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If x and y are connected parametrically by the equations given in Exercises 1 to 10, without eliminating the parameter, Find (dy)/(dx). x= 2at^(2), y= at^(4).

Determine the shortest distance between the following lines and conclude whether they intersect or not (i) vec(r)=(hat(i)-hat(j))+t(2hat(i)+hat(k)),vec(r)=(2hat(i)-hat(j))+s(hat(i)-hat(j)-hat(k)) (ii) (x-1)/(2)=(y+1)/(3)=(z-0)/(1),(x+1)/(5)=(y-2)/(1),z=2

Without expandingthe determinant prove the following. |[x,a,x+a],[y,b,y+b],[z,c,z+c]|=0

y log y dx - x dy = 0

Let there are 4 sections of 25 students each in a coaching class. Now, out of 150 students 100 are to be selected randomly and entrolled in these sections. Then, the probability that the students A and B (both present in 150 students) are selected and placed in the same section, is

Evaluate the definite integrals int_(0)^(pi/2)sin2xdx

Let f(x)={(1-|x|","|x|le1),(0","|x|gt1):} and g(x)=f(x-1)"for all" x in R The graph for g(x) is given by

How many grams of benzene would be produced when 135.5 gms of phenyl magnesium chloride is trated with 224 ml of ethyne at STP ?

If for 5 observations of pairs (x,y), Sigmax=15,Sigmay=25,Sigmay^(2)=135 and Sigmaxy=83, then the value of b_(xy) is

Consider the equation sinx=k, k being a parameter lying in the inverval [-1,1]. Statement-1 : For given k, the equation sinx=k has two solutions in [0,2pi] Statement-2 : sintheta=kiffsin(pi-theta)=k.

Variance (sigma^(2)) and thecoefficient of variation (CV) of two distributions are given below. (i) Compute the standard deviations of the given distributions.(ii) What are their means?

Which of thefollowing reactions is not involved in extraction of Cu from copper pyrite ?

Trachea divide at the level of :-

A circle C is drawn on the line joining the centres of the circles C _(1) :x ^(2) +y ^(2) -4 =0 and C _(2): x ^(2) +y ^(2) -8x + 7=0as a diameter. The length of the intercept made on the common chord of the circles C_(1) and C_(2) by the circle C is

Simplify: cot^(-1)""(1)/(sqrt(x^(2)-1))"for" c lt-1.

IfDelta = {:|( a_11, a_12, a_113),( a_21,a_22,a_23) ,(a_31,a_32, a_33) |:} and Aij is Cofactors of a_ijthen value of Deltais given by

If the normals at t_(1) andt_(2) on y^(2) = 4ax meet again on the parabola then t_(1)t_(2)=

Let f _(a)(x) =In x andfor n ge 0 and x gt 0 Let f _(a)(x) = int _(0) ^(x)f _(a) (t) dt then: Value of lim _( x to oo) ((lfloorn )f _(n)(1))/(ln (n)):

Evaluate the following inegrals int sin x sin 2x sin 3x dx

(1)/(1.3)+(1)/(2)((1)/(3.5))+(1)/(3)((1)/(5.7))+....=

Compute theintegrals I = int_(0)^(5) (dx)/( 2 x + sqrt(3 x + 1))

A is2 xx 2matrix whose elements are given by a_(ij) = 2 I_(ij) - j .Then find M_(21) +A_(11) , whereM_(ij)is the minor of a_(ij)and A _(ij)is the co-factor ofa_(ij).

Findthe area of theregionboundedby linex=2and parabolay^2= 8x

Resolve (2x+3)/((x-1)(x^(2)+x+1)) into partial fractions.

Letalphaandbetabe therootsof thequadraticequationax^2+bx+c=0. Observethe lists givenbelow: Letcorrectmatchof List-IfromList -IIis

A triangle is formed by Y-axis, the straight line L passing through the points (3,0)(1, (4)/(3)) and the straight line perpendicular to the line L and passing through the point (8,1). Then, the area of the triangle (in sq units) is

Simplify (1+cosx+isinx)/(1+cosx-isinx).

Select the Correct Option If 3axy=c, then dy/dx=

The probability of guessing correctly atleast 8 out of 10 answers on a true-false type examination is .........

There are 5% defective items in a large bulk of items. What is the probability that a sample of 10 items will include not more than one defective items?

Write{x:x =1 or x=2 or x=3} set in the form of lists?

Let A be the area bounded by the curve y=sin x(0 le x le pi) and x-axis and B is the area bounded by the curves y=sin x(0 le x le pi//2), y=a cosx (0 le x le pi//2) and x-axis (a gt0). If A:B=1, then a is equal to

f: R rarr R be defined as f(x) = |x| + |x^(2) -1|. The total number of points at which f attains either local maximum or a level minimum is

Consider the following relations R_(1) = {(x,y) : x and y are integers and x=ay or y=ax for some integer a} R_(2) ={(x,y) : x and y are integers and ax + by =1for some integers a,b},Then

In the list {2, 3, 0, 8, 11, 1, 4, 7, 8, 2, 1, 3}, what is Q_(3)-Q_(1) ?

The differential equation of the family of circles passing through the origin and having their centres on the x-axis is

Let f(x) = {{:((x((3)/(4))^(1//x)-((3)/(4))^(-1//x))/(((3)/(4))^(1//x)+((3)/(4))^(-1//x))",",x ne 0),(0",",x = 0):}. If P = f'(0^(-)) - f'(0^(+)), then 4 (lim_(x rarr p^(-))((exp((x+2)log 4))[(x+1)/(4)]-16)/(4^(x) - 16)), is...... (where [x] denotes greatest integer function.)

If f is any function, then (1)/(2)[f(x)+f(-x)] is always :

Equation of a line passing through the point whose position vector is 2i-3j+4k and in the direction of the vector 3i+4j-5k is

The sixth term of an A.P. is equal to 2. The value of the common difference of the A.P. which makes the product a_1 a_4 a_5 least is given by :

(1-omega+omega^2)(1-omega^2+omega^4)(1-omega^4+omega^2)….to 2nFactors =2^(2n)

If omega ne 1 is a cube root of unity satisfying (1)/(a + w) + (1)/(b + w) + (1)/(c +w) = 2 w^(2) and (1)/(a + w^(2)) + (1)/(b + w^(2)) + (1) /(c + w^(2)) = 2w then the value of (1)/(a + 1) + (1)/(b + 1) + (1)/(c + 1) is

Evaluate the integrals in exercise. overset(1)underset(0) int(x)/(x^(2)+1)dx

Let p(n) = (n+1)(n+3) (n+5)(n+7)(n+9) . What is the largest integer that is a divisor of p(n) for all positive even integers n ?

v&=2v-1 {:("Quantity A","Quantity B"),((v&)s,4v):}

If bara and barb are unit vectors and barc satisfies 2(hata times hatb)+barc=hatb times hatc then the maximum value of |(bara times barc).barb| is

Let P be a point on the circle S with both coordinates being positive. Let the tangent to S at P intersect the coordinate axes at the points M and N. Then, the mid-point of the line segment MN must lie on the curve.