Prove that:

Question: Prove that:4nC2n:2nCn= [1 3 5 ... (4n 1)] : [1 3 5 ... (2n 1)]2. Solution: $\frac{{ }^{4 n} C_{2 n}}{{ }^{2 n} C_{n}}=\frac{1.3 .5 \ldots(4 n-1)}{[1.3 .5 \ldots(2 n-1)]^{2}}$ $\mathrm{LHS}=\frac{{ }^{4 n} C_{2 n}}{{ }^{2 n} C_{n}}$ $=\frac{(4 n) !}{(2 n) !(2 n) !} \times \frac{n ! n !}{(2 n) !}$ $=\frac{[4 n \times(4 n-1) \times(4 n-2) \times(4 n-3) \ldots \ldots \ldots \ldots \ldots . \ldots 3 \times 2 \times 1] \times(n !)^{2}}{[2 n \times(2 n-1) \times(2 n-2) \ldots \ldots . .3 \tim...

Find the sum of n terms of an A.P. whose nth terms is given by an = 5 − 6n.

Question: Find the sum ofnterms of an A.P. whosenthterms is given by an = 5 6n. Solution: Here, we are given an A.P., whose $n^{\text {th }}$ term is given by the following expression, $a_{n}=5-6 n$ So, here we can find the sum of the $n$ terms of the given A.P., using the formula, $S_{n}=\left(\frac{n}{2}\right)(a+l)$ Where,a= the first term l= the last term So, for the given A.P, The first term (a) will be calculated usingin the given equation fornthterm of A.P. $a=5-6(1)$ $=5-6$ $=-1$ Now, th...

Find the sum to n term of the A.P. 5, 2, −1, −4, −7, ...,

Question: Find the sum to n term of the A.P. 5, 2, 1, 4, 7, ..., Solution: In the given problem, we need to find the sum of the $n$ terms of the given A.P. ${ }^{*} 5,2,-1,-4,-7, \ldots$ ". So, here we use the following formula for the sum ofnterms of an A.P., $S_{n}=\frac{n}{2}[2 a+(n-1) d]$ Where;a= first term for the given A.P. d= common difference of the given A.P. n= number of terms For the given A.P. $(5,2,-1,-4,-7, \ldots)$, Common difference of the A.P. $(d)=a_{2}-a_{1}$ $=2-5$ $=-3$ Num...

If f : R → R is defined by f(x)

Question: If $f: R \rightarrow R$ is defined by $f(x)=x^{2}$, write $f^{-1}(25)$ Solution: Let $f^{-1}(25)=x \quad \ldots(1)$ $\Rightarrow f(x)=25$ $\Rightarrow x^{2}=25$ $\Rightarrow x^{2}-25=0$ $\Rightarrow(x-5)(x+5)=0$ $\Rightarrow x=\pm 5$ $\Rightarrow f^{-1}(25)=\{-5,5\}$ $[$ from (1)]...

For all positive integers n, show that

Question: For all positive integers $n$, show that ${ }^{2 n} C_{n}+{ }^{2 n} C_{n-1}=\frac{1}{2}\left({ }^{2 n+2} C_{n+1}\right)$. Solution: $\mathrm{LHS}={ }^{2 n} C_{n}+{ }^{2 n} C_{n-1}$ $=\frac{(2 n) !}{n ! n !}+\frac{(2 n) !}{(n-1) !(2 n-n+1) !}$ $=\frac{(2 n) !}{n ! n !}+\frac{(2 n) !}{(n-1) !(n+1) !}$ $=\frac{(2 n) !}{n(n-1) ! n !}+\frac{(2 n) !}{(n-1) !(n+1) n !}$ $=\frac{(2 n) !}{n !(n-1) !}\left[\frac{1}{n}+\frac{1}{n+1}\right]$ $=\frac{(2 n) !}{n !(n-1) !}\left[\frac{2 n+1}{n(n+1)}\r...

In each of the figures given below, AB || CD. Find the value of x in each case.

Question: In each of the figures given below,AB||CD. Find the value ofxin each case.\ Solution: (i) Draw $E F\|A B\| C D$. Now, $A B \| E F$ and $B E$ is the transversal. Then, $\angle A B E=\angle B E F \quad[$ Alternate Interior Angles $]$ $\Rightarrow \angle B E F=35^{\circ}$ Again. $E F \| C D$ and $\mathrm{DE}$ is the transversal. Then, $\angle D E F=\angle F E D$ $\Rightarrow \angle F E D=65^{\circ}$ $\therefore x^{\circ}=\angle B E F+\angle F E D$ $=(35+65)^{\circ}$ $=100^{\circ}$ or, $x=...

Write the total number of one-one functions from set A = {1, 2, 3, 4} to set B = {a, b, c}.

Question: Write the total number of one-one functions from setA= {1, 2, 3, 4} to setB= {a,b,c}. Solution: A has 4 elements and $B$ has 3 elements. Also, one-one function is only possible from $A$ to $B$ if $n(A) \leq n(B)$. But, here $n(A)n(B)$. So, the number of one-one functions from $A$ to $B$ is 0 ....

Write the total number of one-one functions from set A = {1, 2, 3, 4} to set B = {a, b, c}.

Question: Write the total number of one-one functions from setA= {1, 2, 3, 4} to setB= {a,b,c}. Solution: A has 4 elements and $B$ has 3 elements. Also, one-one function is only possible from $A$ to $B$ if $n(A) \leq n(B)$. But, here $n(A)n(B)$. So, the number of one-one functions from $A$ to $B$ is 0 ....

Prove that the product of 2n consecutive negative integers is divisible by (2n)!

Question: Prove that the product of 2nconsecutive negative integers is divisible by (2n)! Solution: Let $2 n$ negative integers be $(-r),(-r-1),(-r-2), \ldots, \ldots,(-r-2 n+1)$. Then, product $=(-1)^{2 n}(r)(r+1)(r+2), \ldots \ldots, \ldots(r+2 n-1)$ $=\frac{(r-1) !(r)(r+1)(r+2) \ldots \ldots(r+2 n-1)}{(r-1) !}$ $=\frac{(r+2 n-1) !}{(r-1) !}$ $=\frac{(r+2 n-1) !}{(r-1) !(2 n) !} \times(2 n) !$ $={ }^{r+2 n-1} C_{2 n} \times(2 n) !$ This is divisible by $(2 n) !$....

Find the sum of the following arithmetic progressions:

Question: Find the sum of the following arithmetic progressions: (i) 50, 46, 42, ... to 10 terms(ii) 1, 3, 5, 7, ... to 12 terms(iii) 3, 9/2, 6, 15/2, ... to 25 terms(iv) 41, 36, 31, ... to 12 terms(v) a + b, a b, a 3b, ... to 22 terms (vi) $(x-y)^{2},\left(x^{2}+y^{2}\right),(x+y)^{2}, \ldots$, to $n$ terms (vii) $\frac{x-y}{x+y} \frac{3 x-2 y}{x+y} \frac{5 x-3 y}{x+y}, \ldots$ to $\mathrm{n}$ terms (viii) $-26,-24,-22, \ldots$ to 36 terms. Solution: In the given problem, we need to find the su...

Solve the following

Question: If=mC2, then find the value ofC2. Solution: ${ }^{\alpha} C_{2}=\frac{\alpha}{2} \times \frac{(\alpha-1)}{1} \times{ }^{\alpha} C_{0} \quad\left[\because{ }^{n} C_{r}=\frac{n}{r} \cdot{ }^{n-1} C_{r-1}\right]$ $=\frac{1}{2} \alpha(\alpha-1) \quad\left[\because{ }^{n} C_{0}=1\right]$ $=\frac{1}{2}\left[{ }^{m} C_{2}\left({ }^{m} C_{2}-1\right)\right]$ $=\frac{1}{2}\left[\frac{m !}{2 !(m-2) !}\left(\frac{m !}{2 !(m-2) !}-1\right)\right]$ $=\frac{1}{2}\left[\frac{m(m-1)}{2}\left(\frac{m(m...

If A = {a, b, c} and B = {−2, −1, 0, 1, 2}, write the total number of one-one functions from A to B.

Question: IfA= {a,b,c} andB= {2, 1, 0, 1, 2}, write the total number of one-one functions fromAtoB. Solution: Let $f: A \rightarrow B$ be a one-one function. Then, $f(a)$ can take 5 values, $f(b)$ can take 4 values and $f(c)$ can take 3 values. Then, the number of one-one functions = 543 = 60...

Solve the following

Question: If16Cr=16Cr+ 2, findrC4. Solution: Given; ${ }^{16} C_{r}={ }^{16} C_{r+2}$ $16=r+r+2 \quad\left[\because\right.$ Property $5:{ }^{n} C_{x}={ }^{n} C_{y} \Rightarrow x=y$ or $\left.x+y=n\right]$ $\Rightarrow 2 r+2=16$ $\Rightarrow 2 r=14$ $\Rightarrow r=7$ Now, ${ }^{r} C_{4}={ }^{7} C_{4}$ $\Rightarrow{ }^{7} C_{4}={ }^{7} C_{3} \quad\left[\because{ }^{n} C_{r}={ }^{n} C_{n-r}\right]$ $\Rightarrow{ }^{7} C_{4}={ }^{7} C_{3}=\frac{7}{3} \times \frac{6}{2} \times \frac{5}{1} \times{ }^{...

If A = {1, 2, 3} and B = {a, b}, write the total number of functions from A to B.

Question: IfA= {1, 2, 3} andB= {a,b}, write the total number of functions fromAtoB. Solution: Formula: If set $A$ has $m$ elements and set $B$ has $n$ elements, then the number of functions from $A$ to $B$ is $n^{m}$. Given: $A=\{1,2,3\}$ and $B=\{a, b\}$ $\Rightarrow n(A)=3$ and $n(B)=2$ $\therefore$ Number of functions from $A$ to $B=2^{3}=8$...

Solve the following

Question: If2nC3:nC2= 44 : 3, findn. Solution: Given: $2 n_{C_{3}}: n_{C_{2}}=44: 3$ $\frac{2 n_{C_{3}}}{n_{C_{2}}}=\frac{44}{3}$ $\Rightarrow \frac{2 n !}{3 !(2 n-3) !} \times \frac{2 !(n-2) !}{n !}=\frac{44}{3}$ $\Rightarrow \frac{2 n(2 n-1)(2 n-2)}{3 n(n-1)}=\frac{44}{3}$ $\Rightarrow(2 n-1)(2 n-2)=22(n-1)$ $\Rightarrow 4 n^{2}-6 n+2=22 n-22$ $\Rightarrow 4 n^{2}-28 n+24=0$ $\Rightarrow n^{2}-7 n+6=0$ $\Rightarrow n^{2}-6 n-n+6=0$ $\Rightarrow n(n-6)-1(n-6)=0$ $\Rightarrow(n-1)(n-6)=0$ $\Righ...

Which of the following graphs represents a one-one function?

Question: Which of the following graphs represents a one-one function? Solution: In the graph of (b), different elements on thex-axis have different images on they-axis. But in (a), the graph cuts thex-axis at 3 points, which means that 3 points on thex-axis have the same image as 0 and hence, it is not one-one....

Which one of the following graphs represents a function?

Question: Which one of the following graphs represents a function? Solution: In graph (b), 0 has more than one image, whereas every value ofxin graph (a) has a unique image.Thus, graph (a) represents a function.So, the answer is (a)....

Which one of the following graphs represents a function?

Question: Which one of the following graphs represents a function? Solution: In graph (b), 0 has more than one image, whereas every value ofxin graph (a) has a unique image.Thus, graph (a) represents a function.So, the answer is (a)....

Solve the following

Question: IfnC4,nC5andnC6are in A.P., then findn. Solution: SincenC4,nC5andnC6are in AP. $\therefore$ 2. ${ }^{n} C_{5}={ }^{n} C_{4}+{ }^{n} C_{6}$ $\Rightarrow 2 \times \frac{n !}{5 !(n-5) !}=\frac{n !}{4 !(n-4) !}+\frac{n !}{6 !(n-6) !}$ $\Rightarrow \frac{2}{5 \times 4 !(n-5)(n-6) !}=\frac{1}{4 !(n-5)(n-4)(n-6) !}+\frac{1}{6 \times 5 \times 4 !(n-6) !}$ $\Rightarrow \frac{2}{5(n-5)}=\frac{1}{(n-5)(n-4)}+\frac{1}{30}$ $\Rightarrow \frac{2}{5(n-5)}-\frac{1}{(n-5)(n-4)}=\frac{1}{30}$ $\Rightarr...

Solve the following

Question: IfnC4,nC5andnC6are in A.P., then findn. Solution: SincenC4,nC5andnC6are in AP. $\therefore$ 2. ${ }^{n} C_{5}={ }^{n} C_{4}+{ }^{n} C_{6}$ $\Rightarrow 2 \times \frac{n !}{5 !(n-5) !}=\frac{n !}{4 !(n-4) !}+\frac{n !}{6 !(n-6) !}$ $\Rightarrow \frac{2}{5 \times 4 !(n-5)(n-6) !}=\frac{1}{4 !(n-5)(n-4)(n-6) !}+\frac{1}{6 \times 5 \times 4 !(n-6) !}$ $\Rightarrow \frac{2}{5(n-5)}=\frac{1}{(n-5)(n-4)}+\frac{1}{30}$ $\Rightarrow \frac{2}{5(n-5)}-\frac{1}{(n-5)(n-4)}=\frac{1}{30}$ $\Rightarr...

Let f = {(1, 2), (3, 5), (4, 1)) and g = {(2, 3), (5, 1), (1, 3)}.

Question: Letf= {(1, 2), (3, 5), (4, 1)) andg= {(2, 3), (5, 1), (1, 3)}. Then,gof= __________ andfog= __________. Solution: Given:f= {(1, 2), (3, 5), (4, 1)) andg= {(2, 3), (5, 1), (1, 3)} $f o g(x)=f(g(x))$ $f o g(1)=f(g(1))$ $=f(3)$ $=5$ $f o g(2)=f(g(2))$ $=f(3)$ $=5$ $f o g(5)=f(g(5))$ $=f(1)$ $=2$ Hence, $f o g=\{(1,5),(2,5),(5,2)\}$ $g o f(x)=g(f(x))$ $g o f(1)=g(f(1))$ $=g(2)$ $=3$ $g o f(3)=g(f(3))$ $=g(5)$ $=1$ $g o f(4)=g(f(4))$\ $=g(1)$ $=3$ Hence, $g \circ f=\{(1,3),(3,1),(4,3)\}$ He...

Show that (a − b)2, (a2 + b2) and (a + b)2 are in A.P.

Question: Show that $(a-b)^{2},\left(a^{2}+b^{2}\right)$ and $(a+b)^{2}$ are in A.P. Solution: Here, we are given three terms and we need to show that they are in A.P., First term $\left(a_{1}\right)=(a-b)^{2}$ Second term $\left(a_{2}\right)=\left(a^{2}+b^{2}\right)$ Third term $\left(a_{3}\right)=(a+b)^{2}$ So, in an A.P. the difference of two adjacent terms is always constant. So, to prove that these terms are in A.P. we find the common difference, we get, $d=a_{2}-a_{1}$ $d=\left(a^{2}+b^{2}...

Solve the following

Question: If28C2r:24C2r 4= 225 : 11, findr. Solution: We have, ${ }^{28} C_{2 r}:{ }^{24} C_{2 r-4}=225: 11$ $\Rightarrow \frac{{ }^{28} C_{2 r}}{{ }^{24} C_{2 r-4}}=\frac{225}{11}$ $\Rightarrow \frac{28 !}{2 r !(28-2 r) !} \times \frac{(2 r-4) !(28-2 r) !}{24 !}=\frac{225}{11}$ $\Rightarrow \frac{28 \times 27 \times 26 \times 25}{2 r(2 r-1)(2 r-2)(2 r-3)}=\frac{225}{11}$ $\Rightarrow 2 r(2 r-1)(2 r-2)(2 r-3)=\frac{28 \times 27 \times 26 \times 25 \times 11}{225}$ $\Rightarrow 2 r(2 r-1)(2 r-2)(...

In the given figure, AB || CD. Find the values of x, y and z.

Question: In the given figure,AB||CD. Find the values ofx,yandz. Solution: $A B \| C D$ and let EF and EG be the transversals. Now, $A B \| C D$ and EF is the transversal. Then, $\angle A E F=\angle E F G \quad$ [Alternate Angles] $\Rightarrow y=75^{\circ}$ $\Rightarrow y=75$ Also, $\angle E F C+\angle E F D=180^{\circ} \quad$ [Since CFD is a straight line] $\Rightarrow x+y=180$ $\Rightarrow x+75=180$ $\Rightarrow x=105$ And, $\angle E G F+\angle E G D=180^{\circ} \quad[$ Since CFGD is a straigh...

Question: Let $f=\{(1,2),(3,5),(4,1))$ and $g=\{(2,3),(5,1),(1,3)\}$. Then, gof $=$__________andfog =__________. Solution: Given:f= {(1, 2), (3, 5), (4, 1)) andg = {(2, 3), (5, 1), (1, 3)} $f o g(x)=f(g(x))$ $f o g(1)=f(g(1))$ $=f(3)$ $=5$ $f o g(2)=f(g(2))$ $=f(3)$ $=5$ $f o g(5)=f(g(5))$ $=f(1)$ $=2$ Hence, $f o g=\{(1,5),(2,5),(5,2)\}$ $g o f(x)=g(f(x))$ $g o f(1)=g(f(1))$ $=g(2)$ $=3$ $g o f(3)=g(f(3))$ $=g(5)$ $=1$ $g o f(4)=g(f(4))$\ $=g(1)$ $=3$ Hence, $g \circ f=\{(1,3),(3,1),(4,3)\}$ He...