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Federal (FBISE) · Full book test (board pattern) · Free

1st Year Mathematics Full Book Paper

100 marks · 3 hours (Section A: 1 hour 40 minutes) · 84 questions. Covers: 9 chapters (1. Complex Numbers to 9. Trigonometric Functions).

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1st Year Mathematics Full Book Paper

1st Year · Mathematics · Federal (FBISE)

Name: ____________Roll No: ________Class/Section: ________Date: ________Total marks: 100Time: 3 hours (Section A: 1 hour 40 minutes)

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

  1. What is the position of the particle from mean position when amplitude is 0.0004mm and angle is π3\frac{\pi}{3}
    1. 1500(1+3i)\frac{1}{500}(1 +\sqrt{3i})
    2. 15000(1+3i)\frac{1}{5000}(1 +\sqrt{3i})
    3. 15000(3+i)\frac{1}{5000}(\sqrt{3} + i)
    4. 1500(3+i)\frac{1}{500}(\sqrt{3} + i)
  2. The sum of the series - 3 + ( - 1 ) + 1 + 3 + 5 + .............. + a16a_{16} is :
    1. 290
    2. 190
    3. 192
    4. None of these
  3. 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 !
  4. The projection of i‾−2j‾+ak‾ \underline{i} - 2\underline{j} + a\underline{k} along 2i‾−3j‾+23k‾ 2\underline{i} - 3\underline{j} + 2\sqrt{3}\underline{k} is 4 then α \alpha is :
    1. 3\sqrt{3}
    2. 232\sqrt{3}
    3. -3\sqrt{3}
    4. 333\sqrt{3}
  5. If n is even positive integer then the middle term in the expansion of (a + b)ⁿ is.
    1. (n2)thterm\left( \frac{n}{2} \right)^{\text{th}} \text{term}
    2. (n+12)thterm\left( \frac{n + 1}{2} \right)^{\text{th}} \text{term}
    3. (n−12)thterm\left( \frac{n - 1}{2} \right)^{\text{th}} \text{term}
    4. (n2+1)thterm\left( \frac{n}{2} + 1 \right)^{\text{th}} \text{term}
  6. For the matrices A=[23−51] A = \begin{bmatrix} 2 & 3 \\ -5 & 1 \end{bmatrix} and B=[4−113] B = \begin{bmatrix} 4 & -1 \\ 1 & 3 \end{bmatrix} , the value of matrix X in the matrix equations 2X - A = B will be :
    1. [124−88] \begin{bmatrix} 12 & 4 \\ -8 & 8 \end{bmatrix}
    2. [62−44] \begin{bmatrix} 6 & 2 \\ -4 & 4 \end{bmatrix}
    3. [31−22] \begin{bmatrix} 3 & 1 \\ -2 & 2 \end{bmatrix}
    4. [93−33] \begin{bmatrix} 9 & 3 \\ -3 & 3 \end{bmatrix}
  7. If f(x) f(x) is a periodic function and its period is π\pi , then f(x) f(x) could be :
    1. sin⁡2x \sin 2x
    2. tan⁡2x \tan 2x
    3. cos⁡2x \cos 2x
    4. All
  8. If cos⁡θ=−310 \cos \theta = -\frac{3}{\sqrt{10}} and 90∘<θ<180∘ 90^\circ < \theta < 180^\circ than tan⁡2θ= \tan 2\theta =
    1. 34\frac{3}{4}
    2. ±34\pm\frac{3}{4}
    3. −43\frac{-4}{3}
    4. −34\frac{-3}{4}
  9. What should be added to 3pq+5p2q2+p3 3pq + 5p^2q^2 + p^3 to get p3+2p2q2+4pq p^3 + 2p^2q^2 + 4pq ?
    1. p2q2−pq p^{2}q^{2} - pq
    2. 2p2q2+pq 2p^{2}q^{2} + pq
    3. −3p2q2+pq -3p^{2}q^{2} + pq
    4. 5p2q2+pq 5p^{2}q^{2} + pq
  10. 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
  11. The sum of first 100 positive integers :
    1. 5050
    2. 5051
    3. 5052
    4. None of these
  12. 10! = ?
    1. 5! + 5!
    2. 8! + 2!
    3. 10! + 0!
    4. None of these
  13. 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
  14. (110)+(112)+(114)+⋯+(1110) \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
  15. The matrix in reduced echelon form is :
    1. [01−120610234−500010.401100]\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. [13−61000014−300011200000]\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. [130000014120010−300000]\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. [130000010−300011200000]\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}
  16. If x, - x belongs to domain of f(x) f(x) ,then functionf(x) f(x) is said to be an even function if :
    1. f(−x) f(-x) =f(x) f(x)
    2. f(−x) f(-x) =-f(x) f(x)
    3. f(−x) f(-x) \neqf(x) f(x)
    4. f(−x) f(-x)
  17. The exact value of sin⁡7.5∘cos⁡7.5∘cos⁡15∘ \sin 7.5^\circ \cos 7.5^\circ \cos 15^\circ is ?
    1. 14\frac{1}{4}
    2. 12\frac{1}{2}
    3. 18\frac{1}{8}
    4. 116\frac{1}{16}
  18. ax+2xy+3 ax + 2xy + 3 is a ______ polynomial :
    1. Linear
    2. Cubic
    3. Constant
    4. Quadratic
  19. Algebraic form of z=2(cos⁡315∘+isin⁡315∘)z = \sqrt{2} (\cos 315^\circ + i \sin 315^\circ) is
    1. 1+i1 + i
    2. 1−i1 - i
    3. i i
    4. 2+i2 + i
  20. How many multiples of 4 between 14 and 523 is :
    1. 126
    2. 127
    3. 128
    4. 227
  21. 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
  22. If a‾=i‾−2k‾\underline{a} = \underline{i} - 2\underline{k} and b‾=3i‾−2j‾\underline{b} = 3\underline{i} - 2\underline{j}, then a‾×b‾=?\underline{a} \times \underline{b} = ?
    1. 2(i‾−3j‾+k‾)2\left( \underline{i} - 3\underline{j} +\underline{k}\right)
    2. −2(2i‾+3j‾+k‾)-2\left(2 \underline{i}+ 3\underline{j} +\underline{k}\right)
    3. 2(i‾+3j‾−k‾)2\left( \underline{i} + 3\underline{j} -\underline{k}\right)
    4. 2i‾+3j‾+4k‾2\underline{i} + 3\underline{j} +4\underline{k}
  23. Middle term (s) in the expansion of (2x2−15x)11 \left( 2x^2 - \frac{1}{5x} \right)^{11} is :
    1. T₅ , T₆
    2. T₄ , T₅
    3. T₆ , T₇
    4. T₇ , T₈
  24. The principal diagonal / main diagonal elements in the matrix A=[1−3124−1356] 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
  25. The graph of which of the trigonometric function does not crosses the x - axis?
    1. sin⁡x{\sin x}
    2. cos⁡x{\cos x}
    3. tanx{tanx}
    4. secx{secx}
  26. sin⁡(π2−θ)sin⁡(π2+θ)=\sin\left(\frac{\pi}{2} - \theta\right) \sin\left(\frac{\pi}{2} + \theta\right) =
    1. 1+cos⁡2θ1 +\cos 2\theta
    2. sin⁡2θ\sin 2\theta
    3. 1+cos⁡2θ2\frac{1 +\cos 2\theta}{2}
    4. 12sin⁡2θ\frac{1}{ 2}\sin2\theta
  27. 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
  28. If z=(3+4i)(2−3i)z = ( 3 + 4i ) ( 2 - 3i) then ∣z∣|{z}|
    1. 727\sqrt{2}
    2. 5135\sqrt{13}
    3. 13513\sqrt{5}
    4. 272\sqrt{7}
  29. ∑k=1103 \sum_{k=1}^{10} {3} is equal to:
    1. 10
    2. 103
    3. 300
    4. 30
  30. How many lines can be drawn through 21 points on a circle ?
    1. 310
    2. 210
    3. 410
    4. 570
  31. If AB→=CD→\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 )
  32. 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
  33. If A=[123−3] A = \begin{bmatrix} 1 & 2 \\ 3 & - 3 \end{bmatrix} and B=[−16] B= \begin{bmatrix} - 1 \\ 6 \end{bmatrix} , given that AX =B , then X =
    1. [11] \begin{bmatrix} 1 \\ 1 \end{bmatrix}
    2. [−11] \begin{bmatrix} -1 \\ 1 \end{bmatrix}
    3. [1−1] \begin{bmatrix} 1 \\ -1 \end{bmatrix}
    4. [−1−1] \begin{bmatrix} -1 \\ -1 \end{bmatrix}
  34. Which of the two trigonometric functions have the domain x=R−{(2n+1)π2},n∈Z? x = R - \left\{ (2n + 1) \frac{\pi}{2} \right\}, n \in \mathbb{Z} ?
    1. sin⁡θ\sin\theta and cos⁡θ\cos\theta
    2. tan⁡θ\tan\theta and sec⁡θ\sec\theta
    3. cot⁡θ\cot\theta and cosecθcosec\theta
    4. tan⁡θ\tan\theta and cosecθcosec\theta
  35. If sin⁡A=15 \sin A = \frac{1}{5} and sin⁡B=13 \sin B = \frac{1}{3} and 0<A,B<π2 0 < A, B < \frac{\pi}{2} , then cos⁡(A+B)= \cos(A + B) =
    1. 83−115\frac{8\sqrt{3} - 1}{15}
    2. −83+115-\frac{8\sqrt{3} + 1}{15}
    3. 83+115\frac{8\sqrt{3} + 1}{15}
    4. −83+115\frac{-8\sqrt{3} + 1}{15}
  36. Use of letters to represent an unknown quantity was introduced by :
    1. Muhammad Bin Musa
    2. Rene Descartes
    3. Euler
    4. James Sylvester
  37. Write z=−2−2iz = - 2 - 2i in polar form.
    1. 22(cos⁡3π4+isin⁡3π4)2\sqrt{2} \left( \cos\frac{3\pi}{4} + i \sin\frac{3\pi}{4} \right)
    2. 22(cos⁡3π4−isin⁡3π4)2\sqrt{2} \left( \cos\frac{3\pi}{4} - i \sin\frac{3\pi}{4} \right)
    3. 22(−cos⁡3π4+isin⁡3π4)2\sqrt{2} \left( -\cos\frac{3\pi}{4} + i \sin\frac{3\pi}{4} \right)
    4. 22(−cos⁡3π4−isin⁡3π4)2\sqrt{2} \left( -\cos\frac{3\pi}{4} - i \sin\frac{3\pi}{4} \right)
  38. Which term of the sequence - 6 , - 2 , 2 , ............ is 7 0 ?
    1. 19 th
    2. 20 th
    3. 21 th
    4. None of these
  39. 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
  40. A vector having magnitude one is called :
    1. Null vector
    2. Unit vector
    3. Position vector
    4. None of these
  41. (n1)+2(n2)+3(n3)+⋯+n(nn)=? \binom{n}{1} + 2\binom{n}{2} + 3\binom{n}{3} + \cdots + n\binom{n}{n} = ?
    1. n.2nn.2^n
    2. n.2n−1n.2^{n -1}
    3. n.2n+1n.2^{n+1}
    4. 2n+12^{n+1}
  42. If A and B are two matrices and O ( A + B ) = 2×32 \times3 , then O ( B A ) =
    1. 2×22 \times 2
    2. 3×33\times 3
    3. 3×23\times 2
    4. Not possible
  43. The range of the −3sin⁡3θ-3\sin 3\theta , is :
    1. Set of real numbers
    2. [ -3, 3 ]
    3. [ 3, - 3 ]
    4. [ - 1 , 1 ]
  44. Using product to sum identities, the exact value of −cos⁡195∘sin⁡45∘ -\cos 195^\circ \sin 45^\circ is :
    1. 33−34\frac{3\sqrt{3} - 3}{4}
    2. 33+34\frac{3\sqrt{3} + 3}{4}
    3. 3+14\frac{\sqrt{3} + 1}{4}
    4. None of these
  45. 2x+72x + 7 is a factor of 2x3+5x2−11x−142x^3 + 5x^2 - 11x - 14, then factorize form of the expression will be:
    1. (2x−7)(x−2)(x+1)(2x-7)(x-2)(x+1)
    2. (2x+7)(x−2)(x+1)(2x+ 7)(x-2)(x+1)
    3. (2x−7)(x−2)(x−1)(2x-7)(x-2)(x-1)
    4. None of these
  46. If z1=2−3i z₁= 2 - 3i and [ z₁z₂] = 13 then value of [−z2ˉ][-\bar{z₂} ] is
    1. 13
    2. 13\sqrt{13}
    3. 1313\frac{\sqrt{13}}{13}
    4. 113 \frac{1}{13}
  47. Predict the general term for the sequence.
    1. 43n−2\frac{4}{3^{n - 2} }
    2. 43n−1\frac{4}{3^{n - 1} }
    3. 43n\frac{4}{3^{n } }
    4. 43n+1\frac{4}{3^{n+ 1} }
  48. If 18Cr=18Cr+2 18C_r = 18C_{r+2} then rC5=? rC_5 = ?
    1. 24
    2. 56
    3. 72
    4. 120
  49. The vector from the point A to the origin where AB‾=4i‾−2j‾\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 )
  50. In the expansion of (a+b)n (a + b)^n , Tr+1=(nr)an−rbr T_{r+1} = \binom{n}{r} a^{n-r} b^r is ______ term:
    1. rthr^{th}
    2. (r+1)th( r + 1 )^{ th}
    3. (r−1)th( r - 1 )^{th}
    4. (r+2)th( r + 2 )^{th}
  51. For a system of non-homogeneous equations with three variables system will have unique solution if.
    1. Rank A < 3
    2. Rank AbA_{b} < 3
    3. Rank A = Rank AbA_{b} = 3
    4. Rank A = Rank AbA_{b} < 3
  52. From the graph of a trigonometric function , we can find its ___________.
    1. Domain
    2. Range
    3. Period
    4. All
  53. tan⁡(π2+θ)=…\tan\left(\frac{\pi}{2} + \theta\right) = \ldots
    1. cot θ
    2. −cot⁡θ-\cot\theta
    3. tan θ
    4. −tan⁡θ-\tan\theta
  54. The volume of a cuboid is 105x7105{x}^7 with length and height as 5x5x and respectively , then breadth is :
    1. 3x3x
    2. 3x23x^2
    3. 21x21x
    4. 21x221x^2
  55. The argument θ of a complex number that lies in the interval −π<θ≤π-\pi < \theta \leq \pi is called
    1. Principal argument
    2. Limited argument
    3. Minimum argument
    4. Maximum argument
  56. 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
  57. 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
  58. Which of the following is Lagrange , identity ?
    1. ∣a×b∣2=∣a∣2∣b∣2−a⋅b|a \times b|^2 = |a|^2 |b|^2 - a \cdot b
    2. ∣a×b∣2=∣a∣2∣b∣2+(a⋅b)2|a \times b|^2 = |a|^2 |b|^2 + (a \cdot b)^2
    3. ∣a×b∣2=∣a∣2∣b∣2−(a⋅b)2|a \times b|^2 = |a|^2 |b|^2 - (a \cdot b)^2
    4. ∣a×b∣2=∣a∣2∣b∣2+a⋅b|a \times b|^2 = |a|^2 |b|^2 + a \cdot b
  59. (a+x)n=∑r=0n(nr)an−rxr where n∈Z(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
  60. 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
  61. The period of the trigonometric function f(x)=sin⁡2x f(x) = \sin^2 x is :
    1. π\pi
    2. 2π2\pi
    3. π2\frac{\pi}{2}
    4. π2\pi^2
  62. 2cos⁡αsin⁡β= 2 \cos \alpha \sin \beta =
    1. sin⁡(α+β)+sin⁡(α−β){\sin(\alpha + \beta)}+{\sin(\alpha - \beta)}
    2. sin⁡(α+β)−sin⁡(α−β){\sin(\alpha + \beta)}-{\sin(\alpha - \beta)}
    3. cos⁡(α+β)+cos⁡(α−β){\cos(\alpha + \beta)}+{\cos(\alpha - \beta)}
    4. cos⁡(α−β)−cos⁡(α+β){\cos(\alpha - \beta)}-{\cos(\alpha +\beta)}
  63. If 2 , 5 , - 2 are zero of a polynomial then polynomial is :
    1. x3−5x−6x x^3 - 5x - 6x
    2. x3−5x2+2x−2 x^3 - 5x^2 + 2x - 2
    3. x3−5x2−4x+20 x^3 - 5x^2 - 4x + 20
    4. None of these
  64. For any two complex numbers z₁, z₂ which of the following can be true?
    1. z₁> z₂
    2. z₁ < z₂
    3. z1ˉ>z2ˉ\bar{z₁}>\bar{z₂}
    4. [ z₁] > [ z₂]

Q2. Short questions (Attempt all. Each question has an OR choice · 7 × 4 = 28)

  1. Find a₁ for geometric series. aₙ = 324 , r = 3 , Sₙ = 484

    OR

    Find the moment of the force F⃗=3i^=2j^−5k^\vec{F} = 3\hat{i} = 2\hat{j} - 5\hat{k} about the point ( 2 , 1 , -1 ) when it is applied at point ( 3 , 0 , 2 ).
  2. Find the value of λ, so that the given matrix are singular.[λ20213λ21]\begin{bmatrix} \lambda & 2 & 0 \\ 2 & 1 & 3 \\ \lambda & 2 & 1 \end{bmatrix}

    OR

    Check whether the function is odd or even? y = 3 Sin x + 4 Cos x
  3. Write the following equation and inequation in term of x and y by taking z = x + 𝑖y. ∣2z−𝑖∣=4| 2z - 𝑖 | = 4

    OR

    Use the product-to-sum formula to change the sum or difference. sin (- u) sin 5u
  4. Show that x - 3 is a factor of x³ - 2x² - 5x + 6.

    OR

    Find n, if n! = 990. (n - 3)!
  5. Find the last one or two digit of the number where applicable. 29¹⁰

    OR

    Find the first three terms of arithmetic series. n = 14 , aₙ = 53 , Sₙ = 378
  6. Find the angle between the vectors i^−2j^+k^\hat{i} -2\hat{j} + \hat{k} and 2i^−3j^+k^2\hat{i} -3\hat{j} + \hat{k} .

    OR

    If A=[120030216] A = \begin{bmatrix} 1 & 2 & 0 \\ 0 & 3 & 0 \\ 2 & 1 & 6 \end{bmatrix} then find ∣A∣=∣At∣ |A| = |A^t|
  7. Find the maximum and minimum values of the following trigonometric function.y=11+3 Sin(2θ−15) y = \frac{1}{1 + 3 \, \text{Sin}(2\theta - 15)}

    OR

    Find the magnitude of the following complex number.(3−−8)(3+−8)\left( \sqrt{3} - \sqrt{-8} \right) \left( \sqrt{3} + \sqrt{-8} \right)

Q3. Long questions (Attempt all. Each question has an OR choice · 3 × 8 = 24)

  1. Find the domain, range, and period of the trigonometric function. 5 sin 3x

    OR

    Using the properties of the determinant prove the following.∣3a12a+12a+11a+2312∣=(a−1)(a−2) \begin{vmatrix} 3a & 1 & 2a+1 \\ 2a+1 & 1 & a+2 \\ 3 & 1 & 2 \end{vmatrix} = (a-1)(a-2)
  2. Verify that 2∣z∣≥∣Re (z)∣+∣Im (z)∣Hint: (Start with (∣x∣−∣y∣)2≥0) \sqrt{2} |z| \geq |\text{Re } (z)| + |\text{Im } (z)| \quad \text{Hint: (Start with } (|x| - |y|)^2 \geq 0)

    OR

    A rectangular solid has a volume of 14 cubic units. The width is twice the height and the length is 2 units more than the width. Find the dimensions of the solid.
  3. OPQR is a trapezium made from three equilateral triangles with OP→=r⃗\overrightarrow{OP} = \vec{r}, OS→=s⃗\overrightarrow{OS} = \vec{s} and M is the mid point of QR‾\overline{QR}. (1) Write PS→\overrightarrow{PS} in terms of r⃗\vec{r} and s⃗\vec{s}. (2) Show that OQ→\overrightarrow{OQ} is parallel to SM→\overrightarrow{SM}.

    OR

    Find the sum of the series: 7 + 12 + 20 + 31 + 45 + 62 + ............ up to n terms.