CBSE
CBSE Study Material
Solve each of the following in equations and represent the solution set on the number line.
. Solve each of the following in equations and represent the solution set on the number line.
Solve each of the following in equations and represent the solution set on the number line. 3 – 2x ≥ 4x – 9, where x ϵ R.
Solve each of the following in equations and represent the solution set on the number line. 3x – 4 > x + 6, where x ϵ R.
Match the following columns:
Column I Column II (a) A man goes 10m due east and then 20m due north. His distance from the starting point is ……m. (p) 25√3 (b) In an equilateral triangle with each side 10cm, the altitude is...
Match the following columns:
Column I Column II (a) In a given ∆ABC, DE║BC and $\frac{{AD}}{{DB}} = \frac{{3}}{{5}}$. If AC = 5.6cm, then AE = ……..cm. (p) 6 (b) If ∆ABC~∆DEF such that 2AB = 3DE and BC = 6cm, then EF = …….cm....
Which of the following is a false statement? (a) If the areas of two similar triangles are equal, then the triangles are congruent. (b) The ratio of the areas of two similar triangles is equal to the ratio of their corresponding sides. (c) The ratio of the areas of two similar triangles is equal to the ratio of squares of their corresponding. (d) The ratio of the areas of two similar triangles is equal to the ratio of squares of their corresponding altitudes.
Correct Answer: (b) The ratio of the areas of two similar triangles is equal to the ratio of their corresponding sides. Explanation: The ratio of the areas of two similar triangles is equal to the...
Which of the following is a true statement? (a) Two similar triangles are always congruent (b) Two figures are similar if they have the same shape and size. (c)Two triangles are similar if their corresponding sides are proportional. (d) Two polygons are similar if their corresponding sides are proportional.
Correct Answer: (c)Two triangles are similar if their corresponding sides are proportional. Explanation: Given, ∆ABC~ ∆DEF $\frac{{AB}}{{DE}} = \frac{{AC}}{{DF}} = \frac{{BC}}{{EF}}$
In ∆ABC, if AB = 16cm, BC = 12cm and AC = 20cm, then ∆ABC is (a) acute-angled (b) right-angled (c) obtuse-angled
Correct Answer: (b) right-angled Explanation: Given, ????????2 + ????????2 = 162 + 122 => 256 + 144 => 400 ????????2 = 202 => 400 ∴ ????????2 + ????????2 = ????????2 ∆ABC is a right-angled...
In an isosceles ∆ABC, if AC = BC and 
, then ∠C = ? (a) 
(b) 
(c) 
(d) 
Correct Answer: (d) ${90^0}$ Explanation: Given, AC = BC ????????2 = 2????????2 = ????????2 + ????????2 = ????????2 + ????????2 Applying Pythagoras theorem, ∆ABC is right angled at C ∠???? =...
In the given figure, O is the point of intersection of two chords AB and CD such that OB = OD and ∠AOC = 
. Then, ∆OAC and ∆ODB are (a) equilateral and similar (b) equilateral but not similar (c) isosceles and similar (d) isosceles but not similar
Correct Answer: (c) isosceles and similar Explanation: In ∆AOC and ∆ODB, ∠???????????? = ∠???????????? (???????????????????????????????????????? ????????????????????????????????...
If ∆ABC~∆QRP, 
, AB = 18cm and BC = 15cm, then PR = ? (a) 18cm (b) 10cm (c) 12 cm (d) 
cm
Correct Answer: (b) 10 cm Explanation: ∆ABC ~ ∆QRP $\begin{array}{l} \frac{{AB}}{{QR}} = \frac{BC}{PR}\\ \end{array}$ $\frac{{ar(\Delta ABC)}}{{ar(\Delta QRP)}} = \frac{9}{4}$ $\begin{array}{l}...
The line segments joining the midpoints of the sides of a triangle form four triangles, each of which is (a) congruent to the original triangle (b) similar to the original triangle (c) an isosceles triangle (d) an equilateral triangle
Correct Answer: (b) similar to the original triangle Explanation: The line segments joining the midpoint of the sides of a triangle form four triangles,...
Two isosceles triangles have their corresponding angles equal and their areas are in the ratio 25: 36. The ratio of their corresponding heights is (a) 25 : 36 (b) 36 : 25 (c) 5 : 6 (d) 6: 5
Correct Answer: (c) 5:6 Explanation: Let x and y be the corresponding heights of the two triangles. The corresponding angles of the triangles are equal. The triangles are similar. (By AA criterion)...
∆ABC~∆DEF such that ar(∆ABC) = 36 
and ar(∆DEF) = 49 . Then, the ratio of their corresponding sides is (a) 36 : 49 (b) 6 : 7 (c) 7 : 6 (d) √6 : √7
Correct Answer: (b) 6:7 Explanation: In ∆ABC ~ ∆DEF, $\begin{array}{l} \frac{{AB}}{{DE}} = \frac{{BC}}{{EF}} = \frac{{AC}}{{DF}}\\ \frac{{ar(\Delta ABC)}}{{ar(\Delta DEF)}} =...
In ∆ABC and ∆DEF, we have: 
, then ar(∆ABC) : ∆(DEF) = ? (a) 5 : 7 (b) 25 : 49 (c) 49 : 25 (d) 125 : 343
Correct Answer: (b) 25 :49 Explanation: In ∆ABC and ∆DEF, $\frac{{AB}}{{DE}} = \frac{{BC}}{{EF}} = \frac{{AC}}{{DF}} = \frac{5}{7}$ By SSS criterion, ∆ABC ~ ∆DEF $\begin{array}{l} \frac{{ar(\Delta...
In an equilateral ∆ABC, D is the midpoint of AB and E is the midpoint of AC. Then, ar(∆ABC) : ar(∆ADE) = ? (a) 2 : 1 (b) 4 : 1 (c) 1 : 2 (d) 1 : 4
Correct Answer: (b) 4:1 Explanation: In ∆ABC, D is the midpoint of AB and E is the midpoint of AC. By midpoint theorem and Basic Proportionality Theorem,...
It is given that ∆ABC~∆PQR and 
, then 
Correct Answer: (d)\frac{9}{4}$ Explanation: Given, ∆ABC ~ ∆PQR $\begin{array}{l} \frac{{BC}}{{QR}} = \frac{2}{3}\\ \end{array}$ $\begin{array}{l} \frac{{ar(\Delta PQR)}}{{ar(\Delta ABC)}} =...
Corresponding sides of two similar triangles are in the ratio 4:9 Areas of these triangles are in the ration (a) 2:3 (b) 4:9 (c) 9:4 (d) 16:81
Correct Answer: (d) 16:81 Explanation: If two triangles are similar, then the ratio of their areas is equal to the ratio of the squares of their corresponding sides. $\begin{array}{l} \frac{{area\...
In the given figure, two line segment AC and BD intersect each other at the point P such that PA = 6cm, PB = 3cm, PC = 2.5cm, PD = 5cm, ∠APB = 
and ∠CDP = , then ∠PBA = ? (a) (b) (c) (d) 
Correct Answer: (d) ${100^0}$ Explanation: In ∆ APB and ∆ DPC, In ∆ APB and ∆ DPC, ∠???????????? = ∠???????????? = 500 $\begin{array}{l} \frac{{AP}}{{BP}} = \frac{6}{3} = 2\\...
If in ∆ABC and ∆PQR, we have: 
, then (a) ∆PQR ~ ∆CAB (b) ∆PQR ~ ∆ABC (c) ∆CBA ~ ∆PQR (d) ∆BCA ~ ∆PQR
Correct Answer: (a) ∆PQR ~ ∆CAB Explanation: In ∆ABC and ∆PQR, $\begin{array}{l} \frac{{AB}}{{QR}} = \frac{{BC}}{{PR}} = \frac{{CA}}{{PQ}}\\ \end{array}$ ∆ABC ~ ∆QRP
In ∆ABC and ∆DEF, it is given that ∠B = ∠E, ∠F = ∠C and AB = 3DE, then the two triangles are (a) congruent but not similar (b) similar but not congruent (c) neither congruent nor similar (d) similar as well as congruent
Correct Answer: (b) similar but not congruent Explanation: In ∆ABC and ∆DEF, ∠???? = ∠???? ∠???? = ∠???? Applying AA similarity theorem, ∆ABC - ∆DEF. AB = 3DE AB ≠ DE ∆ABC and ∆DEF are similar but...
If ∆ABC~∆EDF and ∆ABC is not similar to ∆DEF, then which of the following is not true? (a) BC.EF = AC.FD (b) AB.EF = AC.DE (c) BC.DE = AB.EF (d) BC.DE = AB.FD
Correct Answer: (c) BC. DE = AB. EF Explanation: ∆ABC ~ ∆EDF $\begin{array}{l} \frac{{AB}}{{DE}} = \frac{{AC}}{{EF}} = \frac{{BC}}{{DF}}\\ BC.DE \ne AB.EF \end{array}$
Solve each of the following in equations and represent the solution set on the number line. 5x + 2 < 17, where (i) x ϵ Z, (ii) x ϵ R.
Solve each of the following in equations and represent the solution set on the number line. 3x + 8 > 2, where (i) x ϵ Z, (ii) x ϵ R.
Solve each of the following in equations and represent the solution set on the number line. –2x > 5, where (i) x ϵ Z, (ii) x ϵ R.
A laboratory blood test is 
effective in detecting a certain disease when it is in fact, present. However, the test also yields a false positive result for 
of the healthy person tested (that is, if a healthy person is tested, then, with probability 
, the test will imply he has the disease). If 
percent of the population actually has the disease, what is the probability that a person has the disease given that his test result is positive?
As per the given question,
Solve each of the following in equations and represent the solution set on the number line. 6x ≤ 25, where (i) x ϵ N, (ii) x ϵ Z.
Fill in the blanks with correct inequality sign (>, <, ≥, ≤).
(i) 5x < 20 ⇒ x4
(ii) –3x > 9 ⇒ x 3
(iii) 4x > –16 ⇒ x 4
(iv) –6x ≤ –18 ⇒ 3
(v) x > –3 ⇒ –2x 6
(vi)a < b and c > 0 ⇒
(vii) p – q = –3 ⇒ pq
(viii)u – v = 2 ⇒ u V
In answering a question on a multiple choice test a student either knows the answer or guesses. Let 3/4 be the probability that he knows the answer and 1/4 be the probability that he guesses. Assuming that a student who guesses at the answer will be correct with probability 1/4. What is the probability that a student knows the answer given that he answered it correctly?
As per the given question,
A speaks the truth 8 times out of 10 times. A die is tossed. He reports that it was 5 What is the probability that it was actually 
As per the given question,
is known to speak truth 3 times out of 5 times. He throws a die and reports that it is 
Find the probability that it is actually
As per the given question,
Let 
be three mutually exclusive diseases. Let 
be the set of observable symptoms of these diseases. A doctor has the following information from a random sample of 5000 patients: 1800 had disease 
has disease 
and the others had disease 
1500 patients with disease 
patients with disease and 900 patients with disease showed the symptom. Which of the diseases is the patient most likely to have?
As per the given question,
A test for detection of a particular disease is not fool proof. The test will correctly detect the disease 
of the time, but will incorrectly detect the disease 
of the time. For a large population of which an estimated 
have the disease, a person is selected at random, given the test, and told that he has the disease. What are the chances that the person actually have the disease?
As per the given question,
Write a unit vector in the direction of 
, where 
and 
are the points 
and 
respectively.
Solution: $\begin{array}{l} \overrightarrow{O P}=\hat{\imath}+3 \hat{\jmath} \\ \overrightarrow{O Q}=4 \hat{\imath}+5 \hat{\jmath}+6 \hat{k} \\ \overrightarrow{P Q}=\overrightarrow{O...
If 
and 
is the given point, find the coordinates of 
.
Solution: $\begin{array}{l} \overrightarrow{A B}=2 \hat{\imath}+\hat{\jmath}-3 \hat{k} \\ \overrightarrow{O A}=\hat{\imath}+2 \hat{\jmath}-\hat{k} \\ \overrightarrow{O B}=? \\ \overrightarrow{A...
By examining the chest 
-ray, probability that T.B is detected when a person is actually suffering is 
. The probability that the doctor diagnoses incorrectly that a person has T.B. on the basis of -ray is 
. In a certain city 1 in 100 persons suffers from T.B. A person is selected at random is diagnosed to have T.B. What is the chance that he actually has T.B.?
As per the given question,
Find the position vector of the mid-point of the vector joining the points 
and 
Solution: $\begin{array}{l} \overrightarrow{O A}=3 \hat{\imath}+2 \hat{\jmath}+6 \hat{k} \\ \overrightarrow{O B}=\hat{\imath}+4 \hat{\jmath}-2 \hat{k} \end{array}$ Let $\mathrm{C}$ be the mid-point...
Find the position vector of a point 
which divides the line joining 
and 
in the ratio 
(i) internally (ii) externally.
Solution: $\begin{array}{l} \overrightarrow{O A}=-2 \hat{\imath}+\hat{\jmath}+3 \hat{k} \\ \overrightarrow{O B}=3 \hat{\imath}+5 \hat{\jmath}-2 \hat{k} \end{array}$ (i) $\mathrm{R}$ divides...
The position vectors of two points and 
are 
and 
respectively. Find the position vector of a point 
which divides 
externally in the ratio 
. Also, show that 
is the mid-point of the line segment 
.
Solution: Given: $\overrightarrow{O A}=(2 \vec{a}+\vec{b})$ $\overrightarrow{O B}=(\vec{a}-3 \vec{b})$ Position vector of $\mathrm{C}$ which divides $\mathrm{AB}$ in the ratio $1: 2$ externally is...
Find the position vector of the point which divides the join of the points 
and 
(i) internally and (ii) externally in the ratio 
Solution: Given: $\overrightarrow{O A}=(2 \vec{a}-3 \vec{b})$ $\overrightarrow{O B}=(3 \vec{a}-2 \vec{b})$ (i) Let $\mathrm{P}$ be the point that divides $\mathrm{A}, \mathrm{B}$ internally in the...
If a machine is correctly set up, it produces acceptable items. If it is incorrectly set up, it produces only 
acceptable item s. Past experience shows that 
of the set ups are correctly done. If after a certain set up, the machine produces 2 acceptable items, find the probability that the machine is correctly setup.
As per the given question,
Using vector method, show that the points 
and 
are the vertices of a rightangled triangle.
Solution: $\begin{array}{l} \overrightarrow{O A}=\hat{\imath}-\hat{\jmath} \\ \overrightarrow{O B}=4 \hat{\imath}-3 \hat{\jmath}+\hat{k} \\ \overrightarrow{O C}=2 \hat{\imath}-4 \hat{\jmath}+5...
Show that the points A, B and C having position vectors 
and 
respectively, form the vertices of a right-angled triangle.
Solution: $\begin{array}{l} \overrightarrow{O A}=3 \hat{\imath}-4 \hat{\jmath}-4 \hat{k} \\ \overrightarrow{O B}=2 \hat{\imath}-\hat{\jmath}+\hat{k} \\ \overrightarrow{O C}=\hat{\imath}-3...
If the position vectors of the vertices 
and of a 
be 
and 
, respectively, prove that is equilateral.
Solution: $\begin{array}{l} \overrightarrow{O A}=\hat{\imath}+2 \hat{\jmath}+3 \hat{k} \\ \overrightarrow{O B}=2 \hat{\imath}+3 \hat{\jmath}+\hat{k} \\ \overrightarrow{O C}=3...
Coloured balls are distributed in four boxes as shown in the following table ![\[\begin{tabular}{cc|c|c|c} \hline Box & \multicolumn{3}{c}{ Colour } \\ & Black & White & Red & Blue \\ \hline I & 3 & 4 & 5 & 6 \\ \hline II & 2 & 2 & 2 \\ \hline III & 1 & 2 & 1 \\ \hline IV & 4 & 3 & 5 \\ \hline \end{tabular}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-22c35adcd58fd61079fed1526e84cfa7_l3.png "Rendered by QuickLaTeX.com")
A box is selected at random and then a ball is randomly drawn from the selected box. The colour of the ball is black, what is the probability that ball drawn is from the box III.
As per the given question,
The position vectors of the points and are 
and 
respectively. Show that the points A, B and C are collinear.
Solution: $\begin{array}{l} \overrightarrow{O A}=2 \hat{\imath}+\hat{\jmath}-\hat{k} \\ \overrightarrow{O B}=3 \hat{\imath}-2 \hat{\jmath}+\hat{k} \\ \overrightarrow{O C}=\hat{\imath}+4...
Show that the points 
and 
having position vectors 
and 
respectively, are collinear.
Solution: $\begin{array}{l} \overrightarrow{O A}=\hat{\imath}+2 \hat{\jmath}+7 \hat{k} \\ \overrightarrow{O B}=2 \hat{\imath}+6 \hat{\jmath}+2 \hat{k} \\ \overrightarrow{O C}=3 \hat{\imath}+10...
Find the direction ratios and the direction cosines of the vector joining the points 
and 
.
Solution: $\begin{array}{l} \overrightarrow{O A}=2 \hat{\imath}+\hat{\jmath}-2 \hat{k} \\ \overrightarrow{O B}=3 \hat{\imath}+5 \hat{\jmath}-4 \hat{k} \\ \overrightarrow{A B}=\overrightarrow{O...
Find the direction ratios and direction cosines of the vector 
.
Solution: $\vec{a}=5 \hat{\imath}-3 \hat{\jmath}+4 \hat{k}$ Direction ratios are ratios of $\mathrm{x}, \mathrm{y}$ and $\mathrm{z}$ component of the vector while direction cosine are cosines of...
Assume that the chances of the patient having a heart attack are . It is also assumed that a meditation and yoga course reduce the risk of heart attack by 
and prescription of certain drug reduces its chances by 
. At a time a patient can choose any one of the two options with equal probabilities. It is given that after going through one of the two options the patient selected at random suffers a heart attack. Find the probability that the patient followed a course of meditation and yoga?
As per the given question,
If 
and 
are two given points, find a unit vector in the direction of 
.
Solution: $\mathrm{A}(-2,1,2)$ $\mathrm{B}(2,-1,6)$ Points given in such ways represents position vectors, $\overrightarrow{O A}$ and $\overrightarrow{O B}$ $\begin{array}{l} \overrightarrow{O A}=-2...
If 
and 
, find 
Solution: $\begin{array}{l} \vec{a}=\hat{\imath}-2 \hat{\jmath} \\ \vec{b}=2 \hat{\imath}-3 \hat{\jmath} \\ \vec{c}=2 \hat{\imath}+3 \hat{k} \\ \vec{a}+\vec{b}+\vec{c}=(\hat{\imath}-2...
In ∆DEF and ∆PQR, it is given that ∠D = ∠Q and ∠R = ∠E, then which of the following is not true? 
Correct Answer: (b) $\frac{{DE}}{{PQ}} = \frac{{EF}}{{RP}}$ Explanation: In ∆DEF and ∆PQR, ∠???? = ∠???? ∠???? = ∠???? Applying AA similarity theorem, ∆DEF ~ ∆QRP $\frac{{DE}}{{PQ}} =...
In ∆ABC and ∆DEF, it is given that 
, then (a) ∠B = ∠E (b) ∠A = ∠D (c) ∠B = ∠D (d) ∠A = ∠F
Correct Answer: (c)∠???? = ∠D Explanation: ∆ ABC − EDF The corresponding angles, ∠???? ???????????? ∠???? ???????????????? ???????? ????????????????????. ∠???? = ∠D
Find a vector of magnitude 21 units in the direction of the vector 
.
Solution: $\vec{A}=2 \hat{\imath}-3 \hat{\jmath}+6 \hat{k}$ Given magnitude of required vector is 21 units $\begin{array}{l} \mathrm{I} \overrightarrow{\mathrm{I}}...
In ∆ABC, AB = 6√3 , AC = 12 cm and BC = 6cm. Then ∠B is
Answer: Given, ???????? = 6√3???????? ????????2 = 108 ????????2 AC = 12 cm ????????2 = 144 ????????2 BC = 6 cm ????????2 = 36 ???????? ∴ ????????2 = ????????2 + ????????2 The square of the longest...
Find a vector of magnitude 8 units in the direction of the vector 
.
Solution: $\vec{A}=5 \hat{\imath}-\hat{\jmath}+2 \hat{k}$ Magnitude of required vector is 8 units $\mathrm{I} \vec{A} \mathrm{I}=\sqrt{5^{2}+1^{2}+2^{2}}=\sqrt{25+1+4}=\sqrt{30}$ $\begin{array}{l}...
In the given figure, ∠BAC = and AD⊥BC. Then, (a) BC.CD = 
(b) AB.AC = (c) BD.CD = 
(d) AB.AC =
Correct Answer: (c) BD.CD = $A{D^2}$ Explanation: In ∆ BDA and ∆ADC, ∠???????????? = ∠???????????? = 900 ∠???????????? = 900 − ∠???????????? => 900 − (900 − ∠????????????)...
There are three coins. One is two headed coin (having head on both faces), another is a biased coin that comes up heads 
of the time and third is an unbiased coin. One of the three coins is chosen at random and tossed, it shows heads, what is the probability that it was the two headed coin?
As per the given question,
It is given that ∆ABC~∆DFE. If ∠A = , ∠C = , , AB = 5cm, AC = 8cm and DF = 7.5cm, then which of the following is true? (a) DE = 12cm, ∠F = , (b) DE = 12cm, ∠F = , (c) DE = 12cm, ∠D = , (d) EF = 12cm, ∠D = ,
Correct Answer: (b) DE = 12cm, ∠F = ${100^0}$ Explanation: Given, In triangle ABC, ∠???? + ∠???? + ∠???? = 1800 ∠???? = 180 − 30 − 50 => 1000 ∵ ∆ABC ~ ∆DFE ∠???? = ∠???? = 300 ∠???? = ∠???? =...
Find a vector of magnitude 9 units in the direction of the vector 
.
Solution: $\vec{A}=-2 \hat{\imath}+\hat{\jmath}+2 \hat{k}$ $\mathrm{I} \vec{A} \mathrm{I}=\sqrt{(-2)^{2}+1^{2}+2^{2}}=\sqrt{4+1+4}=\sqrt{9}=3$ Note: Any vector $\vec{X}$ is given by $\vec{X}=$...
ABC and BDE are two equilateral triangles such that D is the midpoint of BC. Ratio of these area of triangles ABC and BDE is (a) 2 : 1 (b) 1 : 4 (c) 1 : 2 (d) 4 : 1
Correct Answer: (d) 4 : 1 Explanation: Given, ABC and BDE are two equilateral triangles D is the midpoint of BC and BDE is also an equilateral triangle. E is also the midpoint of AB. D and E are the...
Differentiate the following functions with respect to x: ![\[{{\sin }^{-1}}\left( \frac{{{2}^{x+1}}}{1+{{4}^{x}}} \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-bc02cd2d3e108e34d60e8eea3d42a4b1_l3.png "Rendered by QuickLaTeX.com")
In ∆ABC, it is given that AB = 9cm, BC = 6cm and CA = 7.5cm. Also, ∆DEF is given such that EF = 8cm and ∆DEF~∆ABC. Then, perimeter of ∆DEF is (a) 22.5cm (b) 25cm (c) 27cm (d) 30cm
Correct Answer: (d) 30 cm Explanation: Perimeter of ∆ABC = AB + BC + CA => 9 + 6 + 7.5 => 22.5 cm $\begin{array}{l} \frac{{Perimeter(\Delta ABC)}}{{Perimeter(\Delta DEF)}} =...
If 
and 
then find a unit vector parallel to 
.
Solution: $\begin{array}{l} \vec{a}=\hat{\imath}+2 \hat{\jmath}-3 \hat{k} \\ \vec{b}=2 \hat{\imath}+4 \hat{\jmath}+9 \hat{k} \end{array}$ Then $\vec{a}+\vec{b}=(\hat{\imath}+2 \hat{\jmath}-3...
∆ABC~∆DEF such that AB = 9.1cm and DE = 6.5cm. If the perimeter of ∆DEF is 25cm, what is the perimeter of ∆ABC? (a) 35cm (b) 28cm (c) 42cm (d) 40cm
Correct Answer: (a) 35 cm Explanation: Given, ∆ABC ~ ∆DEF $\begin{array}{l} \frac{{Perimeter(\Delta ABC)}}{{Perimeter(\Delta DEF)}} = \frac{{AB}}{{DE}}\\ \frac{{Perimeter(\Delta ABC)}}{{25}} =...
For given y, prove the following ![\[y={{\tan }^{-1}}\{\frac{\sqrt{1+x}-\sqrt{1-x}}{\sqrt{1+x}+\sqrt{1-x}}\},\frac{dy}{dx}=\frac{1}{2\sqrt{1-{{x}^{2}}}}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-4551a30e309647f6bd453bf98044a0b7_l3.png "Rendered by QuickLaTeX.com")
∆ABC~∆DEF and the perimeters of ∆ABC and ∆DEF are 30cm and 18cm respectively. If BC = 9cm, then EF = ? (a) 6.3cm (b) 5.4cm (c) 7.2cm (d) 4.5cm
Correct Answer: (b) 5.4 cm Explanation: Given, ∆ABC ~ ∆DEF $\begin{array}{l} \frac{{Perimeter(\Delta ABC)}}{{Perimeter(\Delta DEF)}} = \frac{{BC}}{{EF}}\\ \frac{{30}}{{18}} = \frac{9}{{EF}}\\ EF =...
If 
and 
then find a unit vector in the direction of 
.
Solution: $\begin{array}{l} \vec{a}=3 \hat{\imath}+\hat{\jmath}-5 \hat{k} \\ \vec{b}=\hat{\imath}+2 \hat{\jmath}-\hat{k} \end{array}$ Then $\vec{a}-\vec{b}=(3 \hat{\imath}+\hat{\jmath}-5...
In ∆ABC, DE ║ BC such that 
. If AC = 5.6cm, then AE = ? (a) 4.2cm (b) 3.1cm (c) 2.8cm (d) 2.1cm
Correct Answer: (d) 2.1 cm Explanation: Given, DE || BC. Applying Thales’ theorem, $\begin{array}{l} \frac{{AD}}{{DB}} = \frac{{AE}}{{EC}}\\ \end{array}$ AE be x cm. EC = (5.6 –...
For given y, prove the following ![\[y=\sin \{2{{\tan }^{-1}}\sqrt{\frac{1-x}{1+x}}\},\frac{dy}{dx}=-\frac{x}{\sqrt{1-{{x}^{2}}}}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-e42e04c75bab33bcd5b143ab0c9eb080_l3.png "Rendered by QuickLaTeX.com")
If 
and 
then find the unit vector in the direction of .
Solution: $\vec{a}=-\hat{\imath}+\hat{\jmath}-\hat{k}$ $\vec{b}=2 \hat{\imath}-\hat{\jmath}+2 \hat{k}$ Then $(\vec{a}+\vec{b})=(-\hat{\imath}+\hat{\jmath}-\hat{k})+(2 \hat{\imath}-\hat{\jmath}+2...
In ∆ABC, DE ║ BC so that AD = (7x – 4) cm, AE = (5x – 2) cm, DB = (3x + 4) cm and EC = 3x cm. Then, we have: (a) x = 3 (b) x = 5 (c) x = 4 (d) x = 2.5
Correct Answer: (c) x = 4 Explanation: Given, DE || BC Applying Thales’ theorem, $\begin{array}{l} \frac{{AD}}{{BD}} = \frac{{AE}}{{EC}}\\ \frac{{7x-4}}{{3x+4}} =...
For given y, prove the following ![\[y={{\sec }^{-1}}\left( \frac{x+1}{x-1} \right)+{{\sin }^{-1}}\left( \frac{x-1}{x+1} \right),\frac{dy}{dx}=0\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-f39971f1a420c94a512fa017049b6920_l3.png "Rendered by QuickLaTeX.com")
If 
then find the value of 
so that 
may be a unit vector.
Solution: $\vec{a}=2 \hat{\imath}-4 \hat{\jmath}+5 \hat{k}$ Given: $(\lambda \vec{a})$ is a unit vector Since $(\lambda \vec{a})$ is a unit vector, so I $\lambda \vec{a} \mathrm{I}=1$...
In a ∆ABC, if DE is drawn parallel to BC, cutting AB and AC at D and E respectively such that AB = 7.2cm, AC = 6.4cm and AD = 4.5cm. Then, AE = ? (a) 5.4cm (b) 4cm (c) 3.6cm (d) 3.2cm
Correct Answer: (b) 4cm Explanation: Given, DE || BC Applying basic proportionality theorem, $\begin{array}{l} \frac{{AD}}{{AB}} = \frac{{AE}}{{AC}}\\ \frac{{4.5}}{{AB}} =...
Find a unit vector in the direction of the vector:
A. 
B. 
C. 
D. 
Solution: If $\vec{a}=\mathrm{a}_{1} \hat{\imath}+\mathrm{a}_{2} \hat{\jmath}+\mathrm{a}_{3} \hat{k}$, then Unit vector in the direction of $\vec{a}$ can be given by $\hat{a}=\frac{\vec{a}}{I...
For given y, prove the following ![\[y={{\sin }^{-1}}\left( \frac{2x}{1+{{x}^{2}}} \right)+{{\sec }^{-1}}\left( \frac{1+{{x}^{2}}}{1-{{x}^{2}}} \right),\frac{dy}{dx}=\frac{4}{1+{{x}^{2}}}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-21d795e16afe301f59ddb8698013982b_l3.png "Rendered by QuickLaTeX.com")
In ∆ABC, DE║BC so that AD = 2.4cm, AE = 3.2cm and EC = 4.8cm. Then, AB = ? (a) 3.6cm (b) 6cm (c) 6.4cm (d) 7.2cm
Correct Answer: (b) 6 cm Explanation: Given, DE || BC Applying basic proportionality theorem, $\begin{array}{l} \frac{{AD}}{{BD}} = \frac{{AE}}{{EC}}\\ \frac{{2.4}}{{BD}} =...
In ∆ABC, it is given that 
. If ∠B = 
and ∠C = , then ∠BAD = ? (a) (b) 
(c) (d)
Correct Answer: (a) ${30^0}$ Explanation: Given, $\begin{array}{l} \frac{{AB}}{{AC}} = \frac{{BD}}{{DC}}\\ \end{array}$ Applying angle bisector theorem, AD bisects ∠????. In...
Of the students in a college, it is known that 
reside in a hostel and do not reside in hostel. Previous year results report that of students residing in hostel attain grade and 
of ones not residing in hostel attain grade in their annual examination. At the end of the year, one students is chosen at random from the college and he has an A grade. What is the probability that the selected student is a hosteller?
As per the given question,
In the given figure, ABCD is a trapezium whose diagonals AC and BD intersect at O such that OA = (3x – 1) cm, OB = (2x + 1)cm, OC = (5x – 3)cm and OD = (6x – 5)cm. Then, x = ? (a) 2 (b) 3 (c) 2.5 (d) 4
Correct Answer: (a) 2 Explanation: Given, The diagonals of a trapezium are proportional. $\begin{array}{l} \frac{{OA}}{{OC}} = \frac{{OB}}{{OD}}\\ \frac{{3X-1}}{{5X-3}} =...
If the bisector of an angle of a triangle bisects the opposite side, then the triangle is (a) scalene (b) equilateral (c) isosceles (d) right-angled
Correct Answer: (c) isosceles Explanation: Let AD be the angle bisector of angle A in triangle ABC. Applying angle bisector theorem, $\begin{array}{l} \frac{{AB}}{{AC}} =...
Write down the magnitude of each of the following vectors:
A. 
B. 
C. 
D. 
Solution: A. $\vec{a}=\hat{\imath}+2 \hat{\jmath}+5 \hat{k}$ If $\vec{a}=\mathrm{a}_{1} \hat{\imath}+\mathrm{a}_{2} \hat{\jmath}+\mathrm{a}_{3} \hat{k}$, then $\mathrm{I} \vec{a}...
For given y, prove the following ![\[y={{\tan }^{-1}}\left( \frac{ax-b}{a+bx} \right),\frac{dy}{dx}=\frac{1}{1+{{x}^{2}}}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-3e1659cc6e3e1daaaec261c54e2f70d1_l3.png "Rendered by QuickLaTeX.com")
The line segments joining the midpoints of the adjacent sides of a quadrilateral form (a) parallelogram (b) trapezium (c) rectangle (d) square
Correct Answer: (a) parallelogram Explanation: The line segments joining the midpoints of the adjacent sides of a quadrilateral form a parallelogram.
If the diagonals of a quadrilateral divide each other proportionally, then it is a (a) parallelogram (b) trapezium (c) rectangle (d) square
Correct Answer: (b) trapezium Explanation: Diagonals of a trapezium divide each other proportionally.
Differentiate the following functions with respect to x: ![\[{{\tan }^{-1}}\left( \frac{2x}{1+15{{x}^{2}}} \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-a6cd86822f24da96ed4ade0364cd4e49_l3.png "Rendered by QuickLaTeX.com")
An insurance company insured 2000 scooters and 300 motorcycles. The probability of an accident involving a scooter is 
and that of a motorcycle is 
. An insured vehicle met whith an accident. Find the probability that the accident vehicle was a motorcycle
As per the given question,
The lengths of the diagonals of a rhombus are 24cm and 10cm. The length of each side of the rhombus is (a) 12cm (b) 13cm (c) 14cm (d) 17cm
Correct Answer: (b) 13 cm Explanation: Let ABCD be the rhombus with diagonals AC and BD intersecting each other at O. AC = 24 cm BD = 10 cm Diagonals of a rhombus bisect each...
In a rhombus of side 10cm, one of the diagonals is 12cm long. The length of the second diagonal is (a) 20cm (b) 18cm (c) 16cm (d) 22cm
Correct Answer: (c) 16 cm Explanation: Let ABCD be the rhombus with diagonals AC and BD intersecting each other at O. Also, diagonals of a rhombus bisect each other at...
Differentiate the following functions with respect to x: ![\[{{\tan }^{-1}}\left( \frac{5x}{1-6{{x}^{2}}} \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-ae8be0c9136357a6337c075fae16c653_l3.png "Rendered by QuickLaTeX.com")
In an equilateral triangle ABC, if AD ⊥ BC, then which of the following is true? 
Correct Answer: (c) 3???????? 2 = 4???????? 2 Explanation: Applying Pythagoras theorem, In right-angled triangles ABD and ADC, $\begin{array}{l} A{B^2} = A{D^2} + B{D^2}\\ A{B^2} = {\left(...
Differentiate the following functions with respect to x: ![\[{{\tan }^{-1}}\left( \frac{3-2x}{1+6x} \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-b5ddc8b349634beba58684d306a5ad8f_l3.png "Rendered by QuickLaTeX.com")
For and the chances of being selected as the manager of a firm are in the ratio 
respectively. The respective probabilities for them to introduce a radical change in marketing startegy are 
and 
. If the change does take place, find the probability that it is due to the appointment of or .
As per the given question,
Differentiate the following functions with respect to x: ![\[{{\tan }^{-1}}\left( \frac{\sqrt{a}+\sqrt{x}}{1-\sqrt{ax}} \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-36d29b86f33ba0f144d47ede490b44c5_l3.png "Rendered by QuickLaTeX.com")
In a triangle, the perpendicular from the vertex to the base bisects the base. The triangle is (a) right-angled (b) isosceles (c) scalene (d) obtuse-angled
Correct Answer: (b) Isosceles Explanation: In an isosceles triangle, the perpendicular from the vertex to the base bisects the base.
In a ∆ABC, it is given that AD is the internal bisector of ∠A. If AB = 10cm, AC = 14cm and BC = 6cm, then CD = ? (a) 4.8cm (b) 3.5cm (c) 7cm (d) 10.5cm
Correct Answer: (b) 3.5cm Explanation: Using angle bisector in ∆ABC, $\begin{array}{l} \frac{{AB}}{{AC}} = \frac{{BD}}{{DC}}\\ \frac{10}{14} = \frac{6-x}{x}\\...
Differentiate the following functions with respect to x: ![\[{{\tan }^{-1}}\left( \frac{\sqrt{x}-x}{1+{{x}^{3/2}}} \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-42233773be8bcde653a8e026a233d4c0_l3.png "Rendered by QuickLaTeX.com")
In ∆ABC, it is given that AD is the internal bisector of ∠A. If BD = 4cm, DC = 5cm and AB = 6cm, then AC = ? (a) 4.5cm (b) 8cm (c) 9cm (d) 7.5cm
Correct Answer: (d) 7.5 cm Explanation: Given, AD bisects angle A Applying angle bisector theorem, $\begin{array}{l} \frac{{BD}}{{DC}} = \frac{{AB}}{{AC}}\\ \frac{4}{5} =...
Differentiate the following functions with respect to x: ![\[{{\cos }^{-1}}(2x)+1{{\cos }^{-1}}(\sqrt{1-4{{x}^{2}}})\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-af2a5e3b3a955a9b298e41a6c27dc619_l3.png "Rendered by QuickLaTeX.com")
In a certain college, 
of boys and of girls are taller than 
meters. Further more, of the students in the college are girls. A student selected at random from the college is found to be taller than meters. Find the probability that the selected student is girl.
As per the given question,
Differentiate the following functions with respect to x: ![\[{{\tan }^{-1}}\left( \frac{{{e}^{2x}}+1}{{{e}^{2x}}-1} \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-5be39f5d5f3b4862002e3a36a158f686_l3.png "Rendered by QuickLaTeX.com")
In a ∆ABC, it is given that AB = 6cm, AC = 8cm and AD is the bisector of ∠A. Then, BD : DC = ? (a) 3 : 4 (b) 9 : 16 (c) 4 : 3 (d) √3 : 2
Correct Answer: (a) 3 : 4 Explanation: In ∆ ABD and ∆ACD, ∠???????????? = ∠???????????? $\begin{array}{l} \frac{{BD}}{{DC}} = \frac{{AB}}{{AC}}\\ \frac{6}{8} = \frac{3}{4}\\ BD:DC =...
Differentiate the following functions with respect to x: ![\[{{\sin }^{-1}}\left( \frac{{{x}^{2}}}{\sqrt{{{a}^{4}}+{{x}^{4}}}} \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-51ec26adcb17ae190a3d4616ec483129_l3.png "Rendered by QuickLaTeX.com")
Differentiate the following functions with respect to x: ![\[{{\tan }^{-1}}\left( \frac{\sqrt{1+{{a}^{2}}{{x}^{2}}}-1}{ax} \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-2b31568800750eec9955126b975c8244_l3.png "Rendered by QuickLaTeX.com")
Differentiate the following functions with respect to x: ![\[{{\sin }^{-1}}(2ax\sqrt{1-{{a}^{2}}{{x}^{2}}})\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-5d928dae54669e3b4cc8f63521ac3c61_l3.png "Rendered by QuickLaTeX.com")
A bag contains 1 white and 6 red balls, and a second bag contains 4 white and 3 red balls. One of the bags is picked up at random and a ball is randomly drawn from it, and is found to be white in colour. Find the probability that the drawn ball was from the first bag.
As per the given question,
Differentiate the following functions with respect to x: ![\[{{\tan }^{-1}}\left( \frac{x}{\sqrt{{{a}^{2}}-{{x}^{2}}}} \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-98c017b51b32f0a425e73af69ab6bafe_l3.png "Rendered by QuickLaTeX.com")
Differentiate the following functions with respect to x: ![\[{{\cos }^{-1}}\left( \frac{1-{{x}^{2n}}}{1+{{x}^{2n}}} \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-e7ccaa494edacbb3d273f9af0ddb647b_l3.png "Rendered by QuickLaTeX.com")
Differentiate the following functions with respect to x: ![\[{{\sec }^{-1}}\left( \frac{{{x}^{2}}+1}{{{x}^{2}}-1} \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-529116632c327e3fc38c318cb9e0965b_l3.png "Rendered by QuickLaTeX.com")
Differentiate the following functions with respect to x: ![\[{{\sin }^{-1}}\left( \frac{1}{\sqrt{1+{{x}^{2}}}} \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-da52dcffbfc8c11fb150692e6dcd0375_l3.png "Rendered by QuickLaTeX.com")
A factory has three machines and , which produce 100,200 and 300 items of a particular type daily. The machines produœe 
and 
defective items respectively. One day when the production was over, an item was picked up randomly and it was found to be defective. Find the probability that it was produced by machine .
As per the given question,
In a group of 400 people, 160 are smokers and non-vegetarian, 100 are smokers and vegetarian and the remaining are nonsmokers and veqetarian. The probabilities of aettina a special chest disease are 
and 
respectively. A person is chosen from the group at random and is found to be suffering from the disease. What is the probability that the selected person is a smoker and non-vegetarian?
As per the given question,
∆ABC is an isosceles triangle with AB = AC = 13cm and the length of altitude from A on BC is 5cm. Then, BC = ? (a) 12cm (b) 16cm (c) 18cm (d) 24cm
Correct Answer: (d) 24 cm Explanation: In triangle ABC, Let the altitude from A on BC meets BC at D. AD = 5 cm AB = 13 cm D is the midpoint of BC Applying Pythagoras theorem in...
Three urns and contain 6 red and 4 white; 2 red and 6 white; and 1 red and 5 white balls redpectively. an urn is chosen at random and a ball is drawn. If the ball drawn is found to be red, find the probability that the ball was drawn from urn .
As per the given question,
The height of an equilateral triangle having each side 12cm, is (a) 6√2 cm (b) 6√3m (c) 3√6m (d) 6√6m
Correct Answer: (b) 6√3????m Explanation: Let ABC be the equilateral triangle with AD as its altitude from A. In right-angled triangle ABD, ????????2 = ????????2 + ????????2...
The hypotenuse of a right triangle is 25cm. The other two sides are such that one is 5cm longer than the other. The lengths of these sides are (a) 10cm, 15cm (b) 15cm, 20cm (c) 12cm, 17cm (d) 13cm, 18cm
Correct Answer: (b) 15 cm, 20 cm Explanation: Given, Length of hypotenuse = 25 cm Let the other two sides be x cm and (x−5) cm. Applying Pythagoras theorem, 252 = ????2 + (???? − 5 ) 2 625 = ????2 +...
In the given figure, O is the point inside a ∆MNP such that ∆MOP = .OM = 16 cm and OP = 12 cm if MN = 21cm and ∆NMP = then NP=?
Answer: In right triangle MOP, By using Pythagoras theorem, ????????2 = ????????2 + ????????2 => 122 + 162 => 144 + 256 => 400 MO = 20 cm In right triangle MPN, By using...
A ladder 25m long just reaches the top of a building 24m high from the ground. What is the distance of the foot of the ladder from the building? (a) 7m (b) 14m (c) 21m (d) 24.5m
Correct Answer: (a) 7 m Explanation: Let the ladder BC reaches the building at C. Let the height of building where the ladder reaches be AC. BC = 25 m AC = 24 m In right-angled...
The shadow of a 5m long stick is 2m long. At the same time the length of the shadow of a 12.5m high tree (in m) is (a) 3.0 (b) 3.5 (c) 4.5 (d) 5.0
Correct Answer: (d) 5.0 Explanation: Suppose DE is a 5 m long stick and BC is a 12.5 m high tree. Suppose DA and BA are the shadows of DE and BC. In ∆ABC and ∆ADE...
A vertical pole 6m long casts a shadow of length 3.6m on the ground. What is the height of a tower which casts a shadow of length 18m at the same time? (a) 10.8m (b) 28.8m (c) 32.4m (d) 30m
Correct Answer: (d) 30m Explanation: Let AB and AC be the vertical pole and its shadow, AB = 6 m AC = 3.6 m Let DE and DF be the tower and its shadow. DF = 18 m DE =? In...
A company has two plants to manufacture bicycles. The first plant manufactures of the bicycles and the second plant . Out of that of the bicycles are rated of standard quality at the first plant and of standard quality at the second plant. A bicycle is picked up at random and found to be standard quality. Find the probability that it comes from the second plant.
As per the given question,
A vertical stick 1.8m long casts a shadow 45cm long on the ground. At the same time, what is the length of the shadow of a pole 6m high? (a) 2.4m (b) 1.35m (c) 1.5m (d) 13.5m
Correct Answer: (c) 1.5m Explanation: Let AB and AC be the vertical stick and its shadow, AB = 1.8 m AC = 45 cm => 0.45 m Let DE and DF be the pole and its shadow, DE = 6 m...
Two poles of height 13m and 7m respectively stand vertically on a plane ground at a distance of 8m from each other. The distance between their tops is (a) 9m (b) 10m (c) 11m (d) 12m
Correct Answer: (b) 10 m Explanation: Let AB and DE be the two poles. AB = 13 m DE = 7 m Distance between their bottoms = BE = 8 m Draw a perpendicular DC to AB from D,...
A man goes 24m due west and then 10m due both. How far is he from the starting point? (a) 34m (b) 17m (c) 26m (d) 28m
Correct Answer: (c) 26 m Explanation: The man starts from point A and goes 24 m due west to point B. From here, he goes 10 m due north and stops at C. In right triangle...
For the following statement state whether true(T) or false (F). The sum of the squares on the sides of a rhombus is equal to the sum of the squares on its diagonals
Answer: ABCD is a rhombus having AC and BD its diagonals. The diagonals of a rhombus perpendicular bisect each other. AOC is a right-angled triangle. In right triangle AOC, By using Pythagoras...
For each of the following statements state whether true(T) or false (F) (i) the ratio of the perimeter of two similar triangles is the same as the ratio of their corresponding medians. (ii) if O is any point inside a rectangle ABCD then 
Answers: (i) True Given, ∆ABC ~ ∆DEF ∠???????????? = ∠???????????? ∠???? = ∠???? (∠???????????? ~ ∆????????????) By AA criterion, ∆ABP and ∆DEQ $\frac{A B}{D E}=\frac{A P}{D Q}$...
For each of the following statements state whether true(T) or false (F) (i) In a ABC , AB = 6 cm, A 
and AC = 8 cm and in a DEF , DF = 9 cm D = and DE= 12 cm, then ABC ~ DEF. (ii) the polygon formed by joining the midpoints of the sides of a quadrilateral is a rhombus.
Answers: (i) False In ∆ABC, AB = 6 cm ∠???? = 450 ???????? = 8 ???????? I???? ∆????????????, ???????? = 9 ???????? ∠???? = 450 ???????? = 12 ???????? ∆???????????? ~ ∆???????????? (ii) False...
In a factory, machine produces of the total output, machine produces and the machine produces the remaining output. If defective items produced by machines and are 
respectively. Three machines working output and found to be defective. Find the probability that it was produced by machine ?
As per the given question,
For each of the following statements state whether true(T) or false (F) (i) if two triangles are similar then their corresponding angles are equal and their corresponding sides are equal (ii) The length of the line segment joining the midpoints of any two sides of a triangles is equal to half the length of the third side.
Answers: (i) False If two triangles are similar, their corresponding angles are equal and their corresponding sides are proportional. (ii) True ABC is a triangle with M, N DE is...
For each of the following statements state whether true(T) or false (F) (i) Two circles with different radii are similar. (ii) any two rectangles are similar
Answers: (i) False Two rectangles are similar if their corresponding sides are proportional. (ii) True Two circles of any radii are similar to each other.
Find the length of each side of a rhombus are 40 cm and 42 cm. find the length of each side of the rhombus.
Answer: ABCD is a rhombus. The diagonals of a rhombus perpendicularly bisect each other. ∠???????????? = 900 ???????? = 20 ???????? ???????? = 21 ???????? In right...
In the given figure, ∠ AMN = ∠ MBC = 
. If p, q and r are the lengths of AM, MB and BC respectively then express the length of MN of terms of P, q and r.
Answer: In ∆AMN and ∆ABC, ∠???????????? = ∠???????????? =$76^{\circ}$ ∠???? = ∠???? (????????????????????????) By AA similarity criterion, ∆AMN ~ ∆ABC If two triangles...
If the lengths of the sides BC, CA and AB of a ∆ ABC are a, b and c respectively and AD is the bisector ∠ A then find the lengths of BD and DC
Answer: Let, DC = x BD = a - x Using angle bisector there in ∆ ABC, $\frac{A B}{A C}=\frac{B D}{D C}$ $\frac{c}{b}=\frac{a-x}{x}$ $c x=a b-b x$ $x(b+c)=a b$ $x=\frac{a...
A man goes 12m due south and then 35m due west. How far is he from the starting point?
Answer: In right-angled triangle SOW, Using Pythagoras theorem, ????????2 = ????????2 + ????????2 => 352 + 122 => 1225 + 144 => 1369 ???????? = 37 ???? Hence,...
There are three coins. One is two-headed coin (having head on both faces), another is biased coin that comes up heads of the times and third is also a biased coin that comes up tail of the times. One of the three coins is chosen at random and tossed, and it shows heads. What is the probability that it was the two-headed coin?
As per the given question,
An item is manufactured by three machine A, B and C. out of the total number of items manufactured durina a specified period. 
are manufacture on machine A on and on C. 
of the items produced on and of items produced on are defective and 
of these produced on are defective. All the items stored at one godown. One items is drawn at random and is found to be defective. What is the probability that it was manufactured on machine A?
As per the given question,
A manufacturer has three machine operators and . The first operator A produces defective items, where as the other two operators and produce and 
defective items respectively. is on the job for of the time, is on the job for of the time and is on the job for of the time. A defective item is produced, what is the probability that was produced by ?
As per the given question,
Suppose we have four boxes 
and 
containing coloured marbles as given below: ![\[\begin{tabular}{|c|c|c|c|} \hline Box & \multicolumn{3}{|c|}{ Colour } \\ \cline { 2 - 4 } & Red & White & Black \\ \hline A & 1 & 6 & 3 \\ \hline B & 6 & 2 & 2 \\ \hline C & 8 & 1 & 1 \\ \hline D & 0 & 6 & 4 \\ \hline \end{tabular}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-7e969fd3ae37333f22397872de21f8c2_l3.png "Rendered by QuickLaTeX.com")
one of the boxes has been selected at random and a single marble is drawn from it. If the marble is red, what is the probability that it was drawn from that box ? Box B? Box C?
As per the given question,
An insurance company issued 3000 scooters, 4000 cars and 5000 trucks. The probabilities of the accident involving a scooter, a car and a truck are , 
and 
respectively. One of the insured vehicles meet with an accident Find the probability that it is a truck.
As per the given question,
An insurance company issued 3000 scooters, 4000 cars and 5000 trucks. The probabilities of the accident involving a scooter, a car and a truck are , and respectively. One of the insured vehicles meet with an accident Find the probability that it is a (i) scooter (ii) car
As per the given question,
A factory has three machines X, Y, and Z producing 1000, 2000 and 3000 bolts per day respectively. The machine X produces 1% defective bolts, Y produces 1.5% and Z produces 2% defective bolts. At the end of the day, a bolt is drawn at random and is found to be defective. What is the probability that this defective bolt has been produced by machine?
Total number of bolts produced in day =(1000+2000+3000) =6000 Let E1, E2 and E3 be the events of drawing a bolt produced by machines X, Y and Z respectively. Then, P(E)=1000/6000=1/6,...
Each of the equal sides of an isosceles triangle is 25 cm. Find the length of its altitude if the base is 14 cm.
Answer: The altitude drawn from the vertex opposite to the non-equal side bisects the non-equal side. ABC is an isosceles triangle having equal sides AB and BC. The altitude drawn from the vertex...
In triangle BMP and CNR it is given that PB = 5 cm, MP = 6cm BM = 9 cm and NR = 9cm. If ∆BMP ~ ∆CNR then find the perimeter of ∆CNR.
Answer: When two triangles are similar, then the ratios of the lengths of their corresponding sides are proportional. ∆BMP ~ ∆CNR $\frac{B M}{C N}=\frac{B P}{C R}=\frac{M P}{N R}$ $\quad \frac{B...
In the given figure MN|| BC and AM: MB= 1: 2. Find 
Answer: Given, AM : MB = 1 : 2 $\frac{M B}{A M}=\frac{2}{1}$ Adding 1 to both sides, $\frac{M B}{A M}+1=\frac{2}{1}+1$ $\frac{M_{B}+A M}{A M}=\frac{2+1}{1}$ $\frac{A B}{A...
Two triangles DEF an GHK are such that 
and 
. If ∆DEF ~ ∆GHK then find the measures of ∠F.
Answer: If two triangles are similar then the corresponding angles of the two triangles are equal. ∆DEF ~ ∆GHK ∴ ∠???? = ∠???? = 570 In ∆ DEF Using the a????????????????...
Mark (√) against the correct answer in the following: The general solution of the 
is
A. 
B. 
C. 
D. None of these
Solution: $\frac{\mathrm{dy}}{\mathrm{dx}}+\frac{\mathrm{y}}{\mathrm{x}}=\mathrm{x}^{2}$ Comparing with $\frac{d y}{d x}+P y=Q$ Integrating factor $=\mathrm{x}$ General solution is $\mathrm{yx}=\int...
Find the length of each side of a rhombus whose diagonals are 24cm and 10cm long.
Answer: ABCD is a rhombus. The diagonals of a rhombus perpendicularly bisect each other. ∠???????????? = 900 ???????? = 12 ???????? ???????? = 5 ???????? In right triangle AOB,...
Mark (√) against the correct answer in the following: The general solution of the DE 
is
A. 
B. 
C. 
D. None of these
Solution: $\frac{d y}{d x}+y \operatorname{Cot} x=2 \operatorname{Cos} x$ Comparing with $\frac{d y}{d x}+P y=Q$ Integrating factor is $e^{\int \cot x d x}=\operatorname{Sin} x$ General solution is...
Mark (√) against the correct answer in the following: The general solution of the 
A. 
B. 
C. 
D. None of these
Solution: $\frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{y} \tan \mathrm{x}=\mathrm{Secx}$ Comparing with $\frac{d y}{d x}+P y=Q$ Integrating factor $e^{\int \tan x d x}=\operatorname{Sec} x$ General...
Mark (√) against the correct answer in the following: The general solution of the DE 
is
A. 
B. 
C. 
D. None of these
Solution: $\frac{d y}{d x}=\frac{y}{x}+\sin \frac{y}{x}$ Let $\mathrm{y}=\mathrm{vx}$ $\begin{array}{l} \mathrm{dy} / \mathrm{dx}=\mathrm{v}+\mathrm{xdv} / \mathrm{d} \mathrm{x} \\...
In an equilateral triangle with side a, prove that area = 
Answer: We know that the altitude of an equilateral triangle bisects the side on which it stands and forms right angled triangles with the remaining sides. ABC is an...
The corresponding sides of two similar triangles are in the ratio 2: 3. If the area of the smaller triangle is 48cm2, find the area of the larger triangle.
Answer: If two triangles are similar, then the ratio of their areas is equal to the squares of their corresponding sides. $\frac{\text { area of } smaller triangle}{\text { area of } larger...
In a trapezium ABCD, it is given that AB║CD and AB = 2CD. Its diagonals AC and BD intersect at the point O such that ar(∆AOB) = 84cm2 . Find ar(∆COD).
Answer: In ∆ AOB and COD ∠???????????? = ∠???????????? (???????????????????????????????? ???????????????????????? ???????? ???????? ∥ ????????) ∠???????????? = ∠????????????...
Mark (√) against the correct answer in the following: The general solution of the DE 
is
A. 
B. 
C. 
D. None of these
Solution: $\begin{array}{r} (x-y) d y+(x+y) d x=0 \\ \qquad \frac{d y}{d x}=\frac{x+y}{y-x} \end{array}$ Let $\mathrm{y}=\mathrm{vx}$ $\begin{array}{l} \mathrm{dy} /...
∆ABC~∆DEF such that ar(∆ABC) = 64 cm2 and ar(∆DEF) = 169cm2. If BC = 4cm, find EF.
Answer: Given, ∆ ABC ~ ∆ DEF If two triangles are similar then the ratio of their areas is equal to the ratio of the squares of their corresponding sides. $\frac{\text { area }(\triangle A B...
Mark (√) against the correct answer in the following: The general solution of the DE 
is
A. 
B. 
C. 
D. None of these
Solution: $\begin{array}{r} 2 \mathrm{xydy}+\left(\mathrm{x}^{2} \quad \mathrm{y}^{2}\right) \mathrm{d} \mathrm{x}=0 \\ \frac{d y}{d x}=\frac{y^{2}-x^{2}}{2 x y} \end{array}$ Let...
Find the length of the altitude of an equilateral triangle of side 2a cm.
Answer: The altitude of an equilateral triangle bisects the side on which it stands and forms right angled triangles with the remaining sides. ABC is an equilateral...
A ladder 10m long reaches the window of a house 8m above the ground. Find the distance of the foot of the ladder from the base of the wall.
Answer: Let AB be A ladder and B is the window at 8 m above the ground C. In right triangle ABC By using Pythagoras theorem, ????????2 = ????????2 + ????????2 102 = 82 +...
Mark (√) against the correct answer in the following: The general solution od the DE 
is
A. 
B. 
C. 
D. None of these
Solution: $\mathrm{x} \frac{d y}{d x}=y+x \tan \frac{y}{x}$ Dividing both sides by $x$, we get, $\frac{d y}{d x}=\frac{y}{x}+\tan \frac{y}{x}$ Let $\mathrm{y}=\mathrm{vx}$ Differentiating both...
In the given figure, DE║BC such that AD = x cm, DB = (3x + 4) cm, AE = (x + 3) cm and EC = (3x + 19) cm. Find the value of x.
Answer: In ∆ADE and ∆ABC, ∠???????????? = ∠???????????? (???????????????????????????????????????????????????? ???????????????????????? ???????? ???????? ∥ ????????)...
Mark (√) against the correct answer in the following: The general solution of the DE 
is.
A. 
B. 
C. 
D. None of these
Solution: $\begin{array}{l} \mathrm{x}^{2} \frac{d y}{d x}=x^{2}+x y+y^{2} \\ \frac{d y}{d x}=1+\frac{y}{x}+\frac{y^{2}}{x^{2}} \end{array}$ Let $\mathrm{y}=\mathrm{vx}$ $\begin{array}{l}...
In ∆ABC~∆DEF such that 2AB = DE and BC = 6cm, find EF.
Answer: When two triangles are similar, then the ratios of the lengths of their corresponding sides are equal. ∆ABC ~ ∆DEF $\therefore \frac{A B}{D E}=\frac{B C}{E F}$ $\frac{A B}{2 A B}=\frac{6}{E...
Two triangles ABC and PQR are such that AB = 3 cm, AC = 6cm, ∠???? = 
, PR = 9cm ∠???? = and PQ = 4.5 cm. Show that ∆ ABC ~ ∆ PQR and state that similarity criterion.
Answer: In ∆ABC and ∆PQR ∠???? = ∠???? = 700 $\frac{A B}{P Q}=\frac{A C}{P R}$ $\frac{3}{4.5}=\frac{6}{9}$ $\frac{1}{1.5}=\frac{1}{1.5}$ By SAS similarity criterion, ∆ABC ~...
Mark (√) against the correct answer in the following: The general solution of the 
is
A. 
B. 
C. 
D. None of these
Solution: $\frac{d y}{d x}=\frac{y^{2}-x^{2}}{2 x y}$ Let $\mathrm{y}=\mathrm{vx}$ $\begin{array}{l} \mathrm{dy} / \mathrm{dx}=\mathrm{v}+\mathrm{xdv} / \mathrm{dx} \\ \frac{x^{2}...
If D, E, F are the respectively the midpoints of sides BC, CA and AB of ∆ABC. Find the ratio of the areas of ∆DEF and ∆ABC.
Answer: Using midpoint theorem, The segment joining two sides of a triangle at the midpoints of those sides is parallel to the third side and is half the length of...
State the converse of Pythagoras theorem.
Converse of Pythagoras theorem: If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.
Mark (√) against the correct answer in the following: The general solution of the 
A. 
B. 
C. 
D. None of these
Solution: $\begin{array}{r} \frac{\mathrm{dy}}{\mathrm{dx}}=\sqrt{1-\mathrm{x}^{2}} \sqrt{1-\mathrm{y}^{2}} \\ \frac{d y}{\sqrt{1-y^{2}}}=\sqrt{1-x^{2}} d x \end{array}$ $\operatorname{Let}...
State Pythagoras theorem
Pythagoras theorem: The square of the hypotenuse is equal to the sum of the squares of the other two sides. Here, the hypotenuse is the longest side and it’s always opposite the right angle.
State the SAS-similarity criterion
SAS-similarity criterion: If one angle of a triangle is equal to one angle of the other triangle and the sides including these angles are proportional then the two triangles are similar.
Mark (√) against the correct answer in the following: The general solution of the DE 
is
A. 
B. 
C. 
D. None of these
Solution: $\log \left(\frac{d y}{d x}\right)=(a x+b y)$ $\begin{array}{l} \frac{d y}{d x}=e^{a x+b y} \\ \frac{d y}{e^{b y}}=e^{a x} d x \end{array}$ On integrating on both sides we get...
State the SSS-similarity criterion for similarity of triangles
SSS-similarity criterion for similarity of triangles: If the corresponding sides of two triangles are proportional then their corresponding angles are equal, and hence the two triangles are...
Mark (√) against the correct answer in the following: The general solution of the 
is
A. 
B. 
C. 
D. None of these
Solution: $\begin{array}{r} x \sqrt{1+y^{2}} \mathrm{dx}+\mathrm{y} \sqrt{1+x^{2}} \mathrm{dy}=0 \\ \frac{y d y}{\sqrt{1+y^{2}}}=\frac{-x d x}{\sqrt{1+x^{2}}} \end{array}$ Let $1+y^{2}=t$ and...
State the AA-similarity criterion
AA-similarity criterion: If two angles are correspondingly equal to the two angles of another triangle, then the two triangles are similar.
State the AAA-similarity criterion
AAA-similarity criterion: If the corresponding angles of two triangles are equal, then their corresponding sides are proportional and hence the triangles are similar.
In a class, of the boys and of the girls have an IQ of more than 
In this class, of the students are boys. If a student is selected at random and found to have an IQ of more than 150 , find the probability that the student is a boy.
As per the given question,
State the midpoint theorem
Midpoint theorem: The line segment connecting the midpoints of two sides of a triangle is parallel to the third side and is equal to one half of the third side.
Mark (√) against the correct answer in the following: the general solution of the DE 
is
A. 
B. 
C. 
D. None of these
Solution: $\left(1+x^{2}\right) d y-x y d x=0$ $\frac{d y}{y}=\frac{x}{1+x^{2}} d x$ Let $1+\mathrm{x}^{2}=\mathrm{t}$ $\begin{array}{r} 2 \mathrm{x} \mathrm{dx}=\mathrm{dt} \\ \frac{d y}{y}=\frac{d...
State and converse of Thale’s theorem.
Thale’s theorem: If a line divides any two sides of a triangle in the same ratio, then the line must be parallel to the third side.
Mark (√) against the correct answer in the following: The solution of the DE 
is
A. 
B. 
C. 
D. none of these
Solution: $\frac{d y}{d x}+y \log y \operatorname{Cot} x=0$ Let $\log \mathrm{y}=\mathrm{t}$ On differentiating we get $\begin{array}{l} \frac{1}{y} d y=d t \\ \frac{d t}{t}=-\operatorname{Cot} x d...
State the basic proportionality theorem.
Basic proportionality theorem: If a line is draw parallel to one side of a triangle intersect the other two sides, then it divides the other two sides in the same ratio.
Mark (√) against the correct answer in the following: the solution of the DE 
is
A. 
B. 
C. 
D. none of these
Solution: $\begin{array}{l} x \operatorname{Cosydy}=\left(x e^{x} \log x+e^{x}\right) d x \\ \operatorname{Cosydy}=\frac{x \operatorname{exlogx}+e x}{x} d x \end{array}$ On integrating on both sides...
State the two properties which are necessary for given two triangles to be similar.
Answer: The two triangles are similar if and only if The corresponding sides are in proportion. The corresponding angles are equal.
Mark (√) against the correct answer in the following: The solution of the 
is
A. 
B. 
C. 
D. none of these
Solution: $\cos x(1+\cos y) d x-\sin y(1+\sin x) d y=0$ Let $1+\cos y=t$ and $1+\sin x=u$ On differentiating both equations, we get $-\sin y d y=d t$ and $\cos x d x=d u$ Substitute this in the...
A letter is known to have come either from LONDON or CLIFTON. On the envelope just two consecutive letters 
are visible. What is the probability that the letter has come from (i) LONDON (ii) CLIFTON?
As per the given question,
Mark (√) against the correct answer in the following: The solution of the 
is
A. 
B. 
C. 
D. None of these
Solution: $\begin{array}{l} \frac{d y}{d x}=\frac{-2 x y}{x^{2}+1} \\ \frac{d y}{y}=\frac{-2 x d x}{x^{2}+1} \end{array}$ Let $\mathrm{x}^{2}+1=\mathrm{t}$ On differentiating on both sides we get $2...
Mark (√) against the correct answer in the following: The solution of the 
is
A. 
B. 
C. 
D. None of these
Solution: $\begin{array}{l} \frac{d y}{d x}=\frac{1-\operatorname{Cos} x}{1+\operatorname{Cos} x} \\ \frac{d y}{d x}=\frac{2 \sin ^{2} \frac{x}{2}}{2 \cos ^{2} \frac{x}{2}}, \\ \frac{d y}{d x}=\tan...
Mark (√) against the correct answer in the following: The solution of the 
is
A. 
B. 
C. 
D. None of these
Solution: $\frac{\mathrm{dy}}{\mathrm{dx}}+\sqrt{\frac{1-\mathrm{y}^{2}}{1-\mathrm{x}^{2}}}=0$ $\frac{-d y}{\sqrt{1-y^{2}}}=\frac{d x}{\sqrt{1-x^{2}}}$ On integrating on both sides, we get $-\sin...
Mark (√) against the correct answer in the following: The solution of the 
is
A. 
B. 
C. 
D. None of these
Solution: $\begin{array}{l} \frac{d y}{d x}=e^{x+y}+x^{2} e^{y} \\ \left(e^{-y}\right) d y=\left(e^{x}+e^{2}\right) d x \end{array}$ On integrating on both sides, we get $\begin{array}{l}...
Mark (√) against the correct answer in the following: The solution of the DE 
is
A. 
B. 
C. 
D. None of these
Solution: $\begin{array}{l} \frac{d y}{d x}=1-x+y-x y \\ \frac{d y}{d x}=1-x+y(1-x) \\ \frac{d y}{d x}=(1+y)(1-x) \\ \frac{d y}{1+y}=(1-x) d x \end{array}$ On integrating on both sides, we get $\log...
Mark (√) against the correct answer in the following: The solution of the 
is.
A. 
B. 
C. 
D. None of these
Solution: $\frac{d y}{d x}=\frac{1+y^{2}}{1+x^{2}}$ On integrating on both sides, we get $\begin{array}{l} \tan ^{-1} y=\tan ^{-1} x+c \\ \frac{y-x}{1+y x}=\mathrm{c} \\...
Mark (√) against the correct answer in the following: The solution of the 
is
A. 
B. 
C. 
D. None of these
Solution: $x \frac{d y}{d x}=\cot y$ Separating the variables, we get, $\begin{array}{c} \frac{d y}{\cot y}=\frac{d x}{x} \\ \tan y d y=\frac{d x}{x} \end{array}$ Integrating both sides, we get,...
Mark (√) against the correct answer in the following: The solution of the 
is
A. 
B. 
C. 
D. None of these
Solution: $\begin{array}{l} \text { Given } x d y+y d x=0 \\ x d y=-y d x \end{array}$ $-\frac{d y}{y}=\frac{d x}{d x}$ On integrating on both sides we get, $\begin{array}{l} -\log y=\log x+c \\...
Mark (√) against the correct answer in the following: The solution of the 
is
A.
B. 
C. 
D. None of these
Solution: $\begin{array}{l} \left(e^{x}+1\right) y d y=(y+1) e^{x} d x \\ \frac{y d y}{y+1}=\frac{e^{x} d x}{\left(e^{x}+1\right)} \end{array}$ On differentiating on both sides we get $e^{x} d x=d...
Suppose 5 men out of 100 and 25 women out of 1000 are good orators. An orator is chosen at random. Find the probability that a male person is selected. Assume that there are equal number of men and women.
As per the given question,
Mark (√) against the correct answer in the following: The solution of the DE 
is
A. 
B. 
C. 
D. None of these
Solution: $\begin{array}{c} \frac{d y}{d x}=2^{x} 2^{y} \\ 2^{-y} d y=2^{x} d x \end{array}$ $\frac{d y}{d x}=2^{x+y}$ On integrating on both sides, we get $\begin{array}{c} -\frac{2^{-y}}{\log...
Mark (√) against the correct answer in the following: The solution of the DE 
is
A. 
B. 
C. 
D. None of these
Solution: $\begin{array}{c} \frac{d y}{d x}=e^{x+y} \\ \frac{d y}{d x}=e^{x} e^{y} \\ e^{-y} d y=e^{x} d x \end{array}$ On integrating on both sides, we get $\begin{array}{c}...