Name: PaperBanao
Price: 1000 PKR

Your school name

1st Year Mathematics Second half Book Paper

1st Year · Mathematics · Punjab (PTB)

Name: ____________Roll No: ________Class/Section: ________Date: ________Total marks: 100Time: 3 hours

Q1. Multiple choice questions (Attempt all · 20 × 1 = 20)

1. e^-1
2. e^-1/2
3. e²
4. e³
The derivative of equals ......:
Period of 2 cos 3x is equal to:
3. 2
4. None of these
If x-a is factor of f(x) then f(x)=0, x=a is called:
1. Root
2. Factor
3. Polynomial
4. Degree
Angle in semi circle is:
1. 30^o
2. 45^o
3. 90^o
4. 60^o
Sin 196^o =......:
1. Sin 16^o
2. Cos 33^o
3. Cos 16^o
4. -Sin 16^o
The expansion of is valid if ......:
1. IxI<1
2. IxI< 1/3
3. IxI>1
4. None of these
......:
is equal to:
1. f(0)
2. f'(a)
3. f'(x)
4. f'(0)
Period of its:
2. 4
3. 2
4. 8
Synthetic division is a process of ......:
1. Addition
2. Multiplication
3. Subtraction
4. Division
A vector with magnitude 1 is called:
1. Null vector
2. Unit vector
3. Zero vector
4. Constant vector
=?
1. Tan
2. - tan
3. - tan
4. Cot
The inequality 4ⁿ>3ⁿ+2^n-1 is true for integral values of n if ......:
1. n>0
2. n>1
3. n>2
4. n<2
1. 40
2. 60
3. 80
4. 120
What is the derivative of with respect to x:
2. 2x-1
3. 1-2x
The trigonometric function are periodic so their curves repeat after a ......:
1. Fixed interval
2. Not fixed interval
3. Both a and b
4. None of these
x³-3x²+2x-6 has factor:
1. x-4
2. x-3
3. x+3
4. x+2
The angle between the vectors is:
1. 30^o
2. 40^o
3. 60^o
4. 90^o
:
1. Sin
2. Cot
3. - Cot
4. Tan

Q2. Short questions (Attempt any 8 of 12 · 8 × 2 = 16)

When the polynomial 4x⁴+2x³+kx²+13 is divided by x+1, the remainder is 16. Find the value of k.
Without using the table, find the value of: cos (-240^o).
Using the binomial theorem, show that 11ⁿ-10n leaves a remainder 1 when divided by 100 for all positive integers n.
Use factor theorem to find the values of a and b if -2 and 2 are the roots of the polynomial 2x³+4x²+ax+b.
If are the angles of a triangle ABC, then prove that:
Suppose you aim to lose weight by reducing your calories intake by 50 calories each week. If you start at 2500 calories, prove that after n weeks, your daily intake is 2500-50n calories.
Find the remainder without performing division when f(x)=x⁴+x³+x²+1 is divided by x+1.
Find distance between the following point: P(cos x, cos y), Q (sin x , sin y)
Expand the following upto 4 term, taking the values of x such that the expansion in the following case valid:
Use synthetic division to show that x is the zero of the polynomial and use the result to factorize the polynomial completely: x³-28x-48, x=-4
Find the values of sin 2, cos2 and tan2, when: .
Find the term independent of x in the following expansion:

Find the maximum and minimum values of the following function:
Find from principle, the derivatives of the following expressions w.r.t their respective independent variables: (3x-2)^-2
The value of an investment grows according to the formula for continuous compounding A(t) = P_oe^rt, where P_o is the initial principal, r is the annual interest rate, and t is the time rate in years. What happens to the value of the investment as .
Find the maximum and minimum values of the following function:
Find the derivative of the following function by definition: f(x)=c
The apparent brightness B(d) of a star decreases as the distance from Earth increases following the inverse square law , where L is the star's luminosity. Find the limit of the brightness as .
Determine whether the following functions are even, odd or neither odd nor even:
Find by definition, the derivatives w.r.t 'x' of the following function defined as:
A company weekly sales (in thousands) follow the function . What is the limit of S(t) as and what does it represent?
A giant Ferris wheel has a diameter of 60 feet. The lowest point of the wheel is located 6 feet above the ground. The wheel completes one full revolution every 80 seconds. Made an equation that represent the height (t) of a rider on the Ferris wheel at any given time t.
Find derivative of with respect to x.
Evaluate:

Let u=3i+2j-5k, v=i-5j-k and w=-4i-j+7k. Find the following: I3v+wI.
Find the moment about the point M(-2,4,6) of the force represented by , where coordinates of points A and B are (1,2,-3) and (3,-4,2) respectively.
Find the vector whose magnitude is 5 and is parallel to 3i+4j-k
Find a real number a so that the vectors u and v are perpendicular: u=ai+2aj-k, v=i+aj+3k
If u= xi + 2j + 3k, v=i+yj-3k and w=2i-3j represent the sides of a triangle. Find the values of x and y.
Find the magnitude of the vector v and write the direction cosines of v. v= - 6i+8j
If IaI=3, IbI=4 and Ia+bI=5. Find the angle between a and b.
Find a scalar a so the the vectors 2i+aj+5k and 3i+j+ak are orthogonal.
Find a unit vector perpendicular to the plane containing a and b. Also find sine of the angle between them. a=i+j+k , b= i - j - k
Show that components of a vector are the projection of that vector along i, j and k respectively.
Compute the cross product axb and bxa. Check your answer by showing that each a and b are perpendicular to axb and bxa. a=2i-2j+k, b= -i+j+3k
If Ia+bI=Ia-bI, then show that a and b are perpendicular.
Compute the cross product axb and bxa. Check your answer by showing that each a and b are perpendicular to axb and bxa. a=-4i+j-2k, b=2i+j+k

A retailer company has developed a polynomial regression model to predict weekly product demand: D(w)=w³-2w²+5w-4, where D(w) represent predicted demand (in units) and w is the week number. Use the demand is 22 units, calculated the prediction error.
An electric shop sells three types of appliances. Fans, Heaters and Ovens. The monthly sales quantities are 500 units of Fans, 300 unit of Heaters and 200 units of Ovens. The profit per unit for each appliance is Rs 500 for Fans, Rs. 400 for Heater and Rs 2,000 for ovens.Calculate the total monthly profit using vector operations.
Prove that:
Differentiate w.r.t 'x':
Use mathematical induction to prove that whenever n is non-negative integer.