-1)(1/8)=pi/4`

Concept: Properties of Inverse Trigonometric Functions

Using properties of determinants prove the following: `|[1,x,x^2],[x^2,1,x],[x,x^2,1]|=(1-x^3)^2`

Concept: Properties of Determinants

Differentiate the following function with respect to *x*: `(log x)^x+x^(logx)`

Concept: Logarithmic Differentiation

If `y=log[x+sqrt(x^2+a^2)] ` show that `(x^2+a^2)(d^2y)/(dx^2)+xdy/dx=0`

Concept: Logarithmic Differentiation

Show that the function `f(x)=|x-3|,x in R` is continuous but not differentiable at *x *= 3.

Concept: Concept of Continuity

If *x* = a sin t and `y = a (cost+logtan(t/2))` ,find `((d^2y)/(dx^2))`

Concept: Second Order Derivative

Evaluate : `intsin(x-a)/sin(x+a)dx`

Concept: Integration Using Trigonometric Identities

Evaluate: `int(5x-2)/(1+2x+3x^2)dx`

Concept: Integrals of Some Particular Functions

Evaluate : ` int x^2/((x^2+4)(x^2+9))dx`

Concept: Integrals of Some Particular Functions

Evaluate : `int_0^4(|x|+|x-2|+|x-4|)dx`

Concept: Evaluation of Definite Integrals by Substitution

If `veca and vecb` are two vectors such that `|veca+vecb|=|veca|,` then prove that vector `2veca+vecb` is perpendicular to vector `vecb`

Concept: Product of Two Vectors - Scalar (Or Dot) Product of Two Vectors

Find the coordinates of the point, where the line `(x-2)/3=(y+1)/4=(z-2)/2` intersects the plane *x − y + z* − 5 = 0. Also find the angle between the line and the plane.

Concept: Angle Between Line and a Plane

Find the vector equation of the plane which contains the line of intersection of the planes `vecr (hati+2hatj+3hatk)-4=0` and `vec r (2hati+hatj-hatk)+5=0` which is perpendicular to the plane.`vecr(5hati+3hatj-6hatk)+8=0`

Concept: Vector and Cartesian Equation of a Plane

A speaks truth in 60% of the cases, while B in 90% of the cases. In what percent of cases are they likely to contradict each other in stating the same fact? In the cases of contradiction do you think, the statement of B will carry more weight as he speaks truth in more number of cases than A?

Concept: Independent Events

A school wants to award its students for the values of Honesty, Regularity and Hard work with a total cash award of Rs 6,000. Three times the award money for Hard work added to that given for honesty amounts to Rs 11,000. The award money given for Honesty and Hard work together is double the one given for Regularity. Represent the above situation algebraically and find the award money for each value, using matrix method. Apart from these values, namely, Honesty, Regularity and Hard work, suggest one more value which the school must include for awards.

Concept: Invertible Matrices

Show that the height of the cylinder of maximum volume, which can be inscribed in a sphere of radius R is `(2R)/sqrt3.` Also find the maximum volume.

Concept: Maxima and Minima

Find the equation of the normal at a point on the curve *x*^{2} = 4y which passes through the point (1, 2). Also find the equation of the corresponding tangent.

Concept: Tangents and Normals

Using integration, find the area bounded by the curve *x*^{2} = 4y and the line *x* = 4y − 2.

Concept: Area of the Region Bounded by a Curve and a Line

Find the equation of the normal at a point on the curve *x*^{2} = 4y which passes through the point (1, 2). Also find the equation of the corresponding tangent.

Concept: Tangents and Normals

Show that the differential equation 2y^{x}^{/y} d*x* + (y − 2*x* e^{x/}^{y}) dy = 0 is homogeneous. Find the particular solution of this differential equation, given that *x* = 0 when y = 1.

Concept: Methods of Solving First Order, First Degree Differential Equations - Homogeneous Differential Equations

Find the vector equation of the plane passing through three points with position vectors ` hati+hatj-2hatk , 2hati-hatj+hatk and hati+2hatj+hatk` . Also find the coordinates of the point of intersection of this plane and the line `vecr=3hati-hatj-hatk lambda +(2hati-2hatj+hatk)`

Concept: Vector and Cartesian Equation of a Plane

A cooperative society of farmers has 50 hectares of land to grow two crops A and B. The profits from crops A and B per hectare are estimated as Rs 10,500 and Rs 9,000 respectively. To control weeds, a liquid herbicide has to be used for crops A and B at the rate of 20 litres and 10 litres per hectare, respectively. Further not more than 800 litres of herbicide should be used in order to protect fish and wildlife using a pond which collects drainage from this land. Keeping in mind that the protection of fish and other wildlife is more important than earning profit, how much land should be allocated to each crop so as to maximize the total profit? Form an LPP from the above and solve it graphically. Do you agree with the message that the protection of wildlife is utmost necessary to preserve the balance in environment?

Concept: Graphical Method of Solving Linear Programming Problems

Assume that the chances of a patient having a heart attack is 40%. Assuming that a meditation and yoga course reduces the risk of heart attack by 30% and prescription of certain drug reduces its chance by 25%. At a time a patient can choose any one of the two options with equal probabilities. It is given that after going through one of the two options, the patient selected at random suffers a heart attack. Find the probability that the patient followed a course of meditation and yoga. Interpret the result and state which of the above stated methods is more beneficial for the patient.

Concept: Conditional Probability