Class 11 Math

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Class 11

Q1: Solve all parts:
(A) If $A = \{2, 3, 5\}$, then the number of subsets of $A$ are:
(a) 3 (b) 32 (c) 6 (d) 8

(B) $\sin 20^\circ \sin 40^\circ \sin 60^\circ \sin 80^\circ =$
(a) $\frac{1}{16}$ (b) $\frac{3}{16}$ (c) $-\frac{3}{16}$ (d) None

(C) Number of terms in the expansion of $(1 + 3x + 3x^2 + x^3)^{10}$ is:
(a) 31 (b) 32 (c) 10 (d) 11

(D) The $m$-th term of an A.P. is $\frac{1}{n}$, and the $n$-th term is $\frac{1}{m}$. Its $(mn)$-th term is:
(a) $mn$ (b) $\frac{1}{mn}$ (c) 1 (d) None

(E) If the slope of a line passing through the points $(x, 1)$ and $(-3, 5)$ is $\frac{4}{3}$, then $x = ?$:
(a) -4 (b) 4 (c) -6 (d) 6

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Q2: Solve all parts:
(a) If $(4x+3, y) = (3n+5, -2)$, find $x$ and $y$.
(b) Prove that $\sin 105^\circ + \cos 105^\circ = \cos 45^\circ$.
(c) $\lim_{x \to 0} \frac{\tan x}{x} = ?$
(d) Convert $\frac{5+4i}{4-5i}$ into the form $A + iB$.
(e) Find the probability of getting a head in a toss of one coin.

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Q3: Solve all parts:
(a) Convert $i^9 + i^{19}$ into the form $a + ib$.
(b) If $n(A) = 8, n(B) = 6$, and $n(A \cap B) = 3$, then find $n(A \cup B)$.
(c) Find the domain and range of $f(x) = \frac{1}{x-2}$.
(d) Evaluate $\tan \frac{13\pi}{12}$.

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Q4: Solve all parts:
(a) Find $\lim_{x \to 1} \frac{x^m - 1}{x^n - 1}$.

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Q5: Solve all parts:
(a) If for $x \neq 0, af(x) + bf\left(\frac{1}{x}\right) = \frac{1}{x} + 5$, $a \neq b$, then find $f(x)$.
(b) Find the domain and range of the real function $f(x) = \sqrt{x - 1}$.
(c) Find the value of $1 + i^5 + i^{10} + i^{15}$.

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Q6: Solve all parts:
(a) Find the modulus of $\frac{1+i}{1-i} - \frac{1-i}{1+i}$.
(b) Solve $-\frac{1}{3} < \frac{\pi}{2} - \frac{4}{3} < \frac{1}{6}$.
(c) If $n_p^4 : n_p^2 = 1:2$, then find the value of $n$.
(d) How many numbers of 7 digits can be formed with the digits $1, 2, 2, 0, 1, 1, 3$?
(e) Prove that $\sum_{r=0}^n 3^r \binom{n}{r} = 4^n$.

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Q7: Solve all parts:
(a) Two subjects are compulsory for a student in an examination.
In how many ways can a student select 5 subjects out of 10 subjects?
(b) Solve for $x$, $x^2 - 37 - (3x+5) \geq 9x - 8(x - 3)$.
(c) If $y = n + 2^n + 3^n + \dots$ and $|n| (1, 1)$, then prove that $n = \frac{5}{12}$.
(d) Find the distance on $x$-axis, which is equidistant from the points $(7, 6)$ and $(3, 4)$.
(e) Find the equation of the hyperbola if foci $(\pm 5, 0)$ and transverse axis is equal to 8.

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Q8: Find the angles between the lines $\sqrt{3}x + y = 1$ and $x + \sqrt{3}y = 1$.

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Q9: Find the value of $\lim_{x \to 0} \frac{\sin nx}{\sin x}$.

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Q10: A fair coin with 1 marked on one face and 6 on the other, and a fair die are both tossed. Find the probability that the sum of numbers that turn up is:
(i) 3
(ii) 12