{"id":625790,"date":"2023-09-08T05:51:09","date_gmt":"2023-09-08T05:51:09","guid":{"rendered":"https:\/\/www.indcareer.com\/schools\/?p=625790"},"modified":"2023-09-08T05:51:25","modified_gmt":"2023-09-08T05:51:25","slug":"kc-sinha-exercise-3-1-mathematics-solution-class-11-chapter-3-phalan","status":"publish","type":"post","link":"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-3-1-mathematics-solution-class-11-chapter-3-phalan\/","title":{"rendered":"KC Sinha: Exercise 3.1 – Mathematics Solution Class 11 Chapter 3 \u092b\u0932\u0928"},"content":{"rendered":"\n\n\n\n\n

Question 1<\/h4>\n\n\n\n

\u091c\u093e\u0901\u091a \u0915\u0940\u091c\u093f\u090f \u0915\u093f \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0938\u0902\u092c\u0902\u0927\u094b \u092e\u0947 \u0915\u094c\u0928 \u092b\u0932\u0928 \u0939\u0948? \u092f\u0926\u093f \u0935\u0947 \u092b\u0932\u0928 \u0939\u0948 \u0924\u094b \u0909\u0928\u0915\u0947 \u092a\u094d\u0930\u093e\u0902\u0924 \u0924\u0925\u093e \u092a\u0930\u093f\u0938\u0930 \u091c\u094d\u091e\u093e\u0924 \u0915\u0940\u091c\u093f\u090f\u0964<\/strong><\/p>\n\n\n\n

(i) f={(1,4),(2,3),(3,4),(4,3)}<\/p>\n\n\n\n

(ii) g={(2,5),(-1,0),(1,6)}<\/p>\n\n\n\n

(iii) h={(1,2),(2,2),(3,2)}<\/p>\n\n\n\n

(iv) \u0444={(1,2),(1,3),(2,5)}<\/p>\n\n\n\n

(v) \u03c8={(2,1),(3,1),(5,2)}<\/p>\n\n\n\n

(vi) u={(2,1),(5,1),(8,1),(11,1),(14,1),(17,1)}<\/p>\n\n\n\n

(vii) v={(0,0),(1,1),(1,-1),(4,2),(4,-2),(9,3),(9,-3),(16,4),(16,-4)}<\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

\"\"\/<\/a><\/figure>\n\n\n\n

(i) f<\/em> \u090f\u0915 \u092b\u0932\u092e\u0928 \u0939\u0948, \u092a\u094d\u0930\u093e\u0902\u0924={1,2,3,4}<\/p>\n\n\n\n

\u092a\u0930\u093f\u0938\u0930={3,4}<\/p>\n\n\n\n

(ii) g \u090f\u0915 \u092b\u0932\u0928 \u0939\u0948, \u092a\u094d\u0930\u093e\u0902\u0924={2,-1,1}<\/p>\n\n\n\n

\u092a\u0930\u093f\u0938\u0930={5,0,6}<\/p>\n\n\n\n

(iii) h \u090f\u0915 \u092b\u0932\u0928 \u0939\u0948, \u092a\u094d\u0930\u093e\u0902\u0924={1,2,3}<\/p>\n\n\n\n

\u092a\u0930\u093f\u0938\u0930={2}<\/p>\n\n\n\n

(iv) \u0444 \u090f\u0915 \u092b\u0932\u0928 \u0928\u0939\u0940 \u0939\u0948\u0964<\/p>\n\n\n\n

\u0905\u0935\u092f\u0935 \u0915\u093e \u0926\u094b \u092a\u094d\u0930\u0924\u093f\u092c\u093f\u0902\u092c 2 \u0924\u0925\u093e 3 \u0939\u0948\u0964<\/p>\n\n\n\n

Question 2<\/h4>\n\n\n\n

\u092e\u093e\u0928 \u0932\u0940\u091c\u093f\u090f \u0915\u093f (Let) A={1,2,3} \u0914\u0930 (and) B={4,5}<\/strong><\/p>\n\n\n\n

\u092e\u093e\u0928 \u0932\u0940\u091c\u093f\u090f \u0915\u093f (Let) f<\/em>={(2,4),(3,5)} \u0915\u094d\u092f\u093e A \u0938\u0947 B \u092e\u0947 f<\/em> \u090f\u0915 \u092b\u0932\u0928 \u0939\u0948?<\/strong><\/p>\n\n\n\n

[If f<\/em> is a function from A into B?]<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

\"\"\/<\/a><\/figure>\n\n\n\n

f<\/em>={(2,4),(3,5)}<\/p>\n\n\n\n

f<\/em>, A \u0938\u0947 B \u092e\u0947 \u090f\u0915 \u092b\u0932\u0928 \u0928\u0939\u0940 \u0939\u0948\u0964<\/p>\n\n\n\n

\u0915\u094d\u092f\u094b\u0915\u093f 1 \u0915\u093e \u0915\u094b\u0908 \u092a\u094d\u0930\u0924\u093f\u092c\u093f\u0902\u092c B \u092e\u0947 \u092e\u094c\u091c\u0941\u0926 \u0928\u0939\u0940 \u0939\u0948<\/p>\n\n\n\n

Question 3<\/h4>\n\n\n\n

\u092e\u093e\u0928 \u0932\u0940\u091c\u093f\u090f \u0915\u093f A={1,2,3},B={3,6,9,10} \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0938\u0902\u092c\u0902\u0927\u094b \u092e\u0947 A \u0938\u0947 B \u092e\u0947 \u0915\u094c\u0928 \u092b\u0932\u0928 \u0939\u0948 ?\u0909\u0928\u0915\u0947 \u092a\u0930\u093f\u0938\u0930 \u092d\u0940 \u091c\u094d\u091e\u093e\u0924 \u0915\u0940\u091c\u093f\u090f \u092f\u0926\u093f \u0935\u0947 \u092b\u0932\u0928 \u0939\u0948 \u0964
f<\/em>={(1,9),(2,3),(3,10)}
<\/strong> Sol :
f<\/em> \u092b\u0932\u0928 \u0939\u0948 \u092a\u0930\u093f\u0938\u0930={3,9,10}<\/p>\n\n\n\n

g={(1,6),(2,10),(3,9),(1,3)}
Sol :
g \u090f\u0915 \u092b\u0932\u0928 \u0928\u0939\u0940 \u0939\u0948 \u0964<\/p>\n\n\n\n

h={(2,6),(3,9)}
Sol :
h \u090f\u0915 \u092b\u0932\u0928 \u0928\u0939\u0940 \u0939\u0948 \u0964<\/p>\n\n\n\n

u={(x,y):y=3x;x\u2208A}
Sol :
u={(1,3),(2,6),(3,9)}
u \u090f\u0915 \u092b\u0932\u0928 \u0939\u0948 \u0964
\u092a\u0930\u093f\u0932\u0930={3,6,9}<\/p>\n\n\n\n

Question 7<\/h4>\n\n\n\n

\u092e\u093e\u0928 \u0932\u0940\u091c\u093f\u090f \u0915\u093f $f=\\left\\{\\left(\\frac{m}{n}, m\\right) ; m \\in Z, n \\in Z, n \\neq 0\\right\\}$ \u0938\u0947 \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 Q\u00d7Z \u090f\u0915 \u0909\u092a\u0938\u092e\u0941\u091a\u094d\u091a\u092f \u0939\u0948 \u0964 \u0915\u094d\u092f\u093e Q \u0938\u0947 Z \u092e\u0947 f<\/em> \u090f\u0915 \u092b\u0932\u0928 \u0939\u0948 \u0964 \u0905\u092a\u0928\u0947 \u0909\u0924\u094d\u0924\u0930 \u0915\u093e \u0914\u091a\u093f\u0924\u094d\u092f \u0926\u0940\u091c\u093f\u090f \u0964
Sol :
f<\/em> \u090f\u0915 \u092b\u0932\u0928 \u0928\u0939\u0940\u0902 \u0939\u0948<\/p>\n\n\n\n

$f\\left(\\frac{2}{1}\\right)=2 \\quad, \\quad f\\left(\\frac{4}{2}\\right)=4$<\/p>\n\n\n\n

$f=\\left\\{\\left(\\frac{2}{1}, 2\\right),\\left(\\frac{4}{2}, 4\\right),\\dots3\\right.$<\/p>\n\n\n\n

f<\/em>={(2,2),(2,4)..}<\/p>\n\n\n\n

Question 8<\/h4>\n\n\n\n

\u092e\u093e\u0928 \u0932\u0940\u091c\u093f\u090f \u0915\u093f N \u092a\u094d\u0930\u093e\u0915\u0943\u0924 \u0938\u0902\u0916\u094d\u092f\u093e\u0913 \u0915\u093e \u0938\u092e\u0941\u091a\u094d\u091a\u092f \u0939\u0948 \u0914\u0930 N \u092a\u0930 \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0938\u0902\u092c\u0902\u0927 R \u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930 \u0939\u0948 \u0915\u093f
R={(x,y):y=2x,x,y\u2208N},R \u0915\u0947 \u092a\u094d\u0930\u093e\u0902\u0924 , \u0938\u0939\u092a\u094d\u0930\u093e\u0902\u0924 \u0914\u0930 \u092a\u0930\u093f\u0938\u0930 \u091c\u094d\u091e\u093e\u0924 \u0915\u0940\u091c\u093f\u090f \u0915\u094d\u092f\u093e \u092f\u0939 \u0938\u0902\u092c\u0902\u0927 \u092b\u0932\u0928 \u0939\u0948 ?
Sol :
R={(x,y):y=2x,x,y\u2208N}<\/p>\n\n\n\n

R={(1,2),(2,4),(3,6),(4,8)….}<\/p>\n\n\n\n

R \u0915\u093e \u092a\u094d\u0930\u093e\u0902\u0924={1,2,3,4..}=N<\/p>\n\n\n\n

R \u0915\u093e \u092a\u0930\u093f\u0938\u0930={2,4,6,8..}=\u090f\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913 \u0915\u093e \u0938\u092e\u0941\u091a\u094d\u091a\u092f<\/p>\n\n\n\n

R \u0915\u093e \u0938\u0939\u092a\u094d\u0930\u093e\u0902\u0924={1,2,3,4,…}<\/p>\n\n\n\n

\u0938\u0902\u092c\u0902\u0927 R \u090f\u0915 \u092b\u0932\u0928 \u0939\u0948\u0964<\/p>\n\n\n\n

\"\"\/<\/a><\/figure>\n\n\n\n

Question 9<\/h4>\n\n\n\n

(i)<\/strong> f<\/em>(x)=2x-5 \u0938\u0947 \u090f\u0915 \u092b\u0932\u0928 f<\/em> \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0939\u0948 , \u091c\u094d\u091e\u093e\u0924 \u0915\u0940\u091c\u093f\u090f f<\/em>(0),f<\/em>(7) \u0914\u0930 f<\/em>(-3)
[A function f is defined by f<\/em>(x)=2x-5, find f<\/em>(0),f<\/em>(7) and f<\/em>(-3)]
Sol :<\/p>\n\n\n\n

(ii)<\/strong> \u092f\u0926\u093f (If) f<\/em>(x)=x2<\/sup>+4,x\u2208R , \u0924\u094b \u091c\u094d\u091e\u093e\u0924 \u0915\u0940\u091c\u093f\u090f (then find) f(2),f(-5) \u0914\u0930 (and)f(t+2),t\u2208R
Sol :<\/p>\n\n\n\n

(iii)<\/strong> \u092f\u0926\u093f(If) f(x)=x2<\/sup>+1,x\u2208R, \u0924\u094b \u091c\u094d\u091e\u093e\u0924 \u0915\u0940\u091c\u093f\u090f (find) f<\/em>(1)\u00d7f<\/em>(4)
Sol :<\/p>\n\n\n\n

(i) f<\/em>(x)=2x-5<\/p>\n\n\n\n

x=0 \u092a\u0930, f<\/em>(0)=2(0)-5=-5<\/p>\n\n\n\n

x=7 \u092a\u0930 , f<\/em>(7)=2(7)-5<\/p>\n\n\n\n

=14-5=9<\/p>\n\n\n\n

x=3 \u092a\u0930, f<\/em>(3)=2(-3)-5<\/p>\n\n\n\n

=-6-5=-11<\/p>\n\n\n\n

(ii)<\/p>\n\n\n\n

f<\/em>(x)=x2<\/sup>+4<\/p>\n\n\n\n

f<\/em>(t+2)=(t+2)2<\/sup>+4<\/p>\n\n\n\n

=t2<\/sup>+2.t.2+22<\/sup>+4<\/p>\n\n\n\n

=t2<\/sup>+4t+8<\/p>\n\n\n\n

f<\/em>(1)=12<\/sup>+1=2<\/p>\n\n\n\n

f<\/em>(4)=42<\/sup>+1=17<\/p>\n\n\n\n

f<\/em>(1)\u00d7f<\/em>(4)=2\u00d717=34<\/p>\n\n\n\n

(i)<\/p>\n\n\n\n

f<\/em>(x)=x2<\/sup>+2x+1<\/p>\n\n\n\n

f<\/em>(-1)=(-1)2<\/sup>+2(-1)+1<\/p>\n\n\n\n

=1-2+1=0<\/p>\n\n\n\n

f<\/em>(1)=12<\/sup>+2(1)+1<\/p>\n\n\n\n

=1+2+1=4<\/p>\n\n\n\n

f<\/em>(-1)\u00d7f<\/em>(1)=04=0<\/p>\n\n\n\n

f<\/em>(-1)+f<\/em>(1)=0+4=4<\/p>\n\n\n\n

f<\/em>(0)=02<\/sup>+2(0)+1=1<\/p>\n\n\n\n

f<\/em>(-1)+f<\/em>(0)\u2260f<\/em>(0)<\/p>\n\n\n\n

Question 11<\/h4>\n\n\n\n

(i)<\/strong> \u092e\u093e\u0928 \u0932\u0940\u091c\u093f\u090f \u0915\u093f f<\/em>={(1,1),(2,3),(0,-1),(-1,3)} \u0915\u0941\u091b \u092a\u0942\u0930\u094d\u0923\u093e\u0915\u094b a \u0914\u0930 b \u0915\u0947 \u0932\u093f\u090f Z \u0938\u0947 Z \u092e\u0947 \u090f\u0915 \u092b\u0932\u0928 f<\/em>(x)=ax+b \u0938\u0947 \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0939\u0948 \u0964 a \u0914\u0930 b \u0915\u0940\u091c\u093f\u090f
Sol :
\u092e\u093e\u0928\u093e y=f<\/em>(x)<\/p>\n\n\n\n

y=ax+b<\/p>\n\n\n\n

(0,-1)\u2208f<\/em><\/p>\n\n\n\n

\u2234-1=a(0)+b<\/p>\n\n\n\n

-1=b<\/p>\n\n\n\n

(1,1)\u2208f<\/em><\/p>\n\n\n\n

1=a(1)+b<\/p>\n\n\n\n

1=a+b<\/p>\n\n\n\n

1=a-1<\/p>\n\n\n\n

2=a<\/p>\n\n\n\n

(ii)<\/strong> \u092e\u093e\u0928 \u0932\u0940\u091c\u093f\u090f \u0915\u093f (Let) f<\/em>={(1,1),(2,3),(0,-1),(-1,-3)} Z \u0938\u0947 Z \u092e\u0947 \u090f\u0915 \u0930\u0947\u0916\u093f\u0915 \u092b\u0932\u0928 \u0939\u0948 \u0924\u094b f<\/em>(x) \u091c\u094d\u091e\u093e\u0924 \u0915\u0940\u091c\u093f\u090f [be a linear function from Z to Z , find f<\/em>(x)]
Sol :
\u092e\u093e\u0928\u093e \u0935\u0939 \u0930\u0947\u0916\u093f\u0915 \u092b\u0932\u0928y=f<\/em>(x)=ax+b \u0939\u0948 \u0964<\/p>\n\n\n\n

y=ax+b<\/p>\n\n\n\n

(0,-1)\u2208f<\/em><\/p>\n\n\n\n

-1=a(0)+b<\/p>\n\n\n\n

-1=b<\/p>\n\n\n\n

(1,1)\u2208f<\/em><\/p>\n\n\n\n

1=a(1)+b<\/p>\n\n\n\n

1=a+b<\/p>\n\n\n\n

1=a-1<\/p>\n\n\n\n

2=a<\/p>\n\n\n\n

\u2234 \u0905\u092d\u093f\u0937\u094d\u091f \u092b\u0932\u0928<\/p>\n\n\n\n

f<\/em>(x)=2x-1<\/p>\n\n\n\n

Question 14<\/h4>\n\n\n\n

\u092e\u093e\u0928 \u0932\u0940\u091c\u093f\u090f \u0915\u093f A={9,10,11,12,13} \u0924\u0925\u093e f<\/em>:A\u2192N,f<\/em>(n)=n \u0915\u093e \u092e\u0939\u0924\u094d\u0924\u0935 \u0905\u092d\u093e\u091c\u094d\u092f \u0917\u0941\u0923\u0928\u0916\u0902\u0921 \u0926\u094d\u0935\u093e\u0930\u093e \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0939\u0948 \u0964 f<\/em> \u0915\u093e \u092a\u0930\u093f\u0938\u0930 \u091c\u094d\u091e\u093e\u0924 \u0915\u0940\u091c\u093f\u090f
Sol :
f<\/em>={(9,3),(10,5),(11,11),(12,3),(13,13)}<\/p>\n\n\n\n

f<\/em> \u0915\u093e \u092a\u0930\u093f\u0938\u0930={3,5,11,13}<\/p>\n\n\n\n

Question 15<\/h4>\n\n\n\n

\u092f\u0926\u093f A={-3,-2,-1,0,1,2,3} \u0914\u0930 f<\/em>(x)=x2<\/sup>-1,f<\/em>:A\u2192R \u0915\u094b \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0915\u0930\u0924\u093e \u0939\u0948 , \u0924\u094b f<\/em> \u0915\u093e \u092a\u0930\u093f\u0938\u0930 \u091c\u094d\u091e\u093e\u0924 \u0915\u0940\u091c\u093f\u090f
Sol :
f<\/em>(-3)=(-3)2<\/sup>-1=8<\/p>\n\n\n\n

f<\/em>(-2)=(-2)2<\/sup>-1=3<\/p>\n\n\n\n

f<\/em>(-1)=(-1)2<\/sup>-1=0<\/p>\n\n\n\n

f<\/em>(0)=(0)2<\/sup>-1=-1<\/p>\n\n\n\n

f<\/em>(1)=(1)2<\/sup>-1=0<\/p>\n\n\n\n

f<\/em>(2)=(2)2<\/sup>-1=3<\/p>\n\n\n\n

f<\/em>(3)=(3)2<\/sup>-1=8<\/p>\n\n\n\n

f<\/em>={(-3,8),(-2,3),(0,-1),(1,0),(2,3),(3,8)}<\/p>\n\n\n\n

\u092a\u0930\u093f\u0938\u0930={-1,0,3,8}<\/p>\n\n\n\n

Question 16<\/h4>\n\n\n\n

\u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u092b\u0932\u0928\u094b \u0915\u093e \u092a\u094d\u0930\u093e\u0902\u0924 \u0924\u0925\u093e \u092a\u0930\u093f\u0938\u0930 \u091c\u094d\u091e\u093e\u0924 \u0915\u0940\u091c\u093f\u090f
Find the domain and range of the following functions
(i)<\/strong> f<\/em>(x)=x
Sol :
\u092a\u094d\u0930\u093e\u0902\u0924=R, \u092a\u0930\u093f\u0938\u0930=R
f<\/em>(x)=x,x\u2208R \u0915\u0947 \u0932\u093f\u090f ,f<\/em>(2) \u0915\u093e \u092e\u0940\u0928 \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u0939\u094b\u0917\u093e<\/p>\n\n\n\n

\u2234\u092a\u094d\u0930\u093e\u0902\u0924=R<\/p>\n\n\n\n

Let y=f<\/em>(x)\u21d2y=x,x \u0915\u0947 \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u092e\u093e\u0928 \u0915\u0947 \u0932\u093f\u090f , y\u2208R \u092a\u0930\u093f\u0938\u0930=R<\/p>\n\n\n\n

(ii)<\/strong> f<\/em>(x)=2-3x
Sol :
f<\/em>(x)=2-3x<\/p>\n\n\n\n

x\u2208R \u092a\u0930 , f<\/em>(x) \u0915\u093e \u092e\u093e\u0928 \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u0939\u094b\u0917\u093e<\/p>\n\n\n\n

\u2234\u092a\u094d\u0930\u093e\u0902\u0924=R<\/p>\n\n\n\n

\u092e\u093e\u0928\u093e y=f<\/em>(x)
y=2-3x\u21d23x=2-y
$x=\\frac{2-y}{3}$<\/p>\n\n\n\n

x \u0915\u0947 \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u092e\u093e\u0928 \u0915\u0947 \u0932\u093f\u090f , y\u2208R
\u092a\u0930\u093f\u0938\u0930=R<\/p>\n\n\n\n

(iii)<\/strong> f<\/em>(x)=x2<\/sup>-1
Sol :
f<\/em>(x)=x2<\/sup>-1<\/p>\n\n\n\n

x\u2208R, \u092a\u0930 f<\/em>(x) \u0915\u093e \u092e\u093e\u0928 \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u0939\u094b\u0917\u093e<\/p>\n\n\n\n

\u092a\u094d\u0930\u093e\u0902\u0924=R<\/p>\n\n\n\n

\u092e\u093e\u0928\u093e y=f<\/em>(x)<\/p>\n\n\n\n

y=x2<\/sup>-1<\/p>\n\n\n\n

y+1=x2<\/sup><\/p>\n\n\n\n

$\\sqrt{y+1}=x$<\/p>\n\n\n\n

x \u0915\u0947 \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u092e\u093e\u0928 \u0915\u0947 \u0932\u093f\u090f<\/p>\n\n\n\n

y+1\u22650<\/p>\n\n\n\n

y\u2265-1<\/p>\n\n\n\n

\u092a\u0930\u093f\u0938\u0930=[-1,\u221e]
(iv)<\/strong> f(x)=x2<\/sup>+2
Sol :
f<\/em>(x)=x2<\/sup>+2
f<\/em>(x) \u0915\u0947 \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u092e\u093e\u0928 \u0915\u0947 \u0932\u093f\u090f ,<\/p>\n\n\n\n

x\u2208R<\/p>\n\n\n\n

\u092a\u094d\u0930\u093e\u0902\u0924=R<\/p>\n\n\n\n

\u092e\u093e\u0928\u093e y=f<\/em>(x)\u21d2y=x2<\/sup>+2<\/p>\n\n\n\n

y-2=x2<\/sup><\/p>\n\n\n\n

x \u0915\u0947 \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u092e\u093e\u0928 \u0915\u0947 \u0932\u093f\u090f<\/p>\n\n\n\n

y-2\u22650<\/p>\n\n\n\n

y\u22652<\/p>\n\n\n\n

\u092a\u0930\u093f\u0938\u0930=[2,\u221e)
(v)<\/strong> f<\/em>(x)=$\\sqrt{x-1}$
Sol :
$f(x)=\\sqrt{x-1}$<\/p>\n\n\n\n

f<\/em>(x) \u0915\u0947 \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u092e\u093e\u0928 \u0915\u0947 \u0932\u093f\u090f,<\/p>\n\n\n\n

x-1\u22650<\/p>\n\n\n\n

x\u22651<\/p>\n\n\n\n

\u092a\u094d\u0930\u093e\u0902\u0924=[1,\u221e)<\/p>\n\n\n\n

\u092e\u093e\u0928\u093e y=f<\/em>(x)\u21d2$y=\\sqrt{x-1}$<\/p>\n\n\n\n

y2<\/sup>=x-1<\/p>\n\n\n\n

y2<\/sup>+1=x<\/p>\n\n\n\n

x  \u0915\u0947 \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u092e\u093e\u0928 \u0915\u0947 \u0932\u093f\u090f,<\/p>\n\n\n\n

y\u2208R+<\/sup><\/p>\n\n\n\n

\u092a\u0930\u093f\u0938\u0930=[0,\u221e)<\/p>\n\n\n\n

Question 17<\/h4>\n\n\n\n

\u092b\u0932\u0928 $f(x)=\\frac{x^{2}+3 x+5}{x^{2}-5 x+4}$ \u0915\u093e \u092a\u094d\u0930\u093e\u0902\u0924 \u091c\u094d\u091e\u093e\u0924 \u0915\u0940\u091c\u093f\u090f
Sol :
f<\/em>(x) \u0915\u0947 \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u092e\u093e\u0928 \u0915\u0947 \u0932\u093f\u090f,<\/p>\n\n\n\n

x\u2208R-{x:x2<\/sup>-5x+4=0}<\/p>\n\n\n\n

x\u2208R-{1,4}<\/p>\n\n\n\n

\u092a\u094d\u0930\u093e\u0902\u0924=R-{1,4}<\/p>\n\n\n\n

Question 18<\/h4>\n\n\n\n

\u092b\u0932\u0928 f<\/em>(x)=2-3x,x>0 \u0915\u093e \u092a\u0930\u093f\u0938\u0930 \u091c\u094d\u091e\u093e\u0924 \u0915\u0940\u091c\u093f\u090f
Find the range of the function f<\/em>(x)=2-3x,x>0
Sol :
\u092e\u093e\u0928\u093e y=f<\/em>(x)<\/p>\n\n\n\n

y=2-3x<\/p>\n\n\n\n

3x=2-y<\/p>\n\n\n\n

$x=\\frac{2-y}{3}>0$, y\u2208(-\u221e,2)<\/p>\n\n\n\n

\u092a\u0930\u093f\u0938\u0930=(-\u221e,2)<\/p>\n\n\n\n

Question 19<\/h4>\n\n\n\n

\u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u092b\u0932\u0928\u094b \u0915\u093e \u092a\u094d\u0930\u093e\u0902\u0924 \u0924\u0925\u093e \u092a\u0930\u093f\u0938\u0930 \u091c\u094d\u091e\u093e\u0924 \u0915\u0940\u091c\u093f\u090f
Find the domain and range of the following functions
(i)<\/strong> {(x,$\\sqrt{9-x^{2}}$):x\u2208R}
Sol :
f<\/em>(x)=$\\sqrt{9-x^{2}}$<\/p>\n\n\n\n

f<\/em>(x) \u0915\u0947 \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u092e\u093e\u0928 \u0915\u0947 \u0932\u093f\u090f<\/p>\n\n\n\n

9-x2<\/sup>\u22650<\/p>\n\n\n\n

-x2<\/sup>\u22640<\/p>\n\n\n\n

x2<\/sup>\u22659<\/p>\n\n\n\n

-3\u2264x\u22643<\/p>\n\n\n\n

\u092a\u094d\u0930\u093e\u0924=[-3,3]<\/p>\n\n\n\n

\u092e\u093e\u0928\u093e y=f(x)<\/p>\n\n\n\n

$y=\\sqrt{9-x^{2}}$<\/p>\n\n\n\n

x2<\/sup>=9-y2<\/sup><\/p>\n\n\n\n

$x=\\sqrt{9-y^{2}}$<\/p>\n\n\n\n

x \u0915\u0947 \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u092e\u093e\u0928 \u0915\u0947 \u0932\u093f\u090f<\/p>\n\n\n\n

9-y2<\/sup>\u22650<\/p>\n\n\n\n

-y2<\/sup>\u2265-9\u21d2y2<\/sup>\u22649<\/p>\n\n\n\n

-3\u2264y\u22643<\/p>\n\n\n\n

\u092a\u0930\u093f\u0938\u0930=[0,3] (\u2235y\u22650)<\/p>\n\n\n\n

(ii)<\/strong> {(x,-|x|):x\u2208R}
Sol :
f<\/em>(x)=-|x|<\/p>\n\n\n\n

f<\/em>(x) \u0915\u0947 \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u092e\u093e\u0928 \u0915\u0947 \u0932\u093f\u090f<\/p>\n\n\n\n

x\u2208R<\/p>\n\n\n\n

\u092a\u094d\u0930\u093e\u0902\u0924=R<\/p>\n\n\n\n

x\u2208R,f<\/em>(x)[-\u221e,0]<\/p>\n\n\n\n

\u092a\u0930\u093f\u0938\u0930=[-\u221e,0]<\/p>\n\n\n\n

Question 20<\/h4>\n\n\n\n

\u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0928\u093f\u092f\u092e\u094b \u0926\u094d\u0935\u093e\u0930\u093e \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u092b\u0932\u0928 \u0915\u093e \u092a\u0930\u093f\u092d\u093e\u0937\u093e, \u092a\u094d\u0930\u093e\u0902\u0924 \u0924\u0925\u093e \u092a\u0930\u093f\u0938\u0930 \u091c\u094d\u091e\u093e\u0924 \u0915\u0940\u091c\u093f\u090f
Find the domain of definition and range of the function defined by the rules:
(i)<\/strong>  f<\/em>(x)=x2<\/sup><\/p>\n\n\n\n

(ii)<\/strong> g(x)=|x|<\/p>\n\n\n\n

(iii)<\/strong> $h(x)=\\frac{1}{3-x^{2}}$
Sol :
$h(x)=\\frac{1}{3-x^{2}}$<\/p>\n\n\n\n

h(x) \u092e\u0947 \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u092e\u093e\u0928 \u0915\u0947 \u0932\u093f\u090f ,<\/p>\n\n\n\n

x\u2208R-{x:3-x2<\/sup>=0}<\/p>\n\n\n\n

x\u2208R-{-\u221a3,3}<\/p>\n\n\n\n

\u092a\u094d\u0930\u093e\u0902\u0924=R-{-\u221a3,\u221a3}<\/p>\n\n\n\n

\u092e\u093e\u0928\u093e y=f<\/em>(x)<\/p>\n\n\n\n

$y=\\frac{1}{3-x^{2}}$<\/p>\n\n\n\n

3y-x2<\/sup>y=1<\/p>\n\n\n\n

3y-1=x2<\/sup>y<\/p>\n\n\n\n

$\\frac{3 y-1}{y}=x^{2}$<\/p>\n\n\n\n

$\\sqrt{\\frac{3 y-1}{y}}=x$<\/p>\n\n\n\n

x \u0915\u0947 \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u092e\u093e\u0928 \u0915\u0947 \u0932\u093f\u090f<\/p>\n\n\n\n

3y-1\u22650,y\u22600<\/p>\n\n\n\n

3y\u22651,y<0<\/p>\n\n\n\n

$y \\geq \\frac{1}{3}$,<\/p>\n\n\n\n

\u092a\u0930\u093f\u0938\u0930=(-\u221e,0)\u222a$\\left[\\frac{1}{3}, \\infty\\right)$<\/p>\n\n\n\n

(iv)<\/strong> u(x)=$\\sqrt{4-x^{2}}$
Sol :<\/p>\n\n\n\n

Question 21<\/h4>\n\n\n\n

\u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0928\u093f\u092f\u092e\u094b \u092a\u0930 \u0935\u093f\u091a\u093e\u0930 \u0915\u0940\u091c\u093f\u090f<\/p>\n\n\n\n

Consider the following rules:
(i)<\/strong> f<\/em>:R\u2192R:f<\/em>(x)=loge<\/sub>x
Sol :
f \u0914\u0930 g \u092b\u0932\u0928 \u0928\u0939\u0940 \u0939\u0948<\/p>\n\n\n\n

(ii)<\/strong> g:R\u2192R:g<\/em>(x)=\u221ax
Sol :
x \u0915\u0947 \u090b\u0923\u093e\u0924\u094d\u092e\u0915 \u092e\u093e\u0928\u094b \u0915\u0947 \u0932\u093f\u090f \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0928\u0939\u0940 \u0939\u0948<\/p>\n\n\n\n

(iii)<\/strong> h:A\u2192R:h(x)$=\\frac{1}{x^{2}-4}$ , \u091c\u0939\u093e\u0901 (where) A=R-{-2,2}\u0909\u0928\u092e\u0947 \u0915\u094c\u0928 \u092b\u0932\u0928 \u0939\u0948 ? \u0909\u0928\u0915\u093e \u092a\u0930\u093f\u0938\u0930 \u091c\u094d\u091e\u093e\u0924 \u0915\u0940\u091c\u093f\u090f \u092f\u0926\u093f \u0935\u0947 \u092b\u0932\u0928 \u0939\u0948 \u0964
Sol :
h(x) \u0915\u0947 \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u092e\u093e\u0928 \u0915\u0947 \u0932\u093f\u090f<\/p>\n\n\n\n

x2<\/sup>-4\u22600<\/p>\n\n\n\n

x2<\/sup>\u22604\u21d2x\u2260-2,2<\/p>\n\n\n\n

h \u090f\u0915 \u092b\u0932\u0928 \u0939\u0948 \u0964<\/p>\n\n\n\n

\u092e\u093e\u0928\u093e y=f<\/em>(x)\u21d2$y=\\frac{1}{x^{2}-4}$<\/p>\n\n\n\n

x2<\/sup>y-4y=1<\/p>\n\n\n\n

x2<\/sup>y=1+4y<\/p>\n\n\n\n

$x^{2}=\\frac{1+4 y}{y}$<\/p>\n\n\n\n

$x=\\sqrt{\\frac{1+4 y}{y}}$<\/p>\n\n\n\n

$x^{2}=\\frac{1+4 y}{y}$<\/p>\n\n\n\n

$x=\\sqrt{\\frac{1+4y}{y}}$<\/p>\n\n\n\n

x \u0915\u0947 \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u092e\u093e\u0928 \u0915\u0947 \u0932\u093f\u090f ,<\/p>\n\n\n\n

1+4y\u22640,y\u22600<\/p>\n\n\n\n

\u092f\u093e y>0<\/p>\n\n\n\n

1+4y\u22640<\/p>\n\n\n\n

4y\u2264-1<\/p>\n\n\n\n

$y \\leq-\\frac{1}{4}$<\/p>\n\n\n\n

\u092a\u0930\u093f\u0938\u0930$=\\left(-\\infty,-\\frac{1}{4}\\right] \\cup(0,\\infty)$<\/p>\n\n\n\n

Question 22<\/h4>\n\n\n\n

\u092e\u093e\u0928 \u0932\u0940\u091c\u093f\u090f $f(x)=\\frac{x^{2}-4}{x-2}$ \u0926\u094d\u0935\u093e\u0930\u093e
f<\/em>:R-{2}\u2192R, \u0924\u0925\u093e g(x)=x+2 g:R\u2192R \u0938\u0947 \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0939\u0948 \u0964 \u091c\u094d\u091e\u093e\u0924 \u0915\u0940\u091c\u093f\u090f f<\/em>=g \u0939\u0948 \u092f\u093e \u0928\u0939\u0940\u0902 \u0964
Sol :
Domain(f<\/em>)=R-{2}<\/p>\n\n\n\n

Domain(g)=R<\/p>\n\n\n\n

f\u2260<\/em>g<\/p>\n\n\n\n

Question 23<\/h4>\n\n\n\n

\u092e\u093e\u0928 \u0932\u0940\u091c\u093f\u090f \u0915\u093f f<\/em>:R\u2192R \u0915\u094d\u0930\u092e\u0936 f<\/em>(x)=x+1,g(x)=2x-3 \u0926\u094d\u0935\u093e\u0930\u093e \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0939\u0948
 f+g,f-g \u0914\u0930 $\\frac{f}{g}$ \u091c\u094d\u091e\u093e\u0924 \u0915\u0940\u091c\u093f\u090f
Sol :
(f<\/em>+g)(x)=f<\/em>(x)+g(x)<\/p>\n\n\n\n

=x+1+2x-3<\/p>\n\n\n\n

=3x-2, \u091c\u0939\u093e f<\/em>+g:R\u2192R<\/p>\n\n\n\n

(f<\/em>-g)(x)=f<\/em>(x)-g(x)<\/p>\n\n\n\n

=x+1-2x+3<\/p>\n\n\n\n

=-x+4 , \u091c\u0939\u093e\u0901 f<\/em>-g:R\u2192R<\/p>\n\n\n\n

$\\left(\\frac{f}{g}\\right)(x)=\\frac{f(x)}{g(x)}$<\/p>\n\n\n\n

$=\\frac{x+1}{2 x-3}$ , 2x-3\u22600<\/p>\n\n\n\n

$=\\frac{x+1}{2 x-3}, \\quad x \\neq \\frac{3}{2}$<\/p>\n\n\n\n

\u091c\u0939\u093e\u0901 $\\frac{f}{g}: R-\\left\\{\\frac{3}{2}\\right\\} \\rightarrow R$<\/p>\n\n\n\n

Question 24<\/h4>\n\n\n\n

\u092e\u093e\u0928 \u0932\u0940\u091c\u093f\u090f \u0915\u093f f<\/em>={(1,1),(2,3),(0,-1),(-1,-3)}Z \u0938\u0947 Z \u092e\u0947 \u090f\u0915 \u0930\u0948\u0916\u093f\u0915 \u092b\u0932\u0928 \u0939\u0948 \u0914\u0930 g(x)=x , \u0924\u094b f+g \u091c\u094d\u091e\u093e\u0924 \u0915\u0940\u091c\u093f\u090f
Sol :
(f+g)={(1,2),(2,5),(0,-1),(-1,-4)}<\/p>\n\n\n\n

Question 25<\/h4>\n\n\n\n

\u091c\u094d\u091e\u093e\u0924 \u0915\u0940\u091c\u093f\u090f (Find) f+g,f-g,f.g,$\\frac{f}{g}$ \u0914\u0930(and) \u03b1f(\u03b1\u2208R) \u092f\u0926\u093f(if)
(i)<\/strong> $f(x)=\\frac{1}{x+4}, x \\neq . .4$ \u0914\u0930(and) g(x)=(x+4)3<\/sup>
Sol :
(f<\/em>+g)(x)=f<\/em>(x)+g(x)<\/p>\n\n\n\n

$=\\frac{1}{x+4}+(x+4)^{3}$<\/p>\n\n\n\n

$=\\frac{1+(x+4)^{4}}{x+4}$,x\u2260-4<\/p>\n\n\n\n

(f<\/em>-g)(x)=f<\/em>(x)-g(x)<\/p>\n\n\n\n

$=\\frac{1}{(x+4)}-(x+4)^{3}$<\/p>\n\n\n\n

$=\\frac{1-(x+4)^{4}}{x+4}$,x\u2260-4<\/p>\n\n\n\n

(f<\/em>.g)(x)=f<\/em>(x).g(x)<\/p>\n\n\n\n

$=\\left(\\frac{1}{x+4}\\right) \\times(x+4)^{3}$<\/p>\n\n\n\n

=(x+4)2<\/sup>
$\\left(\\frac{f}{g} \\right)(x)=\\frac{f(x)}{g(x)}$<\/p>\n\n\n\n

$=\\frac{\\frac{1}{x\\neq4}}{(x+4)^{3}}=\\frac{1}{(x+4)^{4}}$,x\u2260-4<\/p>\n\n\n\n

\u03b1.f<\/em>=$\\alpha \\cdot\\left(\\frac{1}{x+4}\\right)$<\/p>\n\n\n\n

$=\\frac{\\alpha}{x+4}$ ,x\u2260-4<\/p>\n\n\n\n

(ii)<\/strong> f<\/em>(x)=cosx,g(x)=ex<\/sup>
Sol :<\/p>\n\n\n\n

f<\/em>(x)=cosx, g(x)=ex<\/sup><\/p>\n\n\n\n

(f<\/em>+g)(x)=f<\/em>(x)+g(x)<\/p>\n\n\n\n

=cosx+ex<\/sup><\/p>\n\n\n\n

(f<\/em>-g)(x)=f<\/em>(x)-g(x)
=cosx-ex<\/sup><\/p>\n\n\n\n

(f.g)x=f(x).g(x)
=cosx.ex<\/sup><\/p>\n\n\n\n

$\\left(\\frac{f}{g}\\right) x=\\frac{f(x)}{g(x)}$<\/p>\n\n\n\n

$=\\frac{\\cos x}{e^{x}}$<\/p>\n\n\n\n

=e-x<\/sup>.cosx<\/p>\n\n\n\n

\u03b1.f<\/em>(x)=\u03b1.cosx<\/p>\n\n\n\n

(iii)<\/strong> f(x)=$\\sqrt{x-1}$ , g(x)=$\\sqrt{x+1}$
Sol :
(f<\/em>+g)(x)=f<\/em>(x)+g(x)<\/p>\n\n\n\n

$=\\sqrt{x-1}+\\sqrt{x+1}$ ,x\u22651<\/p>\n\n\n\n

(f<\/em>-g)(x)=f<\/em>(x)-g(x)
$=\\sqrt{x-1}-\\sqrt{x+1}$,x\u22651<\/p>\n\n\n\n

(f.g)(x)=$\\sqrt{x-1} \\cdot \\sqrt{x+1}$,x\u22651
$=\\sqrt{x^{2}-1^{2}}$,x\u22651
$=\\sqrt{x^{2}-1}$,x\u22651<\/p>\n\n\n\n

$\\left(\\frac{f}{g}\\right)(x)=\\frac{f(x)}{g(x)}$
$=\\frac{\\sqrt{x-1}}{\\sqrt{x+1}}$,x\u22651<\/p>\n\n\n\n

\u03b1.f<\/em>(x)=\u03b1$\\sqrt{x-1}$,x\u22651<\/p>\n\n\n\n

Question 26<\/h4>\n\n\n\n

\u092f\u0926\u093f (If) f<\/em>(x)=x,g(x)=|x| , \u091c\u094d\u091e\u093e\u0924 \u0915\u0940\u091c\u093f\u090f (find)(f+g)(-2),(f-g)(2),(f.g)(2),$\\frac{f}{g}(2), 5 f(2)$
Sol :
(f<\/em>+g)(x)=f<\/em>(x)+g(x)<\/p>\n\n\n\n

=x+|x|<\/p>\n\n\n\n

(f<\/em>+g)(-2)=-2+|-2|
=-2+2
=0<\/p>\n\n\n\n

(f<\/em>-g)(x)=f<\/em>(x)-g(x)
=x-|x|<\/p>\n\n\n\n

(f<\/em>-g)(2)=2-|2|=2-2=0<\/p>\n\n\n\n

(f<\/em>.g)(x)=f<\/em>(x).g(x)
=x.|x|<\/p>\n\n\n\n

(f<\/em>.g)(2)=2.|2|
=2\u00d72=4<\/p>\n\n\n\n

$\\left(\\frac{f}{g}\\right)(x)=\\frac{f(x)}{g(x)}$<\/p>\n\n\n\n

$=\\frac{x}{|x|}$<\/p>\n\n\n\n

$\\left(\\frac{f}{g}\\right)(-2)=\\frac{-2}{|-2|}$<\/p>\n\n\n\n

$=\\frac{-2}{2}=-1$<\/p>\n\n\n\n

5f<\/em>(2)=5\u00d72=10<\/p>\n\n\n\n

f<\/em>(x)=x
f<\/em>(2)=2<\/p>\n\n\n\n

Question 27<\/h4>\n\n\n\n
\"\"\/<\/a><\/figure>\n\n\n\n
\"\"\/<\/a><\/figure>\n\n\n\n
\"\"\/<\/a><\/figure>\n\n\n\n
\"\"\/<\/a><\/figure>\n\n\n\n

Question 28<\/h4>\n\n\n\n
\"\"\/<\/a><\/figure>\n\n\n\n

Question 29<\/h4>\n\n\n\n
\"\"\/<\/a><\/figure>\n\n\n\n
\"\"\/<\/a><\/figure>\n\n\n\n

Question 30<\/h4>\n\n\n\n
\"\"\/<\/a><\/figure>\n\n\n\n
\"\"\/<\/a><\/figure>\n\n\n\n
\"\"\/<\/a><\/figure>\n\n\n\n
\"\"\/<\/a><\/figure>\n\n\n\n
\n
KC Sinha Solutions<\/a><\/div>\n\n\n\n
KC Sinha Class 11 Solutions<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Question 1 \u091c\u093e\u0901\u091a \u0915\u0940\u091c\u093f\u090f \u0915\u093f \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0938\u0902\u092c\u0902\u0927\u094b \u092e\u0947 \u0915\u094c\u0928 \u092b\u0932\u0928 \u0939\u0948? \u092f\u0926\u093f \u0935\u0947 \u092b\u0932\u0928 \u0939\u0948 \u0924\u094b \u0909\u0928\u0915\u0947 \u092a\u094d\u0930\u093e\u0902\u0924 \u0924\u0925\u093e \u092a\u0930\u093f\u0938\u0930 \u091c\u094d\u091e\u093e\u0924 \u0915\u0940\u091c\u093f\u090f\u0964 (i) f={(1,4),(2,3),(3,4),(4,3)} (ii) g={(2,5),(-1,0),(1,6)} (iii) h={(1,2),(2,2),(3,2)} (iv) \u0444={(1,2),(1,3),(2,5)} (v) \u03c8={(2,1),(3,1),(5,2)} (vi) u={(2,1),(5,1),(8,1),(11,1),(14,1),(17,1)} (vii) v={(0,0),(1,1),(1,-1),(4,2),(4,-2),(9,3),(9,-3),(16,4),(16,-4)} Sol : (i) f \u090f\u0915 \u092b\u0932\u092e\u0928 \u0939\u0948, \u092a\u094d\u0930\u093e\u0902\u0924={1,2,3,4} \u092a\u0930\u093f\u0938\u0930={3,4} (ii) g \u090f\u0915 \u092b\u0932\u0928 \u0939\u0948, \u092a\u094d\u0930\u093e\u0902\u0924={2,-1,1} \u092a\u0930\u093f\u0938\u0930={5,0,6} (iii) h \u090f\u0915 […]<\/p>\n","protected":false},"author":302,"featured_media":625792,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"newspack_featured_image_position":"","newspack_post_subtitle":"","newspack_article_summary_title":"Overview:","newspack_article_summary":"","newspack_hide_updated_date":false,"newspack_show_updated_date":false,"footnotes":""},"categories":[919],"tags":[],"boards":[],"class_list":["post-625790","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-class-11","entry"],"yoast_head":"\nKC Sinha: Exercise 3.1 - Mathematics Solution Class 11 Chapter 3 \u092b\u0932\u0928 - IndCareer Schools<\/title>\n<meta name=\"description\" content=\"Question 1 \u091c\u093e\u0901\u091a \u0915\u0940\u091c\u093f\u090f \u0915\u093f \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0938\u0902\u092c\u0902\u0927\u094b \u092e\u0947 \u0915\u094c\u0928 \u092b\u0932\u0928 \u0939\u0948? \u092f\u0926\u093f \u0935\u0947 \u092b\u0932\u0928 \u0939\u0948 \u0924\u094b \u0909\u0928\u0915\u0947 \u092a\u094d\u0930\u093e\u0902\u0924 \u0924\u0925\u093e \u092a\u0930\u093f\u0938\u0930 \u091c\u094d\u091e\u093e\u0924 \u0915\u0940\u091c\u093f\u090f\u0964 (i) f={(1,4),(2,3),(3,4),(4,3)} (ii)\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-3-1-mathematics-solution-class-11-chapter-3-phalan\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"KC Sinha: Exercise 3.1 - Mathematics Solution Class 11 Chapter 3 \u092b\u0932\u0928\" \/>\n<meta property=\"og:description\" content=\"Question 1 \u091c\u093e\u0901\u091a \u0915\u0940\u091c\u093f\u090f \u0915\u093f \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0938\u0902\u092c\u0902\u0927\u094b \u092e\u0947 \u0915\u094c\u0928 \u092b\u0932\u0928 \u0939\u0948? \u092f\u0926\u093f \u0935\u0947 \u092b\u0932\u0928 \u0939\u0948 \u0924\u094b \u0909\u0928\u0915\u0947 \u092a\u094d\u0930\u093e\u0902\u0924 \u0924\u0925\u093e \u092a\u0930\u093f\u0938\u0930 \u091c\u094d\u091e\u093e\u0924 \u0915\u0940\u091c\u093f\u090f\u0964 (i) f={(1,4),(2,3),(3,4),(4,3)} (ii)\" \/>\n<meta property=\"og:url\" content=\"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-3-1-mathematics-solution-class-11-chapter-3-phalan\/\" \/>\n<meta property=\"og:site_name\" content=\"IndCareer Schools\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/indcareer\" \/>\n<meta property=\"article:published_time\" content=\"2023-09-08T05:51:09+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2023-09-08T05:51:25+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2023\/09\/indcareer-schools-2-4-2-scaled.jpg\" \/>\n\t<meta property=\"og:image:width\" content=\"1600\" \/>\n\t<meta property=\"og:image:height\" content=\"901\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/jpeg\" \/>\n<meta name=\"author\" content=\"Pooja\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:creator\" content=\"@indcareer\" \/>\n<meta name=\"twitter:site\" content=\"@indcareer\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"Pooja\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"9 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-3-1-mathematics-solution-class-11-chapter-3-phalan\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-3-1-mathematics-solution-class-11-chapter-3-phalan\/\"},\"author\":{\"name\":\"Pooja\",\"@id\":\"https:\/\/www.indcareer.com\/schools\/#\/schema\/person\/d6945cf059726f162259ba738092301e\"},\"headline\":\"KC Sinha: Exercise 3.1 – Mathematics Solution Class 11 Chapter 3 \u092b\u0932\u0928\",\"datePublished\":\"2023-09-08T05:51:09+00:00\",\"dateModified\":\"2023-09-08T05:51:25+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-3-1-mathematics-solution-class-11-chapter-3-phalan\/\"},\"wordCount\":3206,\"publisher\":{\"@id\":\"https:\/\/www.indcareer.com\/schools\/#organization\"},\"image\":{\"@id\":\"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-3-1-mathematics-solution-class-11-chapter-3-phalan\/#primaryimage\"},\"thumbnailUrl\":\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2023\/09\/indcareer-schools-2-4-2-scaled.jpg\",\"articleSection\":[\"class 11\"],\"inLanguage\":\"en-US\"},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-3-1-mathematics-solution-class-11-chapter-3-phalan\/\",\"url\":\"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-3-1-mathematics-solution-class-11-chapter-3-phalan\/\",\"name\":\"KC Sinha: Exercise 3.1 - Mathematics Solution Class 11 Chapter 3 \u092b\u0932\u0928 - IndCareer Schools\",\"isPartOf\":{\"@id\":\"https:\/\/www.indcareer.com\/schools\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-3-1-mathematics-solution-class-11-chapter-3-phalan\/#primaryimage\"},\"image\":{\"@id\":\"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-3-1-mathematics-solution-class-11-chapter-3-phalan\/#primaryimage\"},\"thumbnailUrl\":\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2023\/09\/indcareer-schools-2-4-2-scaled.jpg\",\"datePublished\":\"2023-09-08T05:51:09+00:00\",\"dateModified\":\"2023-09-08T05:51:25+00:00\",\"description\":\"Question 1 \u091c\u093e\u0901\u091a \u0915\u0940\u091c\u093f\u090f \u0915\u093f \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0938\u0902\u092c\u0902\u0927\u094b \u092e\u0947 \u0915\u094c\u0928 \u092b\u0932\u0928 \u0939\u0948? 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\u092f\u0926\u093f \u0935\u0947 \u092b\u0932\u0928 \u0939\u0948 \u0924\u094b \u0909\u0928\u0915\u0947 \u092a\u094d\u0930\u093e\u0902\u0924 \u0924\u0925\u093e \u092a\u0930\u093f\u0938\u0930 \u091c\u094d\u091e\u093e\u0924 \u0915\u0940\u091c\u093f\u090f\u0964 (i) f={(1,4),(2,3),(3,4),(4,3)} (ii)","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-3-1-mathematics-solution-class-11-chapter-3-phalan\/","og_locale":"en_US","og_type":"article","og_title":"KC Sinha: Exercise 3.1 - Mathematics Solution Class 11 Chapter 3 \u092b\u0932\u0928","og_description":"Question 1 \u091c\u093e\u0901\u091a \u0915\u0940\u091c\u093f\u090f \u0915\u093f \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0938\u0902\u092c\u0902\u0927\u094b \u092e\u0947 \u0915\u094c\u0928 \u092b\u0932\u0928 \u0939\u0948? \u092f\u0926\u093f \u0935\u0947 \u092b\u0932\u0928 \u0939\u0948 \u0924\u094b \u0909\u0928\u0915\u0947 \u092a\u094d\u0930\u093e\u0902\u0924 \u0924\u0925\u093e \u092a\u0930\u093f\u0938\u0930 \u091c\u094d\u091e\u093e\u0924 \u0915\u0940\u091c\u093f\u090f\u0964 (i) f={(1,4),(2,3),(3,4),(4,3)} (ii)","og_url":"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-3-1-mathematics-solution-class-11-chapter-3-phalan\/","og_site_name":"IndCareer Schools","article_publisher":"https:\/\/www.facebook.com\/indcareer","article_published_time":"2023-09-08T05:51:09+00:00","article_modified_time":"2023-09-08T05:51:25+00:00","og_image":[{"width":1600,"height":901,"url":"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2023\/09\/indcareer-schools-2-4-2-scaled.jpg","type":"image\/jpeg"}],"author":"Pooja","twitter_card":"summary_large_image","twitter_creator":"@indcareer","twitter_site":"@indcareer","twitter_misc":{"Written by":"Pooja","Est. reading time":"9 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-3-1-mathematics-solution-class-11-chapter-3-phalan\/#article","isPartOf":{"@id":"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-3-1-mathematics-solution-class-11-chapter-3-phalan\/"},"author":{"name":"Pooja","@id":"https:\/\/www.indcareer.com\/schools\/#\/schema\/person\/d6945cf059726f162259ba738092301e"},"headline":"KC Sinha: Exercise 3.1 – Mathematics Solution Class 11 Chapter 3 \u092b\u0932\u0928","datePublished":"2023-09-08T05:51:09+00:00","dateModified":"2023-09-08T05:51:25+00:00","mainEntityOfPage":{"@id":"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-3-1-mathematics-solution-class-11-chapter-3-phalan\/"},"wordCount":3206,"publisher":{"@id":"https:\/\/www.indcareer.com\/schools\/#organization"},"image":{"@id":"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-3-1-mathematics-solution-class-11-chapter-3-phalan\/#primaryimage"},"thumbnailUrl":"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2023\/09\/indcareer-schools-2-4-2-scaled.jpg","articleSection":["class 11"],"inLanguage":"en-US"},{"@type":"WebPage","@id":"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-3-1-mathematics-solution-class-11-chapter-3-phalan\/","url":"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-3-1-mathematics-solution-class-11-chapter-3-phalan\/","name":"KC Sinha: Exercise 3.1 - Mathematics Solution Class 11 Chapter 3 \u092b\u0932\u0928 - IndCareer Schools","isPartOf":{"@id":"https:\/\/www.indcareer.com\/schools\/#website"},"primaryImageOfPage":{"@id":"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-3-1-mathematics-solution-class-11-chapter-3-phalan\/#primaryimage"},"image":{"@id":"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-3-1-mathematics-solution-class-11-chapter-3-phalan\/#primaryimage"},"thumbnailUrl":"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2023\/09\/indcareer-schools-2-4-2-scaled.jpg","datePublished":"2023-09-08T05:51:09+00:00","dateModified":"2023-09-08T05:51:25+00:00","description":"Question 1 \u091c\u093e\u0901\u091a \u0915\u0940\u091c\u093f\u090f \u0915\u093f \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0938\u0902\u092c\u0902\u0927\u094b \u092e\u0947 \u0915\u094c\u0928 \u092b\u0932\u0928 \u0939\u0948? 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