eSaral - JEE | NEET | Class 8-10 Exam Prep App

Using binomial theorem evaluate each of the following:

Question: Using binomial theorem evaluate each of the following: (i) (96)3 (ii) (102)5 (iii) (101)4 (iv) (98)5 Solution: (i) $(96)^{3}$ $=(100-4)^{3}$ $={ }^{3} C_{0} \times 100^{3} \times 4^{0}-{ }^{3} C_{1} \times 100^{2} \times 4^{1}+{ }^{3} C_{2} \times 100^{1} \times 4^{2}-{ }^{3} C_{3} \times 100^{0} \times 4^{3}$ $=1000000-120000+4800-64$ $=884736$ (ii) $(102)^{5}$ $=(100+2)^{5}$ $={ }^{5} C_{0} \times 100^{5} \times 2^{0}+{ }^{5} C_{1} \times 100^{4} \times 2^{1}+{ }^{5} C_{2} \times 100...

In the given figure, AB || CD and a transversal t cuts them at E and F respectively.

Question: In the given figure,AB||CDand a transversaltcuts them at E and F respectively. If EG and FG are the bisectors ofBEF and EFD respectively, prove that EGF = 90. Solution: It is given that,AB||CDandtisa transversal.BEF + EFD = 180 .....(1) (Sum of the interior angles on the same side of a transversal is supplementary)EG is the bisector of BEF. (Given) $\therefore \angle B E G=\angle G E F=\frac{1}{2} \angle B E F$ ⇒BEF = 2GEF .....(2)Also, FG is the bisector of EFD. (Given) $\therefore \a...

Write the domain of the real function

Question: Write the domain of the real function $f(x)=\sqrt{[x]-x}$. Solution: $[x]$ is the greatest integer function. $[x] \leq x, \forall x \in R$ $\Rightarrow[x]-x \leq 0, \forall x \in R$ $\Rightarrow \sqrt{[x]-x}$ does not exist for any $x \in R$. Domain $=\phi$...

Find the sum of first 22 terms of an A.P. in which d = 22 and a22 = 149.

Question: Find the sum of first 22 terms of an A.P. in which d = 22 and a22= 149. Solution: In the given problem, we need to find the sum of first 22 terms of an A.P. Let us take the first term asa. Here, we are given that, $a_{22}=149$...........(1) $d=22$...............(2) Also, we know, $a_{n}=a+(n-1) d$ For the 22ndterm (n =22),' $a_{22}=a+(22-1) d$ $149=a+21(22) \quad$ (Using 1 and 2 ) $a=149-462$ $a=-313$..........(3) So, as we know the formula for the sum ofnterms of an A.P. is given by, ...

Write the domain of the real function

Question: Write the domain of the real function $f(x)=\sqrt{x-[x]}$. Solution: $[\mathrm{x}]$ is the greatest integral function. So, $0 \leq x-[x]1$ $\Rightarrow \sqrt{x-[x]}$ exists for every $x \in R$. $\Rightarrow$ Domain $=R$...

Solve the following

Question: Find $(x+1)^{6}+(x-1)^{6}$. Hence, or otherwise evaluate $(\sqrt{2}+1)^{6}+\sqrt{2}-1^{6}$. Solution: The expression $(x+1)^{6}+(x-1)^{6}$ can be written as $(x+1)^{6}+(x-1)^{6}$ $=2\left[{ }^{6} C_{0} x^{6}+{ }^{6} C_{2} x^{4}+{ }^{6} C_{4} x^{2}+{ }^{6} C_{6} x^{0}\right]$ $=2\left[x^{6}+15 x^{4}+15 x^{2}+1\right]$ By taking $x=\sqrt{2}$, we get: $(\sqrt{2}+1)^{6}+(\sqrt{2}-1)^{6}=2\left[(\sqrt{2})^{6}+15(\sqrt{2})^{4}+15(\sqrt{2})^{2}+1\right]$ $=2[8+15 \times 4+15 \times 2+1]$ $=2 ...

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

Question: In the given figure,AB||CDandEF||GH. Find the values ofx,y,zandt. Solution: In the given figure, $x=60^{\circ}$ [Vertically-Opposite Angles] $\angle P R Q=\angle S Q R$ [Alternate Angles] $y=60^{\circ}$ $\angle A P R=\angle P Q S \quad$ [Corresponding Angles] $\Rightarrow 110^{\circ}=\angle P Q R+60^{\circ} \quad[\because \angle P Q S=\angle P Q R+\angle R Q S]$ $\Rightarrow \angle P Q R=50^{\circ}$ $\angle P Q R+\angle R Q S+\angle B Q S=180^{\circ} \quad[$ Since AB is a straight line...

Solve the following

Question: Find $(a+b)^{4}-(a-b)^{4}$. Hence, evaluate $(\sqrt{3}+\sqrt{2})^{4}-(\sqrt{3}-\sqrt{2})^{4}$. Solution: The expression $(a+b)^{4}-(a-b)^{4}$ can be written as $(a+b)^{4}-(a-b)^{4}=2\left[{ }^{4} C_{1} a^{3} b^{1}+{ }^{4} C_{3} a^{1} b^{3}\right]$ $=2\left[4 a^{3} b+4 a b^{3}\right]$ $=8\left(a^{3} b+a b^{3}\right)$ Putting $a=\sqrt{3}$ and $b=\sqrt{2}$, we get: $(\sqrt{3}+\sqrt{2})^{4}-(\sqrt{3}-\sqrt{2})^{4}=8\left[(\sqrt{3})^{3} \times \sqrt{2}+\sqrt{3} \times(\sqrt{2})^{3}\right]$ ...

If 12th term of an A.P. is −13 and the sum of the first four terms is 24

Question: If 12th term of an A.P. is 13 and the sum of the first four terms is 24, what is the sum of first 10 terms. Solution: In the given problem, we need to find the sum of first 10 terms of an A.P. Let us take the first termaand the common difference asd Here, we are given that, $a_{12}=-13$ $S_{4}=24$ Also, we know, $a_{n}=a+(n-1) d$ For the $12^{\text {th }}$ term $(n=12)$, $a_{12}=a+(12-1) d$ $-13=a+11 d$ $a=-13-11 d$...........(1) So, as we know the formula for the sum ofnterms of an A....

In the given figure, AB || CD. Prove that p + q − r = 180.

Question: In the given figure,AB||CD. Prove thatp+qr= 180. Solution: Draw $P F Q\|A B\| C D$. Now, $P F Q \| A B$ and EF is the transversal. Then, $\angle A E F+\angle E F P=180^{\circ} \ldots \ldots(1)$ [Angles on the same side of a transversal line are supplementary] Also, $P F Q \| C D$. $\angle P F G=\angle F G D=r^{\circ}[$ Alternate Angles $]$ and $\angle E F P=\angle E F G-\angle P F G=q^{\circ}-r^{\circ}$ putting the value of $\angle E F P$ in eqn. (i) we get, $p^{\circ}+q^{\circ}-r^{\ci...

Let A = {1, 2, 3, 4} and B = {a, b} be two sets. Write the total number of onto functions from A to B.

Question: LetA= {1, 2, 3, 4} andB= {a,b} be two sets. Write the total number of onto functions fromAtoB. Solution: Formula:When two setsAandBhavemandnelements respectively, then the number of onto functions fromAtoBis $\left\{\begin{array}{l}\sum_{r=1}^{n}(-1)^{r} n C_{r} r^{m}, \text { if } m \geq n \\ o, \text { if } mn\end{array}\right.$ Here, number of elements inA= 4 =mNumber of elements inB= 2 =n So, $m\mathrm{n}$ Number of onto functions $=\sum_{r=1}^{2}(-1)^{r} 2 C_{r} r^{4}$ $=(-1)^{1} ...

Evaluate the following:

Question: Evaluate the following: (i) $(\sqrt{x+1}+\sqrt{x-1})^{6}+(\sqrt{x+1}-\sqrt{x-1})^{6}$ (ii) $\left(x+\sqrt{x^{2}-1}\right)^{6}+\left(x-\sqrt{x^{2}-1}\right)^{6}$ (iii) $(1+2 \sqrt{x})^{5}+(1-2 \sqrt{x})^{5}$ (iv) $(\sqrt{2}+1)^{6}+(\sqrt{2}-1)^{6}$ (v) $(3+\sqrt{2})^{5}-(3-\sqrt{2})^{5}$ (vi) $(2+\sqrt{3})^{7}+(2-\sqrt{3})^{7}$ (vii) $(\sqrt{3}+1)^{5}-(\sqrt{3}-1)^{5}$ (viii) $(0.99)^{5}+(1.01)^{5}$ (ix) $(\sqrt{3}+\sqrt{2})^{6}-(\sqrt{3}-\sqrt{2})^{6}$ (x) $\left\{a^{2}+\sqrt{a^{2}-1}\...

Evaluate the following:

Question: Evaluate the following: (i) $(\sqrt{x+1}+\sqrt{x-1})^{6}+(\sqrt{x+1}-\sqrt{x-1})^{6}$ (ii) $\left(x+\sqrt{x^{2}-1}\right)^{6}+\left(x-\sqrt{x^{2}-1}\right)^{6}$ (iii) $(1+2 \sqrt{x})^{5}+(1-2 \sqrt{x})^{5}$ (iv) $(\sqrt{2}+1)^{6}+(\sqrt{2}-1)^{6}$ (v) $(3+\sqrt{2})^{5}-(3-\sqrt{2})^{5}$ (vi) $(2+\sqrt{3})^{7}+(2-\sqrt{3})^{7}$ (vii) $(\sqrt{3}+1)^{5}-(\sqrt{3}-1)^{5}$ (viii) $(0.99)^{5}+(1.01)^{5}$ (ix) $(\sqrt{3}+\sqrt{2})^{6}-(\sqrt{3}-\sqrt{2})^{6}$ (x) $\left\{a^{2}+\sqrt{a^{2}-1}\...

Let f be an invertible real function. Write

Question: Letfbe an invertible real function. Write Solution: Given thatf is an invertible real function. $f^{-1} o f=I$, where $\mathrm{I}$ is an identity function. So, $\left(f^{-1} o f\right)(1)+\left(f^{-1} o f\right)(2)+\ldots+\left(f^{-1} o f\right)(100)$ $=I(1)+I(2)+\ldots+I(100)$ $=1+2+\ldots+100($ As $I(x)=x, \forall x \in R)$ $=\frac{100(100+1)}{2}\left[\right.$ Sum of first n natural numbers $\left.=\frac{n(n+1)}{2}\right]$ $=5050$...

The first term of an A.P. is 2 and the last term is 50.

Question: The first term of an A.P. is 2 and the last term is 50. The sum of all these terms is 442. Find the common difference. Solution: In the given problem, we have the first and the last term of an A.P. along with the sum of all the terms of A.P. Here, we need to find the common difference of the A.P. Here, The first term of the A.P (a) = 2 The last term of the A.P (l) = 50 Sum of all the terms Let the common difference of the A.P. bed. So, let us first find the number of the terms (n) usin...

The third term of an A.P. is 7 and the seventh term exceeds three times the third term by 2.

Question: The third term of an A.P. is 7 and the seventh term exceeds three times the third term by 2. Find the first term, the common difference and the sum of first 20 terms. Solution: In the given problem, let us take the first term asaand the common difference asd. Here, we are given that, $a_{3}=7$...........(1) $a_{7}=3 a_{3}+21$.........(2) So, using (1) in (2), we get, $a_{7}=3(7)+2$ $=21+2$ $=23$...........(3) Also, we know, $a_{n}=a+(n-1) d$ For the $3^{\text {th }}$ term $(n=3)$, $a_{...

Let f : R → R be defined as

Question: Let $f: R \rightarrow R$ be defined as $f(x)=\frac{2 x-3}{4} .$ Write $f o f^{-1}$ (1). Solution: Let $f^{-1}(x)=y$ ...(1) $\Rightarrow f(y)=x$ $\Rightarrow \frac{2 y-3}{4}=x$ $\Rightarrow 2 y-3=4 x$ $\Rightarrow 2 y=4 x+3$ $\Rightarrow y=\frac{4 x+3}{2}$ $\Rightarrow f^{-1}(x)=\frac{4 x+3}{2} \quad[$ from $(1)]$ $\Rightarrow f^{-1}(x)=\frac{4 x+3}{2}$ $\therefore\left(f o f^{-1}\right)(1)=f\left(\frac{4(1)+3}{2}\right)=f\left(\frac{7}{2}\right)=\frac{2\left(\frac{7}{2}\right)-3}{4}=\f...

In the given figure, AB || CD. Prove that ∠BAE − ∠DCE = ∠AEC.

Question: In the given figure,AB||CD. Prove that BAE DCE= AEC. Solution: Draw $E F\|A B\| C D$ through $\mathrm{E}$. Now, $E F \| A B$ and $\mathrm{AE}$ is the transversal. Then, $\angle B A E+\angle A E F=180^{\circ} \quad$ [Angles on the same side of a transversal line are supplementary] Again, $E F \| C D$ and $\mathrm{CE}$ is the transversal. Then $\angle D C E+\angle C E F=180^{\circ} \quad$ [Angles on the same side of a transversal line are supplementary] $\Rightarrow \angle D C E+(\angle ...

Let f : R → R, g : R → R be two functions defined by

Question: Let $f: R \rightarrow R, g: R \rightarrow R$ be two functions defined by $f(x)=x^{2}+x+1$ and $g(x)=1-x^{2}$. Write fog $(-2)$. Solution: $(f o g)(-2)=f(g(-2))$ $=f\left(1-(-2)^{2}\right)$ $=f(-3)$ $=(-3)^{2}+(-3)+1$ $=9-3+1$ $=7$...

Let f : R−{−35}→R be a function defined as

Question: Let $f: R-\left\{-\frac{3}{5}\right\} \rightarrow R$ be a function defined as $f(x)=\frac{2 x}{5 x+3}$. Write $f^{-1}:$ Range of $f \rightarrow R-\left\{-\frac{3}{5}\right\}$. Solution: Let $f^{-1}(x)=y$ ...(1) $\Rightarrow f(y)=x$ $\Rightarrow \frac{2 y}{5 y+3}=x$ $\Rightarrow 2 y=5 x y+3 x$ $\Rightarrow 2 y-5 x y=3 x$ $\Rightarrow y(2-5 x)=3 x$ $\Rightarrow y=\frac{3 x}{2-5 x}$ $\Rightarrow f^{-1}(x)=\frac{3 x}{2-5 x} \quad[$ from (1) $]$...

The first and the last terms of an A.P. are 17 and 350 respectively.

Question: The first and the last terms of an A.P. are 17 and 350 respectively. If the common difference is 9, how many terms are there and what is their sum? Solution: In the given problem, we have the first and the last term of an A.P. along with the common difference of the A.P. Here, we need to find the number of terms of the A.P. and the sum of all the terms. Here, The first term of the A.P (a) = 17 The last term of the A.P (l) = 350 The common difference of the A.P. = 9 Let the number of te...

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: $\angle A D C=\angle D A B \quad$ [Alternate Interior Angles] $\Rightarrow z=75^{\circ}$ $\angle A B C=\angle B C D \quad$ [Alternate Interior Angles] $\Rightarrow x=35^{\circ}$ We know that the sum of the angles of a triangle is $180^{\circ}$. $\Rightarrow 35^{\circ}+y+75^{\circ}=180^{\circ}$ $\Rightarrow y=70^{\circ}$ $\therefore x=35^{\circ}, y=70^{\circ}$ and $z=75^{\circ} .$...

In the given figure, AB || CD. Find the value of x.

Question: In the given figure,AB||CD. Find the value ofx. Solution: $A B \| C D$ and $\mathrm{PQ}$ is the transversal. Then, $\angle P E F=\angle E G H \quad[$ Corresponding Angles $]$ $\Rightarrow \angle E G H=85^{\circ}$ And, $\angle E G H+\angle Q G H=180^{\circ}$ $\Rightarrow 85^{\circ}+\angle Q G H=180^{\circ}$ $\Rightarrow \angle Q G H=95^{\circ}$ Also, $\angle C H Q+\angle G H Q=180^{\circ} \quad$ [Since CD is a straight line] $\Rightarrow 115^{\circ}+\angle G H Q=180^{\circ}$ $\Rightarro...

Using binomial theorem, write down the expansions of the following:

Question: Using binomial theorem, write down the expansions of the following: (i) $(2 x+3 y)^{5}$ (ii) $(2 x-3 y)^{4}$ (iii) $\left(x-\frac{1}{x}\right)^{6}$ (iv) $(1-3 x)^{7}$ (v) $\left(a x-\frac{b}{x}\right)^{6}$ (vi) $\left(\frac{\sqrt{x}}{a}-\sqrt{\frac{a}{x}}\right)^{6}$ (vii) $(\sqrt[3]{x}-\sqrt[3]{a})^{6}$ (viii) $\left(1+2 x-3 x^{2}\right)^{5}$ (ix) $\left(x+1-\frac{1}{x}\right)$ (x) $\left(1-2 x+3 x^{2}\right)^{3}$ Solution: (i) $(2 x+3 y)^{5}$ $={ }^{5} C_{0}(2 x)^{5}(3 y)^{0}+{ }^{5}...

Using binomial theorem, write down the expansions of the following:

Question: Using binomial theorem, write down the expansions of the following: (i) $(2 x+3 y)^{5}$ (ii) $(2 x-3 y)^{4}$ (iii) $\left(x-\frac{1}{x}\right)^{6}$ (iv) $(1-3 x)^{7}$ (v) $\left(a x-\frac{b}{x}\right)^{6}$ (vi) $\left(\frac{\sqrt{x}}{a}-\sqrt{\frac{a}{x}}\right)^{6}$ (vii) $(\sqrt[3]{x}-\sqrt[3]{a})^{6}$ (viii) $\left(1+2 x-3 x^{2}\right)^{5}$ (ix) $\left(x+1-\frac{1}{x}\right)$ (x) $\left(1-2 x+3 x^{2}\right)^{3}$ Solution: (i) $(2 x+3 y)^{5}$ $={ }^{5} C_{0}(2 x)^{5}(3 y)^{0}+{ }^{5}...

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