Sol. Correct option is (D).$-3 \vec{i}-8 \vec{j}$
If the position vectors of the points A and B be respectively 
and 
then 
(A) 
(B) 
(C) 
(D) 
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Maths
If the position vectors of the points A and B be respectively
(A) 
(B) 
(C) 2 (D) 
Sol. 6 Correct option is (D).$\sqrt{14}$
The position vector of the point 
is (A) 
(B) 
(C) 
(D) 
Sol. Correct option is(D).$x \vec{i}+y \vec{j}+z \vec{k}$
The degree of the equation 
is (A) 0 (B) 1 (C) 2 (D) 3
The degree of the equation $\left(\frac{d^{2} y}{d x^{2}}\right)^{2}-x\left(\frac{d y}{d x}\right)^{3}=y^{3}$ is (A) 0 (B) 1 (C) 2 (D) 3 Sol. Correct option is(C).2
The order of the differental equation 
is (A) 0 (B) 1 (C) 2 (D) 3
Sol. Correct option is (B).1
The integrating factor of the linear differential equation 
(A) 
(B) 
(C) 
(D) 
Sol. Correct option is (A) $\int_{e} P d y$
The solution of the different equation 
is (A) 
(B) 
(C) 
(D) 
Sol. Correct option is (A)$e^{x}+e^{-y}+k=0$.
(A) 
(B) 
(C) 
(D) 
Sol. Correct option is (B).$\frac{a^{3}-b^{3}}{3}$
(A) 
(B) 
(C) 
(D) 
answer (B) $\tan x+\sec x+k$
The integration of 
with respect to 
is : (A) 0 (B) 
(C) 
(D) 
Sol. Correct option is (B).$k$
(A) 
(B) 
(C) 
(D) 
Sol. Correct option is (I)).$\frac{x^{9}}{9}+k$
If 
, then 
(A) 
(B) 
(C) 6 (D) 0
Sol. Correct option is(B). $6 x$
If 
, then 
(A) 
(B) 
(C) 
(D) 
Sol. 3 Correct option is (B).$\frac{-\sin (\log x)}{x}$
(A) 
(B) 0 (C) 1 (D) 2
Sol. Correct option is (C). 1
(A) 
(B) 
(C) 
(D) 
Sol. Correct option is (C).$\frac{-1}{\sqrt{1-x^{2}}}$
![A=\left[\begin{array}{ll}3 & 6 \\ 5 & -4\end{array}\right], B=\left[\begin{array}{ll}7 & 8 \\ 5 & 6\end{array}\right] \Rightarrow 2 A+3 B=](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-25b0066843bea0f9a8f6c5d79004468c_l3.png "Rendered by QuickLaTeX.com")
(A) ![\left[\begin{array}{ll}27 & 24 \\ 25 & 10\end{array}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-0244b575c30b8bb5208b42dfdcc25484_l3.png "Rendered by QuickLaTeX.com")
(B) ![\left[\begin{array}{ll}27 & 36 \\ 25 & 10\end{array}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-267b7e311bb74ec1afb12dab22e1d744_l3.png "Rendered by QuickLaTeX.com")
(C) ![\left[\begin{array}{ll}27 & 36 \\ 25 & 15\end{array}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-132988bf5f327adb8995b779bc66bf3a_l3.png "Rendered by QuickLaTeX.com")
(D) ![\left[\begin{array}{ll}27 & 36 \\ 35 & 10\end{array}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-9f4283ade214880dae7d14028a71004c_l3.png "Rendered by QuickLaTeX.com")
Sol. Correct option is (B). $\left[\begin{array}{ll}27 & 36 \\ 25 & 10\end{array}\right]$
(A) 
(B) 
(C) 
(D) 
Sol. Correct option is (D).$(a-b)(b-c)(c-a)$
(A) 
(B) 
(C) 
(d) 
Sol. Correct option is (B). $\frac{\pi}{2}$
The principal value of 
is 1 (A) 
(B) (C) (D)
Sol.Correct option is (C). $\frac{2 \pi}{3}$
The height of a TV transmission tower at any place on the surface of the earth is 
. The maximum distance up to which transmission of tower will reach is- (A) (B) 
(C) 
(D) 
Answer: Option (C)$56 \mathrm{~km}$
Boolean expression for NAND gate is- (A) 
(B) 
(C) (D) 
Answer: Option (A)$\overline{A \cdot B}=\gamma$
Diode is used as- (A) An amplifier (B) An 0scillator (C) A modulator (D) A rectifier
Answer: Option (D) A rectifier
Diode is used as- (A) An amplifier (B) An 0scillator (C) A modulator (D) A rectifier
Answer: Option (D) A rectifier
For n-type Germanium, impurity doped in Germanium is- (A) Trivalent (B) Tetravalent (C) Pentavalent (D) None of these
Answer: Option (C)Pentavalent
– Rays are deflected in- (A) Gravitational field (B) Only in magnetic field (C) Decay constant (D) Time period
Answer: Option (B)Only in magnetic field
Time during which the amount of radioactive substance becomes half of its initial nount is called. (A) Average life (B) Half-life (C) Decay constant (D) Time period
Answer: Option (B) Half-life
The energy of electron in first Bohr orbit of hydrogen atom is- 
. (A) 
(B) 
(C) 
(D) 
Answer: Option (A) $-3.4 \mathrm{eV}$
Which series of hydrogen spectrum does not lie in infrared region? (A) Humphreys series (B) Pfund series (c) Bracket series (D) Lyman series
Answer: Option (b) Pfund series and (C) Bracket series
Which one of following in charge less? (A) Alpha particle (B) Beta particle (C) Photon particle (D) Proton
Answer: Option (D)Proton
The energy of emitted photo electron depends up on- (A) Intensity of light (B) Wave length of light (C) Work-function of metal (D) None of these
Answer: Option (C) Work-function of metal
The unit of ratio of magnetic field 
and electrical field 
is- (A) 
(B) 
(C) 
(D) 
Answer: Option (B) $s m^{-1}$
The direction of transmission of electromagnetic wave is- (A) Parallel to 
(B) Parallel to 
(C) Parallel to 
(D) Parallel to 
Answer: Option (D)Parallel to $\vec{E} \times \vec{B}$
The focal length of a lens 
in air is 
. Its focal length is medium of refractive index 
is- (A) 
(B) 
(C) 
(D) 
Answer: Option (A)$20 \mathrm{~cm}$
Transverse nature of light is shown by- (A) Interference (B) Reflection (C) Polarisation (D) Dispersion
Transverse nature of light is shown by- (A) Interference (B) Reflection (C) Polarisation (D) Dispersion Answer: Option (C) Polarisation
14. The angle of minimum deviation for thin prism of refractive index 
is (A) 
(B) 
(C) 
(D) 
Answer: Option (B) $(\mu-1) A$
The critical angle of light passing from glass to air is minimum for- (A) Red colour (B) Green colour (C) Yellow colour (D) Violet colour
Answer: Option (D)Violet colour
A short sighted person uses for clear vision- (A) Convex Lens (B) Concave Lens (C) Cylindrical Lens (D) Bi-focal Lens
Answer: Option (B)Concave Lens
The relation between peak current 
and root mean square current I rms is- (A) 
(B) 
(C) 
(D) 
Answer: Option (A)$I_{0}=\sqrt{2} I_{r m s}$
The unit of reactance is- (A) Ohm (B) Fared (C) Ampere (D) Mho 2
Answer: Option (A)Ohm
If the equation of an electric current is 
, the frequency of electric current is- (A) 
(B) 50 (D) Ampere (D) Mho
Answer: Option (B)50
Wheat stone’s bridge is used in measuring – (A) High resistance (B) Low resistance (C) Both high and low resistance (D) Potential difference
Answer: Option (C) Both high and low resistance
Absorbed electrical energy is- (A) Proportional to the potential difference (B) Inversely proportional to the potential difference (C) Proportional to the square of the potential difference (1)) None of these
Answer: Option (C) Proportional to the square of the potential difference
Which element is used in electric heater? (A) C (B) Platinum (C) Tungsten (D) Nichrome
Answer: Option (D)Nichrome
S.I unit of pole strength is- (A) 
(B) N/A-m (C) A-m (D) 
Answer: Option (C) A-m
Permeability of a ferromagnetic substance- (a) 
(B) 
(C) 
(D) 
Answer: Option (A)$\mu>>1$
Three capacitors each of capacitors each of capacity 
are connected in series. The resultant capacity will be- (A) 
(B) 
(C) 
(D) 
Answer: Option (C)$C / 3$ 1
The electrical intensity inside 
(C) zero (D) 
Answer: Option (C)zero
If 
(A) 
(B) 
(C) 
(D) 
Sol. Correct answer option is (A) $\frac{1}{2 a} \log \frac{x-a}{x+a}+k$
For any unit matrix 1 (A) 
(B) 
(C) 
(D) 
Sol. Correct answer option is (A)$I^{2}=I$
(A) 
(B) 
(C) 
(D) 
Sol. Correct answer option is (C)$2 \tan ^{-1} x$
The inverse of ![A=\left[\begin{array}{cc}2 & 3 \\ 5 & \mathrm{k}\end{array}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-75f546e8870d632c3ca55afe74593675_l3.png "Rendered by QuickLaTeX.com")
will not be obtained if 
has the value (A) 2 (B) 
(C) 
(D) 
Sol. Correct answer option is (D)$\frac{15}{2}$
(A) 40 (B) 
(C) 42 (D) 15
Sol. Correct answer option is (C)42
If 
and are two independent event, then (A) 
(B) 
(C) 
(D) 
Sol. Correct answer option is (A) $P(A \cup B)=1-P\left(A^{\prime}\right) P\left(B^{\prime}\right)$
The distance of the plane 
from the point 
(A) 4 (B) 3 (C) 2 (D) 
Sol. Correct answer option is (C)2
If the planes 
and 
(A) 
(B) 
(C) 
(D)
Sol. Correct answer option is (C) $a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}=0$
If the line 
is parallel to the plane 
, then (A) 
(B) 
(C) 
(D) 
Sol. Correct answer option is (B) $a l+b m+c z=0$
The direction ratios of the normal to the plane 
are 8 (A) 
(B) 
(C) 
(D) 
Sol. Correct answer option is (B)$7,4,-2$
If the direction cosines of the two straight lines are 
and 
then the cosine of the angle 
between them or 
(A) 
(B) 
(C) 
(D) 
Sol. Correct answer option is (C) $l_{1} l_{2}+m_{1} m_{2}+n_{1} n_{2}$
The co-ordinates of the midpoint of the line segment joining the points 
and 
are (A) 
(B) 
(C) 
, (D) 
Sol. Correct answer option is (D)$(5,0,6)$
The direction ratios of the line joining the points 
and 
are (A) 
(B) ![\[\sqrt{\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}+\left(z_{1}-z_{2}\right)^{2}}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-5372bf743bfeed3f7650ef403b875b17_l3.png "Rendered by QuickLaTeX.com")
(C) 
(D) 
Sol. Correct answer option is (D)$x_{1}-x_{2}, y_{2}-y_{1}, z_{2}-z_{1}$
The direction cosines of the 
-axis are 7 (A) 
(B) 
(C) 
(D) 
Sol. Correct answer option is (C) $(0,1,0)$
(A) 0 (B) 1 (C) 
(D) 
Sol. Correct answer option is (C)$|\vec{a}|^{2}$
(A) 0 (B) 1 (C) 
(D) 
Sol. Correct answer option is (D)$-\vec{i}$
(A) 0 (B) 1 (C) 
(D) 
Correct answer option is (A)0
If 
then (A) 
(B) 
(C) 
(D) 
If $\vec{a} \cdot \vec{b}=0$ then (A) $\vec{a} \perp \vec{b}$ (B) $\vec{a} \| \vec{b}$ (C) $\vec{a}+\vec{b}=\overrightarrow{0}$ (D) $\vec{a}-\vec{b}=\overrightarrow{0}$ Sol. Correct answer option is...
If 
be the origin and 
and 
then 
is equal to (A) 
(B) 
(C) 
(D) 
Sol. Correct answer option is (D)$3 \hat{i}+\hat{j}+\hat{k}$
The modulus of 
(A) 
(B) 
(C) 
(D) 6
Sol. Correct answer option is (C)$3 \sqrt{6}$
The positive vector of the point 
is (A) 
(B) 
(C) 
(D) 
Sol. Correct answer option is (D)$\vec{i}+2 \vec{k}$
The order of the differential equation 
is (A) 1 (B) 2 (C) 3 (D) 4
Sol. Correct answer option is (B) 2
The degree of the differential equation 
is (A) 1 (B) 2 (C) 3 (D) 4
Sol. Correct answer option is (A)
The integrating factor of the linear differential equation 
(A) 
(B) 
(C) 
(D) 
Sol. Correct answer option is (B)$e^{\tan x}$
The solution of the differential equation 
is (A) 
(B) 
(B) 
(D) 
Sol. Correct answer option is (B)
(A) 
(B) 
(C) 
(D) 
Sol. Correct answer option is (B)$\frac{b^{6}-a^{6}}{6}$
(A) 
(B) 
(C) 
(D) 
Sol. Correct answer option is (A)
(A) 
(B) 
(C) (D) 
Sol. Correct answer option is (B)
(A) (B) 
(C) 
(D) 
Sol. Correct answer option is (A)
, then (A) 
(B) 
(C) 
(D) 
Sol. Correct answer option is (D)
If 
(A) 
(B) 
(C) 
(D) 
Sol. Correct answer option is (D)
(A) 0 (B) 1 (C) (D)
Sol. Correct answer option is (C)
If ![\left[\begin{array}{ll}2 x-y & 5 \\ 3 & \mathrm{y}\end{array}\right]=\left[\begin{array}{ll}6 & 5 \\ 3 & -2\end{array}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-4b795a404144fdb77f9a46aab4e01fbf_l3.png "Rendered by QuickLaTeX.com")
, then 
(A) 3 (B) 4 (C) 5 (D) 8
Sol. Correct answer option is (none of these)
(A) 
(B) 
(C) (D) 
Sol. Correction answer option is (D)
(A) 
(B) 
(C) 
(D) 
Sol. Correct answer option is (C)
The principal value of 
is (A) (B) 
(C) 
(D)
Sol. Correct answer option is (D)
If 
, then the function 
(A) One-one (B) constant (C) onto (D) many one
Sol. Correct answer option is (A) One-one
Show that the line joining the point (4,7,8),(2,3,4) is parallel to the line joining the points (2,4,10),( 2, 4,2)
Prove by Vector method, that in any triangle ABC following situation is valid
Find the area of the Smaller portion of the circle under the following conditions
prove the following
Evaluate the following
Find the maximum and minimum values of the given equation
Odds are 8:5 against a man, who is 55 years old, living till he is 75 and 4:3 against his wife who is now 48, living till she is 68. Find the probability that the Couple will be alive 20 years hence.
If A and B are independent events then prove the following condition
Find the distance of the point (4, 5,6) from the given plane
Prove by direction numbers, that the point (1, 1,3) , (2, 4,5) and (5, 13,11) are in a straight line.
Find the value of P, if situation is given
solve the following
Prove by vector method, that the angle inscribed in a semi-circle is right angle.
Evaluate the following
solve the following
evaluate the following
prove the following
integrate the following
if y is given than find dy/dx in the following
if y is given than find dy/dx in the following
Find dy/dx if x= y log (xy)
Find the value of x under given conditions
prove the given condition
prove the given condition
Find the values of x and y when following condition is given
prove the following
prove the following
solve the following question
A speak the truth in 75% cases and dB B in 80% of the cases. In what percentage of the cases are they likely to contradict each other in stating the same fact
evaluate the following
solve the following
solve dy/dx in the following
if y is given than find dy/dx
evaluate the following
Evaluate the following
Integrate the following
if y is given than find dy/dx
if y is given than find dy/dx
if a and b is given than find the given in the following
evaluate the following
minimize the following
Find the co-ordinates of the point where the line joining the points P(1, -2, 3) and Q(4, 7, 8) cuts the xy-plane.
prove the following
solve the following
solve the following
What is the chance of getting 7 or 11 with two dice
if a and b are given then find multiplication of a and b
Integrate the following
5. If y is given then find dy/dx
Solve the following for x
evaluate the following
evaluate the following
Prove the following
Minimize the given terms
Find the equation of the straight line perpendicular to the two lines and passing through their point of intersection of the given terms
Solve the differential equation
Evaluate the following
A speak the truth in 75% cases and dB B in 80% of the cases. In what percentage of the cases are they likely to contradict each other in stating the same fact
find multiplication of a and b in the following
Integrate the following
find dy/dx by first principle in the following
find dy/dx by first principle in the following
Find the value of X and Y in the following
Evaluate the following
Prove the following
(a) In the figure given below, P and Q are centers of two circles intersecting at B and C. ACD is a straight line. Calculate the numerical value of x.
(b) In the figure given below, O is the circumcenter of triangle ABC in which AC = BC. Given that ∠ACB = 56°, calculate (i)∠CAB (ii)∠OAC Solution: Given that (a) Arc AB subtends ∠APB at the center...
(a) In the figure (i) given below, AB is a diameter of the circle APBR. APQ and RBQ are straight lines, ∠A = 35°, ∠Q = 25°. Find (i) ∠PRB (ii) ∠PBR (iii) ∠BPR. (b) In the figure (ii) given below, it is given that ∠ABC = 40° and AD is a diameter of the circle. Calculate ∠DAC.
Solution (a) (i) ∠PRB = ∠BAP (Angles in the same segment of the circle) ∴ ∠PRB = 35° (∵ ∠BAP = 35° given)
(a)In the figure (i) given below, O is the centre of the circle and ∠PBA = 42°. Calculate the value of ∠PQB (b) In the figure (ii) given below, AB is a diameter of the circle whose centre is O. Given that ∠ECD = ∠EDC = 32°, calculate (i) ∠CEF (ii) ∠COF.
Solution: In ∆APB, ∠APB = 90° (Angle in a semi-circle) But ∠A + ∠APB + ∠ABP = 180° (Angles of a triangle) ∠A + 90° + 42°= 180° ∠A + 132° = 180° ⇒ ∠A = 180° – 132° = 48° But ∠A = ∠PQB (Angles in the...
In the figure (i) given below, calculate the values of x and y. (b) In the figure (ii) given below, O is the centre of the circle. Calculate the values of x and y.
(a) ABCD is cyclic Quadrilateral ∠B + ∠D = 1800 Y + 400 + 45o = 180o (y + 85o = 180o) Y = 180o – 85o = 95o ∠ACB = ∠ADB xo = 40 (a) Arc ADC Subtends ∠AOC at the centre and ∠ ABC at the remaining part...
The line through A (– 2, 3) and B (4, b) is perpendicular to the line 2x – 4y = 5. Find the value of b.
Solution: The slope of the line passing through A (-2, 3) and B (4, b) will be m1 = (b – 3)/ (4 + 2) = (b – 3)/ 6 Now, the gradient of the given line 2x – 4y = 5 is 4y = 2x + 5 y = (2/4) x + 5/4 y =...
Find the equation of a straight line whose inclination is 60° and which passes through the point (0, – 3).
Solution: Given, Inclination of a straight line is 60o So, the slope = tan 60o = √3 = m And, the equation of line passes through the point (0, -3) = (x1, y1) Hence, the equation of line is given by...
The line segment joining A (2, 3) and B (6, – 5) is intercepted by the x-axis at the point K. Write the ordinate of the point k. Hence, find the ratio in which K divides AB. Also, find the coordinates of the point K.
Solution: Since the point K is on X axis, its y co-ordinate is zero. Let the point K be (x,0). Let the point K divides the line segment joining A(2,3) and B(6,-5) in the ratio m:n. Here x1 = 2 ,...
One card is drawn from a well shuffled deck of 52 cards. Find the probability of getting: (v) a diamond or a spade
(v) Number of favorable outcomes for a diamond or a spade = 13 + 13 = 26 So, number of favorable outcomes = 26 Hence, P(getting a diamond or a spade) = 26/52 = 1/2
One card is drawn from a well shuffled deck of 52 cards. Find the probability of getting: (iii) the jack or the queen of the hearts (iv) a diamond
(iii) Favorable outcomes for jack or queen of hearts = 1 jack + 1 queen So, the number of favorable outcomes = 2 Hence, P(jack or queen of hearts) = 2/52 = 1/26 (iv) Number of favorable outcomes for...
One card is drawn from a well shuffled deck of 52 cards. Find the probability of getting: (i) a queen of red color (ii) a black face card
Solution: We have, Total possible outcomes = 52 (i) Number queens of red color = 2 Number of favorable outcomes = 2 Hence, P(queen of red color) = 2/52 (ii) Number of black cards = 26 Number of...
A game consists of spinning arrow which comes to rest pointing at one of the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12; as shown below.If the outcomes are equally likely, find the probability that the pointer will point at: (v) a number less than or equal to 9 (vi) a number between 3 and 11
(v) Favorable outcomes for a number less than or equal to 9 are 1, 2, 3, 4, 5, 6, 7, 8, 9 So, number of favorable outcomes = 9 Hence, P(the pointer will be at a number less than or equal to 9) =...
A game consists of spinning arrow which comes to rest pointing at one of the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12; as shown below.If the outcomes are equally likely, find the probability that the pointer will point at: (iii) a prime number (iv) a number greater than 8
(iii) Favorable outcomes for a prime number are 2, 3, 5, 7, 11 So, number of favorable outcomes = 5 Hence, P(the pointer will be at a prime number) = 5/12 (iv) Favorable outcomes for a number...
A game consists of spinning arrow which comes to rest pointing at one of the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12; as shown below.If the outcomes are equally likely, find the probability that the pointer will point at: (i) 6 (ii) an even number
Solution: We have, Total number of possible outcomes = 12 (i) Number of favorable outcomes for 6 = 1 Hence, P(the pointer will point at 6) = 1/12 (ii) Favorable outcomes for an even number are 2, 4,...
A bag contains twenty Rs 5 coins, fifty Rs 2 coins and thirty Re 1 coins. If it is equally likely that one of the coins will fall down when the bag is turned upside down, what is the probability that the coin: (i) will be a Re 1 coin? (ii) will not be a Rs 2 coin? (iii) will neither be a Rs 5 coin nor be a Re 1 coin?
(iii) Number of favourable outcomes for neither Re 1 nor Rs 5 coins = Number of favourable outcomes for Rs 2 coins = 50 = n(E) Hence, probability (neither Re 1 nor Rs 5 coin) = n(E)/ n(S) = 50/100 =...
A bag contains twenty Rs 5 coins, fifty Rs 2 coins and thirty Re 1 coins. If it is equally likely that one of the coins will fall down when the bag is turned upside down, what is the probability that the coin: (i) will be a Re 1 coin? (ii) will not be a Rs 2 coin?
Solution: We have, Total number of coins = 20 + 50 + 30 = 100 So, the total possible outcomes = 100 = n(S) (i) Number of favourable outcomes for Re 1 coins = 30 = n(E) Probability (Re 1 coin) =...
A bag contains 10 red balls, 16 white balls and 8 green balls. A ball is drawn out of the bag at random. What is the probability that the ball drawn will be: (iii) white or green?
(iii) Number of favorable outcomes for white or green ball = 16 + 8 = 24 = n(E) Hence, probability for drawing a white or green ball = n(E)/ n(S) = 24/34 = 12/17
A bag contains 10 red balls, 16 white balls and 8 green balls. A ball is drawn out of the bag at random. What is the probability that the ball drawn will be: (i) not red? (ii) neither red nor green?
Solution: Total number of possible outcomes = 10 + 16 + 8 = 34 balls So, n(S) = 34 (i) Favorable outcomes for not a red ball = favorable outcomes for white or green ball So, number of favorable...
The probability that two boys do not have the same birthday is 0.897. What is the probability that the two boys have the same birthday?
Solution: We know that, P(do not have the same birthday) + P(have same birthday) = 1 0.897 + P(have same birthday) = 1 Thus, P(have same birthday) = 1 – 0.897 P(have same birthday) =...
A bag contains a certain number of red balls. A ball is drawn. Find the probability that the ball drawn is: (i) black (ii) red
Solution: We have, Total possible outcomes = number of red balls. (i) Number of favourable outcomes for black balls = 0 Hence, P(black ball) = 0 (ii) Number of favourable outcomes for red balls =...
If P(E) = 0.59; find P(not E)
Solution: We know that, P(E) + P(not E) = 1 So, 0.59 + P(not E) = 1 Hence, P(not E) = 1 – 0.59 = 0.41
Which of the following cannot be the probability of an event? (iii) 37% (iv) -2.4
(iii) As 0 ≤ 37 % = (37/100) ≤ 1 Thus, 37 % can be a probability of an event. (iv) As -2.4 < 0 Thus, -2.4 cannot be a probability of an event.
Which of the following cannot be the probability of an event? (i) 3/7 (ii) 0.82
Solution: We know that probability of an event E is 0 ≤ P(E) ≤ 1 (i) As 0 ≤ 3/7 ≤ 1 Thus, 3/7 can be a probability of an event. (ii) As 0 ≤ 0.82 ≤ 1 Thus, 0.82 can be a probability of an...
Two dice are thrown at the same time. Find the probability that the sum of the two numbers appearing on the top of the dice is:(iii) less than or equal to 12
(iii) All the outcomes are favourable to the event E = ‘sum of two numbers ≤ 12’. Thus, P(E) = n(E)/ n(S) = 36/36 = 1
Two dice are thrown at the same time. Find the probability that the sum of the two numbers appearing on the top of the dice is: (i) 8 (ii) 13
Solution: We have, the number of possible outcomes = 6 × 6 = 36. (i) The outcomes favourable to the event ‘the sum of the two numbers is 8’ = E = {(2, 6), (3, 5), (4, 4), (5, 3), (6, 2)} The number...
In a bundle of 50 shirts, 44 are good, 4 have minor defects and 2 have major defects. What is the probability that: (i) it is acceptable to a trader who accepts only a good shirt? (ii) it is acceptable to a trader who rejects only a shirt with major defects?
Solution: We have, Total number of shirts = 50 Total number of elementary events = 50 = n(S) (i) As, trader accepts only good shirts and number of good shirts = 44 Event of accepting good shirts =...
In a musical chairs game, a person has been advised to stop playing the music at any time within 40 seconds after its start. What is the probability that the music will stop within the first 15 seconds?
Solution: Total result = 0 sec to 40 sec Total possible outcomes = 40 So, n(S) = 40 Favourable results = 0 sec to 15 sec Favourable outcomes = 15 So, n(E) = 15 Hence, the probability that the music...
All the three face cards of spades are removed from a well shuffled pack of 52 cards. A card is then drawn at random from the remaining pack. Find the probability of getting:(iii) a black card
(iii) Number of black cards left = 23 cards (13 club + 10 spade) Event of drawing a black card = E = 23 So, n(E) = 23 Hence, probability of drawing a black card = n(E)/ n(S) = 23/49
All the three face cards of spades are removed from a well shuffled pack of 52 cards. A card is then drawn at random from the remaining pack. Find the probability of getting: (i) a black face card (ii) a queen
Solution: We have, Total number of cards = 52 If 3 face cards of spades are removed Then, the remaining cards = 52 – 3 = 49 = number of possible outcomes So, n(S) = 49 (i) Number of black face cards...
A box contains 7 red balls, 8 green balls and 5 white balls. A ball is drawn at random from the box. Find the probability that the ball is: (i) white (ii) neither red nor white.
Solution: We have, Total number of balls in the box = 7 + 8 + 5 = 20 balls Total possible outcomes = 20 = n(S) (i) Event of drawing a white ball = E = number of white balls = 5 So, n(E) = 5 Hence,...
A and B are friends. Ignoring the leap year, find the probability that both friends will have: (i) different birthdays? (ii) the same birthday?
Solution: Out of the two friends, A’s birthday can be any day of the year. Now, B’s birthday can also be any day of 365 days in the year. We assume that these 365 outcomes are equally likely. So,...
In a match between A and B: (i) the probability of winning of A is 0.83. What is the probability of winning of B? (ii) the probability of losing the match is 0.49 for B. What is the probability of winning of A?
Solution: (i) We know that, The probability of winning of A + Probability of losing of A = 1 And, Probability of losing of A = Probability of winning of B Therefore, Probability of winning of A +...
From a well shuffled deck of 52 cards, one card is drawn. Find the probability that the card drawn is:(v) a card with number less than 8 (vi) a card with number between 2 and 9
(v) Numbers less than 8 = { 2, 3, 4, 5, 6, 7} Event of drawing a card with number less than 8 = E = {6H cards, 6D cards, 6S cards, 6C cards} So, n(E) = 24 Thus, probability of drawing a card with...
From a well shuffled deck of 52 cards, one card is drawn. Find the probability that the card drawn is: (iii) a queen of black card (iv) a card with number 5 or 6
(iii) Event of drawing a queen of black colour = {Q(spade), Q(club)} = E So, n(E) = 2 Thus, probability of drawing a queen of black colour = n(E)/ n(S) = 2/52 = 1/26 (iv) Event of drawing a card...
From a well shuffled deck of 52 cards, one card is drawn. Find the probability that the card drawn is: (i) a face card (ii) not a face card
Solution: We have, the total number of possible outcomes = 52 So, n(S) = 52 (i) No. of face cards in a deck of 52 cards = 12 (4 kings, 4 queens and 4 jacks) Event of drawing a face cards = E = (4...
A dice is thrown once. What is the probability of getting a number: (i) greater than 2? (ii) less than or equal to 2?
Solution: The number of possible outcomes when dice is thrown = {1, 2, 3, 4, 5, 6} So, n(S) = 6 (i) Event of getting a number greater than 2 = E = {3, 4, 5, 6} So, n(E) = 4 Thus, probability of...
A bag contains 3 red balls, 4 blue balls and 1 yellow ball, all the balls being identical in shape and size. If a ball is taken out of the bag without looking into it; find the probability that the ball is: (iii) not yellow (iv) neither yellow nor red
(iii) Probability of not drawing a yellow ball = 1 – Probability of drawing a yellow ball Thus, probability of not drawing a yellow ball = 1 – 1/8 = (8 – 1)/ 8 = 7/8 (iv) Neither yellow ball nor red...
A bag contains 3 red balls, 4 blue balls and 1 yellow ball, all the balls being identical in shape and size. If a ball is taken out of the bag without looking into it; find the probability that the ball is: (i) yellow (ii) red
Solution: The total number of balls in the bag = 3 + 4 + 1 = 8 balls So, the number of possible outcomes = 8 = n(S) (i) Event of drawing a yellow ball = {Y} So, n(E) = 1 Thus, probability of drawing...
If two coins are tossed once, what is the probability of getting: (iii) both heads or both tails.
(iii) E = event of getting both heads or both tails = {HH, TT} n(E) = 2 Hence, probability of getting both heads or both tails = n(E)/ n(S) = 2/4 = ½
If two coins are tossed once, what is the probability of getting: (i) both heads. (ii) at least one head.
Solution: We know that, when two coins are tossed together possible number of outcomes = {HH, TH, HT, TT} So, n(S) = 4 (i) E = event of getting both heads = {HH} n(E) = 1 Hence, probability of...
A pair of dice is thrown. Find the probability of getting a sum of 10 or more, if 5 appears on the first die.
Solution: In throwing a dice, total possible outcomes = {1, 2, 3, 4, 5, 6} So, n(S) = 6 For two dice, n(S) = 6 x 6 = 36 Favorable cases where the sum is 10 or more with 5 on 1st die = {(5, 5), (5,...
A book contains 85 pages. A page is chosen at random. What is the probability that the sum of the digits on the page is 8?
Solution: We know that, Number of pages in the book = 85 Number of possible outcomes = n(S) = 85 Out of 85 pages, pages that sum up to 8 = {8, 17, 26, 35, 44, 53, 62, 71, 80} So, pages that sum up...
A die is thrown once. Find the probability of getting a number: (iii) less than 8 (iv) greater than 6
(iii) On a dice, numbers less than 8 = {1, 2, 3, 4, 5, 6} So, n(E) = 6 Hence, probability of getting a number less than 8 = n(E)/ n(S) = 6/6 = 1 (iv) On a dice, numbers greater than 6 = 0 So, n(E) =...
A die is thrown once. Find the probability of getting a number: (i) less than 3 (ii) greater than or equal to 4
Solution: We know that, In throwing a dice, total possible outcomes = {1, 2, 3, 4, 5, 6} So, n(S) = 6 (i) On a dice, numbers less than 3 = {1, 2} So, n(E) = 2 Hence, probability of getting a number...
From 25 identical cards, numbered 1, 2, 3, 4, 5, ……, 24, 25: one card is drawn at random. Find the probability that the number on the card drawn is a multiple of: (iii) 3 and 5 (iv) 3 or 5
(iii) From numbers 1 to 25, there is only one number which is multiple of 3 and 5 i.e. {15} So, favorable number of events = n(E) = 1 Hence, probability of selecting a card with a multiple of 3 and...
A bag contains 3 white, 5 black and 2 red balls, all of the same shape and size. A ball is drawn from the bag without looking into it, find the probability that the ball drawn is: (i) a black ball. (ii) a red ball
Solution: Total number of balls = 3 + 5 + 2 = 10 So, the total number of possible outcomes = 10 (i) There are 5 black balls So, the number of favourable outcomes = 5 Thus, P(getting a black ball) = ...