Name: PaperBanao
Price: 1000 PKR

Your school name

1st Year Mathematics Full 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)

Multiplicative inverse of (1,0) is:
1. (-1,0)
2. (0,1)
3. (0,-1)
4. (1,0)
The improper fraction can be changed into proper fraction by:
1. Addition
2. Subtraction
3. Multiplication
4. Division
The (0,0) solution of the system a₁x+b₁y=0 and a₂x+b₂y=0 is called ......:
1. Consistent
2. Inconsistent
3. Non-trivial
4. Trivial
The expansion of is valid if ......:
1. IxI<1
2. IxI< 1/3
3. IxI>1
4. None of these
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
1. e^-1
2. e^-1/2
3. e²
4. e³
ⁿC_n-r is equal to:
1. ^rp_n
2. ^rC_n
3. ⁿp_r
4. ⁿC_r
Period of 2 cos 3x is equal to:
3. 2
4. None of these
If f: is a function such that rang , then f is called a/an ...... function:
1. Onto
2. Injective
3. Bijective
4. Into
The derivative of equals ......:
: (or sum of first n term):
If x-a is factor of f(x) then f(x)=0, x=a is called:
1. Root
2. Factor
3. Polynomial
4. Degree
To solve absolute value quadratic function we must illuminate roots which are:
1. Real
2. Imaginary
3. Extraneous
4. Double
If
1. -2-3
2. 2-3
3. -2+3
4. 2+3
Conditional equation 3x-1=0 is truly only is ......:
1. x=3
2. x=-3
For homogeneous linear equations, system AX=0 non-trivial solution is possible if IAI = ......:
1. Non-zero
2. Zero
3. Negative
4. Positive
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
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

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

Find the domain and range of .
Solve the following:
Without expansion show that: =0
Resolve into partial fractions.
If then find: z₁+z₂
Given f(x)=x³-2x²+4x-1 find: f(-2)
Solve the following:
If them compute A²B.
Resolve into partial fraction.
If then find the following and express the result into x+y form: z₁+z₂
Given f(x)=x³-2x²+4x-1 find: f(0)
Find the inverse of . Also find its domain and range.

Q3. Short questions (Attempt any 8 of 12 · 8 × 2 = 16)

If ⁿP_r = 120 and ⁿC_r=20, find r.
Using the binomial theorem, show that 11ⁿ-10n leaves a remainder 1 when divided by 100 for all positive integers n.
A sum of Rs. 10400 is paid off in 40 installment such that each installment is Rs. 10 more than the preceding installment. Calculate the value of the first installment.
Without using the table, find the value of: cos (-240^o).
When the polynomial 4x⁴+2x³+kx²+13 is divided by x+1, the remainder is 16. Find the value of k.
In how many ways can 5 couples sit around a round table if no two women are sitting together?
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.
Write down the n^th term of the following sequence:
If are the angles of a triangle ABC, then prove that:
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.
Write the following in factorial form: n(n-1)(n-2)...(n-r+1)
Expand the following upto 4 term, taking the values of x such that the expansion in the following case valid:

Q4. Short questions (Attempt any 9 of 13 · 9 × 2 = 18)

Find the maximum and minimum values of the following function:
Let u=3i+2j-5k, v=i-5j-k and w=-4i-j+7k. Find the following: I3v+wI.
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 from principle, the derivatives of the following expressions w.r.t their respective independent variables: (3x-2)^-2
Find the maximum and minimum values of the following function:
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.
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 .
Find the derivative of the following function by definition: f(x)=c
Determine whether the following functions are even, odd or neither odd nor even:
Find the vector whose magnitude is 5 and is parallel to 3i+4j-k
A company weekly sales (in thousands) follow the function . What is the limit of S(t) as and what does it represent?
Find by definition, the derivatives w.r.t 'x' of the following function defined as:
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.

Q5. Long questions (Attempt any 3 of 5 · 3 × 10 = 30)

Find the number of the diagonals of a 6-sided figure.
Resolve the following into partial fraction:
Use mathematical induction to prove that whenever n is non-negative integer.
Differentiate w.r.t 'x':
Prove that: