Soumya Garg:

Q1 integration of 3x*cos(x^2-10) ans= 3/2*sin(x^2-10)

Q2 integration of e^e ans=e^e*x
Q3 integration of sin(ax+b)cos(ax+b) ans= sin^2(ax+b)/2*a
Q4 solve the differential equation dy/dx=3y2-y
Akriti Patel
Q5 solve the differential equation xdy=ydx
Q6 differentiate y=(2x-1)(4x+3) ans=16x+2
Q7 If P=at^2-bt , find the rate of change of P with respect to t. Ans dP/dt= 2at-b
Q8 A Cubic curve has equation
f(x)= 2x^3- 7x^2+6x+1
The point P(2,1) lies on the curve
Find equation of tangent at P
Ans y=2x-3
Q9 f(x)= 4x^3-9x+2
Calculate f’(x) at x= -1 ans =3

Vedant Jindal
Q10:half of the area of the curve y=cosx with the x axis from x=0 to x=pi/2 ?
Ans ½ units

Q11:lim x tends to zero ((1-cosx)/x^2)?

Ans. ½

Q12:differentiation of 2x.sgn(150) wrt x

Ans. 2

Q13:integration of frac(x) wrt dx over the limits x=0 to x=1 , where frac (x) denotes fractional
part of x which is equal to x -[x]
And it's range is [0,1) and integration of x^n dx=x^n+1/n+1
Ans.½

Q14:For x belong to R tan inverse x is discontinuous at

A)0. B)pi/2. C)-pi/2. D)Always continous

VAIBHAV TYAGI

Q 15. Evaluate the limit as x approaches 2 for the function f(x) = (x^2 - 4)/(x - 2).
Ans: 4
Q 16. Evaluate the definite integral ∫(2x + 1)dx from x = 1 to x = 4
Ans: 13/2
Q 17: Evaluate the limit as x tends to 0
sin(3x)/x
Ans: 3
Q 18: If a rectangle has a length of 6 units and a width of 4 units, and the length is
decreasing at a rate of 2 units per minute, find the rate at which the area is changing?
Ans: -8 units^2/ minute
Q 19: A spherical balloon is being inflated. If the radius is increasing at a rate of 2 cm/s, find
the rate at which the volume is changing when the radius is 3 cm?
Ans: 72*pi
Q 20: Find the second derivative of e^2x at x=0.
Ans: 4

ANCHAL JAISWAL
1 𝑥
2 2
Q 21: 𝑓𝑖𝑛𝑑 ∫ ∫ (𝑥 + 𝑦 )dydx (ans=44/105)
0 0

2 2
Q 22: (𝑥 + 𝑦 + 1)𝑑𝑥 − 2𝑥𝑦𝑑𝑦 = 0 is exact differential equation or not? (ans=no)

2𝑥 −𝑥
Q 23: What is the integrating factor of 𝑥𝑑𝑦/𝑑𝑥 + (1 − 𝑥)𝑦 = 𝑒 : 𝑥 > 0 (ans=𝑥𝑒 )

2 2
Q 24: Find the general solution of 𝑑 𝑦/𝑑𝑥 + 𝑑𝑦/𝑑𝑥 − 2𝑦 = 0
𝑥 −2𝑥
(ans is 𝑦(𝑥) = 𝑐1𝑒 + 𝑐2𝑒 )

3 3
Q 25: What is the order and degree of the differential equation 𝑑 𝑦/𝑑𝑥 + 𝑠𝑖𝑛(𝑑𝑦/𝑑𝑥) = 0
(ans is order 3 and degree not defined)

2
𝑑𝑦 𝑑𝑦 −𝑥
Q 26: Solve the IVP: 2 +2 𝑑𝑥
+ 2𝑦 = 0 , y(0)=1,y’(0)=-1 (ans is y=𝑒 𝑐𝑜𝑠𝑥 )
𝑑𝑥

2
𝑑𝑦 2 −𝑥/2 𝑥 −𝑥/2
Q 27: Find the particular integral of (2 𝑑𝑥
+ 𝑦) = 4𝑒 (ans is PI= 2
𝑒 )

2 2 2 2
Q 28: Solve (𝑥 + 𝑦 )𝑑𝑥 + 2𝑥𝑦𝑑𝑦 = 0 (ans is 𝑥(𝑥 + 3𝑦 ) = 𝐶 )

2 1 1
Q 29: Find the Particular Integral of (𝐷 + 𝐷 − 2)𝑦 = 𝑥 (ans is PI =− 2
(𝑥 + 2
))

3 𝑥 𝑑
Q 30: Find the Particular Integral of (𝐷 + 2)(𝐷 − 1) 𝑦 = 𝑒 ; D= 𝑑𝑥
1 3 𝑥
(ans is PI= 18
𝑥𝑒 )

𝑡𝑎𝑛 3𝑥
Q 31:For what value of K, f(x)= { 𝑠𝑖𝑛 2𝑥
𝑤ℎ𝑒𝑛 𝑥 ≠ 0 𝑎𝑛𝑑 2𝐾 𝑤ℎ𝑒𝑛 𝑥 = 0} is continuous ∀𝑥ϵ𝑅.
(ans is K=¾)
 𝑙𝑜𝑔3(3𝑥+5) 𝑑𝑦 𝑙𝑜𝑔3(3𝑥+5) 3
Q 32: Find the derivative of 𝑒 with respect to x. (ans is 𝑑𝑥
=𝑒 (3𝑥+5)𝑙𝑜𝑔3
)

1
Q 33: solve 𝑦 = ∫ 2 2 𝑑𝑥 (ans is y=tanx-cotx+c)
𝑠𝑖𝑛 𝑥𝑐𝑜𝑠 𝑥

2 2 2 2
Q 34:Find the area enclosed by 𝑥 + 𝑦 = 𝑎 (ans is Π𝑎 )

Q 35:Find the minimum and maximum value of f(x)=sin2x+5 (ans is min=4 and max=6)

𝑚+𝑛 𝑚 𝑛
Q 36: If (𝑥 + 𝑦) =𝑥 𝑦 then find dy/dx. (ans is dy/dx=y/x)

2 2 2
Q 37: Evaluate ∫ 𝑦 𝑑𝑥 − 2𝑥 𝑑𝑦 along the parabola y=𝑥 from(0,0) to (2,4). (ans is 48/5)

2𝑠𝑖𝑛 𝑥−𝑠𝑖𝑛 2𝑥
Q 38: Evaluate lim 3 (ans is 1)
𝑥→𝑜 𝑥

1−𝑐𝑜𝑠 𝑚𝑥 2 2
Q 39: Evaluate lim 1−𝑐𝑜𝑠 𝑛𝑥
(ans is 𝑚 /𝑛 )
𝑥→0

𝑠𝑖𝑛 𝑥
Q 40: Evaluate lim (ans is 1)
𝑥+1 − 1−𝑥
𝑥→0

MEHAK GOYAL
1)If f'(x) = 2x - 3, what is f(x)?
a) x^2 - 3x + C
b) x^2 - 3x^2 + C
c) x^2 - 3x^3 + C
d) x^2 - 3x^4 + C
2)Evaluate the limit as x approaches 2 for (x^2 - 4)/(x - 2).
a) 0
b) 2
c) 4
d) Does not exist
3)What is the critical point of the function f(x) = 2x^3 - 6x^2 + 4x + 1?
a) x = 1
b) x = 2
c) x = 4
d) x = 3
4)What is the derivative of ln(x^2 + 1) with respect to x?
a) (2x)/(x^2 + 1)
b) (2x)/(x^2 - 1)
c) (2x)/(2x^2 + 1)
d) (2x)/(2x^2 - 1)
5)Find the area under the curve y = x^2 from x = 1 to x = 3.
a) 8
b) 10
c) 12
d) 14
6)If f(x) = e^(3x), what is the second derivative of f(x)?
a) 9e^(3x)
b) 6e^(3x)
c) 3e^(3x)
d) 2e^(3x)
7)Evaluate the integral ∫(3x^2 + 2x - 1) dx from x = 0 to x = 2.
a) 14
b) 22
c) 18
d) 26
8)Determine the absolute minimum value of the function f(x) = x^3 - 3x^2 + 2x + 1 on
the interval [0, 3].
a) -4
b) -1
c) 0
d) 2
9)If g(x) = 2x^3 - 6x^2 + 4x - 1, what is the inflection point of g(x)?
a) x = 0
b) x = 1
c) x = 2
d) x = 3
10)Calculate the derivative of cos(2x) with respect to x.
a) -2sin(2x)
b) -sin(2x)
c) 2cos(2x)
d) -2cos(2x)
11)Find the limit as x approaches 0 for (e^x - 1)/x.
a) 0
b) 1
c) e
d) Does not exist
12)Determine the area between the curves y = x^2 and y = 2x in the interval [0, 2].
a) 1
b) 2
c) 3
d) 4
13)If f(x) = ln(x) - x, what is the critical point of f(x)?
a) x = 0
b) x = 1
c) x = e
d) x = 2
14)Evaluate the integral ∫(2cos(x) + 3sin(x)) dx.
a) 2sin(x) - 3cos(x) + C
b) 2sin(x) + 3cos(x) + C
c) -2sin(x) - 3cos(x) + C
d) -2sin(x) + 3cos(x) + C
15)Find the maximum value of the function f(x) = x^2 - 4x + 5 in the interval [1, 3].
a) 5
b) 6
c) 7
d) 8
16)Calculate the limit as x approaches infinity for (3x^2 - 2x)/(x^2 + 1).
a) 1
b) 2
c) 3
d) Does not exist
17)If h(x) = √(x^2 + 1), what is the domain of h(x)?
a) x > 0
b) x < 0
c) All real numbers
d) x ≠ 0
18)Determine the absolute maximum value of the function f(x) = e^(-x) - x^2 on the
interval [-1, 1].
a) 1
b) 2
c) e
d) -e

CHETAN MITTAL
Q1) Which three positive numbers a ,b,c(a<b<c) have the same answer ,whether they are
multiplied or added?
Hence solve (ans=20)

Q2)Find integrating factor of given differential equation.

(ans=xsinx)

Q3)Find the area of the region bounded by the curve y^2=x and the lines x=1 ,x=4 and the
x-axis in the first quadrant. (ans=14/3)

Q4) Evaluate

Q5) Check whether the limit exist or not? (ans=limit exist)

DIA
Question: Find the derivative of the function f(x) = ln(x^2 + 1).
Answer: f'(x) = 2x / (x^2 + 1).

Question: Calculate the integral of the function g(x) = e^x * cos(x).

Answer: ∫g(x) dx = (e^x * sin(x) + e^x * cos(x)) + C.

Question: Determine the derivative of h(x) = (3x^2 - 2x + 1) / (x^2 + 1).

Answer: h'(x) = (6x^3 - 4x^2 + 2x) / (x^2 + 1)^2.

Question: Find the integral of the function k(x) = 1 / (x^3 + 1).

Answer: ∫k(x) dx = (1/3) * arctan((2x - 1) / √3) + C.

Question: Calculate the derivative of the function m(x) = x^2 * ln(x).

Answer: m'(x) = 2x * ln(x) + x.

Question: Determine the integral of the function n(x) = (cos(x))^2.

Answer: ∫n(x) dx = (x/2) + (1/4) * sin(2x) + C.

Question: Find the derivative of the function p(x) = e^(2x) * sin(x).

Answer: p'(x) = 2e^(2x) * sin(x) + e^(2x) * cos(x).

Question: Calculate the integral of the function q(x) = x * e^x.

Answer: ∫q(x) dx = (x - 1) * e^x + C.

Question: Determine the derivative of the function r(x) = √(x) * ln(x).

Answer: r'(x) = (1/2) * (ln(x) + 1) / √(x).

Question: Find the integral of the function s(x) = (4x + 3) / (x^2 + 1).
Answer: ∫s(x) dx = 2ln(x^2 + 1) + 3arctan(x) + C.

Question: Find the derivative of the function f(x) = x^2 * e^x * sin(x).
Answer: f'(x) = x^2 * e^x * (sin(x) + cos(x)) + 2x * e^x * sin(x).

Question: Calculate the integral of the function g(x) = x * ln(x).

Answer: ∫g(x) dx = x^2 * (ln(x) - 1/2) + C.

Question: Determine the derivative of h(x) = √(x^2 + 1) / (x^2 - 1).

Answer: h'(x) = (x / √(x^2 + 1) - 2x / (x^2 - 1)^2) / 2.

Question: Find the integral of the function k(x) = e^(2x) * cos(3x).

Answer: ∫k(x) dx = (1/13) * e^(2x) * (2cos(3x) + 3sin(3x)) + C.

Question: Calculate the derivative of the function m(x) = ln(x^2 + 1)^3.

Answer: m'(x) = 6x / (x^2 + 1) * 3 * (ln(x^2 + 1))^2.

Question: Determine the integral of the function n(x) = (x^2 + 1) / (x^3 + x).
Answer: ∫n(x) dx = ln|x| - ln|x + 1| + arctan(x) + C.
Question: Find the derivative of the function p(x) = cos(x)^2 * sin(x) * ln(x).
Answer: p'(x) = cos(x)^2 * ln(x) * (2sin(x) - sin(2x) / (x) + cos(x) * sin(x) / (x).

Question: Calculate the integral of the function q(x) = e^(x^2) * x.

Answer: ∫q(x) dx = (1/2) * √(π) * erf(x) + C.

Question: Determine the derivative of the function r(x) = (x^2 - 1) * e^x / (x^2 + 1).
Answer: r'(x) = (2x * e^x * (x^2 + 1) - (x^2 - 1) * e^x * 2x) / (x^2 + 1)^2.

Question: Find the integral of the function s(x) = x^3 * ln(x).

Answer: ∫s(x) dx = (1/4) * x^4 * ln(x) - (1/16) * x^4 + C.