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Let f(x) = 4x^(3) + 1. If the area of the region bounded by f(x) and x-axis from x = 0 to x = a is same as the area of region bounded by f(x) and x-axis from x = a to x = 3 then a lies in (a > 0)

Differential equation x^(5)(dy)/(dx) = - y^(5)

Number of ................

If f:{x|xge1,"x" inR}rarr{x|xge2,x inR} f(x) =x+1/(x)then f^(-1)(x) = .........

Find underset(-pi//2)overset(pi//2)int sin^(2)x cos^(4)x dx

Construct a 2xx3 matrix having element:a_(ji)=i/j

If 2veca, 3vecb,2(veca xx vecb) are position vectors of the vectors A,B,C, of triangleABC and |veca|=|vecb|=1,vec(OA).vec(OB)=-3 (where O is the origin), then

If a tangent to the curve y = 6x -''x^(2) is parallel to the line 4x - 2y - 1 =0, then the point of tangency on the curve is

Fill int the blanks choosing correct answer from the bracket. If a cosB = b cosA, then cosB = _____.

If truth-values of statements p andq are F and T respectively. Then the truth-value of

Evaluate the following integrals intsqrt(x)logxdx

sin 12^(@)sin 48^(@)sin 54^(@)=

Using principle of mathematical induction, prove that 7^(4^(n)) -1 is divisible by 2^(2n+3) for any naturalnumber n.

Differentiate sin x^(2) + sin^(2)x + sin^(2) (x^(2))

If Rgergt0 and dgt0, then 0lt(d^(2)+R^(2)-r^(2))/(2dR)le1

The shortest distance between the lines 2x+y+z-1=0=3x+y+2z-2 and x=y=z, is

y= sin x- cos x and f(x)= (d^(17)y)/(dx^(17)) " then " f((pi)/(4))= ……..[a]√2[b]1/√2[c](√2)^17[d]0

Using properties evaluate the following definite integrals, evaluate the following: int_0^1 x(1-x)^n dx

Find the area of the circle4x^(2) + 4y^(2) = 9 which is interior of the parabola x^(2) = 4y.

Find the number of ways in which an examiner can assign 30 marks to 10 questions in a question paper. (fractional marks are not allowed)

Find the values of the following correct to five decimals.sqrt(3.96)

If the line ax +by+c = 0 where a, b, cinR and a, b,c notin 0 is a normal to the curve y=x^(2), then x^(3)+2a^(2)x+4a^(2)c = 0has

If a, b, c are distinct real numbers and P, Q, R are three points whose position vectors are respectively ahati + b hatj+ c hatk , b hati + c hatJ + a hatk and c hai + a hatj + b hatk, then angle QPR =

Find the correct pair from the following(i)In a system of equations if Delta ne 0, Delta_(x)ne 0, Delta_(y)ne 0, Delta_(z)ne 0 then it has unique solution.(ii) Interchange of rows and columns is an elementary operation.(iii) (AB)^(-1)=B^(-1)A^(-1)(iv)(B)^(T)=A^(T)B^(T)

Two dice are thrown simultaneously. If X denotes the number of sixes, find the expectation of X.

Find the direction cosines of the line (x-2)/(2)=(2y-5)/(-3),z=-1. Also find the vector equation of the line.

By using the properties of definite integrals, evaluate the integrals int_(0)^(pi/2)2cos^(2)xdx

Which of the following sentences are propositions and which are not ? Write with reason : x^2-x+1=0

If x_(n) + iy_(n) = (1 + i)^(n) then x_(n - 1) y_(n) - x_(n) y_(n- 1) =

A certain sample of cuprous sulphide is found to have composition Cu_(1.8) S, because of presence of some Cu_(2+) ions in the lattice. What is the mole % of Cu_(2+) in crystal ?

Which of the following curve is not symmetrical about both the axis?

There are 10 points in a plane no three of which are collinear except 4 of them which lie on a line. The number of straight lines determined by them is

Examine the continuity of the following functions at indicated points.f(x)={(sinfrac{1}{x}if xnea),(0 if x=0 atx=0):}

A telegraphic wire suspended between two poles of height 15 mts is in the shape of a parabola. The distance between the poles is 20 mts and maximum sag of the cable wire is 4mt , then the height of the cable at a distance of 5 mt from one end is

Assertion (A), The equation x^(2)+ 2|x|+3=0 has no real root.Reason (R): In a quadratic equation ax^(2)+bx+c=0, a,b,c in R discriminant is less than zero then the equation has no real root.

Find the eccentricity, length of a latus rectum, equations ofthe latus rectum of the hyperbola(x^(2))/(16)-(y^(2))/(9) =1.

The value of lim_(xto0^(+))(-1+sqrt((tanx-sinx)+sqrt((tanx-sinx)+sqrt((tanx-sinx)+…oo))))/(-1+sqrt(x^(3)+sqrt(x^(3)+sqrt(x^(3)+…oo)))) is

A person goes 2 km east, then 3 km north, then 4km west and then 1 km north, starting from the origin. This point is taken as vector vec(A). The vector vec(B) such that 3vec(A)+5vec(B)=(9,32), is

Find the number of distinct terms in the expansion of (x+(2)/(x)+1)^(20)

Correct statement about I and II

I : The number of solutions of the equation z^(2) + |z|^(2) = 0 is 2 II : The number of solutions of z^(2) + |z| = 0 is 3 Which of the statements are true

Ifxisnumericallysosmall sothatx ^ 2andhigherpowersofxcanbeneglected,then(1 +(2x )/(3)) ^(3//2). (32 +5x) ^(-1//5)isapproximatelyequalto

1 mole of a real gas changes it state from state-A(2bar, 3L, 100 K) to state -B (2bar, 5L, 200 K) at constant pressure and finally to state-C (3bar, 10 L, 300 K). If DeltaU_(BC) = 110 J and C_(Pm) ofgas =3R = 3 xx 8.3 JK^(-1)mol^(-1) then thoose the correct option(s) :

Evaluate int_(0)^(pi/2) x sin x dx.

Two intersecting circles have their radii 1 metre and sqrt(3) metre. The distance between their centres is 2 metre Then the overlapping area (in square metre ) is-

For f(x)=4x^(3)+3x^(2)-x-1, the range of vaues of (f(x_(1))-f(x_(2)))/(x_(1)-x_(2))is

Two lines of regressions are represented by 4x + 10y = 9 and 6x + 3y = 4. Find the line of regression y on x.