1st Year Mathematics Second half Book Paper

Q1. Multiple choice questions (Attempt all · 56 × 0.75 = 42)

1. 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
2. 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. $\frac{-3}{4}$
3. 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. $$\left( \frac{n}{2} + 1 \right)^{\text{th}} \text{term}$$
4. 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 !
5. 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. $f(-x)$
6. 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. $\frac{1}{16}$
7. $\left( \begin{array}{c} 11 \\ 0 \end{array} \right) + \left( \begin{array}{c} 11 \\ 2 \end{array} \right) + \left( \begin{array}{c} 11 \\ 4 \end{array} \right) + \cdots + \left( \begin{array}{c} 11 \\ 10 \end{array} \right)$ is equal to
   1. 2¹¹
   2. 2¹²
   3. 2¹⁰
   4. 2¹¹- 1
8. 10! = ?
   1. 5! + 5!
   2. 8! + 2!
   3. 10! + 0!
   4. None of these
9. The graph of which of the trigonometric function does not crosses the x - axis?
   1. ${\sin x}$
   2. ${\cos x}$
   3. ${tanx}$
   4. ${secx}$
10. $\sin\left(\frac{\pi}{2} - \theta\right) \sin\left(\frac{\pi}{2} + \theta\right) =$
    1. $1 +\cos 2\theta$
    2. $\sin 2\theta$
    3. $\frac{1 +\cos 2\theta}{2}$
    4. $\frac{1}{ 2}\sin2\theta$
11. 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
12. 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
13. 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 $cosec\theta$
14. 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. $$\frac{-8\sqrt{3} + 1}{15}$$
15. 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
16. How many lines can be drawn through 21 points on a circle ?
    1. 310
    2. 210
    3. 410
    4. 570
17. The range of the $-3\sin 3\theta$ , is :
    1. Set of real numbers
    2. [ -3, 3 ]
    3. [ 3, - 3 ]
    4. [ - 1 , 1 ]
18. 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
19. $\binom{n}{1} + 2\binom{n}{2} + 3\binom{n}{3} + \cdots + n\binom{n}{n} = ?$
    1. $n.2^n$
    2. $n.2^{n -1}$
    3. $n.2^{n+1}$
    4. $2^{n+1}$
20. 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
21. From the graph of a trigonometric function , we can find its ___________.
    1. Domain
    2. Range
    3. Period
    4. All
22. $\tan\left(\frac{\pi}{2} + \theta\right) = \ldots$
    1. cot θ
    2. $-\cot\theta$
    3. tan θ
    4. $-\tan\theta$
23. 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. $( r + 2 )^{th}$
24. If $18C_r = 18C_{r+2}$ then $rC_5 = ?$
    1. 24
    2. 56
    3. 72
    4. 120
25. The period of the trigonometric function $f(x) = \sin^2 x$ is :
    1. $\pi$
    2. $2\pi$
    3. $\frac{\pi}{2}$
    4. $\pi^2$
26. $2 \cos \alpha \sin \beta =$
    1. $${\sin(\alpha + \beta)}+{\sin(\alpha - \beta)}$$
    2. $${\sin(\alpha + \beta)}-{\sin(\alpha - \beta)}$$
    3. $${\cos(\alpha + \beta)}+{\cos(\alpha - \beta)}$$
    4. $${\cos(\alpha - \beta)}-{\cos(\alpha +\beta)}$$
27. $(a + x)^n = \sum_{r=0}^n \binom{n}{r} a^{n-r} x^r \text{ where } n \in \mathbb{Z}$
    1. Integers
    2. Positive integers
    3. Non - negative
    4. Real numbers
28. 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
29. The range of the$2 \cos 3\theta$ is :
    1. Set of real numbers
    2. ( 2 , - 2 )
    3. ( - 3 , 3 )
    4. ( - 2 , 2 )
30. $\cos 23^\circ \sin 17^\circ =$
    1. $$\sin 40^\circ - \sin 6^\circ$$
    2. $$\sin 40^\circ + \sin 6^\circ$$
    3. $$\frac{1}{2}[\sin 40^\circ - \sin 6^\circ ]$$
    4. $$\frac{1}{2}[\sin 40^\circ + \sin 6^\circ ]$$
31. The coefficient of the 3rd last term in the expansion of (1 + x)³⁰⁰ is:
    1. 277
    2. 44850
    3. 303
    4. 4305600
32. 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
33. The maximum value of the trigonometric function The maximum value of the trigonometric function $y =6 + \frac{3}{5} \cos(2\theta - 1)$ is :
    1. $\frac{38}{5}$
    2. $\frac{32}{5}$
    3. $\frac{33}{5}$
    4. $\frac{38}{3}$
34. $\sin 40^\circ \cos 20^\circ + \cos 40^\circ \sin 20^\circ =$
    1. $\frac{\sqrt{3}}{2}$
    2. sin60°
    3. sin30°
    4. All of these
35. $T_{r+1} = \frac{n(n-1)(x-2) \cdots (n-r+1)}{r!} x^r$ is the ______ of binomial series.
    1. Binomial term
    2. General term
    3. $n^{th}$ term
    4. $7^{th}$ term
36. ${}^n C_r :{}^n C_{r +1} = ?$
    1. $$\frac{n+1}{r+1}$$
    2. $$\frac{r+1}{n-r}$$
    3. $$\frac{n}{r+1}$$
    4. $$\frac{r+1}{n+r}$$
37. For a function $f(x)$, if we have $f(p + x) = f(x)$ where $x \in \text{domain } f(x)$, then $p > 0$ is called:
    1. Domain of $f(x)$
    2. Range of $f(x)$
    3. Period of $f(x)$
    4. Frequency of $f(x)$
38. $\cos u + \cos 3u + \cos 5u + \cos 7u =$
    1. $\cos u \cos 2u \cos 4u$
    2. $\sin u \sin 2u \sin 4u$
    3. $-4\sin u \sin 2u \sin 4u$
    4. $4\cos u \cos 2u \cos 4u$
39. 1² + 2² + 3²+......+ n² = _____________
    1. $$\frac{n(n+1)(n+2)}{6}$$
    2. $$\frac{n(n+1)(2n+1)}{3}$$
    3. $$\frac{n(n+1)(2n+1)}{6}$$
    4. $$\frac{n(n+1)(n+2)}{3}$$
40. How many numbers can be formed from 1 , 2 , 3 , 4 , 5 , if even digit occupy even places .(each digit used at once )
    1. 72
    2. 48
    3. 24
    4. 12
41. The domain of the function $f(x) = \tan \lambda x$ equals to :
    1. $${\lambda} (Domain of tan x)$$
    2. $$\frac{ (Domain of tan x)}{\lambda}$$
    3. $$(Domain of {\lambda} tan x)$$
    4. (Domain of tan x)
42. If $\theta$ is a basic angle, then angles of measures $\frac{\pi}{2} \pm \theta, \pi \pm \theta,$ $\frac{3\pi}{2} \pm \theta, 2\pi \pm \theta$ are called:
    1. Allied angles
    2. Quardrental angles
    3. Angles in standard position
    4. None of these
43. Augustus De Morgan provided a meticulous description of the process and named it :
    1. Mathematical Induction
    2. Binomial theorem
    3. Binomial series
    4. None of these
44. How many different 5 - digit even numbers are possible from digits 1 , 2 , 4 , 6 , 8 .When no digit is repeated :
    1. $3 \times 4 !$
    2. 4!
    3. 5!
    4. $4 \times 4!$
45. Express 2 sin 3x sin 7x as a sum or difference:
    1. cos 4x - cos 10x
    2. cos 10x - cos 4x
    3. cos 4x + cos 10x
    4. cos 10x + cos 4x
46. $\sec\left(\frac{3\pi}{2} - \theta\right) =$
    1. $\cos\theta$
    2. $cosec\theta$
    3. -$cosec\theta$
    4. $\sec\theta$
47. 2ⁿ < ( n + 1 ) ! holds for all :
    1. $n \geq 1$
    2. $n \geq 2$
    3. $n \geq 3$
    4. $n \geq 0$
48. If $\frac{1}{5!} + \frac{1}{6!} = \frac{x}{7!}$ then $x = ?$
    1. 100
    2. 36
    3. 49
    4. 64
49. The trigonometric function $y= \tan x$is not defined at :
    1. $$x = (2n+1)\frac{\pi}{2}, n \in \mathbb{Z}$$
    2. $$[-\infty, +\infty]$$
    3. $$x = n{\pi}, n \in \mathbb{Z}$$
    4. $$x =\frac{n\pi}{2}, n \in \mathbb{Z}$$
50. $(1 + \cos \theta) \tan \frac{\theta}{2} =$
    1. $\sin \theta$
    2. $\sin \frac{\theta}{2}$
    3. $\cos \theta$
    4. $\cos \frac{\theta}{2}$
51. ¹⁴ C ₁ + ¹⁴ C₂ + ¹⁴C₃+ .....+ ¹⁴ C₁₃ =?
    1. 2¹⁴ - 1
    2. 2¹⁴ - 2
    3. 2¹⁴- 3
    4. 2¹⁴- 4
52. The product of r consecutive positive numbers is divisible by:
    1. r!
    2. (r + 1)!
    3. r! + 1
    4. 2r!
53. The graph of sine function is symmetric about :
    1. x - axis
    2. y - axis
    3. Origin
    4. All
54. $\frac{\sin 2\theta}{1 + \cos 2\theta} =$
    1. $\tan \theta$
    2. $\cot\theta$
    3. $\sin \theta$
    4. $\cos \theta$
55. In the binomial expansion of ( a + x )ⁿ the exponent of a __________ from index to zero .
    1. Increase
    2. Decrease
    3. Remains Same
    4. Varies
56. A team of four has to be selected from six boys and four girls .How many different teams can be selected with two boys .
    1. 70
    2. 90
    3. 120
    4. 210