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If int_(0)^(10)f(x)dx=5, then sum_(K=1)^(10) int_(0)^(1) f(K-1+x)dx is equal to

Evaluate: int(dx)/(x-sqrt(x))

Show that the points A(1,-2,-8),B(5,0,-2)andC(11,3,7) are collinear , and find the ratio in which B divides AC.

Integrate the following functions x^2e^x

Which of the following statement is true Statement - I 2lt 1/(1!)+1/(2!)+1/(3!)+1/(4!)+ .....oo lt 3 Statement - II 1/2lt 1/(1!)+1/(2!)+1/(3!)+1/(4!)+ .....oo lt 1

As shown in the (x,y,z) coordinate space below, the cube with vertices L through S has edges that are2 coordinate units long. The coordinates ofQ are (0,0,0) and S is on the positive x-axis . What are the coordinatesof O ?

All the numbers that can be formed using 1, 2, 3, 4, 5 are arranged in the decreasing order. Rank of 35241 is

If (""^(n)C_(r)+3""^(n)C_(r+1)+3""^(n)C_(r+2)+""^(n)C_(r)+3)/(""^(n)C_(r)+4""^(n)C_(r+1)+6""^(n)C_(r+2)+4""^(n)C_(r+2)+4""^(n)C_(r+3)+""^(n)C_(r)+4)=(r+k)/(n+k) ,then the value of k equals

Evaluate lim_(n to oo) (2^(k) +4^k +6^k + ….. + (2n)^(k))/(n^(k+1)) by using the method of finding definite integral as the limit of a sum.

If P(A)=(6)/(11), P(B)=(5)/(11) and P(A cup B)=(7)/(11),find P(A cap B)

Three normals are drawn (k,0) to the parabola y(2)=8x one of the normal is the axis and the remaining two normals are perependicular to each other, then find the value of k.

The probability of a gun man hitting a target is 1/3, he fires ten shots, the random variable X is the number of hits. Find the mean and variance of X.

Let veca+i+2j+j, vecb=i-j+k, vecv=i+h-k. A vector in the plane of veca and vecb has projection 1/sqrt3" on "vecc. Then, one such vector is

Let E and F events with P(E) = (3)/(5), P(F) = (3)/(10) and P(Ecap F) = (1)/(5)*Are E and F independent ?

Find the number of possible common tangents that exist for the following pairs of circles. x^(2) + y^(2) + 4x - 6y - 3 = 0 x^(2) + y^(2) + 4x - 2y + 4 = 0.

Prove the following : 4/pi lt int_(pi/4)^(pi/3) (tanx)/(x) lt (3sqrt(3))/(pi)

Use differential to approximate sqrt(36.6).

If theta is a variable parameter, then the locus represented by x=3 (cot theta - tan theta) is-

Let A = {(x,y) : x^(2) + y^(2) = 36} and B={(x,y) : x^(2) + 9y^(2) = 100}, Then

The value of sum_(j=1)^(20)sum_(i=1)^(20)tan^(-1)((i)/(j)) is :

Integrate the functions 1/(cos(x+a)cos(x+b))

Prove that sum_(r=0)^(2n) r.(""^(2n)C_(r))^(2)= 2.""^(4n-1)C_(2n-1).

Let A={1,2,3} and Let R={(1,1),(2,2),(3,3),(1,2),(2,1),(2,3),(3,2)}. then ,R is

A black and red dice are rolled. Find the conditional probability of obtaining a sum8, given that the red die resulted in a number less than 4

Integrate the functions in exercise. (x-1)/(sqrt(x^(2)-1))dx

Locus of midpoint of chord of circle x^(2)+y^(2)=1 which subtend right angle at (2,-1) is

If the line 3x-2y + 6=0 meets X-axis and Y-axis respectively at A and B, then the equalion of the circle with radius AB and centre at A. is

Let L be the line belonging to the family of straight lines (a+2b) x+(a-3b)y+a-8b =0, a, b in R, which is the farthest from the point (2, 2). If L is concurrent with the lines x-2y+1=0 and 3x-4y+ lambda=0, then the value of lambda is

There are 8 buses running from Kota to Jaipur and 10 buses running from Jaipur to Delhi. In how many ways a person can travel from Kota to Delhi via Jaipur by bus?

Column-I

Construct truth tables for the following and indicate which of these are tautologiesprarrp^^q.

If [sin^(-1)x]^(2)-2[sin^(-1)x]+1le0 (where, [.] represents the greatest integral part of x), then

Find the area of the Delta le formed by the straight line 2x+y=2 and the coordinate axes using integration.

It is given that at x = 1, the function x^(4) – 62x^(2) + ax + 9 attains its maximum value, on the interval [0, 2]. Find the value of a.

Evaluate int _(-3//2) ^(2) f (x)dx, where f (x) is given by f _(x) = max _(-3//2le t le x) (|t -1| -|t|+ t+1)

Find the area of the ellipse (x^(2))/(a^(2))+ (y^(2))/(b^(2))=1by the method of integration.

If A= [(0,1),(1,0)], then A^(2) equal to______

How many of them are divisible by 2

Show that area of the parallelogram whose diagonals are given by vec(a) and vec(b) is (1)/(2).|bar(a)xx bar(b)|. Also, find the area of the parallelogram, whose diagonals are 2 bar(i)-bar(j)+bar(k) and bar(i)+3bar(j)-bar(k).

Evaluate the following integrals. int(1)/(x^(2)+x+1)dx

Let the sides of a parallelogram be U=a, U=b,V=a' and V=b', where U=lx+my+n, V=l'x+m'y+n'. Show that the equation of the diagonal through the point of intersection of U=a, V=a' and U=b, V=b' " is given by " |{:(U,V,1),(a,a',1),(b,b',1):}| =0.

If x^2 - 4x + 3 equal 0, then what is the value of (x - 2)^(2) ?

If the length of the tangent from (2,5) to the circle x^(2) + y^(2) - 5x +4y + k= 0 is sqrt(37)then find k.

The solution of (1 + y^(2)) dx = (Tan^(-1) y -x) dy is

Find the number of solutions of the equation x+y+z=6, where x, y, z in W

Solve the following inequation . (xx) log_(5x+4)x^2lelog_(5x+4)(2x+3)

Integrate the function x e^(-x)

Of all the closed cylindrical cans (right circular), of a given volume of 100 cubic centimetres, find the dimensions of the can which has the minimum surface area?

int_0^4sqrt(x^2+9)