In the adjoining figure, AOB is a straight line. If x : y : z = 4 : 5 : 6, then y = ?
Question: In the adjoining figure,AOBis a straight line. Ifx:y:z= 4 : 5 : 6, theny= ?(a) 60(b) 80(c) 48(d) 72 Solution: (a) 60 Let $\angle A O B=x^{\circ}=(4 a)^{\circ}, \angle C O B=y^{\circ}=(5 a)^{\circ}$ and $\angle B O D=z^{\circ}=(6 a)^{\circ}$ Then, we have: $\angle A O B+\angle C O B+\angle B O D=180^{\circ} \quad$ [Since $A O B$ is a straight line] $\Rightarrow 4 a+5 a+6 a=180^{\circ}$ $\quad \Rightarrow 15 a=180^{\circ}$ $\Rightarrow a=12^{\circ}$ $\therefore y=5 \times 12^{\circ}=60^{...
Read More →Find the domain of f(x)
Question: Find the domain of $f(x)=\cos ^{-1} x+\cos x$. Solution: For $\cos ^{-1} x$ to be defined. $-1 \leq x \leq 1$ Now, $\cos x$ is defined for all real values. So, domain of $\cos x$ is $\mathrm{R}$. Domain of $f(x)$ is $\mathrm{R} \cap[-1,1]=[-1,1]$....
Read More →Find the domain of f(x)
Question: Find the domain of $f(x)=2 \cos ^{-1} 2 x+\sin ^{-1} x$. Solution: For $2 \cos ^{-1} 2 x$ to be defined. $-1 \leq 2 x \leq 1$ $\Rightarrow-\frac{1}{2} \leq x \leq \frac{1}{2} \quad \ldots . .$ (i) For $\sin ^{-1} x$ to be defined. $-1 \leq x \leq 1 \quad \ldots$ (ii) Domain of $f(x)=\left[-\frac{1}{2}, \frac{1}{2}\right] \cap[-1,1]$ $=\left[-\frac{1}{2}, \frac{1}{2}\right]$....
Read More →Find the coefficient of:
Question: Find the coefficient of: (i) $x^{10}$ in the expansion of $\left(2 x^{2}-\frac{1}{x}\right)^{20}$ (ii) $x^{7}$ in the expansion of $\left(x-\frac{1}{x^{2}}\right)^{40}$ (iii) $x^{-15}$ in the expansion of $\left(3 x^{2}-\frac{a}{3 x^{3}}\right)^{10}$ (iv) $x^{9}$ in the expansion of $\left(x^{2}-\frac{1}{3 x}\right)^{9}$ (v) $x^{m}$ in the expansion of $\left(x+\frac{1}{x}\right)^{n}$ (vi) $x$ in the expansion of $\left(1-2 x^{3}+3 x^{5}\right)\left(1+\frac{1}{x}\right)^{8}$. (vii) $a^...
Read More →In an A.P., if the 5th and 12th terms are 30 and 65 respectively,
Question: In an A.P., if the 5th and 12th terms are 30 and 65 respectively, what is the sum of first 20 terms? Solution: In the given problem, let us take the first term asaand the common differenced Here, we are given that, $a_{5}=30$...........(1) $a_{12}=65$.........(2) Also, we know, $a_{n}=a+(n-1) d$ For the 5thterm (n =5), $a_{5}=a+(5-1) d$ $30=a+4 d$ (Using 1) $a=30-4 d$ ................(3) Similarly, for the 12thterm (n =12), $a_{12}=a+(12-1) d$ $65=a+11 d$ (Using 2) $a=65-11 d$ ...........
Read More →Find the domain of definition of f(x)
Question: Find the domain of definition of $f(x)=\cos ^{-1}\left(x^{2}-4\right)$. Solution: For $\cos ^{-1}\left(x^{2}-4\right)$ to be defined $-1 \leq x^{2}-4 \leq 1 \Rightarrow 3 \leq x^{2} \leq 5 \Rightarrow x \in[-\sqrt{5},-\sqrt{3}] \cup[\sqrt{3}, \sqrt{5}]$ Hence, the domain of $f(x)$ is $[-\sqrt{5},-\sqrt{3}] \cup[\sqrt{3}, \sqrt{5}]$....
Read More →Find the coefficient of:
Question: Find the coefficient of: (i) $x^{10}$ in the expansion of $\left(2 x^{2}-\frac{1}{x}\right)^{20}$ (ii) $x^{7}$ in the expansion of $\left(x-\frac{1}{x^{2}}\right)^{40}$ (iii) $x^{-15}$ in the expansion of $\left(3 x^{2}-\frac{a}{3 x^{3}}\right)^{10}$ (iv) $x^{9}$ in the expansion of $\left(x^{2}-\frac{1}{3 x}\right)^{9}$ (v) $x^{m}$ in the expansion of $\left(x+\frac{1}{x}\right)^{n}$ (vi) $x$ in the expansion of $\left(1-2 x^{3}+3 x^{5}\right)\left(1+\frac{1}{x}\right)^{8}$. (vii) $a^...
Read More →An angle is one-fifth of its supplement. The measure of the angle is
Question: An angle is one-fifth of its supplement. The measure of the angle is(a) 15(b) 30(c) 75(d) 150 Solution: (b) 30 Let the measure of the required angle be $x^{\circ}$ Then, the measure of its supplement will be $(180-x)^{\circ}$ $\therefore x=\frac{1}{5}\left(180^{\circ}-x\right)$ $\Rightarrow 5 x=180^{\circ}-x$ $\Rightarrow 6 x=180^{\circ}$ $\Rightarrow x=30^{\circ}$...
Read More →Which of the following statements is false?
Question: Which of the following statements is false?(a) Through a given point, only one straight line can be drawn.(b) Through two given points, it is possible to draw one and only one straight line.(c) Two straight lines can intersect at only one point.(d) A line segment can be produced to any desired length. Solution: (a) Through a given point, only one straight line can be drawn.Clearly, statement (a) isfalsebecause we can draw infinitely many straight lines through a given point....
Read More →In the given figure, AOB is a straight line. If ∠AOC = (3x − 10)°, ∠COD = 50° and ∠BOD = (x + 20)°, then ∠AOC = ?
Question: In the given figure,AOBis a straight line. If AOC= (3x 10),COD= 50 andBOD= (x+ 20), thenAOC= ?(a) 40(b) 60(c) 80(d) 50 Solution: (c) 80 We have : $\angle A O C+\angle C O D+\angle B O D=180^{\circ} \quad$ [Since $A O B$ is a straight line ] $\Rightarrow 3 x-10+50+x+20=180$ $\Rightarrow 4 x=120$ $\Rightarrow x=30$ $\therefore \angle A O C=[3 \times 30-10]^{\circ}$ $\Rightarrow \angle A O C=80^{\circ}$...
Read More →In an A.P., if the first term is 22,
Question: In an A.P., if the first term is 22, the common difference is 4 and the sum tonterms is 64, findn. Solution: In the given problem, we need to find the number of terms of an A.P. Let us take the number of terms asn. Here, we are given that, $a=22$ $d=-4$ $S_{n}=64$ So, as we know the formula for the sum ofnterms of an A.P. is given by, $S_{n}=\frac{n}{2}[2 a+(n-1) d]$ Where;a= first term for the given A.P. d= common difference of the given A.P. n= number of terms So, using the formula w...
Read More →In the given figure, AOB is a straight line. If ∠AOC = (3x + 10)° and ∠BOC (4x − 26)°, then ∠BOC = ?
Question: In the given figure,AOBis a straight line. If AOC= (3x+ 10) and BOC(4x 26), then BOC= ?(a) 96(b) 86(c) 76(d) 106 Solution: (b) 86 We have $\angle A O C+\angle B O C=180^{\circ} \quad$ [Since $A O B$ is a straight line ] $\Rightarrow 3 x+10+4 x-26=180^{\circ}$ $\Rightarrow 7 x=196^{\circ}$ $\Rightarrow x=28^{\circ}$ $\therefore \angle B O C=[4 \times 28-26]^{\circ}$ Hence, $\angle B O C=86^{\circ}$....
Read More →Find the sum of all integers between 100 and 550, which are divisible by 9.
Question: Find the sum of all integers between 100 and 550, which are divisible by 9. Solution: (i) In this problem, we need to find the sum of all the multiples of 9 lying between 100 and 550. So, we know that the first multiple of 9 after 100 is 108 and the last multiple of 9 before 550 is 549. Also, all these terms will form an A.P. with the common difference of 9. So here, First term (a) = 108 Last term (l) = 549 Common difference (d) = 9 So, here the first step is to find the total number o...
Read More →If f(x)
Question: If $f(x)=4-(x-7)^{3}$, then write $f^{-1}(x)$. [NCERT EXEMPLAR] Solution: We have, $f(x)=4-(x-7)^{3}$ Let $y=4-(x-7)^{3}$ $\Rightarrow(x-7)^{3}=4-y$ $\Rightarrow x-7=\sqrt[3]{4-y}$ $\Rightarrow x=7+\sqrt[3]{4-y}$ $\Rightarrow f^{-1}(y)=7+\sqrt[3]{4-y}$ $\therefore f^{-1}(x)=7+\sqrt[3]{4-x}$...
Read More →In the given figure, AOB is a straight line.
Question: In the given figure,AOBis a straight line. If AOC= 4x and BOC= 5x, then AOC= ?(a) 40(b) 60(c) 80(d) 100 Solution: (c) 80 We have $\angle A O C+\angle B O C=180^{\circ} \quad$ [Since $A O B$ is a straight line] $\Rightarrow 4 x+5 x=180^{\circ}$ $\Rightarrow 9 x=180^{\circ}$ $\Rightarrow x=20^{\circ}$ $\therefore \angle A O C=4 \times 20^{\circ}=80^{\circ}$...
Read More →If a function g = {(1, 1), (2, 3), (3, 5), (4, 7)} is described by g(x) = αx+β,
Question: If a function $g=\{(1,1),(2,3),(3,5),(4,7)\}$ is described by $g(x)=\alpha x+\beta$, then find the values of $\alpha$ and $\beta$. [NCERT EXEMPLAR] Solution: We have, A function $g=\{(1,1),(2,3),(3,5),(4,7)\}$ is described by $g(x)=\alpha x+\beta$ As, $g(1)=1$ and $g(2)=3$ So, $\alpha(1)+\beta=1$ $\Rightarrow \alpha+\beta=1 \quad \ldots \ldots(\mathrm{i})$ and $\alpha(2)+\beta=3$ $\Rightarrow 2 \alpha+\beta=3 \quad \ldots .(\mathrm{ii})$ $(\mathrm{ii})-(\mathrm{i})$, we get $2 \alpha-\...
Read More →Two complementary angles are such that twice the measure of one is equal to three times the measure of the other.
Question: Two complementary angles are such that twice the measure of one is equal to three times the measure of the other. The measure of larger angle is(a) 72(b) 54(c) 63(d) 36 Solution: (b) 54 Let the measure of the required angle be $x^{\circ}$ Then, the measure of its complement will be $(90-x)^{\circ}$. $\therefore 2 x=3(90-x)$ $\Rightarrow 2 x=270-3 x$ $\Rightarrow 5 x=270$ $\Rightarrow x=54$...
Read More →Find the 7th term from the end in the expansion
Question: Find the 7 th term from the end in the expansion of $\left(2 x^{2}-\frac{3}{2 x}\right)^{8}$ Solution: LetTr+1be the 7th term from the end in the given expression. Then, we have: Tr+1= (9 7 + 1) = 3rd term from the beginning Now, $T_{3}=T_{2+1}$ $={ }^{8} C_{2}\left(2 x^{2}\right)^{8-2}\left(-\frac{3}{2 x}\right)^{2}$ $=\frac{8 \times 7}{2 \times 1}\left(64 x^{12}\right) \frac{9}{4 x^{2}}$ $=4032 x^{10}$...
Read More →If the mapping f : {1, 3, 4} → {1, 2, 5} and g : {1, 2, 5} → {1, 3}, given by
Question: If the mapping $f:\{1,3,4\} \rightarrow\{1,2,5\}$ and $g:\{1,2,5\} \rightarrow\{1,3\}$, given by $f=\{(1,2),(3,5),(4,1)\}$ and $g=\{(2,3),(5,1),(1,3)\}$, then write fog. [NCERT EXEMPLAR] Solution: We have, $f:\{1,3,4\} \rightarrow\{1,2,5\}$ and $g:\{1,2,5\} \rightarrow\{1,3\}$, are given by $f=\{(1,2),(3,5),(4,1)\}$ and $g=\{(2,3),(5,1),(1,3)\}$, respectively As, $f o g(2)=f(g(2))=f(3)=5$ $f o g(5)=f(g(5))=f(1)=2$ $f o g(1)=f(g(1))=f(3)=5$ So, $f o g:\{1,2,5\} \rightarrow\{1,2,5\}$ is ...
Read More →The measure of an angle is five times its complement. The angle measures
Question: The measure of an angle is five times its complement. The angle measures(a) 25(b) 35(c) 65(d) 75 Solution: (d) $75^{\circ}$ Let the measure of the required angle be $x^{\circ}$. Then, the measure of its complement will be $(90-x)^{\circ}$. $\therefore x=5(90-x)$ $\Rightarrow x=450-5 x$ $\Rightarrow 6 x=450$ $\Rightarrow x=75$...
Read More →Find the 4th term from the end in the expansion
Question: Find the 4 th term from the end in the expansion of $\left(\frac{4 x}{5}-\frac{5}{2 x}\right)^{8}$. Solution: Let Tr+1be the4th term from the end of the given expression. Then, Tr+1is (10 4 + 1)th term, i.e., 7th term, from the beginning. Thus, we have: $T_{7}=T_{6+1}$ $={ }^{9} C_{6}\left(\frac{4 x}{5}\right)^{9-6}\left(\frac{5}{2 x}\right)^{6}$ $=\frac{9 \times 8 \times 7}{3 \times 2}\left(\frac{64}{125} x^{3}\right)\left(\frac{125 \times 125}{64 x^{6}}\right)$ $=\frac{10500}{x^{3}}$...
Read More →An angle which measures more than 180° but less than 360°, is called
Question: An angle which measures more than 180 but less than 360, is called(a) an acute angle(b) an obtuse angle(c) a straight angle(d) a reflex angle Solution: An angle which measures more than180 but less than 360 is called a reflex angle.Hence, the correct answer is option (d)....
Read More →Find the sum of all even integers between 101 and 999.
Question: Find the sum of all even integers between 101 and 999. Solution: In this problem, we need to find the sum of all the even numbers lying between 101 and 999. So, we know that the first even number after 101 is 102 and the last even number before 999 is 998. Also, all these terms will form an A.P. with the common difference of 2. So here, First term (a) = 102 Last term (l) = 998 Common difference (d) = 2 So, here the first step is to find the total number of terms. Let us take the number...
Read More →If two angles are complements of each other, then each angle is
Question: If two angles are complements of each other, then each angle is(a) an acute angle(b) an obtuse angle(c) a right angle(d) a reflex angle Solution: (a) an acute angleIf two angles are complements of each other, that is, the sum of their measures is9090, then each angle is anacute angle....
Read More →Find the 4th term from the beginning and 4th term from the end in the expansion of
Question: Find the 4 th term from the beginning and 4 th term from the end in the expansion of $\left(x+\frac{2}{x}\right)^{9}$. Solution: LetTr+1be the 4th term from the end. Then,Tr+1is (10 4 + 1)th, i.e., 7th, term from the beginning. $\therefore T_{7}=T_{6+1}$ $={ }^{9} C_{6}\left(x^{9-6}\right)\left(\frac{2}{x}\right)^{6}$ $=\frac{9 \times 8 \times 7}{3 \times 2}\left(x^{3}\right)\left(\frac{64}{x^{6}}\right)$ $=\frac{5376}{x^{3}}$ 4th term from the beginning $=T_{4}=T_{3+1}$ $\therefore T_...
Read More →