1 Crore+ students have signed up on EduRev. Have you? 
If f(x) = then what is the value of _{0}∫^{π/2} f(x) dx = (π/4 + 8/15)?
(dy/dx) = (dx/dy)^{1}
So, d^{2}y/dx^{2} = (dx/dy)^{2} d/dx(dx/dy)
= (dy/dx)^{2}(d^{2}x/dy^{2})(dy/dx)
= d^{2}y/dx^{2} + (dy/dx)^{3} d^{2}y/dx^{2} = 0
Here, we have,
Now, replacing C_{3} = C_{3} – 3C_{1}, we get,
Or, (x – 1)
Now, replacing R_{1} = R_{1} + 3R_{3}, we get,
Or, (x – 1)
Or, (x – 1)[1 {(11 – x)(5 – x) – 28}] = 0
Or, (x – 1)(55 – 11x – 5x + x^{2} – 28)
Or, (x – 1)(x^{2} – 16x + 27) = 0
Thus, either x – 1 = 0 i.e. x = 1 or x^{2} – 16x + 27 = 0
Therefore, solving x^{2} – 16x + 27 = 0 further, we get,
x = 8 ± √37
Find the area of the triangle with the vertices (2,3), (4,1), (5,0).
The area of the triangle with vertices (2,3), (4,1), (5,0) is given by
Applying R_{2}→R_{2}R_{3}
Expanding along R_{2}, we get
Δ=(1/2){(1)(30)+1(25)}
Δ=(1/2) (00)=0.
For which of the elements in the determinant Δ= the cofactor is 37.
Consider the element 3 in Δ=
The cofactor of the element 3 is given by
A_{22}=(1)^{2+2} M_{22}
M_{22}= =1(5)(6)(7)=542=37
A_{22}=(1)^{2+2} (37)=37.
Given that, A=
A =
A=cosθ (cosθ )cotθ(tanθ)
A=cos^{2}θ+1=sin^{2}θ.
The above matrix is a skew symmetric matrix and its order is odd
And we know that for any skew symmetric matrix with odd order has determinant = 0
Therefore, the value of the given determinant = 0
What is the relation between the two determinants f(x) = and g(x) =
Let, D =
Expanding D by the 1^{st} row we get,
D = – c
= – c(0 – ab) + b(ac – 0)
= 2abc
Now, we have adjoint of D = D’
Or, D’ =
Or, D’ = D^{2}
Or, D’ = D^{2} = (2abc)^{2}
Find the equation of the line joining A(5,1), B(4,0) using determinants.
Let C(x,y) be a point on the line AB. Thus, the points A(5,1), B(4,0), C(x,y) are collinear. Hence, the area of the triangle formed by these points will be 0.
⇒ Δ = (1/2)
Applying R_{1}→R_{1}R_{2}
Expanding along R_{1}, we get
=(1/2) {1(0y)1(4x)}=0
=(1/2){y4+x}=0
⇒ xy = 4.
For which of the following elements in the determinant Δ= the minor of the element is 2?
Consider the element 7 in the determinant Δ=
The minor of the element 7 can be obtained by deleting R_{2} and C_{2}
∴ M_{22} = 2
Hence, the minor of the element 7 is 2.
Expanding along R_{1}, we get
Δ=5(2424)0+5(80)
Δ=00+40=40.
Here, C_{1} and C_{3} becomes equal when we put p = x^{n}
And R_{1} and R_{3} becomes equal when we put p = n + 1
And R1 and R3 becomes equal when we put p = n + 1
Find the value of k for which the points (3, 2), (1, 2), (5, k) are collinear.
Given that the vertices are (3,2), (1,2), (5,k)
Therefore, the area of the triangle with vertices (3,2), (1,2), (5,k) is given by
Δ=(1/2)
Applying R_{1}→R_{1}R_{2}, we get
1/2
Expanding along R_{1}, we get
(1/2) {2(2k)0+0} = 0
2k = 0
k = 2 .
For which of the following element in the determinant Δ= the minor and the cofactor both are zero.
Consider the element 2 in the determinant Δ=
The minor of the element 2 is given by
∴ M_{22} = = 4040 = 0
⇒ A_{22} = (1)^{2+2} (0) = 0.
Which of the following conditions holds true for a system of equations to be consistent?
If a given system of equations has one or more solutions then the system is said to be consistent.
Differentiate (log2x)^{sin3x} with respect to x.
Consider y=(log2x)^{sin3x}
Applying log on both sides, we get
logy=log(log2x)^{sin3x}
logy=sin3x log(log2x)
Differentiating with respect to x, we get
By using chain rule, we get
dy/dx =y(3 cos3x log(log2x)+
∴ dy/dx=log2x^{sin3x}(3cos3xlog(log2x)+(sin3x/xlog2x))
Find the second order derivative of y=9 log t^{3}.
Given that, y=9 logt^{3}
To solve:y=(8e^{x}+2e^{x})
Differentiating w.r.t x we get,
(dy/dx)= 8(e^{x}+2e^{x})
∴ (dy/dx)= 2e^{x}  8e^{x}.
If the rate of change of radius of a circle is 6 cm/s then find the rate of change of area of the circle when r=2 cm.
The rate of change of radius of the circle is dr/dt = 6 cm/s
The area of a circle is A=πr^{2}
Differentiating w.r.t t we get,
(dA/dt = d/dt) (πr^{2}) = 2πr (dr/dt) =2πr(6)=12πr.
dA/dt _{r=2}=24π= 24×3.14=75.36 cm^{2}/s
A given systems of equations is said to be inconsistent if _____
If a given system of equations has no solutions, then the system is said to be inconsistent.
Consider
Applying log on both sides, we get
logy=log
logy=log4+ (∵logab =loga+logb)
Differentiating both sides with respect to x, we get
Given that, y=tan^{2}x+3 tanx
dy/dx =2 tanx sec^{2}x+3 sec^{2}x=sec^{2}x (2 tanx+3)
By using the u.v rule, we get
(sec^{2}x).(2 tanx+3)+ (d/dx) (2 tanx+3).sec^{2}x
=2 sec^{2}x tanx (2 tanx+3)+sec^{2}x (2 secx tanx)
=2 sec^{2}x tanx (2 tanx+secx+3).
Consider y=8e^{cos2x}
Differentiating w.r.t x by using chain rule, we get
The edge of a cube is increasing at a rate of 7 cm/s. Find the rate of change of area of the cube when x=6 cm.
Let the edge of the cube be x. The rate of change of edge of the cube is given by dx/dt =7cm/s.
The area of the cube is A=6x^{2}
Find the value of x and y for the given system of equations.
3x+2y=6
5x+y=2
By using the matrix method, the given equations can be expressed in the form of the equation AX=B, where
To find the value of x and y, we need to solve the matrix X
X = A^{1} B
Consider y=9^{tan3x}
Applying log on both sides, we get
logy=log9^{tan3x}
Differentiating both sides with respect to x, we get
(∵ Using u.v = u′v + uv′)
(dy/dx) = y(3 sec^{2}x.log9+0)
(dy/dx) = 9^{tan3x} (3 log9 sec^{2}x)
Consider y=3 sin^{1}(e^{2x})
dy/dx = d/dx (3 sin^{1}(e^{2x}))
The rate of change of area of a square is 40 cm^{2}/s. What will be the rate of change of side if the side is 10 cm.
Let the side of the square be x.
A = x^{2}, where A is the area of the square
Given that,
Consider y=(cos3x)^{3x}
Applying log on both sides, we get
logy=log(cos3x)^{3x}
logy=3x log(cos3x)
Differentiating both sides with respect to x, we get
By using u.v=u’ v+uv’, we get
(3x)log(cos3x)+(d/dx)(log(cos3x)).3x
(dy/dx)=y(3 log(cos3x) +(cos3x).3x)
(dy/dx)=y(3 log(cos3x) +
(dy/dx)=y(3 log(cos3x) +
(dy/dx) = y(3 log(cos3x) – 9x tan3x)
(dy/dx) = (cos3x)^{3x} (3 log(cos3x) – 9x tan3x)
Find the second order derivative of y=2e^{2x}3 log(2x3).
Given that, y=2e^{2x}3 log(2x3)
Consider y=(log(log(x^{5})))
The total cost P(x) in rupees associated with a product is given by P(x)=0.4x2+2x10. Find the marginal cost if the no. of units produced is 5.
The Marginal cost is the rate of change of revenue w.r.t the no. of units produced, we get
= 0.8x+2
cost(MC)= =0.8x+2=0.8(5)+2=4+2=6.
Find the value of x, y, z for the given system of equations.
2x+3y+2z=50
x+4y+3z=40
3x+3y+5z=60
The given system of equations can be expressed in the form of AX=B, where
X = A^{1} B
∴ A^{1} = (1/A) adj A
X = A^{1} B
Consider y=
logy=log
logy=log7+
logy=log7+2e^{2x} logx
Differentiating with respect to x on both sides, we get
(logx)2e^{2x} (using u.v=u’ v+uv’)
Given that, y=2 sin^{1}(cosx)
Consider
At what rate will the lateral surface area of the cylinder increase if the radius is increasing at the rate of 2 cm/s when the radius is 5 cm and height is 10 cm?
Let r be the radius and h be the height of the cylinder. Then, dr/dt =2 cm/s
The area of the cylinder is given by A=2πrh
Consider
Applying log on both sides, we get
Differentiating with respect to x, we get
Given that, y=log(2x^{3})
Consider tanx
Differentiating w.r.t x by using chain rule, we get
If the circumference of the circle is changing at the rate of 5 cm/s then what will be rate of change of area of the circle if the radius is 6cm.
The circumference of the circle is given by C=2πr, where r is the radius of the circle.
Use Code STAYHOME200 and get INR 200 additional OFF

Use Coupon Code 








