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

In the given figure, BOC is a diameter of a circle with centre O.

Question: In the given figure,BOCis a diameter of a circle with centreO. IfABandCDare two chords such thatAB||CD. IfAB= 10 cm, thenCD= ?(a) 5 cm(b) 12.5 cm(c) 15 cm(d) 10 cm Solution: (d) 10 cm Draw OE AB and OF CD.In Δ OEB and ΔOFC, we have:OB = OC (Radius of a circle)BOE = COF (Vertically opposite angles)OEB = OFC (90 each) ΔOEB ΔOFC (By AAS congruency rule) OE = OFChords equidistant from the centre are equal. CD = AB = 10 cm...

If the mid-point of the line joining (3, 4) and (k, 7)

Question: If the mid-point of the line joining (3, 4) and (k, 7) is (x, y) and 2x + 2y + 1 = 0 find the value ofk. Solution: We have two points A (3, 4) and B (k, 7) such that its mid-point is. It is also given that point P lies on a line whose equation is $2 x+2 y+1=0$ In general to find the mid-point $\mathrm{P}(x, y)$ of two points $\mathrm{A}\left(x_{1}, y_{1}\right)$ and $\mathrm{B}\left(x_{2}, y_{2}\right)$ we use section formula as, $\mathrm{P}(x, y)=\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{...

In the given figure, O is the centre of a circle and diameter AB bisects the chord CD at a point E such that CE = ED = 8 cm and EB = 4 cm.

Question: In the given figure,Ois the centre of a circle and diameterABbisects the chordCDat a pointEsuch thatCE=ED= 8 cm andEB= 4 cm. The radius of the circle is(a) 10 cm(b) 12 cm(c) 6 cm(d) 8 cm Solution: (a) 10 cmLet the radius of the circle bercm.Let OD = OB =rcm.Then OE = (r- 4) cm and ED = 8 cmNow, in right ΔOED, we have: $\mathrm{OD}^{2}=\mathrm{OE}^{2}+\mathrm{ED}^{2}$ $\Rightarrow(r)^{2}=(r-4)^{2}+8^{2}$ $\Rightarrow r^{2}=r^{2}+16-8 r+64$ ⇒ 8r= 80⇒r= 10 cmHence, the required radius of ...

(i) In what ratio is the line segment joining the points (−2,−3) and (3, 7)

Question: (i) In what ratio is the line segment joining the points (2,3) and (3, 7) divided by the y-axis? Also, find the coordinates of the point of division.(ii) In what ratio is the line segment joining (3, 1) and (8, 9) divided at the point (5, 21/5)? Solution: (i) The ratio in which the $y$-axis divides two points $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$ is $\lambda: 1$ The co-ordinates of the point dividing two points $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\r...

In a parliament election, a political party hired a public relations firm to promote its candidates in three ways − telephone,

Question: In a parliament election, a political party hired a public relations firm to promote its candidates in three ways telephone, house calls and letters. The cost per contact (in paisa) is given in matrixAas $A=\left[\begin{array}{c}140 \\ 200 \\ 150\end{array}\right] \begin{gathered}\text { Telephone } \\ \text { House calls } \\ \text { Letters }\end{gathered}$ The number of contacts of each type made in two citiesXandYis given in the matrix B as Find the total amount spent by the party ...

In the given figure, CD is the diameter of a circle with centre O and CD is perpendicular to chord AB.

Question: In the given figure,CDis the diameter of a circle with centreOandCDis perpendicular to chordAB. IfAB= 12 cm andCE= 3 cm, then radius of the circles is(a) 6 cm(b) 9 cm(c) 7.5 cm(d) 8 cm Solution: (c) 7.5 cmLet OA = OC =rcm. Then $\mathrm{OE}=(r-3) \mathrm{cm}$ and $\mathrm{AE}=\frac{1}{2} \mathrm{AB}=6 \mathrm{~cm}$ Now, in right ΔOAE, we have: $\mathrm{OA}^{2}=\mathrm{OE}^{2}+\mathrm{AE}^{2}$ $\Rightarrow(r)^{2}=(r-3)^{2}+6^{2}$ $\Rightarrow r^{2}=r^{2}+9-6 r+36$ ⇒ 6r= 45 $\Rightarrow ...

In a G.P. of even number of terms, the sum of all terms is five times the sum of the odd terms.

Question: In a G.P. of even number of terms, the sum of all terms is five times the sum of the odd terms. The common ratio of the G.P. is (a) $-\frac{4}{5}$ (b) $\frac{1}{5}$ (c) 4 (d) none of these Solution: (c) 4 Let there be $2 n$ terms in a G.P. Let $a$ be the first term and $r$ be the common ratio. $\because S_{2 n}=5\left(S_{\text {odd terms }}\right)$ $\Rightarrow \frac{a\left(r^{2 n}-1\right)}{(r-1)}=5\left(a+a r^{2}+a r^{4}+a r^{6}+\ldots a r^{(2 n-1)}\right)$ $\Rightarrow \frac{a\left(...

If a vertex of a triangle be (1, 1) and the

Question: If a vertex of a triangle be (1, 1) and the middle points of the sides through it be (2,3) and (5 2) find the other vertices. Solution: Let ain which P and Q are the mid-points of sides AB and AC respectively. The coordinates are: A (1, 1); P (2, 3) and Q (5, 2). We have to find the co-ordinates of $\mathrm{B}\left(x_{1}, y_{1}\right)$ and $\mathrm{C}\left(x_{2}, y_{2}\right)$. In general to find the mid-point $\mathrm{P}(x, y)$ of two points $\mathrm{A}\left(x_{1}, y_{1}\right)$ and $...

AB and CD are two equal chords of a circle with centre O such that ∠AOB = 80°, then ∠COD = ?

Question: ABandCDare two equal chords of a circle with centreOsuch thatAOB= 80, then COD= ?(a) 100(b) 80(c) 120(d) 40 Solution: (b) 80Given: AB = CDWe know that equal chords of a circle subtend equal angles at the centre.COD = AOB=80...

In the given figure, AOB is a diameter of a circle with centre O such that AB = 34 cm and CD is a chord of length 30 cm.

Question: In the given figure,AOBis a diameter of a circle with centreOsuch thatAB= 34 cm andCDis a chord of length 30 cm. Then the distance ofCDfromABis(a) 8 cm(b) 15 cm(c) 18 cm(d) 6 cm Solution: (a) 8 cmJoin OC. Then OC = radius = 17 cm $\mathrm{CL}=\frac{1}{2} \mathrm{CD}=\left(\frac{1}{2} \times 30\right) \mathrm{cm}=15 \mathrm{~cm}$ In right ΔOLC, we have: $\mathrm{OL}^{2}=\mathrm{OC}^{2}-\mathrm{CL}^{2}=(17)^{2}-(15)^{2}=(289-225)=64$ $\Rightarrow \mathrm{OL}=\sqrt{64}=8 \mathrm{~cm}$ Dis...

Given that x > 0, the sum

Question: Given that $x0$, the sum $\sum_{n=1}^{\infty}\left(\frac{x}{x+1}\right)^{n-1}$ equals (a)x (b)x+ 1 (c) $\frac{x}{2 x+1}$ (d) $\frac{x+1}{2 x+1}$ Solution: (b)x + 1 $\sum_{n=1}^{\infty}\left(\frac{x}{x+1}\right)^{(n-1)}=1+\left(\frac{x}{x+1}\right)+\left(\frac{x}{x+1}\right)^{2}+\left(\frac{x}{x+1}\right)^{3}+\left(\frac{x}{x+1}\right)^{4}+\ldots \infty$ $=\frac{1}{1-\left(\frac{x}{x+1}\right)} \quad\left[\because\right.$ it is a G.P. with $a=1$ and $\left.\mathrm{r}=\left(\frac{\mathrm...

In the given figure, O is the centre of a circle. If ∠OAB = 40°

Question: In the given figure,Ois the centre of a circle. IfOAB= 40 andCis a point on the circle, then ACB= ?(a) 40(b) 50(c) 80(d) 100 Solution: (b) 50OA = OB⇒OBA = OAB = 40Now,AOB = 180 - (40 + 40) = 100 $\therefore \angle \mathrm{ACB}=\frac{1}{2} \angle \mathrm{AOB}=\left(\frac{1}{2} \times 100\right)^{\circ}=50^{\circ}$...

Solve the following

Question: Ifx= (43) (46) (46) (49) .... (43x) = (0.0625)54, the value ofxis (a) 7 (b) 8 (c) 9 (d) 10 Solution: (b) 8 $\left(4^{3}\right)\left(4^{6}\right)\left(4^{9}\right)\left(4^{12}\right) \ldots\left(4^{3 x}\right)=(0.0625)^{-54}$ $\Rightarrow 4^{(3+6+9+12+\ldots+3 \mathrm{x})}=\left(\frac{625}{10000}\right)^{-54}$ $\Rightarrow 4^{3(1+2+3+4+\ldots+x)}=\left(\frac{1}{16}\right)^{-54}$ $\Rightarrow 4^{3\left(\frac{x(x+1)}{2}\right)}=\left(\frac{1}{16}\right)^{-54}$ $\Rightarrow 4^{3\left(\frac...

Solve the following

Question: Ifx= (43) (46) (46) (49) .... (43x) = (0.0625)54, the value ofxis (a) 7 (b) 8 (c) 9 (d) 10 Solution: (b) 8 $\left(4^{3}\right)\left(4^{6}\right)\left(4^{9}\right)\left(4^{12}\right) \ldots\left(4^{3 x}\right)=(0.0625)^{-54}$ $\Rightarrow 4^{(3+6+9+12+\ldots+3 \mathrm{x})}=\left(\frac{625}{10000}\right)^{-54}$ $\Rightarrow 4^{3(1+2+3+4+\ldots+x)}=\left(\frac{1}{16}\right)^{-54}$ $\Rightarrow 4^{3\left(\frac{x(x+1)}{2}\right)}=\left(\frac{1}{16}\right)^{-54}$ $\Rightarrow 4^{3\left(\frac...

If x is positive, the sum to infinity of the series

Question: Ifxis positive, the sum to infinity of the series $\frac{1}{1+x}-\frac{1-x}{(1+x)^{2}}+\frac{(1-x)^{2}}{(1+x)^{3}}-\frac{(1-x)^{3}}{(1+x)^{4}}+\ldots$ is (a) 1/2 (b) 3/4 (c) 1 (d) none of these Solution: (a) $\frac{1}{2}$ Let $S=\frac{1}{(1+x)}-\frac{(1-x)}{(1+x)^{2}}+\frac{(1-x)^{2}}{(1+x)^{3}}-\frac{(1-x)^{3}}{(1+x)^{4}}+\ldots \infty$ It is clear that it is a G.P. with $a=\frac{1}{(1+x)}$ and $r=-\frac{(1-x)}{(1+x)}$. $\therefore S=\frac{a}{(1-r)}$ $\Rightarrow S=\frac{\frac{1}{(1+x...

If p, q be two A.M.'s and G be one G.M. between two numbers,

Question: Ifp,qbe two A.M.'s andGbe one G.M. between two numbers, thenG2= (a) (2pq) (p 2q) (b) (2pq) (2qp) (c) (2pq) (p+ 2q) (d) none of these Solution: (a) $(2 p-q)(p-2 q)$ Let the two numbers be $a$ and $b$. $a, p, q$ and $b$ are in A.P. $\therefore p-a=q-p=b-q$ $\Rightarrow p-a=q-p$ and $q-p=b-q$ $\Rightarrow a=2 p-q$ and $b=2 q-p$ ...(i) Also, $a, G$ and $b$ are in G.P. $\therefore G^{2}=a b$ $\Rightarrow G^{2}=(2 p-q)(2 q-p)$...

In the given figure, O is the centre of a circle and ∠ACB = 30°.

Question: In the given figure,Ois the centre of a circle andACB= 30. Then, AOB= ?(a) 30(b) 15(c) 60(d) 90 Figure Solution: (c) 60We know that the angle at the centre of a circle is twice the angle at any point on the remaining part of the circumference.Thus, AOB = (2 ACB) = (2 30) = 60...

In the given figure, BOC is a diameter of a circle and AB = AC. Then, ∠ABC = ?

Question: In the given figure,BOCis a diameter of a circle andAB=AC. Then,ABC= ?(a) 30(b) 45(c) 60(d) 90 Solution: (b) 45Since an angle in a semicircle is a right angle, BAC = 90 ABC + ACB = 90Now, AB = AC (Given)⇒ ABC = ACB = 45...

There are 2 families A and B. There are 4 men, 6 women and 2 children in family A, and 2 men, 2 women and 4 children in family B.

Question: There are 2 familiesAandB. There are 4 men, 6 women and 2 children in familyA, and 2 men, 2 women and 4 children in familyB. The recommend daily amount of calories is 2400 for men, 1900 for women, 1800 for children and 45 grams of proteins for men, 55 grams for women and 33 grams for children. Represent the above information using matrix. Using matrix multiplication, calculate the total requirement of calories and proteins for each of the two families. What awareness can you create amo...

There are 2 families A and B. There are 4 men, 6 women and 2 children in family A, and 2 men, 2 women and 4 children in family B.

Question: There are 2 familiesAandB. There are 4 men, 6 women and 2 children in familyA, and 2 men, 2 women and 4 children in familyB. The recommend daily amount of calories is 2400 for men, 1900 for women, 1800 for children and 45 grams of proteins for men, 55 grams for women and 33 grams for children. Represent the above information using matrix. Using matrix multiplication, calculate the total requirement of calories and proteins for each of the two families. What awareness can you create amo...

A chord is at a distance of 8 cm from the centre of a circle of radius 17 cm.

Question: A chord is at a distance of 8 cm from the centre of a circle of radius 17 cm. The length of the chord is(a) 25 cm(b) 12.5 cm(c) 30 cm(d) 9 cm Solution: (c) 30 cmLet AB be the chord of the given circle with centre O and a radius of 17 cm.From O, draw OM perpendicular to AB.Then OM = 8 cm and OB = 17 cm From the right ΔOMB, we have: $\mathrm{OB}^{2}=\mathrm{OM}^{2}+\mathrm{MB}^{2}$ $\Rightarrow 17^{2}=8^{2}+\mathrm{MB}^{2}$ $\Rightarrow 289=64+\mathrm{MB}^{2}$ $\Rightarrow \mathrm{MB}^{2...

The radius of a circle is 13 cm and the length of one of its chords is 10 cm.

Question: The radius of a circle is 13 cm and the length of one of its chords is 10 cm. The distance of the chord from the centre is (a) 11.5 cm(b) 12 cm (c) $\sqrt{69} \mathrm{~cm}$ (d) 23 cm Solution: (b) 12 cmLet AB be the chord of the given circle with centre O and a radius of 13 cm.Then, AB = 10 cm and OB = 13 cm From O, draw OM perpendicular to AB.We know that the perpendicular from the centre of a circle to a chord bisects the chord. $\therefore B M=\left(\frac{10}{2}\right) \mathrm{cm}=5...

To promote making of toilets for women,

Question: To promote making of toilets for women, an organisation tried to generate awarness through (i) house calls, (ii) letters, and (iii) announcements. The cost for each mode per attempt is given below: (i) ₹50 (ii) ₹20 (iii) ₹ 40 The number of attempts made in three villagesX,YandZ are given below: $\begin{array}{lclr} \text { (i) } \text { (ii) } \text { (iii) } \\ X 400 300 100 \\ Y 300 250 75 \\ Z 500 400 150\end{array}$ Find the total cost incurred by the organisation for three village...

If the coordinates of the mid-points of the sides of a triangle are

Question: If the coordinates of the mid-points of the sides of a triangle are (1, 1) (2, 3) and (3, 4), find the vertices of the triangle. Solution: The co-ordinates of the midpoint $\left(x_{m}, y_{m}\right)$ between two points $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$ is given by, $\left(x_{m}, y_{m}\right)=\left(\left(\frac{x_{1}+x_{2}}{2}\right),\left(\frac{y_{1}+y_{2}}{2}\right)\right)$ Let the three vertices of the triangle be $A\left(x_{A}, y_{A}\right), B\left(x_{B}, y_...

ABCD is a quadrilateral such that A is the centre of the circle passing through B, C and D.

Question: $A B C D$ is a quadrilateral such that $A$ is the centre of the circle passing through $B, C$ and $D$. Prove that $\angle C B D+\angle C D B=\frac{1}{2} \angle B A D$. Solution: In the given figure,ABCDis a quadrilateral such thatAis the centre of the circle passing throughB, CandD. JoinACandBD. We know that the angle subtended by an arc of a circle at the centre is double the angle subtended by it on the remaining part of the circle.Here, arcCDsubtends CADat the centre and CBDatBon th...

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