Roll No.
Z-7-Z
[Total No. of Questions: 31]
[ Total No. of Printed Pages : 8]
{$$11^{\text {th }}$$ ARM(SZ)JKUT2024}
1207-Z
MATHEMATICS
[Time : 3.00 Hours ]
[ Maximum Marks : 80
General Instructions :
(i) This question paper contains 4 Sections A, B, C and D. Each Section is compulsory.
(ii) Section-A : Q. No. 1 to 10 comprises of 10 questions of 1 mark each.
(iii) Section-B : Q. No. 11 to $$\mathbf{2 0}$$ comprises of 10 Very Short Answer (V.S.A.) type questions of 2 marks each.
(iv) Section-C: Q. No. 21 to 28 comprises of 8 Short Answer (S.A.) type questions of 4 marks each.
(v) Section-D : Q. No. 29 to 31 comprises of 3 Long Answer (L.A.) type questions of 6 marks each.
$$Z-7-Z$$
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11^{\text {th }} A R M(S Z) J K U T 2024-1207-Z
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{Section-A}
(Objective Type Questions)
1. Set builder form of $$\{5,25,125,625\}$$ is :
(A) $$\left\{x / x \in N\right.$$ and $$\left.x=n^{2}\right\}$$
(B) $$\left\{x / x=5^{n}, n \in N\right.$$ and $$\left.1 \leq n \leq 4\right\}$$
(C) $$\left\{x / x=n^{5}, n \in N\right.$$ and $$\left.1 \leq n \leq 6\right\}$$
(D) $$\left\{x / x=n^{2}, n \in N\right.$$ and $$\left.0 \leq n \leq 6\right\}$$
2. Let $$f(x)=2 x-5$$, then $$f(-3)$$ is equal to :
(A) -11
(B) 11
(C) 0
(D) 18
3. $$\tan 3 x$$ is equal to :
(A) $$\frac{1-3 \tan x}{\tan ^{2} x}$$
(B) $$\frac{3 \tan x-\tan ^{3} x}{1-3 \tan ^{2} x}$$
(C) $$\frac{1-4 \tan ^{2} x}{4 \sin x}$$
(D) $$\frac{3 \tan x+\tan ^{3} x}{1+3 \tan ^{2} x}$$
4. The value of $$i^{9}+i^{19}$$ is equal to:
(A) $$0+i .0$$
(B) $$9 i$$
(C) $$i$$
(D) $$-i$$
5. Value of
$$
\frac{8!}{6!\times 2!}
$$
is equal to :
(A) 82
(B) 28
(C) 20
(D) 21
6. The e (eccentricity) of hyperbola is :
(A) e=1
(B) e>1
(C) e<1
(D) e=2
7. Equation of circle with centre
$$
\left(\frac{1}{2}, \frac{1}{4}\right)
$$
and radius
$$
\frac{1}{12}
$$
is :
(A) $$x^{2}+y^{2}+20 x+20 y=0$$
(B) $$x^{2}+y^{2}-9 x=0$$
(C) $$36 x^{2}+36 y^{2}-36 x-18 y+11=0$$
(D) $$36 x^{2}+18 y^{2}-18 x+36 y+11=0$$
8. The value of $$24 x<106$$ when $$x$$ is an integer is :
(A) $$\{1,2,3,4\}$$
(B) $$\{\ldots \ldots . .-3,-2,-1,0,1,2,3,4\}$$
(C) $$\{\ldots \ldots \ldots .-4,-3,1$$ $$\qquad$$
(D) $$\{1,2,3,4 \ldots \ldots .$$.
9. The mean of first 10 natural numbers is :
(A) 5
(B) 6
(C) 5.5
(D) 8
10. The probability of 3 heads in a toss of coin thrice is :
(A) $$\frac{1}{8}$$
(B) $$\frac{1}{16}$$
(C) $$\frac{1}{4}$$
(D) $$\frac{1}{2}$$
{Section-B}
(Very Short Answer Type Questions)
2 each
11. Find the subsets of the set $$\{-1,1,2\}$$.
12. If
$$
B=\{3,4,5,6\}, C=\{5,6,7,8\}, D=\{7,8,9,10\}
$$
then find $$B \cup C \cup D$$.
13. Reduce the equation $$3 y+2=0$$ into intercept form and find intercepts on the axes.
14. Find the centre and radius of the circle $$x^{2}+y^{2}-8 x+10 y-12=0$$.
15. Find the equation of parabola with vertex $$(0,0)$$ and focus $$(-2,0)$$.
16. Find the degree measure corresponding to $$\frac{5 \pi}{3}$$ radians. (Use’ $$\pi=\frac{22}{7}$$ )
17. Find the sample space when a coin is tossed once and dice is thrown once.
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\text { 11 th } A R M(S Z) J K U T 2024-1207-Z
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18. Find the mean deviation about the median for the data as under:
13, 17, 16, 14, 14, 13, 10, 16, 141, 18, 12, 17 .
19.
$$
\frac{1}{6!}+\frac{1}{7!}=\frac{x}{8!}
$$
then find $$x$$.
20. Which term of the sequence $$\sqrt{3}, 3,3 \sqrt{3}$$ is 729 .
{Section-C}
(Short Answer Type Questions)
21. Find the domain and range of the real function :
$$
f(x)=\sqrt{9-x^{2}}
$$
22. Prove that :
$$
\cos \left(\frac{\pi}{4}-x\right) \cos \left(\frac{\pi}{4}-y\right)-\sin \left(\frac{\pi}{4}-x\right) \sin \left(\frac{\pi}{4}-y\right)=\sin (x+y) .
$$
23. Find the multiplicative inverse of $$\sqrt{5}+3 i$$.
24. Solve the inequality:
$$
\frac{x}{2} \geq \frac{5 x-2}{3}-\frac{7 x-3}{5}
$$
and show the graph of the solution on number line.
25. If $$p$$ is the length of perpendicular from the origin to the line whose Intercepts on the axes are ‘ $$a$$ ‘ and ‘ $$b$$ ‘, then show that:
$$
\frac{1}{p^{2}}=\frac{1}{a^{2}}+\frac{1}{b^{2}}
$$
26. Find the derivative of :
$$
f(x)=\frac{2 x+3}{x-2}
$$
from the first principle.
27. Find the derivative of:
$$
\left(5 x^{3}+3 x-1\right)(x-1)
$$
28. if Find $$n$$ if
$$
{ }^{n-1} p_{3}:{ }^{n} p_{4}=1: 9
$$
{Section-D}
(Long Answer Type Questions),
29. Prove that:
$$
\frac{\cos 4 x+\cos 3 x+\cos 2 x}{\sin 4 x+\sin 3 x+\sin 2 x}=\cot 3 x
$$
Or
Prove that :
$$
\cos 4 x=1-8 \sin ^{2} x \cos ^{2} x
$$
30. In how many ways can one select a cricket team of 11 from 17 players in which only 5 players can bowl if each cricket team of 11 must include exactly 4 bowlers.
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11^{\text {th }} \mathrm{ARM}(\mathrm{SZ}) J K U T 2024-1207-Z^{\prime}
$$
or
In how many ways can a student choose a programme of 5 courses if 9 courses are available and 2 specific courses are compulsory for every student.
31.Find the mean deviation about the mean for the following data :
Income Per Day (in ₹) & No. of Persons
$$0-100$$ & 4
$$100-200$$ & 8
$$200-300$$ & 9
$$300-400$$ & 10
$$400-500$$ & 7
$$500-600$$ & 5
$$600-700$$ & 4
$$700-800$$ & 3
Or
In Class XI of a school 40% of the students study Mathematics and 30% students study Biology. $$10 \%$$ of the class study both Mathematics and Biology. If a student is selected at random from the class, find the probability that he will be studying Mathematics or Biology or both.
{With ARM(SZ)JKUT2024-1207-Z}
