1st Year Mathematics First half Book Paper

Q1. Multiple choice questions (Attempt all · 64 × 0.75 = 48)

1. The sum of the series - 3 + ( - 1 ) + 1 + 3 + 5 + .............. + $a_{16}$ is :
   1. 290
   2. 190
   3. 192
   4. None of these
2. What is the position of the particle from mean position when amplitude is 0.0004mm and angle is $\frac{\pi}{3}$
   1. $\frac{1}{500}(1 +\sqrt{3i})$
   2. $\frac{1}{5000}(1 +\sqrt{3i})$
   3. $\frac{1}{5000}(\sqrt{3} + i)$
   4. $\frac{1}{500}(\sqrt{3} + i)$
3. For the matrices $A = \begin{bmatrix} 2 & 3 \\ -5 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} 4 & -1 \\ 1 & 3 \end{bmatrix}$, the value of matrix X in the matrix equations 2X - A = B will be :
   1. $\begin{bmatrix} 12 & 4 \\ -8 & 8 \end{bmatrix}$
   2. $\begin{bmatrix} 6 & 2 \\ -4 & 4 \end{bmatrix}$
   3. $\begin{bmatrix} 3 & 1 \\ -2 & 2 \end{bmatrix}$
   4. $\begin{bmatrix} 9 & 3 \\ -3 & 3 \end{bmatrix}$
4. What should be added to $3pq + 5p^2q^2 + p^3$ to get $p^3 + 2p^2q^2 + 4pq$?
   1. $$p^{2}q^{2} - pq$$
   2. $$2p^{2}q^{2} + pq$$
   3. $$-3p^{2}q^{2} + pq$$
   4. $$5p^{2}q^{2} + pq$$
5. The projection of $\underline{i} - 2\underline{j} + a\underline{k}$ along $2\underline{i} - 3\underline{j} + 2\sqrt{3}\underline{k}$ is 4 then $\alpha$ is :
   1. $\sqrt{3}$
   2. $2\sqrt{3}$
   3. -$\sqrt{3}$
   4. $3\sqrt{3}$
6. The sum of first 100 positive integers :
   1. 5050
   2. 5051
   3. 5052
   4. None of these
7. Part of complex number which is along y-axis is called
   1. Real part
   2. Imaginary part
   3. Complex part
   4. Both a and c
8. The matrix in reduced echelon form is :
   1. $$\begin{bmatrix} 0 & 1 & -\frac{1}{2} & 0 & 6 \\ 1 & 0 & \frac{2}{3} & 4 & -5 \\ 0 & 0 & 0 & 1 & 0.4 \\ 0 & 1 & 1 & 0 & 0 \end{bmatrix}$$
   2. $$\begin{bmatrix} 1 & 3 & -6 & 10 & 0 \\ 0 & 0 & 1 & 4 & -3 \\ 0 & 0 & 0 & 1 & \frac{1}{2} \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}$$
   3. $$\begin{bmatrix} 1 & 3 & 0 & 0 & 0\\ 0 & 0 & 1 & 4 & \frac{1}{2} \\ 0 & 0 & 1 & 0 & -3 \\ 0 & 0 & 0 & 0 & 0\end{bmatrix}$$
   4. $$\begin{bmatrix} 1 & 3 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & -3 \\ 0 & 0 & 0 & 1 & \frac{1}{2} \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}$$
9. $ax + 2xy + 3$ is a ______ polynomial :
   1. Linear
   2. Cubic
   3. Constant
   4. Quadratic
10. The line segment joining the midpoints of the diagonals of a trapezium is parallel to the parallel sides and its length is half the ____________ lengths of the parallel sides.
    1. Sum
    2. Difference
    3. Product
    4. Quotient
11. How many multiples of 4 between 14 and 523 is :
    1. 126
    2. 127
    3. 128
    4. 227
12. Algebraic form of $z = \sqrt{2} (\cos 315^\circ + i \sin 315^\circ)$ is
    1. $1 + i$
    2. $1 - i$
    3. $i$
    4. $2 + i$
13. The principal diagonal / main diagonal elements in the matrix $A=\begin{bmatrix} 1 & -3 & 1 \\ 2 & 4 & -1 \\ 3 & 5 & 6 \end{bmatrix}$ are:
    1. 1 , - 3 , 1
    2. 1 , 4 , 3
    3. 1 , 4 , 6
    4. 3 , 5 , 6
14. If 3x³ - 2x² + 5 is divided by x + 1, then x + 1 will be its.
    1. divisor as well as factor
    2. dividend
    3. quotient
    4. remainder
15. If $\underline{a} = \underline{i} - 2\underline{k}$ and $\underline{b} = 3\underline{i} - 2\underline{j}$, then $\underline{a} \times \underline{b} = ?$
    1. $2\left( \underline{i} - 3\underline{j} +\underline{k}\right)$
    2. $-2\left(2 \underline{i}+ 3\underline{j} +\underline{k}\right)$
    3. $2\left( \underline{i} + 3\underline{j} -\underline{k}\right)$
    4. $2\underline{i} + 3\underline{j} +4\underline{k}$
16. $\sum_{k=1}^{10} {3}$is equal to:
    1. 10
    2. 103
    3. 300
    4. 30
17. If $z = ( 3 + 4i ) ( 2 - 3i)$ then $|{z}|$
    1. $7\sqrt{2}$
    2. $5\sqrt{13}$
    3. $13\sqrt{5}$
    4. $2\sqrt{7}$
18. If $A = \begin{bmatrix} 1 & 2 \\ 3 & - 3 \end{bmatrix}$ and $B= \begin{bmatrix} - 1 \\ 6 \end{bmatrix}$ , given that AX =B , then X =
    1. $\begin{bmatrix} 1 \\ 1 \end{bmatrix}$
    2. $\begin{bmatrix} -1 \\ 1 \end{bmatrix}$
    3. $\begin{bmatrix} 1 \\ -1 \end{bmatrix}$
    4. $\begin{bmatrix} -1 \\ -1 \end{bmatrix}$
19. Use of letters to represent an unknown quantity was introduced by :
    1. Muhammad Bin Musa
    2. Rene Descartes
    3. Euler
    4. James Sylvester
20. If $\overrightarrow{AB}=\overrightarrow{CD}$, then coordinates of point A is when B ( 1 , 2 ) , C ( - 2 , 5 ) and D ( 4 , 11 ) are.
    1. A ( 5 , 4 )
    2. A ( - 5 , 4 )
    3. A ( - 5 , - 4 )
    4. A ( 5 , - 4 )
21. Which term of the sequence - 6 , - 2 , 2 , ............ is 7 0 ?
    1. 19 th
    2. 20 th
    3. 21 th
    4. None of these
22. Write $z = - 2 - 2i$ in polar form.
    1. $$2\sqrt{2} \left( \cos\frac{3\pi}{4} + i \sin\frac{3\pi}{4} \right)$$
    2. $$2\sqrt{2} \left( \cos\frac{3\pi}{4} - i \sin\frac{3\pi}{4} \right)$$
    3. $$2\sqrt{2} \left( -\cos\frac{3\pi}{4} + i \sin\frac{3\pi}{4} \right)$$
    4. $$2\sqrt{2} \left( -\cos\frac{3\pi}{4} - i \sin\frac{3\pi}{4} \right)$$
23. If A and B are two matrices and O ( A + B ) = $2 \times3$ , then O ( B A ) =
    1. $2 \times 2$
    2. $3\times 3$
    3. $3\times 2$
    4. Not possible
24. $2x + 7$ is a factor of $2x^3 + 5x^2 - 11x - 14$, then factorize form of the expression will be:
    1. $$(2x-7)(x-2)(x+1)$$
    2. $$(2x+ 7)(x-2)(x+1)$$
    3. $$(2x-7)(x-2)(x-1)$$
    4. None of these
25. A vector having magnitude one is called :
    1. Null vector
    2. Unit vector
    3. Position vector
    4. None of these
26. Predict the general term for the sequence.
    1. $$\frac{4}{3^{n - 2} }$$
    2. $$\frac{4}{3^{n - 1} }$$
    3. $$\frac{4}{3^{n } }$$
    4. $$\frac{4}{3^{n+ 1} }$$
27. If $z₁= 2 - 3i$ and [ z₁z₂] = 13 then value of $[-\bar{z₂} ]$ is
    1. 13
    2. $\sqrt{13}$
    3. $\frac{\sqrt{13}}{13}$
    4. $\frac{1}{13}$
28. For a system of non-homogeneous equations with three variables system will have unique solution if.
    1. Rank A < 3
    2. Rank $A_{b}$ < 3
    3. Rank A = Rank $A_{b}$ = 3
    4. Rank A = Rank $A_{b}$ < 3
29. The volume of a cuboid is $105{x}^7$ with length and height as $5x$ and respectively , then breadth is :
    1. $3x$
    2. $3x^2$
    3. $21x$
    4. $21x^2$
30. The vector from the point A to the origin where $\overline{AB} =4\underline{i} - 2\underline{j}$ and B is the point ( - 2 , 5 ) is
    1. ( 6 , 7 )
    2. ( - 6 , 7 )
    3. ( 6 , - 7 )
    4. ( - 6 , - 7 )
31. Next two terms of geometric sequence 4 , 12 and 36 .
    1. 108 and 324
    2. 106 and 322
    3. 324 and 108
    4. None of these
32. The argument θ of a complex number that lies in the interval $-\pi < \theta \leq \pi$ is called
    1. Principal argument
    2. Limited argument
    3. Minimum argument
    4. Maximum argument
33. The number of non - zero rows of a matrix in the echelon form is called :
    1. Adjoint of a matrix
    2. Inverse of a matrix
    3. Rank of the matrix
    4. Transpose of a matrix
34. If 2 , 5 , - 2 are zero of a polynomial then polynomial is :
    1. $$x^3 - 5x - 6x$$
    2. $$x^3 - 5x^2 + 2x - 2$$
    3. $$x^3 - 5x^2 - 4x + 20$$
    4. None of these
35. Which of the following is Lagrange , identity ?
    1. $$|a \times b|^2 = |a|^2 |b|^2 - a \cdot b$$
    2. $$|a \times b|^2 = |a|^2 |b|^2 + (a \cdot b)^2$$
    3. $$|a \times b|^2 = |a|^2 |b|^2 - (a \cdot b)^2$$
    4. $$|a \times b|^2 = |a|^2 |b|^2 + a \cdot b$$
36. How many terms of the sequence 18, 15, 12, ......... are needed to give a sum of 45?
    1. up to 7th
    2. up to 10th
    3. up to 6th
    4. up to 5th
37. For any two complex numbers z₁, z₂ which of the following can be true?
    1. z₁> z₂
    2. z₁ < z₂
    3. $\bar{z₁}>\bar{z₂}$
    4. [ z₁] > [ z₂]
38. The matrix $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 4 \end{bmatrix}$ is a:
    1. Symmetric matrix
    2. Skew- symmetric matrix
    3. Identity matrix
    4. Null matrix
39. What should be subtracted to the polynomial $x^{2} - 16x + 30$, so that $x - 15$ is the factor of the resulting polynomial?
    1. 30
    2. -15
    3. 15
    4. 16
40. Which of the following is a vector quantity:
    1. Distance
    2. Volume
    3. Time
    4. Weight
41. A sequence in which any term is divided by its immediate previous term gives a constant is called :
    1. Arithmetic sequence
    2. Geometric sequence
    3. Complex sequence
    4. Real sequence
42. $\text{Im} \left( \frac{x_1 + y_1 i}{x_2 + y_2 i} \right)^{-1} =$
    1. $$\frac{x_1 y_2 - x_2 y_1}{x_1^2 + y_1^2} \,$$
    2. $$\frac{x_1 y_2 - x_2 y_1}{x_1^2 + y_1^2} \, i$$
    3. $$\frac{x_1 y_2 - x_2 y_1}{x_2^2 + y_2^2} \, i$$
    4. $$\frac{x_1 y_2 - x_2 y_1}{x_2^2 + y_2^2} \,$$
43. For the matrix , $A = \begin{bmatrix} 2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7 \end{bmatrix}, \, M_{13} + A_{22} =$
    1. 30
    2. -19
    3. -49
    4. 11
44. Algebraic expressions are of :
    1. One kind
    2. Two kinds
    3. Three kinds
    4. Four kinds
45. If $\underline{a} = \underline{b}$ then $\underline{a} \cdot \underline{b}$ is equal to
    1. $|\underline{a}|$
    2. $|\underline{a}|^2$
    3. $2|\underline{a}|$
    4. $\frac{1}{2}|\underline{a}|$
46. nth term of H . P is :
    1. $$\frac{1}{a + d}$$
    2. $$\frac{1}{a_1 + nd}$$
    3. $$\frac{1}{a_1 + (n - 1)d}$$
    4. $${a_1 + (n - 1)d}$$
47. Factors of 2 z³+ 9 z² - 11 z - 30 are
    1. ( z - 2 ) ( z + 5 ) ( z - 5 )
    2. ( z - 2 ) ( z + 2 ) ( z + 5 )
    3. ( z - 2 ) ( z + 5 ) ( 2 z + 3 )
    4. ( z - 3 ) ( z + 5 ) ( 2 z + 3 )
48. Suppose P and Q are two different matrices of order $3 \times n$ and $n\times p$ respectively , then order of the matrix PQ will be :
    1. $p \times 3$
    2. $3 \times p$
    3. $n \times n$
    4. PQ is not defined
49. If $3x^{5} - 5x^{3} + 2nx + 3, \forall n \in \mathbb{Z}$ is divided by $x - 1$, then remainder is:
    1. Odd number
    2. Even number
    3. Rational number
    4. Real number
50. The moment of force $2\underline{i} + 3\underline{j} + 7\underline{k}$ about the point ( 1 , 2 , 3 ) when applied at the point ( - 1 , 2 , 0 )
    1. $9\underline{i} + 8\underline{j} - 6\underline{k}$
    2. $-9\underline{i} - 8\underline{j} + 6\underline{k}$
    3. $9\underline{i} + 8\underline{j} + 6\underline{k}$
    4. $9\underline{i} - 8\underline{j} - 6\underline{k}$
51. If a , A , b are in A.P then 2A is :
    1. a - b
    2. $\frac{ a+ b}{2}$
    3. a + b
    4. b - a
52. e¹⁰ is known as
    1. Euler's formula
    2. Complex formula
    3. Exponential formula
    4. Cauchy's formula
53. If A is a row matrix of order 1 × n then order of $A^tA$ is.
    1. 1 × n
    2. n × 1
    3. 1 × 1
    4. n × n
54. The first Muslim mathemetics who introduced Algebra was :
    1. Muhammad Bin Musa
    2. Jabir Ibn Hayyan
    3. Ibn Al Haytham
    4. Bu Ali Sina
55. If $\underline{u} = x\underline{i} + y\underline{j}$ and $\underline{u} \cdot \underline{i} = 2$, $\underline{u} \cdot \underline{j} = 3$ then $\underline{u}$ is
    1. $\underline{i} +2\underline{j}$
    2. $2\underline{i} +3\underline{j}$
    3. $3\underline{i} +2\underline{j}$
    4. $\underline{i} +\underline{j}$
56. The next term of the sequence - 1 , 2 , 12 , 40 , ........ is :
    1. 110
    2. 112
    3. 76
    4. 88
57. For any complex Number z , R e ( z ) = ( U )
    1. $Im ( iz )$
    2. -$Im ( iz )$
    3. -$Im ( z )$
    4. Im ( z )
58. If A and B are two invertible matrices , then (A B )⁻¹ = :
    1. AB
    2. A⁻¹ B⁻¹
    3. B ⁻¹A⁻¹
    4. $\frac{1}{AB}$
59. If x - b is the factor of q(x), then q(b) is:
    1. factor
    2. divisor
    3. remainder
    4. dividend
60. The angle in _________ is a right angle
    1. Circle
    2. Semi - circle
    3. Triangle
    4. None of these
61. $\sum_{K=1}^{n}{K^2}= ?$
    1. $$\frac{n(n-1)}{2}$$
    2. $$\frac{n(n + 1)}{2}$$
    3. $$\frac{n(n+1)(2n + 1)}{2}$$
    4. $$\left(\frac{n(n+1)}{2}\right)^2$$
62. If ( 2 + 3 ) x + ( 1 + 3 ) y + 2 = 0 , then value of x and y
    1. x = 2 , y = 2
    2. x = - 2 , y = 2
    3. x = 3 , y = 2
    4. x = - 3 , y = - 2
63. For a unique solution of system rank of matrix A must be equal to :
    1. Rank of $A_b$
    2. Rank of A¹
    3. $|A_b|$
    4. $|A^t|$
64. Which of the following is a linear polynomial ?
    1. $$\frac{x^{2}}{2} - \frac{2}{x^{2}}$$
    2. $$\sqrt{2x} - 1$$
    3. $$x^2 + \frac{3x^{\frac{3}{2}}}{\sqrt{x}}$$
    4. $$\frac{x - 1}{x + 1}$$