The two successive terms in the expansion of
Question: The two successive terms in the expansion of (1 +x)24whose coefficients are in the ratio 1 : 4 are (a) 3rdand 4th (b) 4thand 5th (c) 5thand 6th (d) 6thand 7th Solution: For (1 +x)24two successive terms have coefficients in ration 1 : 4 Let the two successive terms be (r+ 1)thand (r+ 2) terms i. e. $T_{r+1}={ }^{24} C_{r} x^{r}$ and $T_{r+2}={ }^{24} C_{r+1} x^{r+1}$ Given that $\frac{{ }^{24} C_{r}}{{ }^{24} C_{r+1}}=\frac{1}{4}$ i. e. $\frac{(24) !}{r !(24-r) !} \times \frac{(r+1) !(2...
Read More →In the given figure, if x = y and AB = CB, then prove that AE = CD.
Question: In the given figure, ifx=yandAB=CB, then prove thatAE=CD. Solution: Consider the triangles AEB and CDB. $\angle E B A=\angle D B C$ (Common angle) ...(i) Further, we have: $\angle \mathrm{BEA}=180-\mathrm{y}$ $\angle \mathrm{BDC}=180-\mathrm{x}$ Since $\mathrm{x}=\mathrm{y}$, we have $:$ $180-\mathrm{x}=180-\mathrm{y}$ $\Rightarrow \angle \mathrm{BEA}=\angle \mathrm{BDC} \quad \ldots$ (ii) $\mathrm{AB}=\mathrm{CB} \quad$ (Given) ...(iii) From (i), (ii) and (iii), we have: $\triangle B ...
Read More →If the sum of three consecutive terms of an increasing A.P.
Question: If the sum of three consecutive terms of an increasing A.P. is 51 and the product of the first and third of these terms is 273, then the third term is(a) 13(b) 9(c) 21(d) 17 Solution: In the given problem, the sum of three consecutive terms of an A.P is 51 and the product of the first and the third terms is 273. We need to find the third term. Here, Let the three terms bewhere,ais the first term anddis the common difference of the A.P So, $(a-d)+a+(a+d)=51$ $3 a=51$ $a=\frac{51}{3}$ $a...
Read More →Solve this
Question: $\cot \left(\frac{\pi}{4}-2 \cot ^{-1} 3\right)=$ (a) 7 (b) 6 (c) 5 (d) none of these Solution: (a) 7 Let $2 \cot ^{-1} 3=y$ Then, $\cot \frac{y}{2}=3$ $\cot \left(\frac{\pi}{4}-2 \cot ^{-1} 3\right)=\cot \left(\frac{\pi}{4}-y\right)$ $=\frac{\cot \pi / 4 \cot y+1}{\cot y-\cot \pi / 4}$ $=\frac{\cot y+1}{\cot y-1}$ $=\frac{\frac{\cot ^{2} \frac{y}{2}-1}{2 \cot \frac{y}{2}}+1}{\frac{\cot ^{2} \frac{y}{2}-1}{2 \cot \frac{y}{2}}-1}$ $=\frac{\cot ^{2} \frac{y}{2}+2 \cot \frac{y}{2}-1}{\cot...
Read More →Given the integers r > 1, n > 2, and coefficient of
Question: Given the integersr 1,n 2, and coefficient of (3r)thand (r+ 2)ndterms in the binomial expansion of (1 +x)2nare equal, then (a)n= 2r (b)n= 3r (c)n= 2r+ 1 (d) none of these Solution: Givenr 1 andn 2 and coefficient ofT3r= coefficient ofTr+2is expansion of (1 +x)2n i.e. ${ }^{2 n} C_{3 r-1}={ }^{2 n} C_{r+1}$ i. e. $3 r-1=r+1$ (using property of combination) and $2 n-3 r+1=r+1$ i. e. $2 r=2$ and $2 n=4 r$ i. e. $r=1$ and $n=2 r$ Hence, the correct answer is option A....
Read More →ΔABC and ΔDBC are two isosceles triangles on the same base BC and vertices A and D are on the same side of BC.
Question: ΔABCand ΔDBCare two isosceles triangles on the same baseBCand verticesAandDare on the same side ofBC. IfADis extended to intersectBCatE, show that(i) ΔABD ΔACD(ii) ΔABE ΔACE(iii)AEbisects Aas well as D(iv)AEis the perpendicular bisector ofBC. Solution: Given:ΔABCand ΔDBCare two isosceles triangles on the same baseBC.To prove:(i) ΔABD ΔACD(ii) ΔABE ΔACE(iii)AEbisects Aas well as D(iv)AEis the perpendicular bisector ofBCProof:(i) InΔABDand ΔACD,BD=CD (Given,ΔDBCis anisosceles triangles)A...
Read More →If the sum of n terms of an A.P. is 2n2 + 5n, then its nth term is
Question: If the sum of $n$ terms of an A.P. is $2 n^{2}+5 n$, then its $n$th term is (a) 4n 3(b) 3n 4(c) 4n+ 3(d) 3n+ 4 Solution: Here, the sum of firstnterms is given by the expression, $S_{n}=2 n^{2}+5 n$ We need to find thenthterm. So we know that thenthterm of an A.P. is given by, So, $a_{n}=\left(2 n^{2}+5 n\right)-\left[2(n-1)^{2}+5(n-1)\right]$ Using the property, $(a-b)^{2}=a^{2}+b^{2}-2 a b$ We get, $a_{v}=\left(2 n^{2}+5 n\right)-\left[2\left(n^{2}+1-2 n\right)+5(n-1)\right]$ $=\left(...
Read More →If the coefficients of x
Question: If the coefficients ofx2andx3in the expansion of (3 +ax)9are the same, then the value ofais (a) $-\frac{7}{9}$ (b) $-\frac{9}{7}$ (c) $\frac{7}{9}$ (d) $\frac{9}{7}$ Solution: (d) $\frac{9}{7}$ Coefficients ofx2=Coefficients ofx3 ${ }^{9} C_{2} \times 3^{9-2} a^{2}={ }^{9} C_{3} \times 3^{9-3} a^{3}$ $\Rightarrow a=\frac{{ }^{9} C_{2}}{{ }^{9} C_{3}} \times 3$ $=\frac{9 ! \times 3 ! \times 6 ! \times 3}{2 ! \times 7 ! \times 9 !}$ $=\frac{9}{7}$...
Read More →The value of sin
Question: The value of $\sin \left(\frac{1}{4} \sin ^{-1} \frac{\sqrt{63}}{8}\right)$ is (a) $\frac{1}{\sqrt{2}}$ (b) $\frac{1}{\sqrt{3}}$ (c) $\frac{1}{2 \sqrt{2}}$ (d) $\frac{1}{3 \sqrt{3}}$ Solution: (C) $\frac{1}{2 \sqrt{2}}$ Let $\sin ^{-1} \frac{\sqrt{63}}{8}=y$ Then, $\sin y=\frac{\sqrt{63}}{8}$ $\cos y=\sqrt{1-\sin ^{2} y}=\sqrt{1-\frac{63}{64}}=\frac{1}{8}$ Now, we have $\sin \left(\frac{1}{4} \sin ^{-1} \frac{\sqrt{63}}{8}\right)=\sin \left(\frac{1}{4} y\right)$ $=\sqrt{\frac{1-\cos \f...
Read More →If the coefficients of x
Question: If the coefficients ofx2andx3in the expansion of (3 +ax)9are the same, then the value ofais (a) $-\frac{7}{9}$ (b) $-\frac{9}{7}$ (c) $\frac{7}{9}$ (d) $\frac{9}{7}$ Solution: (d) $\frac{9}{7}$ Coefficients ofx2=Coefficients ofx3 ${ }^{9} C_{2} \times 3^{9-2} a^{2}={ }^{9} C_{3} \times 3^{9-3} a^{3}$ $\Rightarrow a=\frac{{ }^{9} C_{2}}{{ }^{9} C_{3}} \times 3$ $=\frac{9 ! \times 3 ! \times 6 ! \times 3}{2 ! \times 7 ! \times 9 !}$ $=\frac{9}{7}$...
Read More →In the given figure, BE and CF are two equal altitudes of ΔABC.
Question: In the given figure, BE and CF are two equal altitudes of ΔABC.Show that (i) ΔABE ΔACF, (ii)AB = AC. Solution: In $\triangle \mathrm{ABE}$ and $\triangle \mathrm{ACF}$, we have : $\mathrm{BE}=\mathrm{CF} \quad$ (Given) $\angle \mathrm{BEA}=\angle \mathrm{CFA}=90^{\circ}$ $\angle \mathrm{A}=\angle \mathrm{A}$ (Common) $\triangle \mathrm{ABE} \cong \triangle \mathrm{ACF} \quad$ (AAS criterion) $\mathrm{AB}=\mathrm{AC}(\mathrm{CPCT})$...
Read More →If the sum of n terms of an A.P. is 3n2 + 5n then which of its terms is 164?
Question: If the sum of $n$ terms of an A.P. is $3 n^{2}+5 n$ then which of its terms is $164 ?$ (a) 26 th (b) 27 th (c) 28 th (d) none of these. Solution: Here, the sum of firstnterms is given by the expression, $S_{n}=3 n^{2}+5 n$ We need to find which term of the A.P. is 164. Let us take 164 as thenthterm So we know that thenthterm of an A.P. is given by, $a_{n}=S_{n}-S_{n=1}$ So, $164=S_{n}-S_{n-1}$ $164=3 n^{2}+5 n-\left[3(n-1)^{2}+5(n-1)\right]$ Using the property, $(a-b)^{2}=a^{2}+b^{2}-2...
Read More →AD is an altitude of an isosceles ΔABC in which AB = AC.
Question: ADis an altitude of an isosceles ΔABCin whichAB=AC.Show that (i)ADbisectsBC, (ii)ADbisects A. Solution: Given:ADis an altitude of an isosceles ΔABCin whichAB=AC.To prove:(i)ADbisectsBC, (ii)ADbisects AProof:(i) InΔABDand ΔACD,ADB=ADC= 90 (Given,ADBC)AB=AC (Given)AD=AD (Common side)By RHS congruence criteria,ΔABD ΔACDBD=CD (CPCT)Hence,ADbisectsBC.(ii)∵∵ΔABD ΔACD [From (i)]BAD=CAD (CPCT)Hence,ADbisects A....
Read More →In a ∆ ABC, if C is a right angle, then
Question: In a $\Delta A B C$, if $C$ is a right angle, then (a) $\frac{\pi}{3}$ (b) $\frac{\pi}{4}$ (c) $\frac{5 \pi}{2}$ (d) $\frac{\pi}{6}$ Solution: (b) $\frac{\pi}{4}$ We know $\tan ^{-1} x+\tan ^{-1} y=\tan ^{-1}\left(\frac{x+y}{1-x y}\right)$ $\therefore \tan ^{-1}\left(\frac{a}{b+c}\right)+\tan ^{-1}\left(\frac{b}{c+a}\right)=\tan ^{-1}\left(\frac{\frac{a}{b+c}+\frac{b}{c+a}}{1-\frac{a}{b+c} \times \frac{b}{c+a}}\right)$ $=\tan ^{-1}\left(\frac{\frac{a c+a^{2}+b^{2}+b c}{(b+c)(c+a)}}{\fr...
Read More →Constant term in the expansion of
Question: Constant term in the expansion of $\left(x-\frac{1}{x}\right)^{10}$ is (a) 152 (b) 152 (c) 252 (d) 252 Solution: (c) 252 Suppose (r+ 1)th term is the constant term in the given expansion. Then, we have: $T_{r+1}={ }^{10} C_{r}(x)^{10-r}\left(\frac{-1}{x}\right)^{r}$ $={ }^{10} C_{r}(-1)^{r} x^{10-r-r}$ For this term to be constant, we must have: $10-2 r=0$ $\Rightarrow r=5$ $\therefore$ Required term $=-{ }^{10} C_{5}=-252$...
Read More →The first and last terms of an A.P. are 1 and 11.
Question: The first and last terms of an A.P. are 1 and 11. If the sum of its terms is 36, then the number of terms will be(a) 5(b) 6(c) 7(d) 8 Solution: In the given problem, we need to find the number of terms in an A.P. We are given, First term (a) = 1 Last term (an) = 11 Sum of its terms Now, as we know, $S_{n}=\left(\frac{n}{2}\right)(a+l)$ Where,a= the first term l= the last term So, we get, $36=\left(\frac{n}{2}\right)(1+11)$ $36(2)=12 n$ $n=\frac{36(2)}{12}$ $n=6$ Therefore, the total nu...
Read More →In the given figure, two parallel line l and m are intersected by two parallel lines p and q.
Question: In the given figure, two parallel linelandmare intersected by two parallel linespandq.Show that ΔABC ΔCDA. Solution: Given:In the given figure, two parallel linelandmare intersected by two parallel linespandq.To prove: ΔABC ΔCDAProof:InΔABCand ΔCDA,BAC=DCA (Alternate interior angles,p∥∥q)BCA=DAC (Alternate interior angles,l∥∥m)AC=CA (Common side)By ASA congruence criteria,ΔABC ΔCDA...
Read More →Solve this
Question: If $\cos ^{-1} x\sin ^{-1} x$, then (a) $\frac{1}{\sqrt{2}}x \leq 1$ (b) $0 \leq x\frac{1}{\sqrt{2}}$ (c) $-1 \leq x\frac{1}{\sqrt{2}}$ (d) $x0$ Solution: $\cos ^{-1} x\sin ^{-1} x$ $\Rightarrow \cos ^{-1} x\frac{\pi}{2}-\cos ^{-1} x$ $\Rightarrow 2 \cos ^{-1} x\frac{\pi}{2}$ $\Rightarrow \cos ^{-1} x\frac{\pi}{4}$ $\Rightarrow x\cos \frac{\pi}{4}$ $\Rightarrow x\frac{1}{\sqrt{2}}$ We know that the maximum value of cosine fuction is 1 . $\therefore \frac{1}{\sqrt{2}}x \leq 1$ Hence, th...
Read More →The number of terms with integral coefficients in the expansion of
Question: The number of terms with integral coefficients in the expansion of $\left(17^{1 / 3}+35^{1 / 2} x\right)^{600}$ is (a) 100 (b) 50 (c) 150 (d) 101 Solution: (d) 101 The general term $T_{r+1}$ in the given expansion is given by ${ }^{600} C_{r}\left(17^{1 / 3}\right)^{600-r}\left(35^{1 / 2} x\right)^{r}$ $={ }^{600} C_{r} 17^{200-r / 3} \times 35^{r / 2} x^{r}$ Now, $T_{r+1}$ is an integer if $\frac{r}{2}$ and $\frac{r}{3}$ are integers for all $0 \leq r \leq 600$ Thus, we have $r=0,6,12...
Read More →In the given figure, AD and BC are equal perpendiculars to a line segment AB. Show that CD bisects AB.
Question: In the given figure,ADandBCare equal perpendiculars to a line segmentAB. Show thatCDbisectsAB. Solution: Given:In the given figure,ADandBCare equal perpendiculars to a line segmentAB.To prove:CDbisectsABProof:In ΔAODand ΔBOC,DAO=CBO= 90 (Given)AD=BC (Given)DOA=COB (Vertically opposite angles) By AAS congruence criteria,ΔAOD ΔBOCAO=BO (CPCT)Hence,CDbisectsAB....
Read More →The number of terms with integral coefficients in the expansion of
Question: The number of terms with integral coefficients in the expansion of $\left(17^{1 / 3}+35^{1 / 2} x\right)^{600}$ is (a) 100 (b) 50 (c) 150 (d) 101 Solution: (d) 101 The general term $T_{r+1}$ in the given expansion is given by ${ }^{600} C_{r}\left(17^{1 / 3}\right)^{600-r}\left(35^{1 / 2} x\right)^{r}$ $={ }^{600} C_{r} 17^{200-r / 3} \times 35^{r / 2} x^{r}$ Now, $T_{r+1}$ is an integer if $\frac{r}{2}$ and $\frac{r}{3}$ are integers for all $0 \leq r \leq 600$ Thus, we have $r=0,6,12...
Read More →If the sum of n terms of an A.P. be 3n2 + n and its common difference is 6,
Question: If the sum of $n$ terms of an A.P. be $3 n^{2}+n$ and its common difference is 6 , then its first term is (a) 2(b) 3(c) 1(d) 4 Solution: In the given problem, the sum ofnterms of an A.P. is given by the expression, $S_{n}=3 n^{2}+n$ Here, we can find the first term by substitutingas sum of first term of the A.P. will be the same as the first term. So we get, $S_{n}=3 n^{2}+n$ $S_{1}=3(1)^{2}+(1)$ $=3+1$ $=4$ Therefore, the first term of this A.P is $a=4$. So, the correct option is (d)....
Read More →Solve this
Question: It $\tan ^{-1} \frac{x+1}{x-1}+\tan ^{-1} \frac{x-1}{x}=\tan ^{-1}(-7)$, then the value of $x$ is (a) 0 (b) $-2$ (c) 1 (d) 2 Solution: (d) 2 We know that $\tan ^{-1} x+\tan ^{-1} y=\tan ^{-1}\left(\frac{x+y}{1-x y}\right)$. $\therefore \tan ^{-1}\left(\frac{x+1}{x-1}\right)+\tan ^{-1}\left(\frac{x-1}{x}\right)=\tan ^{-1}(-7)$ $\Rightarrow \tan ^{-1}\left(\frac{\frac{x+1}{x-1}+\frac{x-1}{z}}{1-\frac{x+1}{x-1} \times \frac{x-1}{x}}\right)=\tan ^{-1}(-7)$ $\Rightarrow \tan ^{-1}\left(\fra...
Read More →In the given figure, AB || CD and O is the midpoint of AD.
Question: In the given figure,AB||CDandOis the midpoint ofAD.Show that(i) ΔAOB ΔDOC.(ii)Ois the midpoint ofBC Solution: Given: In the given figure,AB||CDandOis the midpoint ofAD. To prove:(i) ΔAOB ΔDOC.(ii)Ois the midpoint ofBC. Proof:(i) In ΔAOBand ΔDOC,BAO=CDO (Alternate interior angles,AB||CD)AO=DO (Given,Ois the midpoint ofAD)AOB=DOC (Vertically opposite angles) By ASA congruence criteria,ΔAOB ΔDOC(ii)∵ ΔAOB ΔDOC [From (i)]BO=CO (CPCT)Hence,Ois the midpoint ofBC....
Read More →If the sum of P terms of an A.P. is q and the sum of q terms is p
Question: If the sum ofPterms of an A.P. isqand the sum ofqterms isp, then the sum ofp+qterms will be(a) 0(b)pq(c)p+q(d) (p+q) Solution: In the given problem, we are given $S_{p}=q$ and $S_{q}=p$ We need to find $S_{p+q}$ Now, as we know, $S_{n}=\frac{n}{2}[2 a+(n-1) d]$ So, $S_{p}=\frac{p}{2}[2 a+(p-1) d]$ $q=\frac{p}{2}[2 a+(p-1) d]$ $2 q=2 a p+p(p-1) d$ .........(1) Similarly, $S_{p}=\frac{q}{2}[2 a+(q-1) d]$ $p=\frac{q}{2}[2 a+(q-1) d]$ $2 p=2 a q+q(q-1) d$ ..............(2) Subtracting (2) ...
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