In the given figure, l || m and a transversal t cuts them.

Question: In the given figure,l||mand a transversaltcuts them. If 1 : 2 = 2 : 3, find the measure of each of the marked angles. Solution: Let1 = 2kand2= 3k, wherekis some constant.Now,1 and 2 form a linear pair.1 + 2 = 180⇒2k+ 3k= 180⇒ 5k= 180⇒k= 361= 2k= 236 =722= 3k= 336=108Now,3 =1 =72 (Vertically opposite angles)4 =2 =108 (Vertically opposite angles)It is given that,l||mandtisa transversal.5 =1 =72 (Pair of corresponding angles)6 =2 =108 (Pair of corresponding angles)7 =1 =72 (Pair of altern...

Write the total number of words formed by 2 vowels and 3 consonants taken from 4 vowels and 5 consonants.

Question: Write the total number of words formed by 2 vowels and 3 consonants taken from 4 vowels and 5 consonants. Solution: 2 out of 4 vowels and 3 out of 5 consonants can be chosen in ${ }^{4} C_{2} \times{ }^{5} C_{3}$ ways. The total number of letters is 5. These letters can be arranged in 5! ways. $\therefore$ Total number of words $={ }^{4} C_{2} \times{ }^{5} C_{3} \times 5 !$...

Write the number of ways in which 12 boys may be divided into three groups of 4 boys each.

Question: Write the number of ways in which 12 boys may be divided into three groups of 4 boys each. Solution: Number of groups in which 12 boys are to be divided = 3 Now, 4 boys can be chosen out of 12 boys in $\left(C_{4} \times{ }^{8} C_{4} \times{ }^{4} C^{12}{ }_{4}\right)$ ways. These groups can be arranged in 3! ways. $\therefore$ Total number of ways $=\frac{{ }^{12} C_{4} \times{ }^{8} C_{4} \times{ }^{4} C_{4}}{3 !}=\frac{12 ! \times 8 !}{4 ! \times 8 ! \times 4 ! \times 4 ! \times 3 !...

In the given figure, l || m and a transversal t cuts them. If ∠7 = 80°,

Question: In the given figure, $I \| m$ and a transversal $t$ cuts them. If $\angle 7=80^{\circ}$, find the measure of each of the remaining marked angles. Solution: In the given figure,7 and 8 form a linear pair.7 + 8 = 180⇒80 + 8 = 180⇒ 8 = 18080 =100Now,6 =8 =100 (Vertically opposite angles)5 =7 =80 (Vertically opposite angles)It is given that,l||mandtisa transversal.1 =5 =80 (Pair of corresponding angles)2 =6 =100 (Pair of corresponding angles)3 =7 =80 (Pair of corresponding angles)4 = 8 = 1...

The sum of three terms of an A.P. is 21 and the product

Question: The sum of three terms of an A.P. is 21 and the product of the first and the third terms exceeds the second term by 6, find three terms. Solution: In the given problem, the sum of three terms of an A.P is 21 and the product of the first and the third term exceeds the second term by 6. We need to find the three terms. Here, Let the three terms be $(a-d), a,(a+d)$ where, $a$ is the first term and $d$ is the common difference of the A.P So. $(a-d)+a+(a+d)=21$ $3 a=21$ $a=7$ .................

If f

Question: If $f: R^{+} \rightarrow R$ is defined as $f(x)=\log _{3} x$, then $t^{-1}(x)=$ ______________. Solution: Given:f(x) = log3x $f(x)=\log _{3} x$ $\Rightarrow y=\log _{3} x$ $\Rightarrow x=3^{y}$ Thus, $f^{-1}(y)=3^{y}$. Hence, $f^{-1}(x)=\underline{3}^{x}$....

Write the number of ways in which 5 red and 4 white balls can be drawn from a bag containing 10 red and 8 white balls.

Question: Write the number of ways in which 5 red and 4 white balls can be drawn from a bag containing 10 red and 8 white balls. Solution: 4 white and 5 red balls are to be selected from 8 white and 10 red balls. $\therefore$ Required number of ways $={ }^{8} C_{4} \times{ }^{10} C_{5}$...

Question: If $f: R^{+} \rightarrow R$ is defined as $f(x)=\log _{3} x$, then $r^{-1}(x)=$ Solution: Given:f(x) = log3x $f(x)=\log _{3} x$ $\Rightarrow y=\log _{3} x$ $\Rightarrow x=3^{y}$ Thus, $f^{-1}(y)=3^{y}$. Hence, $f^{-1}(x)=\underline{3}^{x}$....

Write the number of parallelograms that can be formed from a set of four parallel lines intersecting another set of three parallel lines.

Question: Write the number of parallelograms that can be formed from a set of four parallel lines intersecting another set of three parallel lines. Solution: A parallelogram is formed by choosing two straight lines from a set of four parallel lines and two straight lines from a set of three parallel lines. Two straight lines from the set of four parallel lines can be chosen in4C2ways and two straight lines from the set of three parallel lines can be chosen in3C2ways. $\therefore$ Number of paral...

In the given figure, l || m and a transversal t cuts them.

Question: In the given figure,l||mand a transversaltcuts them. If 1 = 120, find the measure of each of the remaining marked angles. Solution: We have, $\angle 1=120^{\circ}$. Then, $\angle 1=\angle 5$ [Corresponding angles] $\Rightarrow \angle 5=120^{\circ}$ $\angle 1=\angle 3$ [Vertically-opposite angles] $\Rightarrow \angle 3=120^{\circ}$ $\angle 5=\angle 7$ [Vertically-opposite angles] $\Rightarrow \angle 7=120^{\circ}$ $\angle 1+\angle 2=180^{\circ}$ [Since AFB is a straight line] $\Rightarr...

Which term of the A.P. 3, 15, 27, 39, ... will be 120 more than its 21st term?

Question: Which term of the A.P. 3, 15, 27, 39, ... will be 120 more than its 21stterm? Solution: In the given problem, let us first find the 21stterm of the given A.P. A.P. is 3, 15, 27, 39 Here, First term (a) = 3 Common difference of the A.P. $(d)=15-3=12$ Now, as we know, $a_{n}=a+(n-1) d$ So, for $21^{\text {st }}$ term $(n=21)$, $a_{21}=3+(21-1)(12)$ $=3+20(12)$ $=3+240$ $=243$ Let us take the term which is 120 more than the 21stterm asan. So, $a_{n}=120+a_{21}$ $=120+243$ $=363$ Also, $a_...

Write the maximum number of points of intersection of 8 straight lines in a plane.

Question: Write the maximum number of points of intersection of 8 straight lines in a plane. Solution: We know that two lines are required for one point of intersection. $\therefore$ Number of points of intersection $={ }^{8} C_{2}=\frac{8}{2} \times \frac{7}{1}=28$...

If f(0, ∞) → R is given by

Question: If $f(0, \infty) \rightarrow R$ is given by $f(x)=\log _{10} x$, then $f^{-1}(x)=$ Solution: Given:f(x) = log10x $f(x)=\log _{10} x$ $\Rightarrow y=\log _{10} x$ $\Rightarrow x=10^{y}$ Thus, $f^{-1}(y)=10^{y}$. Hence, $f^{-1}(x)=10^{x}$....

There are 3 letters and 3 directed envelopes.

Question: There are 3 letters and 3 directed envelopes. Write the number of ways in which no letter is put in the correct envelope. Solution: Total number of ways in which the letters can be put = 3! = 6 Suppose, out of the three letters, one has been put in the correct envelope. This can be done in3C1, i.e. 3, ways. Now, out of three, if two letters have been put in the correct envelope, then the last one has been put in the correct envelope as well. This can be done in3C3, i.e. one way. Number...

Prove that the bisectors of two adjacent supplementary angles include a right angle.

Question: Prove that the bisectors of two adjacent supplementary angles include a right angle. Solution: Let $A O B$ denote a straight line and let $\angle A O C$ and $\angle B O C$ be the supplementary angles. Then, we have: $\angle A O C=x^{\circ}$ and $\angle B O C=(180-x)^{\circ}$ Let $O E$ bisect $\angle A O C$ and $O F$ bisect $\angle B O C$. Then, we have: $\angle A O E=\angle C O E=\frac{1}{2} x^{\circ}$ and $\angle B O F=\angle F O C=\frac{1}{2}(180-x)^{\circ}$ Therefore, $\angle C O E+...

Write the value of

Question: Write the value of $\sum_{r=1}^{6}{ }^{56-r} C_{3}+{ }^{50} C_{4}$. Solution: We know: nCr-1+nCr=n+1Cr Now, we have: $\sum_{r=1}^{6}{ }^{56-r} C_{3}+{ }^{50} C_{4}$ $={ }^{55} C_{3}+{ }^{54} C_{3}+{ }^{53} C_{3}+{ }^{52} C_{3}+{ }^{51} C_{3}+{ }^{50} C_{3}+{ }^{50} C_{4}$ $={ }^{55} C_{3}+{ }^{54} C_{3}+{ }^{53} C_{3}+{ }^{52} C_{3}+{ }^{51} C_{3}+{ }^{51} C_{4}$ $={ }^{55} C_{3}+{ }^{54} C_{3}+{ }^{53} C_{3}+{ }^{52} C_{3}+{ }^{52} C_{4}$ $={ }^{55} C_{3}+{ }^{54} C_{3}+{ }^{53} C_{3}...

The number of onto functions from A = {a, b, c} to B = {1, 2, 3, 4} is __________.

Question: The number of onto functions fromA= {a,b,c} toB= {1, 2, 3, 4} is __________. Solution: Given: A function fromA= {a,b,c} toB = {1, 2, 3, 4} If a function from $A$ to $B$ is onto, then number of elements of $A \geq$ number of elements of $B$. But here, number of elements of $A$ number of elements of $B$ Thus, no onto function exist fromA= {a,b,c} toB= {1, 2, 3, 4}.Hence, thenumber of onto functions fromA= {a,b,c} toB= {1, 2, 3, 4} is0....

The sum 4th and 8th terms of an A.P. is 24 and the sum of 6th and 10th terms is 44. Find the A.P.

Question: The sum 4thand 8thterms of an A.P. is 24 and the sum of 6thand 10thterms is 44. Find the A.P. Solution: In the given problem, the sum of 4thand 8thterm is 24 and the sum of 6thand 10thterm is 44. We have to find the A.P We can write this as, $a_{4}+a_{8}=24$............(1) $a_{6}+a_{10}=44$..........(2) We need to find the A.P For the given A.P., let us take the first term asaand the common difference asd As we know, $a_{n}=a+(n-1) d$ For $4^{\text {th }}$ term $(n=4)$, $a_{4}=a+(4-1) ...

If f(x)

Question: If $f(x)=\cos \left[\pi^{2}\right] x+\cos \left[-\pi^{2}\right] x$, then $f\left(\frac{\pi}{2}\right)=$ Solution: Given: $f(x)=\cos \left[\pi^{2}\right] x+\cos \left[-\pi^{2}\right] x$ $f(x)=\cos \left[\pi^{2}\right] x+\cos \left[-\pi^{2}\right] x$ $=\cos 9 x+\cos (-10 x) \quad\left(\because \pi^{2}=9.85\right.$ approx $)$ $=\cos 9 x+\cos 10 x$ $\Rightarrow f\left(\frac{\pi}{2}\right)=\cos 9\left(\frac{\pi}{2}\right)+\cos 10\left(\frac{\pi}{2}\right)$ $=\cos \frac{9 \pi}{2}+\cos 5 \pi$...

Write the expression

Question: Write the expressionnCr+1+nCr 1+ 2 nCrin the simplest form. Solution: ${ }^{n} C_{r+1}+{ }^{n} C_{r-1}+2 \cdot{ }^{n} C_{r}$ $=\left({ }^{n} C_{r+1}+{ }^{n} C_{r}\right)+\left({ }^{n} C_{r}+{ }^{n} C_{r-1}\right) \quad\left[{ }^{n} C_{r}+{ }^{n} C_{r-1}={ }^{n+1} C_{r}\right]$ $={ }^{n+1} C_{r+1}+{ }^{n+1} C_{r} \quad\left[{ }^{n} C_{r}+{ }^{n} C_{r-1}={ }^{n+1} C_{r}\right]$ $={ }^{n+2} C_{r+1}$...

Write the number of diagonals of an n-sided polygon.

Question: Write the number of diagonals of an n-sided polygon. Solution: Ann-sided polygon hasnvertices. By joining any two vertices of the polygon, we obtain either a side or a diagonal of the polygon. Number of line segments obtained by joining the vertices of ann-sided polygon if we take two vertices at a time = Number of ways of selecting 2 out ofn=nC2 Out of these lines,nlines are sides of the polygon. Number of diagonals of the polygon $={ }^{n} C_{2}-n=\frac{n(n-1)}{2}-n=\frac{n(n-3)}{2}$...

Let A = {1, 2, 3} and B = {a, b} be two sets.

Question: LetA= {1, 2, 3} andB= {a,b} be two sets. Then the number of constant functionsfromAtoBis ___________. Solution: Given: SetsA= {1, 2, 3} andB= {a,b}Two constant functions can be formed fromAtoB.i.e., one isf(x) =aand other isf(x) =bHence,the number of constant functions fromAtoBis2....

Solve the following

Question: If35Cn+7=35C4n 2, then write the values ofn. Solution: ${ }^{35} C_{n+7}={ }^{35} C_{4 n-2}$ $n+7+4 n-2=35 \quad\left[\because^{n} C_{x}=^{n} C_{y} \Rightarrow n=x+y\right.$ or $\left.x=y\right]$ $\Rightarrow 5 n=30$ $\Rightarrow n=6$ And, $n+7=4 n-2$ $\Rightarrow n=3$...

Find the 10th term from the end of the A.P. 8, 10, 12, ..., 126.

Question: Find the 10thterm from the end of the A.P. 8, 10, 12, ..., 126. Solution: In the given problem, we need to find the 10thterm from the end for the given A.P. We have the A.P as 8, 10, 12 126 Here, to find the 10thterm from the end let us first find the total number of terms. Let us take the total number of terms asn. So, First term (a) = 8 Last term (an) = 126 Common difference (d) ==2 Now, as we know, So, for the last term, $126=8+(n-1) 2$ $126=8+2 n-2$ $126=6+2 n$ $126-6=2 n$ Further ...

If f(x) = cos [e] x + cos [–e] x,

Question: If $f(x)=\cos [e] x+\cos [-e] x$, then $f(\pi)=$___________. Solution: Given:f(x) = cos[e]x+ cos[e]x $f(x)=\cos [e] x+\cos [-e] x$ $=\cos 2 x+\cos (-3 x) \quad(\because e=2.718$ approx $)$ $=\cos 2 x+\cos 3 x$ $\Rightarrow f(\pi)=\cos 2 \pi+\cos 3 \pi$ $=1-1$ $=0$ Hence, $f(\pi)=\underline{0}$...