11th Class Math Guess Paper 2026 (Punjab Boards)

Last Updated on June 20, 2026 by Ahsa.Pk

11th Class Math Guess Paper is up-to-date, and the most important questions are given according to Punjab boards. These guess papers for 2026 help you to get the highest marks in your papers. Punjab Board guess paper 2026 math is relevant to all chapters, and we have tried to put all necessary questions that help students to score more than seventy percent.

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11th Class Math Guess Paper

11th Class Math Important Short Questions

Short Questions # 2

Find the multiplicative inverse of the following complex number: $(- 4, 7)$
Find the multiplicative inverse of the following complex number: $(\sqrt{2}, - \sqrt{5})$
Separate into real and imaginary parts (write as a simple complex number). $\frac{2 - 7 i}{4 + 5 i}$
Separate into real and imaginary parts (write as a simple complex number). $\frac{(- 2 + 3 i)^{2}}{1 + i}$
Separate into real and imaginary parts (write as a simple complex number). $\frac{i}{1 + i}$
Prove that $\overline{z} = z$ if $z$ is real.
For $z \in C$ , show that: $z^{2} = z \cdot \overline{z}$
If $z_{1} = 2 + i, z_{2} = 3 - 2 i, z_{3} = 1 + 3 i$ then express $\frac{z_{1} z_{2}}{z_{2}}$ in the form of $a + i b$ .
If $z_{1} = 2 + 7 i$ and $z_{2} = - 5 + 3 i$ , then evaluate the following: $2 z_{1} - 4 z_{2}$ .
Show that $i^{n + 1} + i^{n + 2} + i^{n + 3} + i^{n + 4} = 0$ for $a i \in N$
Find the least positive value of $n$ n, if $(\frac{1 + i}{1 - i})^{2 n} = 1$
If $z = \frac{(1 + 2 i)^{2}}{2 - i}$ then evaluate $\overline{z}$ .
Find the real values of $x$ and $y$ in the following: $x + i y + 2 - 3 i = i (5 - i) (3 + 4 i)$
Find the real values of $x$ and $y$ in the following: $(x + i y) (1 - i) = (2 - 3 i) (- 5 + 5 i) (- i \frac{3}{5}) + 1$
Find the real values of $x$ and $y$ in the following: $\frac{x}{2 + i} + \frac{y}{3 - i} = 4 + 5 i$ 2+ix+3−iy=4+5i
Find the real values of $x$ and $y$ if:
Find the real values of $x$ and $y$ if: $(x + i y)^{2} = \frac{2 i - 3}{3 + i}$
If $z_{1} = 2 + 3 i$ z1=2+3i and $z_{2} = 1 - α$ z2=1−α , find the value of a such that $I m (z_{1} z_{2}) = 7$ Im(z1z2)=7
Find the square root of the following complex number: $- 7 - 24 i$ −7−24i
Find the square root of complex number $5 + 12 i$ 5+12i and also represent the square root on an Argand diagram.
Factorize the following: $a^{2} + 4 b^{2} + 1$
Factorize the following: $z^{2} - 2 i z - 1$
Factorize the following: $z^{2} + 6 z + 13$
Factorize the following polynomials into its linear factors: $z^{3} + 8$
Factorize the following polynomials into its linear factors: $z^{4} + 21 z^{2} - 100$
Solve the following complex quadratic equation by completing square method: $2 z^{2} - 3 z + 4 = 0$
Solve the following complex quadratic equation by completing square method: $z^{2} + 4 z + 13 = 0$
Solve the following equation: $2 z^{4} - 32 = 0$
Solve the following equation: $3 z^{5} - 243 z = 0$
Solve the following equation: $z^{3} - 5 z^{2} + z - 5 = 0$
Factorize the polynomial $P (z) = z^{2} + (i - 3) z - 3 i$
Factorize the polynomial $P (z) = z^{3} + (1 + i) z^{2} + i z$
Solve the equation $2 z^{2} - 12 z + 50 = 0$ by completing square method and hence express it as a product of its linear factors.
Find the three cube roots of: 8
Find the three cube roots of: $- 27$
Find the fourth roots of 16,81,625. Also show that their sum is zero in each case.
If $1, w, w^{2}$ are the cube roots of unity, show that $1 + ω^{n} + ω^{2 n} = 3$ where $n$ is a multiple of 3 respectively.
Prove that: $(x^{3} + y^{3}) = (x + y) (x + ω y) (x + ω^{2} y)$
Plot the following points: $(2, 75^{\circ})$
Plot the following points: $(- 3, 120^{\circ})$
Plot the following points: $(2, \frac{π}{6})$
Plot the following points: $(- \frac{5}{2}, \frac{π}{3})$
Plot the following points: $(- 3, - \frac{2 π}{3})$
Given that: (a) $f (x) = x^{2} - 1$ (b) $f (x) = \sqrt{2 x + 3}$ , find (i) $f (- 3)$ , (ii) $f (0)$
Given that: (a) $f (x) = x^{2} - 1$ (b) $f (x) = \sqrt{2 x + 3}$ find (i) $f (x - 2)$ (ii) $f (x^{2} + 3)$
Find $\frac{f (a + h) - f (a)}{h}$ and simplify where: $f (x) = 4 x + 7$
Find $\frac{f (a + h) - f (a)}{h}$ and simplify where: $f (x) = \sin x + 2$
Find $\frac{f (a + h) - f (a)}{h}$ and simplify where: $f (x) = x^{3} + x^{2} - 1$
Express the following: The area $A$ of a square as a function of its perimeter $P$
Express the following: The circumference $C$ of a circle as a function of its area $A$
Find the domain and the range of the function $g$ defined below: $g (x) = 5 - x$
Find the domain and the range of the function $g$ defined below: $g (x) = \sqrt{x + 2}$
Find the domain and the range of the function $g$ defined below: $g (x) = \frac{6 x + 7, x \leq - 2}{4 - 3 x, x > - 2}$
Find the domain and the range of the function $g$ defined below: $g (x) = \frac{x + 2}{3 - x}$
Given $f (x) = x^{3} - a x^{2} + b x + 1.$ If $f (2) = - 3$ and $f (- 1) = 0,$ find the value of $a$ and $b$
Consider the function $f (x) = 3 x - 5.$ . Determine the domain and range of $f (x)$
Consider the function $f (x) = 3 x - 5$ is the function $f$ one-to-one? justify your answer.
Let $f : R \to R$ be defined by $f (x) = \frac{2 x - 3}{x + 1}$ . Find the domain and range of $f (x)$
Let $f : R \to R$ be defined by $f (x) = \frac{2 x - 3}{x + 1}$ . Prove that $f (x)$ is one-to-one.
Given $f (x) = x^{3} - 2 x^{2} + 4 x - 1$ find: $f (\frac{1}{x}), x \neq 0$
Find the domain and range of $f (x) = \frac{x}{x^{2} - 4}$
Find the domain and range of $f (x) = \sqrt{x^{2} - 9}$
Show that the function $f (x) = x + 1,$ where the domain and co-domain are all real numbers, is bijective.
Find the point of intersection of the coordinate axes and the following linear function graphically: $y = - 5 x + 10$
Find the point(s) of intersection of the following function graphically: $f (x) = 2 x + 5, g (x) = - x + 5$
Find the point(s) of intersection of the following function graphically: $f (x) = 3 x - 2, g (x) = 10 - x$
Find the point(s) of intersection of the following function graphically: $f (x) = x - 1, g (x) = x^{2} - 4 x + 3$
Find the point(s) of intersection of the following function graphically: $f (x) = - 2 x - 1, g (x) = x^{2} - 4 x$
Graph the following function: $y = \sqrt{3 x}$
Graph the following function: $y = - \frac{1}{2} \sqrt{x} + 1$
Graph the following function: $y = \sqrt[3]{2 x + 1}$
Sketch and analyze: $y = - x^{2} - 2 x + 3$ Find the maximum and minimum value of the $f (x) = - 2 x^{2} + 4 x + 3$ by completing square.
Find the point of intersection of $y = 3 x + 2$ and $y = - x + 6$ graphically.
Solve: $x^{2} - 4 = 5$
Solve the following: $\frac{x}{x + 1} + \frac{x + 1}{x} = \frac{5}{2}; x \neq - 1, 0$
Solve the following: $\frac{1}{x + 1} + \frac{2}{x + 2} = \frac{7}{x + 5}; x \neq - 1, - 2, - 5$
Solve the following: $\frac{a}{a x - 1} + \frac{b}{b x - 1} = a + b; x \neq \frac{1}{a}, \frac{1}{b}$
Solve the following: $3 x^{2} + 15 x - 2 \sqrt{x^{2} + 5 x + 1} = 2$
Solve the following: $\sqrt{2 x + 8} + \sqrt{x + 5} = 7$
Solve the following: $\sqrt{3 x + 4} = 2 + \sqrt{2 x - 4}$
Solve the following: $\sqrt{x + 7} + \sqrt{x + 2} = \sqrt{6 x + 13}$
If $A = a_{i j}$ $3 \times 4,$ then show that, $I_{3} A = A$
If $A = [\begin{array}{ccc} 0 & - 1 & 2 \\ 3 & 2 & 1 \\ - 1 & 0 & 4 \end{array}], B = [\begin{array}{ccc} 2 & 1 & - 1 \\ 1 & 2 & 4 \\ - 1 & 2 & 1 \end{array}] .$ and $C = [\begin{array}{ccc} 1 & 0 & - 2 \\ - 1 & 5 & 0 \\ 3 & 4 & - 1 \end{array}],$ then find: $A - B$
If $A = [\begin{array}{cc} i & 2 i \\ 1 & - i \end{array}], B = [\begin{array}{cc} - i & 1 \\ 2 i & 1 \end{array}]$ and $C = [\begin{array}{cc} 2 i & 1 \\ - i & 1 \end{array}],$ then show that: $A (B + C) = A B + A C$
If $A$ A and $B$ B are square matrices of the same order, then explain why in general; $(A + B)^{2} \neq A^{2} + 2 A B + B^{2}$
If $A$ A and $B$ B are square matrices of the same order, then explain why in general; $(A - B)^{2} \neq A^{2} - 2 A B + B^{2}$
If $A = [\begin{array}{ccc} - 1 & 2 & 3 \\ 1 & 0 & 2 \\ - 3 & 5 & 3 \end{array}]$ then find $A + A^{t}, A - A^{t}, A A^{t}, A^{t} A$ and $(A^{t})^{t}$ .
Solve the matrix equation $A^{2} - 5 A + 4 I - X = 0$ if $A = [\begin{array}{ccc} 2 & 0 & 1 \\ 2 & 1 & 3 \\ 1 & - 1 & 0 \end{array}]$
If $A$ and $B$ are two matrices such that $A B = B$ and $B A = A$ show that $A^{2} + B^{2} = A + B .$
If $A = [\begin{array}{cccc} 1 & 0 & - 1 & 2 \\ 3 & 1 & 2 & 5 \\ 0 & - 2 & 1 & 6 \end{array}]$ and $B = [\begin{array}{cccc} 2 & - 1 & 3 & 1 \\ 1 & 3 & - 1 & 4 \\ 3 & 1 & 2 & - 1 \end{array}]$ then show that $(A + B)^{t} = A^{t} + B^{t}$
Find $A B$ and $B A$ if $A = [\begin{array}{ccc} 2 & 0 & 1 \\ 1 & 4 & 2 \\ 3 & 0 & 6 \end{array}]$ and $B = [\begin{array}{ccc} 1 & - 1 & 0 \\ 2 & 3 & - 1 \\ 1 & - 2 & 3 \end{array}]$
Evaluate the following determinant: $∣ \begin{array}{ccc} 1 & - 2 & - 4 \\ 3 & - 1 & - 3 \\ - 2 & 3 & 2 \end{array} ∣$
Evaluate the following determinant: $∣ \begin{array}{ccc} a + b & a - b & a \\ a & a + b & a - b \\ a - b & a & a + b \end{array} ∣$
Without expansion show that: $∣ \begin{array}{ccc} 7 & 8 & 9 \\ 5 & 6 & 7 \\ 2 & 3 & 4 \end{array} ∣ = 0$
Without expansion show that: $∣ \begin{array}{ccc} 2 & 2 & 0 \\ 2 & - 8 & 10 \\ - a & 0 & b \end{array} ∣ = 0$
Without expansion show that: $∣ \begin{array}{ccc} 0 & a & - c \\ c & - b & 0 \\ 2 & 1 & 3 x \end{array} ∣ = 0$
Without expansion show that: $∣ \begin{array}{ccc} 2 & 3 & 9 x \\ 3 & 5 & 15 x \\ b c & a & a^{2} \end{array} ∣ = ∣ \begin{array}{ccc} 1 & a^{2} & a^{3} \\ 1 & b^{2} & b^{3} \\ 1 & c^{2} & c^{3} \end{array} ∣$
Without expansion show that: $∣ \begin{array}{ccc} 1 & 2 & - 3 \\ 0 & - 5 & - 2 \\ - 2 & - 2 & 7 \end{array} ∣ = ∣ \begin{array}{ccc} - 5 & - 2 & 5 \\ - 3 & - 1 & 4 i \\ - 2 & - 1 & 2 \end{array} ∣$ then find: $A_{13}, A_{23}, A_{33}$ A1,A23,A33
Find the value of $x$ x if: $∣ \begin{array}{ccc} 1 & x - 1 & 3 \\ - 1 & x + 1 & 2 \\ 2 & - 3 & x \end{array} ∣ = 9$ .
Find the value of $x$ x if: $∣ \begin{array}{ccc} 1 & 1 & 1 \\ 2 & x & 2 \\ 3 & 6 & x \end{array} ∣ = 0$
Find $A A^{t}$ and $A^{t} A : A = [\begin{array}{ccc} - 3 & 2 & - 1 \\ 2 & 1 & 3 \end{array}]$ .
If $A$ is a square matrix of order 3, then show that $∣ k A ∣ = k^{3} ∣ A ∣$
Verify that $(A B)^{t} = B^{t} A^{t}$ if: $A = [\begin{array}{ccc} 1 & - 1 & 2 \\ 0 & - 3 & 1 \end{array}]$ and $B = [\begin{array}{cc} 1 & 1 \\ - 3 & - 2 \\ 0 & 1 \end{array}]$
Verify that $(A B)^{t} = B^{t} A^{t}$ if: $A = [\begin{array}{ccc} 1 & 2 \\ 1 & 4 \\ 2 & 1 \end{array}]$ and $B = [\begin{array}{ccc} 1 & - 3 \\ - 2 & 1 \end{array}]$
Evaluate the determinant if $A = [\begin{array}{ccc} 1 & - 2 & 3 \\ - 2 & 3 & 1 \\ 4 & - 3 & 2 \end{array}]$
Find the cofactor $A_{12}, A_{22}$ and $A_{32}$ of $A = [\begin{array}{ccc} 1 & - 2 & 3 \\ - 2 & 3 & 1 \\ 4 & - 3 & 2 \end{array}]$
Resolve $\frac{7 x + 25}{(x + 3) (x + 4)}$ into partial fractions.
Resolve $\frac{x^{2} + x - 1}{(x + 2)^{3}}$ into partial fraction.

Short Questions # 3

Express $\cos θ + \cos (3 θ) + \cos (5 θ) + \cos (7 θ)$ as a product.
Find the next four terms of the following sequence: 12, 16, 20,…
Write down the first three terms of the following sequence: $a_{n + 1} = 4 a_{n} - 7$ and $a_{1} = 3$
Write down the first three terms of the following sequence: $a_{1} = 1, a_{n + 1} = (3 a_{n} + 2)^{2}$
Write down the nth term of the following sequence: 1, 4, 9, …
Find the common difference and write the next two terms of the following sequence: 9, 16, 23,…
Write the first three terms of the following arithmetic sequence, with given information. $a_{1} = 2$ $d = 13$
Find $a_{n + 1}$ if $a_{n} = 4 + 3 n$
Is 301 a term of the A.P. 5, 11, 17,…?
Which term of the A.P. 3, 8, 13,… is 123?
The $7^{t h}$ and $21^{s t}$ terms of an A.P. are 37 and 107, respectively. Find the A.P. and its $100^{t h}$ term.
How many numbers of three digits are divisible by 7?
Find the $8^{t h}$ term form the end of the A.P 8, 11, 14,..,185.
If the $5^{t h}$ term of an A.P. is 13 and $17^{t h}$ term is 49, find $a_{n}$ and $a_{13}$ .
Find A.M between the given number: $2 + \sqrt{3}$ , $2 - \sqrt{3}$
If 6, 11, 16 are three A.Ms between a and b, find a and b.
The A.M of two numbers is 7 and their product is 45. Find the number.
Sum the series: $3 + 6 + 9 + \dots a_{20}$
Find $S_{n}$ for the following arithmetic series: $a_{1} = 40$ $n = 20$ , $d = - 3$
How many items of series: $96 + 93 + 90 + \dots$ amount to 1071.
Find the 6th term of the G.P: $- 6, - 3, - 3, \dots$
Find the $12^{t h}$ term of $1 + i + 2 i, - 2 + 2 i, \dots$
Find the eight term of a geometric sequence for which $a_{1} = - 3$ and $r = - 2$
Find $a_{n}$ a if $a_{4} = \frac{8}{27}, a_{7} = \frac{- 64}{729}$
Find G.M. between: $- 2 i$ and $8 i$
Insert three G.Ms. between 2 and $\frac{1}{2}$
Sum of n terms the series: $0.2 + 0.22 + 0.222 + \dots$
Find the $9^{t h}$ term of the following harmonic sequence: $\frac{1}{3}, \frac{1}{5}, \frac{1}{7}, \dots$
If 5 is the harmonic mean between 2 and b, find b.
If $a^{2}, b^{2}$ and $c^{2}$ are in A.P., show that $a + b, b + c$ and $c + a$ are in H.P.
Evaluate the following: $\frac{10!}{0! 8!}$
Write the following in factorial form: $n^{3} - n$
Write the following in factorial form: $n (n - 1) (n - 2) \dots (n - r + 1)$
Evaluate the following: $^{10} p_{5}$
Find the value of n when: $^{10} p_{5} = 504$
How many 4 digit number can be formed, with distinct digits, with each digit odd?
How many 5-digits multiples of 5 can be formed from the digits 2,3,5,7,9, when no digit is repeated.
In how may ways can 8 different books including 2 on English be arranged on a shelf in such a way that the English books are never together?
How many different 4-digit number can be formed out from the digits 1, 2, 3, 4, 5, 6, when no digit is repeated?
How many arrangements of the letters of the following word, taken all together can be made? PAKISTAN.
How many permutations of the letters of the word “BANANA” can be made. If B must be the first letter in each arrangement?
In how many different ways can the following persons sit around a round table? (a) 8 persons (b) 7 persons (c) 6 persons.
How many necklaces can be made from 10 beads of different colours?
If $^{3 n} C_{2} :^{n} C_{2} = 15 : 1$ 3, find n.
Find the value of n and r, when: $^{n} C_{r} = 56$ , $^{n} P_{r} = 336$
How many diagonals and triangles can be formed by joining the vertices of the polygon having 15 sides?
In how many ways can a cricket team of 11 players be selected out of 17 players? How many of them will include a particular player?
Find remainder and quotient by simplifying the following: $(5 x^{4} - 3 x^{3} + 2 x^{2} - 1) \div (x^{2} + 4)$ (5x4−3x3+2x2−1)÷(x2+4)
Use the remainder theorem to find the remainder when the first polynomial is divided by the second polynomial: $x^{2} + 5 x + 6$ , $x - 2$ x−2
Use the factor theorem to determine the first polynomial is a factor of the second polynomial: $x - 3$ x−3, $x^{4} - 3 x^{3} + x^{2} - x + 1$
Use synthetic division to show that $x$ is the zero of the polynomial and use the result to factorize the polynomial completely: $x^{3} - 7 x + 6$ x, $x = 2$
Use synthetic division to find the quotient and the remainder when the polynomial $x^{4} - 10 x^{2} - 2 x + 4$ is divided by $x + 3$ .
If $x + 1$ x+1 and $x - 2$ x−2 are factors of $x^{3} - p x^{2} + q x + 2$ Using synthetic division, find the values of p and q.
When the polynomial $4 x^{4} + 2 x^{3} + k x^{2} + 13$ is divided by $x + 1$ , the remainder is 16. Find the value of k.
Use factor theorem to find the values of p and q is $x + 1$ and $x - 2$ are the factors of the polynomial $x^{3} + p x^{2} + q x + 3$
Divide the cube polynomial $3 x^{3} - 10 x^{2} + 13 x - 6$ by the linear polynomial $x - 2$ . Also find the quotient and remainder.
Find the value of k if the polynomial $x^{3} + k x^{2} - 7 x + 6$ has a remainder -4, when divided by $x + 2$ x+2
Show that $x - 2$ is a factor of $f (x) = x^{3} - 7 x + 6$ without factorizing.
If $(x - 2)$ and $(x + 2)$ are factors of $x^{4} - 13 x^{2} + 36$ . Using synthetic division, find the other two factors.
A digital processing system has a transfer function with a numerator $B (z) = z^{2} - z - 2$ Use the factor theorem to find the zeros of the system.
Prove the following: $\sin (180^{\circ} + α) \sin (90^{\circ} - α) = - \sin α \cos α$
Prove the following: $\sin (810^{\circ}) \sin (630^{\circ}) + \cos (135^{\circ}) \sin (225^{\circ}) = - \frac{1}{2}$
Prove the following: $\tan (150^{\circ}) \cot (330^{\circ}) - 2 \sec (135^{\circ}) \csc (225^{\circ}) = - 3$
Prove the following: $\sin (210^{\circ}) + \cos (240^{\circ}) + \tan (225^{\circ}) + \cot (225^{\circ}) = 1$
If $α, β, γ$ are the angles of a triangle ABC, then prove that: $\sin (α + β) = \sin γ$
If $α, β, γ$ are the angles of a triangle ABC, then prove that: $\sec (\frac{α + β}{2}) = \csc \frac{γ}{2}$
If $α, β, γ$ are the angles of a triangle ABC, then prove that: $\tan (α + β) + \tan γ = 0$
Find distance between the following point: $P (\cos x, \cos y)$ , $Q (\sin x, \sin y)$
Prove that: $\sin (45^{\circ} + α) = \frac{1}{\sqrt{2}} (\sin α + \cos α)$
Prove that: $\sin (α + β) \sin (α - β) = \sin^{2} α - \sin^{2} β = \cos^{2} β - \cos^{2} α$
Without using tables, find the values of all trigonometric functions of $105^{\circ}$
Prove that: $\frac{\cos (11^{\circ}) + \sin (11^{\circ})}{\cos (11^{\circ}) - \sin (11^{\circ})} = \tan (56^{\circ})$
Find the values of $\sin (2 α)$ , $\cos (2 α)$ and $\tan (2 α)$ , when: $\sin α = \frac{3}{5}$ where $0 < α < \frac{π}{2}$
Prove that: $\frac{\sin θ + \sin (2 θ)}{1 + \cos θ + \cos (2 θ)} = \tan θ$
Show that: $\sin (2 θ) = \frac{2 \tan θ}{1 + \tan^{2} θ}$
Show that: $\cos (2 θ) = \frac{1 - \tan^{2} θ}{1 + \tan^{2} θ}$
Express the following product as sums or differences: $\cos (x + y) \sin (x - y)$
Express the following product as sums or differences: $\sin (12^{\circ}) \sin (46^{\circ})$
Express the following sums and differences as products: $\cos (12^{\circ}) + \cos (48^{\circ})$ cos(12∘)+cos(48∘)
Express the following sums and differences as products: $\sin (x + 30^{\circ}) + \sin (x - 30^{\circ})$
Prove without using table / calculator, that $\sin (19^{\circ}) \cos (11^{\circ}) + \sin (71^{\circ}) \sin (11^{\circ}) = \frac{1}{2}$
Express $\sin (5 x) + \sin (7 x)$ as a product.

Short Questions # 4

Determine whether the following functions are even, odd or neither odd nor even: $\sin^{2} x$
Determine whether the following functions are even, odd or neither odd nor even: $\tan x + \sec x$
Determine whether the following functions are even, odd or neither odd nor even: $\frac{1}{\csc^{3} x}$
Determine whether the following functions are even, odd or neither odd nor even: $\frac{\sin x + \sin 3 x}{\cos x + \cos 3 x}$
Find the periods of the following function: $\sin 5 x$
Find the periods of the following function: $\cot \frac{x}{2}$
Find the periods of the following function: $\csc (\frac{2 x}{5})$
Find the periods of the following function: $\frac{1}{2} \sin (\frac{3 x}{2} - \frac{π}{2})$
Find the maximum and minimum values of the following function: $\frac{1}{2} + \sin (5 x + π)$
Find the maximum and minimum values of the following function: $\frac{3}{2} + \cos (x - \frac{π}{4})$
Find the maximum and minimum values of the following function: $\frac{1}{10 - 2 \sin 3 x}$
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. Find the maximum height of the rider.
Find the limit of the following sequence if exists: $a_{n} = \frac{2 n + 3}{n + 1}$
Find the limit of the following sequence if exists: $b_{n} = \frac{2 n + 3}{n^{2} + 1}$
Evaluate the following limit by using theorems of limits: $\lim_{x \to 3} (2 x + 4)$
Evaluate the following limit by using theorems of limits: $\lim_{x \to 1} (3 x^{2} - 2 x + 4)$
Evaluate the following limit by using algebraic techniques: $\lim_{x \to - 1} \frac{x^{3} - x}{x + 1}$
Evaluate the following limit by using algebraic techniques: $\lim_{x \to 3} \frac{x^{2} - 5 x + 6}{x^{2} - 2 x - 3}$
Evaluate the following limit using algebraic techniques: $\lim_{x \to 1} \frac{x^{3} - 3 x^{2} + 3 x - 1}{x^{3} - x}$
Evaluate the following limit by using algebraic techniques: $\lim_{h \to 0} \frac{\sqrt{x + h} - \sqrt{x}}{h}$
Evaluate: $\lim_{x \to 3} \frac{x - 3}{\sqrt{x} - \sqrt{3}}$
limit by using algebraic techniques: $\lim_{x \to 2} (\sqrt{x + 2} - \sqrt{6 - x})$
Evaluate the following limit by using algebraic techniques: $\lim_{x \to a} \frac{x^{n} - a^{n}}{x^{m} - a^{m}}$
Evaluate: $\lim_{x \to 1} \frac{x^{2} - 1}{x^{2} - x}$
Evaluate: $\lim_{x \to 3} \frac{x - 3}{\sqrt{x} - \sqrt{3}}$
$\lim_{x \to + \infty} \frac{5 x^{4} - 10 x^{2} + 1}{- 3 x^{3} + 10 x^{2} + 50}$
Evaluate: $\lim_{x \to + \infty} \frac{5 x^{4} - 10 x^{2} + 1}{- 3 x^{3} + 10 x^{2} + 50}$
Evaluate: $\lim_{x \to - \infty} \frac{2 - 3 x}{\sqrt{3} + 4 x^{2}}$
Express the following limit in terms of e. $\lim_{n \to 0} (1 + 2 n)^{\frac{1}{n}}$
Evaluate: $\lim_{θ \to 0} \frac{\sin 7 θ}{θ}$
Evaluate: $\lim_{θ \to 0} \frac{1 - \cos θ}{θ}$
Determine the left hand limit and the right hand limit and then, find limit of the following function when $x \to c$ x→c . $f (x) = 2 x^{2} + x - 5$ , $c = 1$
Discuss the continuity of $f (x)$ at $x = c$ : $f (x) = {\begin{cases} 2 x + 5 & if x \leq 2 \\ 4 x + 1 & if x > 2 \end{cases}$
Discuss continuity of $f (x)$ at $x = 3$ , when $f (x) = {\begin{cases} x - 1 & , x < 3 \\ 2 x + 1 & , x \geq 3 \end{cases}$
Find by definition, the derivatives w.r.t ‘x’ of the following function defined as: $2 - \sqrt{x}$
Find by definition, the derivatives w.r.t ‘x’ of the following function defined as: $\frac{1}{\sqrt{x}}$
Find $\frac{d y}{d x}$ from the first principle and final gradient of the curve at the given point: $\sqrt{x + 2}$ at $x = 6$ .
Find from principle, the derivatives of the following expressions w.r.t their respective independent variables: $(3 x - 2)^{- 2}$
Find the gradient and equation of the tangent line to $y = 3 x^{2} - 4 x + 1$ at $x = 2$ .
Find the gradient of the curve $f (x) = 3 x^{2} + 2 x$ f(x)=3 at $x = 1$ .
The position of a car after t hours is given by: $s (t) = 2 t^{3} - 3 t^{2} + t$ (in kilometres). Find the instantaneous velocity at $t = 2$
A stone is thrown upwards and its height after t seconds is given by: $s (t) = - 16 t^{2} + 32 t + 10$ (in feet). Find the instantaneous velocity at $t = 1$ t=1 .
Find the gradient and an equation of tangent line to the graph of $f (x) = x^{2} - 2$ at the point $P (- 1, 1)$
Find the derivative of the following function by definition: $f (x) = c$
Calculate $\frac{d}{d x} (3 x^{\frac{4}{3}}) = 3 \frac{d}{d x} (x^{\frac{4}{3}})$ dxd(3x34)=3dxd(x34) .
Find the derivative of $y = \frac{3}{4} x^{4} + \frac{2}{3} x^{3} + \frac{1}{2} x^{2} + 2 x + 5$ w.r.t. x.
Find the derivative of $y = (x^{2} + 5) (x^{3} + 7)$ with respect to x.
Find derivative of $y = (2 \sqrt{x} + 2) (x - \sqrt{x})$ with respect to x.
Differentiate $\frac{2 x^{3} - 3 x^{2} + 5}{x^{2} + 1}$ with respect to x.
Let $\underline{u} = 3 \underline{i} + 2 \underline{j} - 5 \underline{k}, \underline{v} = \underline{i} - 5 \underline{j} - \underline{k}$ and $\underline{w} = - 4 \underline{i} - \underline{j} + 7 \underline{k}$ w. Find the following: $\underline{u} + 2 \underline{v} + \underline{w}$
Let $\underline{u} = 3 \underline{i} + 2 \underline{j} - 5 \underline{k}, \underline{v} = \underline{i} - 5 \underline{j} - \underline{k}$ and $\underline{w} = - 4 \underline{i} - \underline{j} + 7 \underline{k}$ w. Find the following: $∣ 3 \underline{v} + \underline{w} ∣$
Find the magnitude of the vector $\underline{v}$ v and write the direction cosines of $\underline{v}, \underline{v} = 3 \underline{i} - 2 \underline{j} + 6 \underline{k}$
Find t so that $∣ 2 \underline{i} + (t - 1) \underline{j} + t \underline{k} ∣ = \sqrt{13}$
Find a unit vector in the direction of $\underline{v} = - \underline{i} + 4 \underline{j} - 8 \underline{k}$
Find the vector whose magnitude is 5 and is parallel to $3 \underline{i} + 4 \underline{j} - \underline{k}$
If $\underline{u} = x \underline{i} + 2 \underline{j} + 3 \underline{k}, \underline{v} = \underline{i} + y \underline{j} - 3 \underline{k}$ and $\underline{w} = 2 \underline{i} - 3 \underline{j}$ represent the sides of a triangle. Find the values of x and y.
The position vectors of the points A,B,C and D are $\underline{u} = \underline{i} + 2 \underline{j} + \underline{k}, \underline{v} = 7 \underline{i} + 8 \underline{j} + 4 \underline{k}, \underline{w} = - \underline{i} + \underline{k}$ and $\underline{z} = \underline{i} + 2 \underline{j} + 2 \underline{k}$ respectively. Show that $\overline{A B}$ is parallel to $\overline{C D}$
Is the following triple can be the direction angles of a single vector? $45^{\circ}, 45^{\circ}, 60^{\circ}$
For the vectors, $\underline{u} = [1, - 2, 3], \underline{v} = [2, 1, 3]$ and $\underline{w} = [- 1, 4, 0]$ , find the following: $\underline{v} + \underline{w}$
Find the unit vectors of $\underline{u} = 2 \underline{i} + 5 \underline{j} - \underline{k}$
If $\underline{u} = 2 \underline{i} + 3 \underline{j} + \underline{k}, \underline{v} = 4 \underline{i} + 6 \underline{j} + 2 \underline{k}$ and $\underline{w} = - 6 \underline{i} - 9 \underline{j} - 3 \underline{k}$ then show that $\underline{u}, \underline{v}$ and $\underline{w}$ w are parallel to each other.
Find the cosines of the angle between $\underline{u}$ and $\underline{v}, \underline{u} = 2 \underline{i} + 3 \underline{j} + \underline{k}, \underline{v} = - \underline{i} + 2 \underline{j} + 2 \underline{k}$
If $\underline{a} + \underline{b} + \underline{c} = \underline{0}$ and $∣ \underline{a} ∣ = 3, ∣ \underline{b} ∣ = 5$ and $∣ \underline{c} ∣ = 7$ . Find the angle between $\underline{a}$ and $\underline{b}$
Calculate the projection of $\underline{a}$ along $\underline{b}$ and projection of $\underline{b}$ along $\underline{a}$ when: $\underline{a} = 2 \underline{i} + 3 \underline{j} - \underline{k}, \underline{b} = \underline{i} - 2 \underline{j} + 4 \underline{k}$
Find a real number a so that the vectors $\underline{u}$ and $\underline{v}$ v are perpendicular: $\underline{u} = a \underline{i} + 3 \underline{j} + \underline{k}, \underline{v} = \underline{i} - 2 \underline{j} + a \underline{k}$
Find the number z so that the triangle with vertices A(3,0, – 2), B(0,3,1) and C(1,1, z) is a right triangle with right angle at C.
If $\underline{u} = 3 \underline{i} - \underline{j} - 2 \underline{k}$ and $\underline{v} = \underline{i} + 2 \underline{j} - \underline{k}$ then find $\underline{u} \cdot \underline{v}$
Find a scalar a so the the vectors $2 \underline{i} + a \underline{j} + 5 \underline{k}$ and $3 \underline{i} + \underline{j} + a \underline{k}$ are orthogonal.
Find the angle between the vectors: $\underline{u} = 2 \underline{i} - \underline{j} + \underline{k}$ and $\underline{v} = - \underline{i} + \underline{j}$
The constant forces $2 \underline{i} + 5 \underline{j} + 6 \underline{k}$ 2i+5j+6k and $- \underline{i} - 2 \underline{j} - \underline{k}$ −i−2j−k act on a body displaced from the position $P (4, - 3, - 2)$ to $Q (6, 1, - 3)$ . Find the total work done.
Compute the cross product $\underline{a} \times \underline{b}$ and $\underline{b} \times \underline{a}$ . Check your answer by showing that each $\underline{a}$ and $\underline{b}$ are perpendicular to $\underline{a} \times \underline{b}$ and $\underline{b} \times \underline{a}, \underline{a} = 2 \underline{i} + \underline{j} - \underline{k}, \underline{b} = \underline{i} - \underline{j} + \underline{k}$
Find a unit vector perpendicular to the plane containing $\underline{a}$ and $\underline{b}$ . Also find sine of the angle between them. $\underline{a} = \underline{i} + 6 \underline{j} - 3 \underline{k}, \underline{b} = 2 \underline{i} + \underline{j} + 3 \underline{k}$
Find the area of the triangle, formed by the points P,Q and R. $P (2, 3, 5); Q (1, 2, 3); R (4, 1, 2)$
Find the area of the parallelogram, whose vertices are: $A (1, 1, 1); B (4, 2, 3); C (5, 6, 7); D (2, 5, 5)$
Which vectors, if any, are perpendicular or parallel $\underline{u} = 5 \underline{i} - \underline{j} + \underline{k}; \underline{v} = \underline{j} - 5 \underline{k}; \underline{w} = - 15 \underline{i} + 3 \underline{j} - 3 \underline{k}$
Use the definition of cross product, for any vectors $\underline{u}, \underline{v}, \underline{w}$ and scalar k, prove that: $\underline{u} \times (\underline{v} + \underline{w}) =$ $(\underline{u} \times \underline{v}) + (\underline{u} \times \underline{w})$
$\underline{u} = 2 \underline{i} - \underline{j} + k$ and $\underline{v} = 4 \underline{i} + 2 \underline{j} - k$ find by determinant formula: $\underline{u} \times \underline{u}$
Find the area of the parallelogram whose vertices are: $P (0, 0, 0), Q (- 1, 2, 4), R (2, - 1, 4)$ and $S (1, 1, 8)$
Find the moment about the point $M (- 2, 4, 6)$ of the force represented by $\overline{A B}$ , where coordinates of points A and B are (1,2,-3) and (3,-4,2) respectively.

Long Questions

Question NO.5

Find the square root of $13 - 20 \sqrt{3} i$ and represent it on an Argand diagram.
Find the real values of u and v if $\frac{u - 2}{2 + i} + \frac{v - 3}{2 - i} = 4 i$
If $z_{1} = 4 + 5 i$ and $z_{2} = α - 2 i$ find the real values of a such that $R e (z_{1} z_{2}) = 20$
Find the roots of $z^{4} + 7 z^{2} - 144 = 0$ z4+7z2−144=0 and hence express it as a product of linear factors.
Find a polynomial $P (z)$ of degree 4 with zeros $2 i, - 2 i, 1, - 1$ and satisfying $P (2) = 240$
Factorize the polynomial $P (z) = z^{3} - 3 z^{2} + z + 5$
Evaluate: ${(\frac{- 1 + \sqrt{- 3}}{2})}^{7} + {(\frac{- 1 - \sqrt{- 3}}{2})}^{7}$
Show that: $(1 - ω + ω^{2}) (1 - ω^{2} + ω^{4}) (1 - ω^{4} + ω^{8}) (1 - ω^{8} + ω^{16})$ … to 2n factors $= 2^{2 n}$
Prove that: ${(\frac{i + \sqrt{3}}{2})}^{8} + {(\frac{i - \sqrt{3}}{2})}^{8} = - 1$
If $ω$ is an imaginary cube root of unity, prove that $\frac{a + b ω^{2} + c ω}{a ω^{2} + b ω + c} = ω$
If $ω$ is a cube root of unity, prove that $\frac{a ω^{12} + b ω^{17} + c ω^{19}}{a ω^{14} + b ω^{22} + c ω^{30}} = ω$
If $z_{1}$ and $z_{2}$ are different complex numbers with $∣ z_{2} ∣ = 1$ , find $∣ \frac{z_{2} - z_{1}}{1 - z_{1} z_{2}} ∣$
An AC source supplies a voltage of $V = 120 (\cos \frac{π}{4} + i \sin \frac{π}{4})$ volts to a circuit with impedance $Z = \frac{1 + i \sqrt{3}}{2}$ ohms. Calculate the current in polar form.
An AC circuit has an impedance of $z = 3 - 6 i$ ohms and is connected to a voltage source of $V = 90 + 30 i$ volts. Find the current in both rectangular and polar forms.
Encrypt the word “Class” by adding the complex number encryption key $k = - 3 + 4 i$ . Then decrypt it back to the original word.
A stone falls from a height of $60 m$ 60m on the ground, the height $h$ after $x$ seconds is approximately given by $h (x) = 40 - 10 x^{2}$ what is the height of stone when: (a) $x = 1$ sec? (b) 1.5 sec (c) $x = 1.7$ sec.
Graph the square root function $y = 2 \sqrt{x} + 1$
Find the maximum and minimum value of the following quadratic function by completing squares: $f (x) = x^{2} + 6 x + 13$
Find the maximum and minimum value of the following quadratic function by completing squares: $f (x) = - x^{2} + 8 x + 13$
Find the maximum and minimum value of the following quadratic function by completing squares: $f (x) = - 2 x^{2} - x + 21$
Find the maximum and minimum point by sketching the following quadratic function. Also find their domain and range: $f (x) = - x^{2} + 2 x - 8$
Find the maximum and minimum point by sketching the following quadratic function. Also find their domain and range: $f (x) = x^{2} + 2 x - 8.3$
Find the inverse of the following quadratic function. Also find their domain and range: $f (x) = x^{2} - 3$ , $x \leq 0$
Find the inverse of the following quadratic function. Also find their domain and range: $f (x) = 2 x^{2} - 8 x + 11$ , $x \geq 2$
Find the inverse of the following quadratic function. Also find their domain and range: $f (x) = 3 x^{2} - 2 x + 6$ , $x \geq 5$
Find the inverse of the following quadratic function. Also find their domain and range: $f (x) = - 3 (x + 4)^{2} - 5$ , $x < - 4$
Solve the following absolute value quadratic equation and inequalities: $∣ x^{2} + 5 x + 4 ∣ = 0$
Solve the following absolute value quadratic equation and inequalities: $∣ 3 x^{2} - 7 x + 2 ∣ = x^{2} - x + 1$
Solve the following absolute value quadratic equation and inequalities: $∣ x^{2} - 5 x + 6 ∣ \leq x + 2$
Solve: $∣ x^{2} - 6 x - 4 ∣ < 3$

Question NO. 6

Using properties of determinants, show that: $∣ \begin{array}{ccc} a + x & a & a \\ a & a + x & a \\ a & a & a + x \end{array} ∣ = x^{2} (3 a + x)$
Using properties of determinants, show that: $∣ \begin{array}{ccc} a + 1 & b + 1 & c + 1 \\ (a + 1)^{2} & (b + 1)^{2} & (c + 1)^{2} \end{array} ∣ = (a - b) (b - c) (c - a)$
Using properties of determinants, show that: $∣ \begin{array}{cccc} a & b & c & c \\ b + c & c + a & a + b & a + b \\ a + b & b & b + c & c + a \\ a + t & a & a & a \\ b & b + t & b & b \\ c & c & c + t \end{array} ∣ = t^{2} (a + b + c + t) .$
Using properties of determinants, show that: $∣ \begin{array}{ccc} a - b - c & 2 a & 2 a \\ 2 b & b - c - a & 2 b \\ 2 c & 2 c & c - a - b \end{array} ∣ = (a + b + c)^{3}$
Using properties of determinants, show that: $∣ \begin{array}{ccc} y + z & z + x & x + y \\ x & y & z \\ x^{2} & y^{2} & z^{2} \end{array} ∣ = (x + y + z) (x - y) (y - z) (z - x) .$
Using properties of determinants, show that: $∣ \begin{array}{ccc} 1 & 1 & 1 \\ a^{2} + 1 & b^{2} + 1 & c^{2} + 1 \\ a^{3} + a & b^{3} + b & c^{3} + c \end{array} ∣ = (a - b) (b - c) (c - a) (a b + b c + c a - 1) .$
Using properties of determinants, show that: $∣ \begin{array}{ccc} 1 + a & 1 & 1 \\ 1 & 1 + b & 1 \\ 1 & 1 & 1 + c \end{array} ∣ = a b c + a b + b c + c a .$
Find the inverse of $A = [\begin{array}{ccc} 1 & 2 & 1 \\ - 5 & 0 & 4 \\ 5 & 4 & 0 \end{array}]$ ; and show that $A^{- 1} A = I_{3}$
Find $A^{- 1}$ if $A = [\begin{array}{ccc} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 1 & - 1 & 1 \end{array}]$ .
Solve the following systems of linear equation by Cramer’s rule: ${\begin{cases} 2 x + y - z = 1 \\ 3 x + 2 y + z = 4 \\ x_{1} + 2 x_{2} - 3 x_{3} = 0 \end{cases}}$ .
Solve the following systems of linear equation by Cramer’s rule: ${\begin{cases} 4 x_{1} - x_{2} + x_{3} = 5 \\ 2 x_{1} + 3 x_{2} + 2 x_{3} = 3 \end{cases}}$ .
Solve the following system of linear equation by matrix inversion method: ${\begin{cases} x - 2 y + z = 1 \\ y - z = 1 \\ x + y = 2 \end{cases}}$ .
Solve the following system of linear equation by matrix inversion method: ${\begin{cases} 2 x - z = 1 \\ y - 3 z = - 1 \end{cases}}$ .
Use matrix inversion method to solve the system: $x_{1} - 2 x_{2} + x_{3} = - 4, 2 x_{1} - 3 x_{2} + 2 x_{3} = - 6, 2 x_{1} + 2 x_{2} + x_{3} = 5$
Resolve the following into partial fraction: $\frac{2 x + 3}{(x + 1) (x + 2) (x + 3)}$
Resolve the following into partial fraction: $\frac{x^{2} + 4 x + 5}{(x + 1) (x^{2} + 5 x + 6)}$
Resolve the following into partial fraction: $\frac{x + 1}{(x - 1)^{2}}$ .
Resolve the following into partial fraction: $\frac{x^{2} + x}{(x^{2} - 1)^{2}}$
Resolve the following into partial fraction: $\frac{3 x^{2} + 4 x - 5}{(x - 1)^{3}}$
Resolve the following into partial fraction: $\frac{1}{x (x + 1)^{3}}$
Resolve $\frac{1}{(x + 1)^{2} (x^{2} - 1)}$ into partial fraction.
Resolve into partial fractions: $\frac{2 x^{2} + 3 x + 3}{(x + 1) (x^{2} + 1)}$
Resolve into partial fractions: $\frac{3 x^{2} + 3}{x^{3} + 1}$
A signal process system has a transfer function $H (z) = \frac{z^{2} + 3 z + 2}{z^{2} - 0.2 z + 0.9}$ Find zero(s) of the transfer function by using factor theorem.
A signal process system has a transfer function $H (z) = \frac{z^{2} - 0.5 z - 0.5}{z^{3} + 1}$ Find zero(s) of the transfer function by using factor theorem.
The denominator of signal processing system’s transfer function equals $A (z) = z^{2} + 1.2 z + 0.35$ Use factor theorem to determine the location of the corresponding poles and assess the stability of the system.

Question NO.7

If $\frac{1}{a - c}, \frac{1}{b - c}, \frac{1}{b - a}$ are in A.P, then show that $\frac{a - b}{a - c} = \frac{a - c}{b - a}$
If $\frac{1}{a}, \frac{1}{b}$ and $\frac{1}{c}$ are in A.P, show that $b = \frac{2 a c}{a + c}$
If $\frac{1}{a}, \frac{1}{b}$ and $\frac{1}{c}$ are A.P., show that the common difference is (equation).
If $a_{k}$ and $a_{m}$ , denotes two different terms of an A.P., show that its nth term is $a_{k} + (n - k) (\frac{a_{k} + a_{m}}{k - m})$
Insert five A.Ms. between $\sqrt{2}$ and $\frac{15}{\sqrt{2}}$
For what value of n, $\frac{a^{n + 1} + b^{n + 1}}{a^{n} + b^{n}}$ is the A.M between a and b, where $a \neq b$
If $\frac{1}{a + b}, \frac{1}{c + a}, \frac{1}{b + c}$ a+b1,c+a1,b+c1 are in A.P. then show that $a^{2}, b^{2}, c^{2}$ a2,b2,c2 are in A.P.
If $\frac{1}{a}, \frac{1}{b}$ and $\frac{1}{c}$ are in G.P. Show that the common ratio is $\pm \sqrt{\frac{a}{c}}$
For what value of $\frac{a^{n} + b^{n}}{a^{n - 1} + b^{n - 1}}$ is the positive geometric mean between a and b?
The A.M of two positive integral numbers exceeds their (positive) G.M. by 2 and their sum is 20, find the numbers.
If the numbers $\frac{1}{k}, \frac{1}{2 k + 1}$ and $\frac{1}{4 k - 1}$ are in harmonic sequence, find k.
Find n so that $\frac{a^{n + 1} + b^{n + 1}}{a^{n} + b^{n}}$ may be H.M between a and b.
If $\frac{b + c - a}{a}, \frac{c + a - b}{b}, \frac{a + b - c}{c}$ are in A.P., show that a, b, c are in H.P.
If between any two numbers there are inserted two A.Ms $A_{1}, A_{2}$ , two G.Ms. $G_{1}, G_{2}$ and two H.Ms. $H_{1}, H_{2}$ show that $\frac{A_{1} + A_{2}}{G_{1} G_{2}} = \frac{H_{1} + H_{2}}{H_{1} H_{2}}$
If the 4th and $7^{t h}$ term of the H.P are $\frac{2}{13}$ and $\frac{2}{25}$ respectively, find the sequence.
Sum the following series upto n terms: $1 \times 3 \times 5 + 2 \times 4 \times 6 + 3 \times 5 \times 7 + \dots$
Sum the following series upto n terms: $2^{2} + 4^{2} + 6^{2} + \dots$
Sum the series: $1^{2} - 2^{2} + 3^{2} - 4^{2} + \dots + (2 n - 1)^{2} - (2 n)^{2}$
Sum the series: $\frac{1^{2}}{1} + \frac{1^{2} + 2^{2}}{2} + \frac{1^{2} + 2^{2} + 3^{2}}{3} + \dots$ to n term.
Find the sum to n term of the series whose $n^{t h}$ term are given: $n^{2} + 2 n - 3$
Given $n^{t h}$ terms of the series, find the sum to 2n terms: $3 n^{2} + 5 n + 2$
Express as a single fraction: $\frac{(n + 2)!}{(r + 2)!} + \frac{(n + 1)!}{(r + 1)!}$
Prove from the first principle that: $^{n} P_{r} = n \cdot^{n - 1} P_{r - 1}$
Prove from the first principle that: $^{n} P_{r} =^{n - 1} P_{r} + r \cdot^{n - 1} P_{r - 1}$
From a standard deck of 52 playing cards, there are 26 black cards and 26 red cards. How many different ways can eight cards be selected if 3 are black and the remaining 5 are red?

Question NO. 8

Prove that: $\frac{\sin θ - \cos θ \tan \frac{θ}{2}}{\cos θ + \sin θ \tan \frac{θ}{2}} = \tan \frac{θ}{2}$
Prove that: $\frac{1 - \tan θ \tan ϕ}{1 + \tan θ \tan ϕ} = \frac{\cos (θ + ϕ)}{\cos (θ - ϕ)}$
Show that $\cos (α + β) \cos (α - β) = \cos^{2} α - \sin^{2} β = \cos^{2} β - \sin^{2} α$
Show that $\frac{\tan α + \tan β}{\tan α - \tan β} = \frac{\sin (α + β)}{\sin (α - β)}$
Show that: $\sin (α + β) = \frac{1 + \cot α \tan β}{\csc α \sec β}$
Show that: $\cot (α + β) = \frac{\cot α \cot β - 1}{\cot α + \cot β}$
If $\sin α = \frac{24}{25}$ and $\cos β = \frac{20}{29}$ where $0 < α < \frac{π}{2}$ and $0 < β < \frac{π}{2}$ show that $\sin (α - β) = \frac{333}{725}$
Prove that: $\frac{\cos 19^{\circ} + \sin 19^{\circ}}{\cos 19^{\circ} - \sin 19^{\circ}} = \tan 64^{\circ}$
Prove that: $\cos (60^{\circ} + θ) \cos (60^{\circ} - θ) + \sin (60^{\circ} + θ) \sin (60^{\circ} - θ) = \cos 2 θ$
If $α, β, γ$ are the angles of a triangle ABC, show that: $\cot \frac{α}{2} + \cot \frac{β}{2} + \cot \frac{γ}{2} = \cot \frac{α}{2} \cot \frac{β}{2} \cot \frac{γ}{2}$
If $α + β + γ = 180$ , show that $\cot α \cot β + \cot β \cot γ + \cot γ \cot α = 1$
If $α, β, γ$ are the angles of $△ A B C$ Prove that: $\tan α + \tan β + \tan γ = \tan α \tan β \tan γ$
If $α, β, γ$ are the angles of $△ A B C$ △ABC Prove that: $\tan \frac{α}{2} \tan \frac{β}{2} + \tan \frac{β}{2} \tan \frac{γ}{2} + \tan \frac{γ}{2} \tan \frac{α}{2} = 1$
Prove the following: $\cot α - \tan α = 2 \cot 2 α$
Prove the following: $\frac{1 - \cos α}{\sin α} = \tan \frac{α}{2}$
Prove the following: $\frac{\cos α - \sin α}{\cos α + \sin α} = \sec 2 α - \tan 2 α$
Prove the following: $\sqrt{\frac{1 + \sin α}{1 - \sin α}} = \frac{\sin \frac{α}{2} + \cos \frac{α}{2}}{\sin \frac{α}{2} - \cos \frac{α}{2}}$
Prove the following: $\frac{\csc θ + 2 \csc θ}{\sec θ} = \cot \frac{θ}{2}$
Prove the following: $\frac{3 + \cos 4 θ}{1 - \cos 4 θ} = \frac{1}{2} (\tan 2 θ + \cot 2 θ)$
Prove the following: $\frac{1 + \sin 2 θ}{1 - \sin 2 θ} = \tan^{2} (\frac{π}{4} + θ)$
Prove the following: $\cos^{2} \frac{π}{8} + \cos^{2} \frac{3 π}{8} + \cos^{2} \frac{5 π}{8} + \cos^{2} \frac{7 π}{8} = 2$
Show that: $2 \cos θ = \sqrt{2 + \sqrt{2 + 2 \cos 4 θ}}$
Prove the following identity: $\frac{\sin 8 x - \sin 2 x}{\cos 8 x + \cos 2 x} = \tan 5 x$
Prove the following identity: $\frac{\sin 80^{\circ} + \sin 40^{\circ}}{\cos 80^{\circ} + \cos 40^{\circ}} = \sqrt{3}$
Prove that: $\sin \frac{π}{9} \sin \frac{2 π}{9} \sin \frac{π}{3} \sin \frac{4 π}{9} = \frac{3}{16}$
Prove that: $\sin 10^{\circ} \sin 30^{\circ} \sin 50^{\circ} \sin 70^{\circ} = \frac{1}{16}$
Show that $\cos 20^{\circ} \cos 40^{\circ} \cos 80^{\circ} = \frac{1}{8}$
Evaluate the following limit: $\lim_{x \to \frac{π}{4}} \frac{\sin x - \cos x}{x - \frac{π}{4}}$
Evaluate the following limit: $\lim_{x \to 0} \frac{\cos a x - \cos b x}{x^{2}}$
Evaluate the following limit: $\lim_{x \to 0} \frac{\cos a x - \cos b x}{\cos c x - \cos d x}$
Express the following limit in term of e: $\lim_{n \to \infty} (1 + \frac{1}{3 n})^{n}$
Express the following limit in term of e: $\lim_{x \to \infty} (\frac{x}{1 + x})^{x}$
Express the following limit in term of e: $\lim_{x \to 0} \frac{e^{\frac{1}{x}} - 1}{e^{\frac{1}{x}} + 1}, x < 0$
If $f (x) = {\begin{cases} 3 x & if x \leq - 2 \\ x^{2} - 1 & if - 2 < x < 2 \\ 3 & if x \geq 2 \end{cases}$ discuss continuity at $x = 2$ and $x = - 2$
Find the values of m and n, so that given function f is continuous at $x = 3$ x=3 . $f (x) =$ f(x)= (equation missing in OCR)
Determine whether $\lim_{x \to 2} f (x)$ and $\lim_{x \to 4} f (x)$ exist, when $f (x) = {\begin{cases} 2 x + 1 & if 0 \leq x \leq 2 \\ 7 - x & if 2 < x < 4 \\ x & if 4 \leq x \leq 6 \end{cases}$
$f (x) = {\begin{cases} \frac{\sqrt{2 x + 5} - \sqrt{x + 7}}{x - 2}, & x \neq 2 \\ k, & x = 2 \end{cases}$ find value of k so that f is continuous at $x = 2$
Discuss the continuity of the function $f (x)$ and $g (x)$ at $x = 3$ $f (x) = {\begin{cases} \frac{x^{2} - 9}{x - 3} & if x \neq 3 \\ 6 & if x = 3 \end{cases}$

Question NO. 9

A particle moves along a line such that its position after t hours is given by $s (t) = 4 t^{2} + 2 t + 1$ (in miles). Find the instantaneous velocity at $t = 3$
Find the derivative of $\sqrt{x}$ at $x = a$ from first principle.
If $y = \frac{1}{x^{2}}$ then find $\frac{d y}{d x}$ at $x = - 1$ by ab-initio method.
Differentiate w.r.t ‘x’: $x^{- 3} + 2 x^{- 3} + 3$ x
Differentiate w.r.t ‘x’: $\frac{(1 + \sqrt{x}) (x - x^{2})}{\sqrt{x}}$
Differentiate w.r.t ‘x’: $(\sqrt{x} - \frac{1}{\sqrt{x}})^{2}$
Differentiate w.r.t ‘x’: $\frac{(x^{2} + 1)^{2}}{x^{2} - 1}$
Differentiate w.r.t ‘x’: $\frac{2 x - 1}{\sqrt{x^{2} + 1}}$
Find $\frac{d y}{d x}$ if $y = \frac{(\sqrt{x} + 1) (x^{2} - 1)}{x^{2}} .$ $(x \neq 1)$
Differentiate $\frac{(\sqrt{x} + 1) (x^{2} - 1)}{x^{2} - x^{2}}$ with respect to x.
If $y = \sqrt{x} - \frac{1}{\sqrt{x}}$ show that $2 x \frac{d y}{d x} + y = 2 \sqrt{x}$
If $y = x^{4} + 2 x^{2} + 2$ prove that $\frac{d y}{d x} = 4 x \sqrt{y - 1}$
Find the direction cosines for the given vector: $\underline{v} = 4 \underline{i} + 2 \underline{j} - 5 \underline{k}$ v
Find the direction cosines for the given vector: $\vec{P Q}$ where $P (9, 3, 13)$ and $Q (11, 6, 19)$
Find the work done, if the point at which the constant force $\underline{F} = 2 \underline{i} + 5 \underline{j} + 3 \underline{k}$ is applied to an object, moves it from $P_{1} (2, - 3, ?)$ to $P_{2} (7, 5, 3)$
A force of magnitude 6 units acting parallel to $4 \underline{i} + 3 \underline{j} - \underline{k}$ displaces the point of application from $A (2, - 1, 3)$ to $B (7, 3, 2)$ . Find the work done.
Show that the vectors $\vec{A B} = 2 \underline{i} - \underline{j} + \underline{k}, \vec{B C} = \underline{i} - 3 \underline{j} - 5 \underline{k}$ and $\vec{A C} = 3 \underline{i} - 4 \underline{j} - 5 \underline{k}$ are the sides of a right triangle.
Prove that: $\underline{a} \times (\underline{b} + \underline{c}) + \underline{b} \times (\underline{c} + \underline{a}) + \underline{c} \times (\underline{a} + \underline{b}) = \underline{0}$
If $\underline{a} + \underline{b} + \underline{c} = \underline{0}$ then prove that $\underline{a} \times \underline{b} = \underline{b} \times \underline{c} = \underline{c} \times \underline{a}$
Find the moment about the point $M (1, - 3, 3)$ of the force represented by $\vec{A B}$ . where the coordinates of points $A (4, 3, - 1)$ and $B (- 1, 3, 7)$ are given.
A force $\vec{F} = 6 \underline{i} + 4 \underline{j} - 4 \underline{k}$ is applied at the point $A (1, - 1, 2)$ . Find the moment of the force about the point $B (3, - 2, 3)$
Given a force $\underline{F} = 2 \underline{i} + \underline{j} - 3 \underline{k}$ acting at a point $A (1, - 2, 1)$ Find the moment of $\vec{F}$ about the point $B (2, 0, 2)$
A force $\underline{F} = - 2 \underline{i} + \underline{k} - 3 \underline{k}$ is applied at $P (- 1, - 3, 2)$ . Find its moment about the point $Q (4, 2, 2)$
If $\underline{a} = 4 \underline{i} + 3 \underline{j} + \underline{k}$ and $\underline{b} = 2 \underline{i} - \underline{j} + 2 \underline{k}$ Find a unit vector perpendicular to both a and b. Also find the sine of the angle between the vectors a and b.
In any triangle ABC, prove that $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$