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If x is a positive real number and x^{2} = 2, then x^{3} =
x^{2} = 2
⇒ x = √2
∴ x^{3} = x^{2} . x
= 2√2
The point (a, a) does not lie on the graph of
Clearly a ≠ a.
Hence, (a, a) doesn't lie on y = x.
In the given figure, BO and CO are the bisectors of ∠B and ∠C respectively. If ∠A = 50°, then ∠BOC = ?
In ΔABC
2x + 2y + ∠A = 180° (Angle sum property)
x + y + (∠A/2) = 90°
x + y = 90° – (A/2) ...1
In BOC, we have
x + y + ∠BOC = 180°
90°  (∠A/2) + ∠BOC = 180° [From (1)]
∠BOC = 180°  90° + (∠A/2)
∠BOC = 90° + (∠A/2)
∠BOC = 90° + 25° = 115°
The area of an equilateral triangle with side 2√3 cm is
Area of equilateral triangle =
=
=
= 3 x 1.732 = 5.196 sq. cm
=
= 6+13
= 4
positive and rational
Which of the following is not a linear equation?
x^{2} + 5 = 3x  5, as it has power 2, then it is a quadratic equation.
In the given figure, ∠BAC = 40°, ∠ACB = 90° and ∠BED = 100°, Then ∠BDE =?
In ΔABC
∠ABC + ∠BAC + ∠ACB = 180° (Angle sum property)
∠ABC = 180°  90°  40°
∠ABC = 50°
In ΔBED
∠BED + ∠EBD + ∠BDE = 180° (Angle sum property)
∠BDE = 180°  50°  100°
∠BDE= 30°
In Figure, AB and CD are parallel lines and transversal EF intersects them at P and Q respectively. If ∠APR = 25°, ∠RQC = 30° and ∠CQF = 65°, then
∠OQP = 180°  ∠OQF
= 180°  (30° + 65°)
⇒ ∠OQP = 85° ...(i)
∠APQ = ∠CQF (Corresponding angles)
⇒ 25° + y° = 65°
⇒ y° = 65°  25°
⇒ y° = 40°
Now in ΔOPQ
∠0 + ∠OPQ + ∠PQO = 180°
⇒ x° + 40° + 85° = 180°
⇒ x° = 180°  85°  40° = 55°
⇒ x = 55°, y = 40°
=
=
=
=
In a grouped frequency distribution, the class intervals are 010, 1020, 2030, .., then the class width is
The class width is the difference between the upper or lower class limits of consecutive classes. In this case, class width equals to the difference between the lower limits of the first two classes.
w = 10  0
So, the class width is 10
The number of angles formed by a transversal with a pair of parallel lines are
As we can see there are 4 angles formed at every point of intersection thus giving a total of 8 angles.
If 4^{x } 4^{x1} = 24, then (2x)^{x} equals
4^{x } 4^{x1} = 24
⇒
⇒
⇒
⇒
4^{x} = 32
(2^{2})^{x} = (2)^{5}
2^{2x} = 2^{5}
Comparing, we get
=
=
=
=
=
=
=
In Fig. if l_{1}  l_{2}, what is x + y in terms of w and z?
Given that,
l_{1} ‖ l_{2}
Let m and n be two transversal cutting them
∠w + ∠x = 180^{o} (Consecutive interior angle)
x = 180^{o} – w (i)
z = y (Alternate angles) (ii)
From (i) and (ii), we get
x + y = 180^{o} – w + z
Write the linear equation such that each point on its graph has an ordinate 5 times its abscissa
y = 5x
at x = 1
y = 5.1 = 5
y = 5
(1,5)
at x = 2
y = 5.2 = 10
y = 10
(2,10)
at x = 3
y = 5.3 = 15
y = 15
(3,15)
The given cumulative frequency distribution shows the class intervals and their corresponding cumulative frequencies. Then the frequency of class interval 2030 is
A cumulative frequency distribution is the sum of the class and all classes below it in a frequency distribution.
Subtract the previous cumulative frequency (c.f) from the cumulative frequency of the current class.
So frequency of the class interval 2030 is 14  5 = 9
The sides of a triangle are 11 m, 60 m and 61 m. The altitude to the smallest side is
Area of Δ = 1/2 x Base x Height
The smallest side is 11 m
Area = 1/2 x 11 x Height.. (i)
Area by Heron's Formula =
s =
Area =
= 330 m^{2}
From eq (i)
330 = 1/2 x 11 x height
Height = 2 x 330/11 = 60 m
In a histogram the area of each rectangle is proportional to
A histogram is a display of statistical information that uses rectangles to show the frequency of data items in successive numerical intervals of equal size. In the most common form of histogram, the independent variable is plotted along the horizontal axis and the dependent variable is plotted along the vertical axis.
Which of the following is rational:
Since 0 is rational number it is in the form of p/q , and where q ≠ 0 as 0/1
In the given figure, AB ║ CD ║ EF, EA ⊥ AB and BDE is the transversal such that ∠DEF = 55°, Then ∠AEB =?
EA ⊥ AB
∠AEF = 90°
∠AEF = ∠BEF + ∠AEB
∠BEF + ∠AEB = 90°
∠BEF = 55°
55° + ∠AEB = 90°
∠AEB = 90°  55°
∠AEB = 55°
The graph of the linear equation 3x – 2y = 6, cuts the xaxis at the point
the linear equation 3x – 2y = 6, cuts the xaxis
when y coordinate is 0
so we put y = 0 in given equation 3x – 2y = 6
3x  2.0 = 6
3x = 6
∴ x = 2
so the coordinates are (2,0)
An isosceles right triangle has area 8cm^{2}. The length of its hypotenuse is
Area of isosceles right triangle = 8 sq. cm
1/2 x Base Base = 8 [Since in isosceles right triangle, base and perpendicular are same]
⇒ (Base)^{2} = 16
⇒ Base = 4 cm
Hypotenuse =
The graph of y = 5 is a line
As, the graph of y = 5 is a line parallel to xaxis i.e. y = 0.
⇒ The line represented by the equation y = 5 is parallel to xaxis and intersects yaxis at y = 5.
So, the graph of y = 5 is a line parallel to the xaxis at a distance of 5 units from the origin making an intercept 5 on the yaxis.
A point of the form (0, b) lies on:
Let P be any point whose coordinate be P(0, b)
Then, if the value of xcoordinate or abscissa is zero then the point P lies in yaxis.
= 8/9
The sides of a triangle are in ratio 3 : 4 : 5. If the perimeter of the triangle is 84 cm, then area of the triangle is :
Let the sides be 3x, 4x and 5x.
Then according to quesiton, 3x + 4x + 5x = 84
⇒ 12x = 84
⇒ x = 7
Therefore, the sides are 3 x 7 = 21, cm, 4 x 7 = 28 cm and 5 x 7 = 35 cm
s = = 42 cm
Area of triangle =
=
=
= 21 7 2 = 294 sq. cm
True, you may have a triangle with two or more acute angles. If a triangle has three acute angles, the triangle is called an Acute Triangle. If a triangle has two acute angles and a single obtuse angle, the triangle is called an Obtuse Triangle.
Which one of the following statements is true?
The sum of two irrational numbers may be a rational number or an irrational number
Eg. a = √2 (which is irrational)
b = 2√2(which is irrational)
a + b = √2 + 2√2 = 3√2 ( which is irrational)
and Let a = 1+√2 (which is irrational)
and b= 1 √2 (which is irrational)
Now, a + b = 1+√2 +1 √2
= 2 (which is rational)
If x < 0 and y < 0, then the point (x, y) lies in
Here,x < 0 (i.e, ve) and y < 0, (i.e, ve)
And we know that,coordinate of the point in 3rd quadrant is (,)i.e, both negative,
So,the given point will lie in 3rd quadrant.
The marks obtained by 17 students in a mathematics test (out of 100) are given below :
91, 82, 100, 100, 96, 65, 82, 76, 79, 90, 46, 64, 72, 68, 66, 48, 49.
Find the range of the data.
Highest Marks = 100
Lowest Marks = 46
Range of data = 100  46 = 54
The base of an isosceles triangle is 8 cm long and each of its equal sides measures 6 cm. The area of the triangle is
Area of isosceles triangle =
Here, a = 6 cm and b = 8 cm
Thus, we have
=
=
=
= 8√5 cm^{2}
Choose the wrong statement:
Every rational number is an integer.
Eg. 1/2 is a rational number but not an integer.
In the adjoining figure, PQ > PR. If OQ and OR are bisectors of ∠Q and ∠R respectively, then
Since PQ > PR then R > Q and hence their bisectors follow the same i.e R/2 > Q/2 and hence OQ>OR
To compare this years result with last years result, teacher went to the class and collected this years number of distinctions from the students. For last years number of distinctions, she opened the result register & wrote the required number of distinctions. The data Collected by her from the students & register respectively, are examples of
The data collected directly from the population is primary data, so the current number of distinctions as confirmed from students is primary data.
The data collected using the old records such as register is called secondary data. So, the data collected from the register is secondary data.
In the adjoining figure, if QP ║ RT, then x is equal to –
∠QPR = ∠PRT = 40° (Alternate interior angles)
In ∠QPR, ∠PQR + ∠QPR + ∠PRQ = 180° (Angle sum property)
65° + 40° + x° = 180°
x° = 180°  40°  65°
x° = 75
If (2, 0) is a solution of the linear equation 2x + 3y = k, then the value of k is
(2, 0) is a solution of the linear equation 2x + 3y = k
⇒ 4 = k
In the adjoining figure, AB = AC and AD is median of ΔABC , then ∠ADC is equal to
As AD is the perpendicular bisector of BC, so ∠ADC = ∠ADB = 90^{o}
multiplying nu nominator and denominator by
√5 + √3 we get
=
= √5 + √3
What is the sum of the angles of a quadrilateral?
For a quadrilateral
Number of sides (n) = 4
Sum of interior angles = (n  2) x 180°
p = (4  2) x 180°
p = 2 x 180°
p = 360°
Find the range of the following data: 25, 20 30, 18, 16, 15
The difference between the maximum and minimum value is called range.
Thus, 30  15 = 15
Read the text carefully and answer the questions:
As shown In the village of Surya there was a big pole PC. This pole was tied with a strong wire of 10 m length.
Once there was a big spark on this pole, thus wires got damaged very badly. Any small fault was usually repaired with the help of a rope which normal board electricians were carrying on bicycles.
This time electricians need a staircase of 10 m so that it can reach at point P on the pole and this should make 60° with line AC.
In the ΔPAC and ΔPBC which side is common?
PC
Read the text carefully and answer the questions:
As shown In the village of Surya there was a big pole PC. This pole was tied with a strong wire of 10 m length.
Once there was a big spark on this pole, thus wires got damaged very badly. Any small fault was usually repaired with the help of a rope which normal board electricians were carrying on bicycles.
This time electricians need a staircase of 10 m so that it can reach at point P on the pole and this should make 60° with line AC.
In the ΔPAC and ΔPBC which angles are given to be equal?
∠B = ∠x
Read the text carefully and answer the questions:
As shown In the village of Surya there was a big pole PC. This pole was tied with a strong wire of 10 m length.
Once there was a big spark on this pole, thus wires got damaged very badly. Any small fault was usually repaired with the help of a rope which normal board electricians were carrying on bicycles.
This time electricians need a staircase of 10 m so that it can reach at point P on the pole and this should make 60° with line AC.
In the figure, ΔPAC and ΔPBC are congruent due to which criteria?
RHS
Read the text carefully and answer the questions:
As shown In the village of Surya there was a big pole PC. This pole was tied with a strong wire of 10 m length.
Once there was a big spark on this pole, thus wires got damaged very badly. Any small fault was usually repaired with the help of a rope which normal board electricians were carrying on bicycles.
This time electricians need a staircase of 10 m so that it can reach at point P on the pole and this should make 60° with line AC.
What is the value of ∠PBC?
45°
Read the text carefully and answer the questions:
As shown In the village of Surya there was a big pole PC. This pole was tied with a strong wire of 10 m length.
Once there was a big spark on this pole, thus wires got damaged very badly. Any small fault was usually repaired with the help of a rope which normal board electricians were carrying on bicycles.
This time electricians need a staircase of 10 m so that it can reach at point P on the pole and this should make 60° with line AC.
The value of ∠x is
30°
Read the text carefully and answer the questions:
In the above picture, one small square is of size 1 km 1 km. From the starting point O(0,0) Deepak started to drive towards his home. He first drives 3km in left then he turned to his left and drove 2 km, there he found a temple. He worshipped there and drove 6km in the left direction, there is a zoo and from the zoo, he drives 2km on the right side, then he reached his home.
From O Sanjay drove for his school, he drove 1km to his right then took a left turn and drives 2km then again took a right turn and drives 2 km. He found a hospital in the way. From Hospital he drove 3 km and finally reached his school.
What are the coordinates of the Hospital?
(3, 2)
Read the text carefully and answer the questions:
In the above picture, one small square is of size 1 km 1 km. From the starting point O(0,0) Deepak started to drive towards his home. He first drives 3km in left then he turned to his left and drove 2 km, there he found a temple. He worshipped there and drove 6km in the left direction, there is a zoo and from the zoo, he drives 2km on the right side, then he reached his home.
From O Sanjay drove for his school, he drove 1km to his right then took a left turn and drives 2km then again took a right turn and drives 2 km. He found a hospital in the way. From Hospital he drove 3 km and finally reached his school.
What is common abscissa of school, Hospital, Zoo and Deepak's home?
3
Read the text carefully and answer the questions:
In the above picture, one small square is of size 1 km 1 km. From the starting point O(0,0) Deepak started to drive towards his home. He first drives 3km in left then he turned to his left and drove 2 km, there he found a temple. He worshipped there and drove 6km in the left direction, there is a zoo and from the zoo, he drives 2km on the right side, then he reached his home.
From O Sanjay drove for his school, he drove 1km to his right then took a left turn and drives 2km then again took a right turn and drives 2 km. He found a hospital in the way. From Hospital he drove 3 km and finally reached his school.
What is the common ordinate of temple and Zoo?
2
Read the text carefully and answer the questions:
In the above picture, one small square is of size 1 km 1 km. From the starting point O(0,0) Deepak started to drive towards his home. He first drives 3km in left then he turned to his left and drove 2 km, there he found a temple. He worshipped there and drove 6km in the left direction, there is a zoo and from the zoo, he drives 2km on the right side, then he reached his home.
From O Sanjay drove for his school, he drove 1km to his right then took a left turn and drives 2km then again took a right turn and drives 2 km. He found a hospital in the way. From Hospital he drove 3 km and finally reached his school.
Deepak Drove in which quadrants?
III and IV
Read the text carefully and answer the questions:
In the above picture, one small square is of size 1 km 1 km. From the starting point O(0,0) Deepak started to drive towards his home. He first drives 3km in left then he turned to his left and drove 2 km, there he found a temple. He worshipped there and drove 6km in the left direction, there is a zoo and from the zoo, he drives 2km on the right side, then he reached his home.
From O Sanjay drove for his school, he drove 1km to his right then took a left turn and drives 2km then again took a right turn and drives 2 km. He found a hospital in the way. From Hospital he drove 3 km and finally reached his school.
Sanjay Drove in which quadrants?
I only
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