1st Year Mathematics Board Pattern Paper 2026

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

1. In how many ways 6 women and 5 men be seated in a row such that male and female occupy alternate seats ?
   1. 11 !
   2. 30 !
   3. 6 ! 5 !
   4. 6 ! 5 ! 2 !
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. If $\cos \theta = -\frac{3}{\sqrt{10}}$ and $90^\circ < \theta < 180^\circ$ than $\tan 2\theta =$
   1. $\frac{3}{4}$
   2. $\pm\frac{3}{4}$
   3. $\frac{-4}{3}$
   4. $\frac{-3}{4}$
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. If $f(x)$ is a periodic function and its period is $\pi$ , then $f(x)$ could be :
   1. $\sin 2x$
   2. $\tan 2x$
   3. $\cos 2x$
   4. All
6. 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. $3\sqrt{3}$
7. The sum of the series - 3 + ( - 1 ) + 1 + 3 + 5 + .............. + $a_{16}$ is :
   1. 290
   2. 190
   3. 192
   4. None of these
8. 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. $\begin{bmatrix} 9 & 3 \\ -3 & 3 \end{bmatrix}$
9. If n is even positive integer then the middle term in the expansion of (a + b)ⁿ is.
   1. $$\left( \frac{n}{2} \right)^{\text{th}} \text{term}$$
   2. $$\left( \frac{n + 1}{2} \right)^{\text{th}} \text{term}$$
   3. $$\left( \frac{n - 1}{2} \right)^{\text{th}} \text{term}$$
   4. $$\left( \frac{n}{2} + 1 \right)^{\text{th}} \text{term}$$
10. 10! = ?
    1. 5! + 5!
    2. 8! + 2!
    3. 10! + 0!
    4. None of these
11. 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
12. The exact value of $\sin 7.5^\circ \cos 7.5^\circ \cos 15^\circ$ is ?
    1. $\frac{1}{4}$
    2. $\frac{1}{2}$
    3. $\frac{1}{8}$
    4. $\frac{1}{16}$
13. $ax + 2xy + 3$ is a ______ polynomial :
    1. Linear
    2. Cubic
    3. Constant
    4. Quadratic
14. If x, - x belongs to domain of $f(x)$ ,then function$f(x)$ is said to be an even function if :
    1. $f(-x)$=$f(x)$
    2. $f(-x)$=-$f(x)$
    3. $f(-x)$\neq$f(x)$
    4. $f(-x)$
15. 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
16. The sum of first 100 positive integers :
    1. 5050
    2. 5051
    3. 5052
    4. None of these
17. 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. $$\begin{bmatrix} 1 & 3 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & -3 \\ 0 & 0 & 0 & 1 & \frac{1}{2} \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}$$
18. $\left( \begin{array}{c} 11 \\ 0 \end{array} \right) + \left( \begin{array}{c} 11 \\ 2 \end{array} \right) + \left( \begin{array}{c} 11 \\ 4 \end{array} \right) + \cdots + \left( \begin{array}{c} 11 \\ 10 \end{array} \right)$ is equal to
    1. 2¹¹
    2. 2¹²
    3. 2¹⁰
    4. 2¹¹- 1
19. In a Chinese restaurant 4 different dishes of rice, 5 different dishes of chicken and 7 different types of salads are available.In how many ways a person can select his meal ?
    1. 140
    2. 150
    3. 16
    4. 14
20. Algebraic form of $z = \sqrt{2} (\cos 315^\circ + i \sin 315^\circ)$ is
    1. $1 + i$
    2. $1 - i$
    3. $i$
    4. $2 + i$
21. $\sin\left(\frac{\pi}{2} - \theta\right) \sin\left(\frac{\pi}{2} + \theta\right) =$
    1. $1 +\cos 2\theta$
    2. $\sin 2\theta$
    3. $\frac{1 +\cos 2\theta}{2}$
    4. $\frac{1}{ 2}\sin2\theta$
22. 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
23. The graph of which of the trigonometric function does not crosses the x - axis?
    1. ${\sin x}$
    2. ${\cos x}$
    3. ${tanx}$
    4. ${secx}$
24. 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. $2\underline{i} + 3\underline{j} +4\underline{k}$
25. How many multiples of 4 between 14 and 523 is :
    1. 126
    2. 127
    3. 128
    4. 227
26. 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
27. Middle term (s) in the expansion of $\left( 2x^2 - \frac{1}{5x} \right)^{11}$ is :
    1. T₅ , T₆
    2. T₄ , T₅
    3. T₆ , T₇
    4. T₇ , T₈
28. How many lines can be drawn through 21 points on a circle ?
    1. 310
    2. 210
    3. 410
    4. 570
29. If $z = ( 3 + 4i ) ( 2 - 3i)$ then $|{z}|$
    1. $7\sqrt{2}$
    2. $5\sqrt{13}$
    3. $13\sqrt{5}$
    4. $2\sqrt{7}$
30. If $\sin A = \frac{1}{5}$ and $\sin B = \frac{1}{3}$ and $0 < A, B < \frac{\pi}{2}$, then $\cos(A + B) =$
    1. $$\frac{8\sqrt{3} - 1}{15}$$
    2. $$-\frac{8\sqrt{3} + 1}{15}$$
    3. $$\frac{8\sqrt{3} + 1}{15}$$
    4. $$\frac{-8\sqrt{3} + 1}{15}$$
31. Use of letters to represent an unknown quantity was introduced by :
    1. Muhammad Bin Musa
    2. Rene Descartes
    3. Euler
    4. James Sylvester
32. Which of the two trigonometric functions have the domain $x = R - \left\{ (2n + 1) \frac{\pi}{2} \right\}, n \in \mathbb{Z} ?$
    1. $\sin\theta$ and $\cos\theta$
    2. $\tan\theta$ and $\sec\theta$
    3. $\cot\theta$ and $cosec\theta$
    4. $\tan\theta$ and $cosec\theta$
33. 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 )
34. $\sum_{k=1}^{10} {3}$is equal to:
    1. 10
    2. 103
    3. 300
    4. 30
35. 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. $\begin{bmatrix} -1 \\ -1 \end{bmatrix}$
36. In the binomial expansion of ( a + x ) ⁿ the exponent of x _________ from zero to index .
    1. Remain Same
    2. Increases
    3. Decrease
    4. Varies
37. 10 persons want to sit at a round table .In how many ways they can be seated , if 3 persons insist on sitting together ?
    1. 5040
    2. 30240
    3. 10080
    4. 2520
38. 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. $$2\sqrt{2} \left( -\cos\frac{3\pi}{4} - i \sin\frac{3\pi}{4} \right)$$
39. Using product to sum identities, the exact value of $-\cos 195^\circ \sin 45^\circ$ is :
    1. $\frac{3\sqrt{3} - 3}{4}$
    2. $\frac{3\sqrt{3} + 3}{4}$
    3. $\frac{\sqrt{3} + 1}{4}$
    4. None of these
40. $2x + 7$ 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
41. The range of the $-3\sin 3\theta$ , is :
    1. Set of real numbers
    2. [ -3, 3 ]
    3. [ 3, - 3 ]
    4. [ - 1 , 1 ]
42. A vector having magnitude one is called :
    1. Null vector
    2. Unit vector
    3. Position vector
    4. None of these
43. Which term of the sequence - 6 , - 2 , 2 , ............ is 7 0 ?
    1. 19 th
    2. 20 th
    3. 21 th
    4. None of these
44. 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
45. $\binom{n}{1} + 2\binom{n}{2} + 3\binom{n}{3} + \cdots + n\binom{n}{n} = ?$
    1. $n.2^n$
    2. $n.2^{n -1}$
    3. $n.2^{n+1}$
    4. $2^{n+1}$
46. If $18C_r = 18C_{r+2}$ then $rC_5 = ?$
    1. 24
    2. 56
    3. 72
    4. 120
47. If $z₁= 2 - 3i$ and [ z₁z₂] = 13 then value of $[-\bar{z₂} ]$ is
    1. 13
    2. $\sqrt{13}$
    3. $\frac{\sqrt{13}}{13}$
    4. $\frac{1}{13}$
48. $\tan\left(\frac{\pi}{2} + \theta\right) = \ldots$
    1. cot θ
    2. $-\cot\theta$
    3. tan θ
    4. $-\tan\theta$
49. 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. $21x^2$
50. From the graph of a trigonometric function , we can find its ___________.
    1. Domain
    2. Range
    3. Period
    4. All
51. 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 )
52. 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. $$\frac{4}{3^{n+ 1} }$$
53. 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
54. In the expansion of $(a + b)^n$, $T_{r+1} = \binom{n}{r} a^{n-r} b^r$ is ______ term:
    1. $r^{th}$
    2. $( r + 1 )^{ th}$
    3. $( r - 1 )^{th}$
    4. $( r + 2 )^{th}$
55. Ten students are participating in a race .In how many ways can the first 3 prizes be won ?
    1. 920
    2. 680
    3. 820
    4. 720
56. 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
57. $2 \cos \alpha \sin \beta =$
    1. $${\sin(\alpha + \beta)}+{\sin(\alpha - \beta)}$$
    2. $${\sin(\alpha + \beta)}-{\sin(\alpha - \beta)}$$
    3. $${\cos(\alpha + \beta)}+{\cos(\alpha - \beta)}$$
    4. $${\cos(\alpha - \beta)}-{\cos(\alpha +\beta)}$$
58. 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
59. The period of the trigonometric function $f(x) = \sin^2 x$ is :
    1. $\pi$
    2. $2\pi$
    3. $\frac{\pi}{2}$
    4. $\pi^2$
60. 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. $$|a \times b|^2 = |a|^2 |b|^2 + a \cdot b$$
61. 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
62. 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
63. $(a + x)^n = \sum_{r=0}^n \binom{n}{r} a^{n-r} x^r \text{ where } n \in \mathbb{Z}$
    1. Integers
    2. Positive integers
    3. Non - negative
    4. Real numbers
64. How many odd numbers less than 1000 can be formed by using the digits 0 , 2 , 5 , 7 ? ( repetition of digits is allowed )
    1. 52
    2. 32
    3. 22
    4. 42