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

If sin A + sin2 A = 1,

Question: If sin A + sin2A = 1, then the value of cos2A + cos4A is (a) 2(b) 1(c) 2(d) 0 Solution: Here the given date is $\sin A+\sin ^{2} A=1$ and We have to find the value of $\cos ^{2} A+\cos ^{4} A$ We know that the given relation is $\sin A+\sin ^{2} A=1 \ldots \ldots(1)$ Now we are going to evaluate the value of $\cos ^{2} A+\cos ^{4} A$ $=\left(\cos ^{2} A\right)+\left(\cos ^{2} A\right)^{2}$ $=\left(1-\sin ^{2} A\right)+\left(1-\sin ^{2} A\right)^{2}$ $=\sin A+\sin ^{2} A$ Here we are us...

Verify associativity of addition of rational numbers i.e., (x + y) + z = x + (y + z), when:

Question: Verify associativity of addition of rational numbers i.e., (x+y) +z=x+ (y+z), when: (i) $x=\frac{1}{2}, y=\frac{2}{3}, z=-\frac{1}{5}$ (ii) $x=\frac{-2}{5}, y=\frac{4}{3}, z=\frac{-7}{10}$ (iii) $x=\frac{-7}{11}, y=\frac{2}{-5}, z=\frac{-3}{22}$ (iv) $x=-2, y=\frac{3}{5}, z=\frac{-4}{3}$ Solution: We have to verify that $(x+y)+z=x+(y+z)$ (i) $x=\frac{1}{2}, y=\frac{2}{3}, z=\frac{-1}{5}$ $(\mathrm{x}+\mathrm{y})+\mathrm{z}=\left(\frac{1}{2}+\frac{2}{3}\right)+\frac{-1}{5}=\left(\frac{3...

A galaxy is moving away from the earth

Question: A galaxy is moving away from the earth at a speed of $286 \mathrm{kms}^{-1}$. The shift in the wavelength of a red line at $630 \mathrm{~nm}$ is $\mathrm{x} \times 10^{-10} \mathrm{~m}$. The value of $\mathrm{x}$, to the nearest integer, is [Take the value of speed of light $c$, as $3 \times 10^{8}$ $\left.\mathrm{ms}^{-1}\right]$ Solution: (6) $\frac{\Delta \lambda}{\lambda} c=v$ $\Delta \lambda=\frac{v}{c} \times \lambda=\frac{286}{3 \times 10^{5}} \times 630 \times 10^{-9}=6 \times ...

The value of k for which the pair of linear equation

Question: The value of k for which the pair of linear equation 4x + 6y 1 = 0 and 2x + ky 7 = 0 represent parallel lines is(a) k = 3(b) k = 2(c) k = 4(d) k = 2 Solution: Given that the pair of linear equation $4 x+6 y-1=0$ and $2 x+k y-7=0$ $a_{1}=4, b_{1}=6, c_{1}=-(-1)=1$ $a_{2}=2, b_{2}=k, c_{2}=-(-7)=7$ It is given that the pair of equations represent parallel lines. $\therefore \frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$ $\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}$ $\Right...

Assertion (A) At point P of a circle with centre O and radius 12 cm,

Question: Assertion (A)At pointPof a circle with centreOand radius 12 cm, a tangentPQof length 16 cm is drawn. Then,OQ= 20 cm.Reason (R)The tangent at any point of a circle is perpendicular to the radius through the point of contact.(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).(c) Assertion (A) is true and Reason (R) is false.(d) A...

A sound wave of frequency 245 Hz

Question: A sound wave of frequency $245 \mathrm{~Hz}$ travels with the speed of $300 \mathrm{~ms}^{-1}$ along the positive $x$-axis. Each point of the wave moves to and fro through a total distance of $6 \mathrm{~cm}$. What will be the mathematical expression of this travelling wave?$\mathrm{Y}(\mathrm{x}, \mathrm{t})=0.03\left[\sin 5.1 \mathrm{x}-\left(0.2 \times 10^{3}\right) \mathrm{t}\right]$$\mathrm{Y}(\mathrm{x}, \mathrm{t})=0.06\left[\sin 5.1 \mathrm{x}-\left(1.5 \times 10^{3}\right) \ma...

The [HCF × LCM] for the numbers 50 and 20 is

Question: The [HCF LCM] for the numbers 50 and 20 is (a) 10(b) 100(c) 1000(d) 50 Solution: Here we have to find $H C F \times L C M$ of 50 and 20 . We know the product of $H C F$ and $L C M$ of two numbers $a$ and $b$ is the product of $a$ and $b$. Therefore $H C F \times L C M$ of 50 and 20 is as follow $H C F \times L C M=50 \times 20$ $=1000$ Hence the correct option is (c)....

Which of the following statements is not true?

Question: Which of the following statements is not true?(a) A line which intersects a circle at two points, is called a secant of the circle.(b) A line intersecting a circle at one point only is called a tangent to the circle.(c) The point at which a line touches the circle is called the point of contact.(d) A tangent to the circle can be drawn from a point inside the circle. Solution: (d) A tangent to the circle can be drawn from a point inside the circle.This statement is false because tangent...

If the mean of the following distribution is 54,

Question: If the mean of the following distribution is 54, find the value ofp: Solution: Consider the table given below: It given that the mean of the distribution is 54. $\therefore$ Mean $=\frac{\sum f_{i} x_{i}}{\sum f_{i}}$ $\Rightarrow \frac{2370+30 p}{39+p}=54$ $\Rightarrow 2370+30 p=2106+54 p$ $\Rightarrow 54 p-30 p=2370-2106$ $\Rightarrow 24 p=264$ $\Rightarrow p=\frac{264}{24}=11$ Hence, the value ofpis 11....

Which of the following statements is not true?

Question: Which of the following statements is not true?(a) A tangent to a circle intersects the circle exactly at one point.(b) The point common to a circle and its tangent is called the point of contact.(c) The tangent at any point of a circle is perpendicular to the radius of the circle through the point of contact.(d) A straight line can meet a circle at one point only. Solution: (d)A straight line can meet a circle at one point only.This statement is not true because a straight line that is...

Verify commutativty of addition of rational numbers for each of the following pairs of rotional numbers:

Question: Verify commutativty of addition of rational numbers for each of the following pairs of rotional numbers: (i) $\frac{-11}{5}$ and $\frac{4}{7}$ (ii) $\frac{4}{9}$ and $\frac{7}{-12}$ (iii) $\frac{-3}{5}$ and $\frac{-2}{-15}$ (iv) $\frac{2}{-7}$ and $\frac{12}{-35}$ (v) 4 and $\frac{-3}{5}$ (vi) $-4$ and $\frac{4}{-7}$ Solution: $C$ ommutativity of the addition of rational numbers means that if $\frac{\mathrm{a}}{\mathrm{b}}$ and $\frac{\mathrm{c}}{\mathrm{d}}$ are two rational numbers, ...

A closed organ pipe of length L

Question: A closed organ pipe of length $\mathrm{L}$ and an open organ pipe contain gases of densities $\rho_{1}$ and $\rho_{2}$ respectively. The compressibility of gases are equal in both the pipes. Both the pipes are vibrating in their first overtone with same frequency. The length of the open pipe is_____ $\frac{x}{3} \mathrm{~L} \sqrt{\frac{\overline{\rho_{1}}}{\rho_{2}}}$ where $x$ is (Round off to the Nearest Integer) Solution: (4) $\mathrm{f}_{\mathrm{c}}=\mathrm{f}_{0}$ $\frac{3 \mathrm...

Which of the following statements is not true?

Question: Which of the following statements is not true?(a) If a pointPlies inside a circle, no tangent can be drawn to the circle passing throughP.(b) If a pointPlies on a circle, then one and only one tangent can be drawn to the circle atP.(c) If a pointPlies outside a circle, then only two tangents can be drawn to the circle fromP.(d) A circle can have more than two parallel tangents parallel to a given line. Solution: (d)A circle can have more than two parallel tangents, parallel to a given ...

A closed organ pipe of length L

Question: A closed organ pipe of length $\mathrm{L}$ and an open organ pipe contain gases of densities $\rho_{1}$ and $\rho_{2}$ respectively. The compressibility of gases are equal in both the pipes. Both the pipes are vibrating in their first overtone with same frequency. The length of the open pipe is_____ $\frac{x}{3} \mathrm{~L} \sqrt{\frac{\overline{\rho_{1}}}{\rho_{2}}}$ where $x$ is (Round off to the Nearest Integer) Solution: (4) $\mathrm{f}_{\mathrm{c}}=\mathrm{f}_{0}$ $\frac{3 \mathrm...

For what value of k will the following

Question: For what value ofkwill the following system of linear equations has no solution:3x+y= 1(2k 1)x+ (k 1)y= 2k+ 1 Solution: The given system of equations is3x+y= 1(2k 1)x+ (k 1)y= 2k+ 1Here,a1= 3,b1= 1,c1= 1a2= 2k 1,b2=k 1,c2= 2k+ 1The given system of linear equations has no solution. $\therefore \frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$ $\Rightarrow \frac{3}{2 k-1}=\frac{1}{k-1} \neq \frac{1}{2 k+1}$ $\Rightarrow \frac{3}{2 k-1}=\frac{1}{k-1}$ and $\frac{1}{k-1} \n...

In the given figure, O is the centre of two concentric circles of radii 5 cm and 3 cm.

Question: In the given figure,Ois the centre of two concentric circles of radii 5 cm and 3 cm. From an external pointPtangentsPAandPBare drawn to these circles. IfPA= 12 cm thenPBis equal to (a) $5 \sqrt{2} \mathrm{~cm}$ (b) $3 \sqrt{5} \mathrm{~cm}$ (c) $4 \sqrt{10} \mathrm{~cm}$ (d) $5 \sqrt{10} \mathrm{~cm}$ Solution: (c) $4 \sqrt{10} \mathrm{~cm}$ Given, $O P=5 \mathrm{~cm}, P A=12 \mathrm{~cm}$ Now, j oin $O$ and $B$. Then, $O B=3 \mathrm{~cm}$. Now, $\angle O A P=90^{\circ}$ (Tangents draw...

In Fig. 1, the value of the median of the data

Question: In Fig. 1, the value of the median of the data using the graph of less than ogive and more than ogive is(a) 5(b) 40(c) 80(d) 15 Solution: Here we have to find the median from the given graph. The following graph is given. From the given graph we can easily see that the total cumulative frequencyN= 80. First we draw a line from the point $\frac{N}{2}=\frac{80}{2}=40$ on the less than ogive graph which is parallel to the $x$-axis and draw a line perpendicular to the $x$-axis from that po...

Δ ABC and Δ PQR are similar triangles

Question: Δ ABC and Δ PQR are similar triangles such that A = 32 and R = 65. Then, B is(a) 83(b) 32(c) 65(d) 97 Solution: It is given that there are two similar triangles, $\triangle A B C$ and $\triangle P Q R$ in which $\angle A=32^{\circ}$ and $\angle R=65^{\circ}$, then we have to find $\angle B$ We have following two similar triangles. We know the relation between angles in the two similar triangles and these are $\angle A=\angle P=32^{\circ}$ $\angle C=\angle R=65^{\circ}$ $\angle B=\angle...

In the given figure, AP, AQ and BC are tangents to the circle.

Question: In the given figure, AP, AQ and BC are tangents to the circle. If AB = 5 cm, AC = 6 cm and BC = 4 cm then the length of AP is (a) 15 cm(b) 10 cm(c) 9 cm(d) 7.5 cm Solution: We know that tangent segments to a circle from the same external point are congruent.Therefore, we haveAP = AQBP = BDCQ = CDNow, AB + BC + AC = 5 + 4 + 6 = 15⇒AB + BD + DC + AC = 15 cm⇒AB + BP + CQ + AC = 15 cm⇒AP + AQ= 15 cm⇒2AP = 15 cm⇒AP = 7.5 cmHence, the correct answer is option(d)...

In the given figure, three circles with centres A, B, C, respectively,

Question: In the given figure, three circles with centresA,B,C, respectively, touch each other externally. IfAB= 5 cm,BC= 7 cm andCA= 6 cm, the radius of the circle with centreAis (a) 1.5 cm(b) 2 cm(c) 2.5 cm(d) 3 cm Solution: (b) 2 cm Given, $A B=5 \mathrm{~cm}, B C=7 \mathrm{~cm}$ and $C A=6 \mathrm{~cm}$. Let, $A R=A P=x \mathrm{~cm}$. $B Q=B P=y \mathrm{~cm}$ $C R=C Q=z \mathrm{~cm}$ (Since the length of tangents drawn from an external point are equal) Then, $A B=5 \mathrm{~cm}$ $\Rightarrow...

The value of p for which the polynomial

Question: The value ofpfor which the polynomial x3+ 4x2 px + 8 is exactly divisible by (x2) is(a) 0(b) 3(c) 5(d) 16 Solution: Here the given polynomial is $x^{3}+4 x^{2}-p x+8$ We have to find the value ofpsuch that the polynomial is exactly divisible by First we have to write equation in basic format of divisibility like this $x^{3}-2 x^{2}$ $6 x^{2}-p x+8$ $6 x^{2}-12 x$ $12 x-p x+8$ $\frac{4 x+8}{16 x-p x}$ After solving, we have seeing here the reminder is = 16xpx So, to find the value ofpwe...

In a Young's double slit experiment 15 fringes

Question: In a Young's double slit experiment 15 fringes are observed on a small portion of the screen when light of wavelength $500 \mathrm{~nm}$ is used. Ten fringes are observed on the same section of the screen when another light source of wavelength $\lambda$ is used. Then the value of $\lambda$ is (in $\mathrm{nm}$ ) Solution: (750) Fringe width, $\beta=\frac{\lambda D}{d}$ where, $\lambda=$ wavelength, $D=$ distance of screen from slits, $d=$ distance between slits ATQ $15 \times----=10 \...

In a Young's double slit experiment 15 fringes

Question: In a Young's double slit experiment 15 fringes are observed on a small portion of the screen when light of wavelength $500 \mathrm{~nm}$ is used. Ten fringes are observed on the same section of the screen when another light source of wavelength $\lambda$ is used. Then the value of $\lambda$ is (in $\mathrm{nm}$ ) Solution: (750) Fringe width, $\beta=\frac{\lambda D}{d}$ where, $\lambda=$ wavelength, $D=$ distance of screen from slits, $d=$ distance between slits ATQ $15 \times________=...

Which of the following numbers has

Question: Which of the following numbers has terminating decimal expansion? (a) $\frac{37}{45}$ (b) $\frac{21}{2^{3} 5^{6}}$ (C) $\frac{17}{49}$ (d) $\frac{89}{2^{2} 3^{2}}$ Solution: Here we have to check terminating decimal expansion. We know that if the numerator can be written in the formwheremandnare non negative positive integer then the fraction will surely terminate. We proceed as follows to explain the above statement $\frac{21}{2^{3} \times 5^{6}}=\frac{2^{2} \times 21}{2^{2} \times 2^...

In a double-slit experiment,

Question: In a double-slit experiment, at a certain point on the screen the path difference between the two interfering waves is $\frac{1}{8}$ th of a wavelength. The ratio of the intensity of light at that point to that at the centre of a bright fringe is:$0.853$$0.672$$0.568$$0.760$Correct Option: 1 Solution: (1) Given, Path difference, $\Delta x=\frac{\lambda}{8}$ Phase differences, $\Delta \phi=\frac{2 \pi}{\lambda} \Delta x$ $=\frac{2 \pi}{\lambda} \times \frac{\lambda}{8}=\frac{\pi}{4}$ $I...

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