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All questions of Mathematics (Maths) for Class 9 Exam

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Read the following text and answer the following questions on the basis of the same:
Rahul and Ravi planned to play Business (board game) in which they were supposed to use two dice.
Ravi got the first chance to roll the dice. What is the probability that he got the sum of the two numbers appearing on the top face of the dice is 8?
  • a)
    1/26
  • b)
    5/36
  • c)
    1/18
  • d)
    0
Correct answer is option 'B'. Can you explain this answer?

Naina Sharma answered
The outcomes when two dice are thrown together are
= (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6)
(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6)
(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6)
(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6)
(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6)
(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)
Total outcomes = 36
No. of outcomes when the sum is 8 = 5
Probability = 5/36

Rainwater harvesting system is a technology that collects and stores rainwater for human use.
Anup decided to do rainwater harvesting. He collected rainwater in the underground tank at the rate of 30 cm3/sec.
Q. Write the equation in standard form.
  • a)
    30x – y + 0 = 0
  • b)
    30x + y + 0 = 0
  • c)
    30x – y – 0 = 0
  • d)
    30x – y = 0
Correct answer is option 'A'. Can you explain this answer?

Let's Tute answered
Rainwater is collected in the  underground tank at the rate of 30 cm³ / sec
 
Water collected in 1 sec  = 30 cm³
Water collected in x secs = 30x  cm³
Water collected in x secs =  y cm³
 
=> y = 30x
a linear equation y = 30x
30x - y = 0 ,30x -y +0=0
is standard form

Nikita has to make her project on 'Monument in India'. She decided to make her project on Gol Gumbaz monument. She already knows following things about it :
  • It is located in a small town in Northern Karnataka.
  • It reaches up to 51 meters in height while the giant dome has an external diameter of 44 meters, making it one of the largest domes ever built.
  • At each of the four corners of the cube is a dome shaped octagonal tower seven stories high with a staircase inside.
Help her in making project by answering the following questions :
Q. What is the height of the cuboidal part?
  • a)
    14 m
  • b)
    7 m
  • c)
    29 m
  • d)
    18 m
Correct answer is option 'C'. Can you explain this answer?

Swati Verma answered
Total height of monument = 51 m
Radius of hemispherical dome part = height of hemispherical dome = 22 m
Height of the cuboidal part = 51 m – 22m
= 29 m

DIRECTION : In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion : There are infinite number of lines which passes through (3, 2).
Reason : A linear equation in two variables has infinitely many solutions.
  • a)
    Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
  • b)
    Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
  • c)
    Assertion (A) is true but reason (R) is false.
  • d)
    Assertion (A) is false but reason (R) is true.
Correct answer is option 'B'. Can you explain this answer?

Swati Verma answered
We know that a linear equation in two variables has infinitely many solutions. So, Reason is correct.
Through a point infinite lines can be drawn.
Through (3, 2) infinite number of lines can be drawn.
Hence, Assertion is also correct.
But reason (R) is not the correct explanation of assertion (A).
Correct option is (b) Both assertion (A) and reason (R) are true and reason (R) is not the correct explanation of assertion (A).

DIRECTION : In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion: x + y = 3 is the equation of a line passing through the origin.
Reason: y = 2x is the equation of a line passing through the origin.
  • a)
    Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
  • b)
    Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
  • c)
    Assertion (A) is true but reason (R) is false.
  • d)
    Assertion (A) is false but reason (R) is true.
Correct answer is option 'D'. Can you explain this answer?

Rohit Sharma answered
For Assertion: The given linear equation is x + y = 3
Since x = 0 and y = 0 is not satisfying x + y = 3, therefore it is not passing through the origin.
So, Assertion is not correct.
Since x = 0 and y = 0 is not satisfying y = 2x, therefore it is passing through the origin.
So, Reason is correct. Correct option is (d) Assertion (A) is false but reason (R) is true.

Direction: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion : (2 - √3) is one zero of the quadratic polynomial then other zero will be (2 + √3).
Reason : Irrational zeros (roots) always occurs in pairs.
  • a)
    Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
  • b)
    Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
  • c)
    Assertion (A) is true but reason (R) is false.
  • d)
    Assertion (A) is false but reason (R) is true.
Correct answer is option 'A'. Can you explain this answer?

Nehal Gupta answered
Assertion and Reasoning

Assertion: (2 - √3) is one zero of the quadratic polynomial then other zero will be (2 + √3).
Reasoning: Irrational zeros (roots) always occur in pairs.

Explanation

Quadratic polynomial is given by ax² + bx + c = 0, where a, b, and c are constants. The quadratic formula is used to find the roots of the quadratic equation.

The roots of the quadratic equation can be real or complex. The roots can be rational or irrational. If a quadratic equation has irrational roots, then they always occur in pairs.

In the given assertion, (2 - √3) is one zero of the quadratic polynomial. Therefore, the other zero will be (2 + √3). This is because the sum of the roots of a quadratic equation is -b/a, and the product of the roots is c/a. In this case, the sum of the roots is (2 - √3) + (2 + √3) = 4, and the product of the roots is (2 - √3)(2 + √3) = 1. Therefore, the other zero must be (2 + √3).

The reasoning provided is correct because irrational roots always occur in pairs. This is because if a quadratic equation has an irrational root, then its conjugate must also be a root. The conjugate of an irrational number is obtained by changing the sign of the radical. In this case, the conjugate of (2 - √3) is (2 + √3).

Conclusion

Hence, both the assertion and reasoning are true, and the reasoning is the correct explanation for the assertion. Therefore, the correct answer is option A.

Direction: In each of the following questions, a statement of Assertion is given followed by a corresponding statement of Reason just below it. Of the statements, mark the correct.
Assertion : If the angles of a quadrilateral are in the ratio 2 : 3 : 7 : 6, then the measure of angles are 40º, 60º, 140º, 120º, respectively.
Reason : The sum of the angles of a quadrilateral is 360º.
  • a)
    Assertion and reason are true and reason is the correct explanation of assertion.
  • b)
    Both assertion and reason are true but reason is not the correct explanation of assertion.
  • c)
    Assertion is true but reason is false.
  • d)
    Assertion is false but reason is true.
Correct answer is option 'A'. Can you explain this answer?

Avinash Patel answered
2x + 3x + 7x + 6x = 360º
=> 18x = 360º
=> x = 360º/18 = 20º
First angle = 2 × 20º = 40º
Second angle = 3 × 20º = 60º
Third angle = 7 x 20º = 140º
Fourth angle = 6 x 20º = 120º

A Ferris wheel (or a big wheel in the United Kingdom) is an amusement ride consisting of a rotating upright wheel with multiple passenger-carrying components (commonly referred to as passenger cars, cabins, tubs, capsules, gondolas, or pods) attached to the rim in such a way that as the wheel turns, they are kept upright, usually by gravity.
After taking a ride in Ferris wheel, Aarti came out from the crowd and was observing her friends who were enjoying the ride . She was curious about the different angles and measures that the wheel will form. She forms the figure as given below.
Q. Find ∠RQP
  • a)
    75
  • b)
    60
  • c)
    30
  • d)
    90
Correct answer is option 'A'. Can you explain this answer?

Aditi Sharma answered
PR = PO [∵ Tangents drawn from an external point are equal
⇒ ∠PRQ = ∠PQR [∵ Angles opposite equal sides are equal
In ∆PQR,
⇒ ∠PRQ + ∠RPQ + ∠POR = 180° [∆ Rule]
⇒ 30° + 2∠PQR = 180°
⇒
= 75° ⇒ SR || QP and QR is a transversal
∵ ∠SRQ = ∠PQR [Alternate interior angle]
∴ ∠SRO = 75° [Tangent is I to the radius through the point of contact]
⇒ ∠ORP = 90°
∴ ∠ORP = ∠ORQ + ∠QRP
90° = ∠ORQ + 75°
∠ORQ = 90° – 75o = 150
Similarly, ∠RQO = 15°
In ∆QOR,
∠QOR + ∠QRO + ∠OQR = 180° [∆ Rule]
∴ ∠QOR + 15° + 15° = 180°
∠QOR = 180° – 30° = 150°
⇒ ∠QSR = 12∠QO
⇒ ∠QSR = 150∘/2 = 750 [Used ∠SRQ = 75° as solved above]
In ARSQ, ∠RSQ + ∠QRS + ∠RQS = 180° [∆ Rule]
∴ 75° + 75° + ∠RQS = 180°
∠RQS = 180° – 150o = 30°

A Ferris wheel (or a big wheel in the United Kingdom) is an amusement ride consisting of a rotating upright wheel with multiple passenger-carrying components (commonly referred to as passenger cars, cabins, tubs, capsules, gondolas, or pods) attached to the rim in such a way that as the wheel turns, they are kept upright, usually by gravity.
After taking a ride in Ferris wheel, Aarti came out from the crowd and was observing her friends who were enjoying the ride . She was curious about the different angles and measures that the wheel will form. She forms the figure as given below.
Q. Find ∠ORP
  • a)
    90
  • b)
    70
  • c)
    100
  • d)
    60
Correct answer is option 'A'. Can you explain this answer?

Nipun Tuteja answered
PR = PO [∵ Tangents drawn from an external point are equal
⇒ ∠PRQ = ∠PQR [∵ Angles opposite equal sides are equal
In ∆PQR,
⇒ ∠PRQ + ∠RPQ + ∠POR = 180° [∆ Rule]
⇒ 30° + 2∠PQR = 180°
⇒
= 75° ⇒ SR || QP and QR is a transversal
∵ ∠SRQ = ∠PQR [Alternate interior angle]
∴ ∠SRO = 75° [Tangent is I to the radius through the point of contact]
⇒ ∠ORP = 90°
∴ ∠ORP = ∠ORQ + ∠QRP
90° = ∠ORQ + 75°
∠ORQ = 90° – 75o = 150
Similarly, ∠RQO = 15°
In ∆QOR,
∠QOR + ∠QRO + ∠OQR = 180° [∆ Rule]
∴ ∠QOR + 15° + 15° = 180°
∠QOR = 180° – 30° = 150°
⇒ ∠QSR = 12∠QO
⇒ ∠QSR = 150∘/2 = 750 [Used ∠SRQ = 75° as solved above]
In ARSQ, ∠RSQ + ∠QRS + ∠RQS = 180° [∆ Rule]
∴ 75° + 75° + ∠RQS = 180°
∠RQS = 180° – 150o = 30°

Direction: In each of the following questions, a statement of Assertion is given followed by a corresponding statement of Reason just below it. Of the statements, mark the correct.
Assertion : A parallelogram consists of two congruent triangles.
Reason : Diagonal of a parallelogram divides it into two congruent triangles.
  • a)
    Assertion and reason are true and reason is the correct explanation of assertion.
  • b)
    Both assertion and reason are true but reason is not the correct explanation of assertion.
  • c)
    Assertion is true but reason is false.
  • d)
    Assertion is false but reason is true.
Correct answer is option 'A'. Can you explain this answer?

Meera Rana answered
In ΔBAC and ΔDCA
∠1 = ∠2
∠3 = ∠4 [alternate pair]
AC = AC (common)
ΔBAC ≅ ΔDCA (By ASA)
Therefore, diagonal of a parallelogram divides it into two congruent triangles.

Direction: In the following questions, a statement of Assertion is given followed by a corresponding statement of Reason just below it. Of the statements, mark the correct
answer as
Assertion : A fair die is rolled. Then the probability of getting an even number is 1/2 and probability of getting an odd number is 1/2.
Reason : Possible outcomes when a fair die is rolled is (1, 2, 3, 4, 5, 6).
  • a)
    Both assertion and reason are true and reason is the correct explanation of assertion
  • b)
    Both assertion and reason are true but reason is not the correct explanation of assertion.
  • c)
    Assertion is true but reason is false.
  • d)
    Assertion is false but reason is true.
Correct answer is option 'A'. Can you explain this answer?

Avinash Patel answered
Possible outcomes when a die is thrown = {1, 2, 3, 4, 5, 6}
P (getting an odd number) = 3/6 = 1/2
P (getting an even number) = 3/6 = 1/2

DIRECTION : In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion : √5 is an irrational number.
Reason : A number is called irrational, if it cannot be written in the form p/q, where p and q are integers and q ≠ 0.
  • a)
    Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
  • b)
    Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
  • c)
    Assertion (A) is true but reason (R) is false.
  • d)
    Assertion (A) is false but reason (R) is true.
Correct answer is option 'A'. Can you explain this answer?

Pooja Shah answered
We know that “A number is called irrational, if it cannot be written in the form p/q, where p and q are integers and q ≠ 0.” So, Reason is correct.
Since, √5 cannot be written in the form of p/q, therefore it is an irrational number. Hence assertion is correct follows from reason.
Correct option is (a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).

Direction: In each of the following questions, a statement of Assertion is given followed by a corresponding statement of Reason just below it. Of the statements, mark the correct.
Assertion : ABCD and PQRC are rectangles and Q is a midpoint of AC . Then DP = PC.
Reason : The line segment joining the midpoint of any two sides of a triangle is parallel to the third side and equal to half of it.
  • a)
    Assertion and reason are true and reason is the correct explanation of assertion.
  • b)
    Both assertion and reason are true but reason is not the correct explanation of assertion.
  • c)
    Assertion is true but reason is false.
  • d)
    Assertion is false but reason is true.
Correct answer is option 'B'. Can you explain this answer?

Meera Rana answered
The correct option is B If both assertion and reason are true but reason is not the correct explanation of assertion
In ΔADC, Q is the mid-point of AC such that PQ∥AD
∴ P is the mid-point of DC
⇒ DP = PC [using converse of mid-point theorem]

∴ Assertion is true, reason is true but reason is not correct explanation of assertion.

DIRECTION : In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion : 0.329 is a terminating decimal.
Reason : A decimal in which a digit or a set of digits is repeated periodically, is called a repeating, or a recurring, decimal.
  • a)
    Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
  • b)
    Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
  • c)
    Assertion (A) is true but reason (R) is false.
  • d)
    Assertion (A) is false but reason (R) is true.
Correct answer is option 'B'. Can you explain this answer?

Ishan Choudhury answered
We know that a decimal in which a digit or a set of digits is repeated periodically, is called a repeating, or a recurring, decimal.
So, Reason is correct.
Also, we know that a decimal that ends after a finite number of digits is called a terminating decimal.
Hence Assertion is correct but reason is not the correct explanation of Assertion Correct option is (b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).

Direction: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion : Zeroes of f(x) = x2 - 4x - 5 are 5, - 1
Reason : The polynomial whose zeroes are 2 + √3, 2 - √3 is x2 - 4x + 7.
  • a)
    Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
  • b)
    Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
  • c)
    Assertion (A) is true but reason (R) is false.
  • d)
    Assertion (A) is false but reason (R) is true.
Correct answer is option 'C'. Can you explain this answer?

Nk Classes answered
Zeroes of f(x) = x2 − 4x − 5 are obtained by solving
x2 − 4x − 5 = 0, which implies
x2 − 5x + x − 5 = 0
or, x(x − 5) + 1(x − 5) = 0
which means x = 5 or x = −1
Thus the assertion is correct.
However, the reason is incorrect. The numbers given are not the zeroes of x2 − 4x + 7.

In a circle of radius 10 cm given below, chord AB and CD are equal. If OE bisects AB and OF bisects CD and OF = 6 cm, then length EB is ________ .
  • a)
    10 cm
  • b)
    8 cm
  • c)
    16 cm
  • d)
    21 cm
Correct answer is option 'B'. Can you explain this answer?

Amit Sharma answered
It is given AB = CD.
So, OF = OE = 6 cm [Equal chords are equidistant from centre] ...... (1)
In ΔOEB
OB = 10 cm [Radius]
OE = OF = 6 cm [from (1)]
Since a line through the center that bisects the chord is perpendicular to the chord, we must have ∠OBE=90∘
∴OB2 = OE2+EB2
[∵∠OEB is 90∘]

EB2=OB2−OE2
EB2 = (10)2−(6)2 = 64
EB = 8 cm

DIRECTION : In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion: x = 3 and y = 2 is a solution of the linear equation 2x + 3y = 12.
Reason: x = 4 and y = 2 is a solution of the linear equation x + 3y = 10.
  • a)
    Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
  • b)
    Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
  • c)
    Assertion (A) is true but reason (R) is false.
  • d)
    Assertion (A) is false but reason (R) is true.
Correct answer is option 'B'. Can you explain this answer?

Swati Verma answered
For Assertion: The given linear equation is 2x + 3y = 12 Substituting x = 3 and y = 2, we get LHS = 2 x 3 + 3 x 2 = 6 + 6 = 12 = RHS So, Assertion is correct.
For Reason: The given linear equation is x + 3y = 10 Substituting x = 4 and y = 2, we get LHS = 4 + 3 x 2 = 4 + 6 = 10 = RHS So, Reason is also correct. But reason (R) is not the correct explanation of assertion (A).
Correct option is (b) Both assertion (A) and reason (R) are true and reason (R) is not the correct explanation of assertion (A).

Direction: In the following questions, a statement of Assertion is given followed by a corresponding statement of Reason just below it. Of the statements, mark the correct answer as
Assertion : Into a circular drum of radius 4.2 m and height 3.5 m, the number of full bags of wheat can be emptied, if the space required for wheat in each bag is 2.1 cu. m, is 92 bags nearly.
Reason : Volume of circular drum is 1/3 πr2h where r is radius and h is height of drum.
  • a)
    Both assertion and reason are true and reason is the correct explanation of assertion.
  • b)
    Both assertion and reason are true but reason is not the correct explanation of assertion.
  • c)
    Assertion is true but reason is false.
  • d)
    Assertion is false but reason is true.
Correct answer is option 'C'. Can you explain this answer?

Radha Iyer answered
Radius of circular drum, r = 4 2. m
Height of circular drum, h = 3 5. m
Volume of circular drum
Space occupied by 1 wheat bag = 2 1. m3
Number of wheat bags
= 92 4. = 92 nearly

Direction: In the following questions, a statement of Assertion is given followed by a corresponding statement of Reason just below it. Of the statements, mark the correct answer as
Assertion : The measure of ∠AOC = 60º
Reason : Angle subtended by an arc of a circle at the centre of the circle is double the angle subtended by arc on the circumference.
  • a)
    Both assertion and reason are true and reason is the correct explanation of assertion.
  • b)
    Both assertion and reason are true but reason is not the correct explanation of assertion.
  • c)
    Assertion is true but reason is false.
  • d)
    Assertion is false but reason is true.
Correct answer is option 'D'. Can you explain this answer?

Meera Rana answered
Join BO.
In ΔAOB , we have
OA = OB [radius]
∠OBA = ∠OAB
[Angle opposite to equal sides of a triangle are equal]
∠OBA = 30º ...(1)
Similarly, in ΔBOC , we get OB = OC
∠OCB = ∠OBC
∠OBC = 40º ...(2)
∠ABC = ∠OBA + ∠OBC
= 30º + 40º = 70º
[Using (1) and (2)]
Since angle subtended by an arc of a circle at the centre of the circle is double the angle subtended by the arc on the circumference.
∠AOC = 2 x ∠ABC
= 2 x 70º = 140º
As the Assertion states that ∠AOC=60∘, it is false.

Direction: In each of the following questions, a statement of Assertion is given followed by a corresponding statement of Reason just below it. Of the statements, mark the correct.
Assertion : In ΔABC , E and F are the midpoints of AC and AB respectively. The altitude AP at BC intersects FE at Q. Then, AQ QP = .
Reason : Q is the midpoint of AP.
  • a)
    Assertion and reason are true and reason is the correct explanation of assertion.
  • b)
    Both assertion and reason are true but reason is not the correct explanation of assertion.
  • c)
    Assertion is true but reason is false.
  • d)
    Assertion is false but reason is true.
Correct answer is option 'B'. Can you explain this answer?

Rohit Sharma answered
In ΔABC , E and F are midpoint of the sides AC
and AB respectively.
FE || BC [By mid-point theorem]
Now, in ΔABP, F is mid-point of AB and FQ || BP.
Q is mid-point of AP
AQ = QP.

DIRECTION : In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion : The points (-1, 2) and (2,- 1) are at different positions in the coordinate plane.
Reason : Point (-1, 2) lies in II-quadrant and (2,- 1) lies in IV quadrant
  • a)
    Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
  • b)
    Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
  • c)
    Assertion (A) is true but reason (R) is false.
  • d)
    Assertion (A) is false but reason (R) is true.
Correct answer is option 'A'. Can you explain this answer?

Ishan Choudhury answered
We know that Point (-1, 2) lies in II-quadrant and (2,- 1) lies in IV quadrant So, Reason is correct.
Hence the points (-1, 2) and (2,- 1) are at different positions in the coordinate plane. So, Assertion is also correct Correct option is (a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).

Direction: In each of the following questions, a statement of Assertion is given followed by a corresponding statement of Reason just below it. Of the statements, mark the correct.
Assertion : In ΔABC , median AD is produced to X such that AD DX = . Then ABXC is a parallelogram.
Reason : Diagonals AX and BC bisect each other at right angles.
  • a)
    Assertion and reason are true and reason is the correct explanation of assertion.
  • b)
    Both assertion and reason are true but reason is not the correct explanation of assertion.
  • c)
    Assertion is true but reason is false.
  • d)
    Assertion is false but reason is true.
Correct answer is option 'C'. Can you explain this answer?

Swati Verma answered
In quadrilateral ABXC , we have
AD = DX [Given]
BD = DC [Given]
So, diagonals AX and BC bisect each other but not
at right angles.
Therefore, ABXC is a parallelogram

DIRECTION : In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion : 113 x 114 = 1112
Reason : If a > 0 be a real number and p and q be rational numbers.
Then ap x aq = ap + q.
  • a)
    Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
  • b)
    Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
  • c)
    Assertion (A) is true but reason (R) is false.
  • d)
    Assertion (A) is false but reason (R) is true.
Correct answer is option 'D'. Can you explain this answer?

Rohit Sharma answered
We know that If a > 0 be a real number and p and q be rational numbers then ap x aq = ap+q.
So, Reason is true.
Now, 113 x 114 = 113+4 = 117
Here assertion is incorrect but reason is correct.
Correct option is (d) Assertion (A) is false but reason (R) is true.

DIRECTION : In the following questions, a statement of assertion (A) is followed by a
statement of reason (R). Mark the correct choice as:
Assertion : A triangle can have two obtuse angles.
Reason : The sum of all the interior angles of a triangle is 1800.
  • a)
    Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
  • b)
    Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
  • c)
    Assertion (A) is true but reason (R) is false.
  • d)
    Assertion (A) is false but reason (R) is true.
Correct answer is option 'D'. Can you explain this answer?

Disha Choudhary answered
Assertion: A triangle can have two obtuse angles.
Reason: The sum of all the interior angles of a triangle is 180°.

The correct answer is option D, which states that Assertion (A) is false but Reason (R) is true.

Explanation:
- A triangle is a closed figure with three sides and three angles.
- The sum of all the interior angles of a triangle is always 180°.
- An obtuse angle is an angle that measures more than 90° but less than 180°.
- In a triangle, the sum of the measures of two angles can never be greater than or equal to 180°. Therefore, it is not possible for a triangle to have two obtuse angles.
- If a triangle has one obtuse angle, the sum of the other two angles will be less than 90°, making the triangle acute-angled.
- If a triangle has three acute angles, the sum of the angles will be less than 180°, making the triangle acute-angled.
- However, a triangle can have one obtuse angle and two acute angles, or three acute angles.

Hence, the assertion that a triangle can have two obtuse angles is false because it contradicts the fact that the sum of all the interior angles of a triangle is always 180°. However, the reason stating that the sum of all the interior angles of a triangle is 180° is true.

Direction: In the following questions, a statement of Assertion is given followed by a corresponding statement of Reason just below it. Of the statements, mark the correct answer as
Assertion : Two diameters of a circle intersect each other at right angles. Then the quadrilateral formed by joining their end-points is a square.
Reason : Equal chords subtend equal angles at the centre.
  • a)
    Both assertion and reason are true and reason is the correct explanation of assertion.
  • b)
    Both assertion and reason are true but reason is not the correct explanation of assertion.
  • c)
    Assertion is true but reason is false.
  • d)
    Assertion is false but reason is true
Correct answer is option 'A'. Can you explain this answer?

Raina Kapoor answered
Assertion: Two diameters of a circle intersect each other at right angles. Then the quadrilateral formed by joining their end-points is a square.

Reason: Equal chords subtend equal angles at the centre.

Explanation:
To understand the given assertion and reason, let's break down the information provided and analyze it step by step.

1. Two diameters of a circle intersect each other at right angles:
- A diameter is a line segment passing through the center of a circle and having both its endpoints on the circle.
- When two diameters intersect each other, they form four angles at their intersection point, also known as the center of the circle.
- Since the diameters intersect at right angles, the four angles formed are all right angles.

2. The quadrilateral formed by joining their end-points is a square:
- A quadrilateral is a polygon with four sides.
- If the four angles of a quadrilateral are all right angles, then it is a square.
- In this case, the four angles formed at the intersection of the diameters are all right angles, so the quadrilateral formed is a square.

3. Equal chords subtend equal angles at the center:
- A chord is a line segment that connects two points on the circumference of a circle.
- If two chords of a circle are equal in length, then the angles they subtend at the center of the circle are also equal.
- This property is useful in determining the congruence of angles formed by chords in a circle.

Now, let's analyze the relationship between the assertion and the reason:

- The assertion states that when two diameters of a circle intersect at right angles, the quadrilateral formed is a square.
- The reason states that equal chords subtend equal angles at the center.
- In this case, the two diameters of the circle intersect at right angles, forming equal chords.
- Since the equal chords subtend equal angles at the center, the four angles formed at the intersection of the diameters are all equal and right angles.
- Therefore, the reason correctly explains why the quadrilateral formed is a square.

Hence, both the assertion and reason are true, and the reason is the correct explanation of the assertion. Therefore, the correct answer is option A.

Direction: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion : (x + 2) is a factor of x3 + 3x2 + 5x + 6 and of 2x + 4.
Reason : If p(x) be a polynomial of degree greater than or equal to one, then (x - a) is a factor of p(x), if p(a) = 0
  • a)
    Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
  • b)
    Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
  • c)
    Assertion (A) is true but reason (R) is false.
  • d)
    Assertion (A) is false but reason (R) is true.
Correct answer is option 'A'. Can you explain this answer?

Meera Rana answered
P(x) = x3 + 3x2 + 5x + 6
g(x) = x + 2
x = -2
put x = -2 in p(x)
p(-2) = (-2)3 + 3(-2)2 + 5(-2) + 6
= -8 + 12 - 10 + 6
= 0
since p(x) is zero
X + 2 is a factor of x3 + 3x2 + 5x + 6
Checking
let x = -2 q(x) = 2x + 4
q(-2) = 2(-2) + 4
= -4 + 4
= 0
since reminder is zero
x + 2 is a factor of 2x + 4

DIRECTION : In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion : A linear equation 3x + 5y = 2 has a unique solution.
Reason : A linear equation in two variables has infinitely many solutions.
  • a)
    Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
  • b)
    Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
  • c)
    Assertion (A) is true but reason (R) is false.
  • d)
    Assertion (A) is false but reason (R) is true.
Correct answer is option 'D'. Can you explain this answer?

Amaira Chakraborty answered
Assertion (A): A linear equation 3x - 5y = 2 has a unique solution.
Reason (R): A linear equation in two variables has infinitely many solutions.

The correct answer is option 'D': Assertion (A) is false but reason (R) is true.

Explanation:
Linear Equation:
A linear equation is an equation in which the degree of each term is 1. It represents a straight line when plotted on a graph. The general form of a linear equation in two variables (x and y) is ax + by + c = 0, where a, b, and c are constants.

Unique Solution:
A linear equation has a unique solution when the solution is a specific pair of values for x and y that satisfy the equation. In other words, there is only one point of intersection between the line represented by the equation and the coordinate plane.

Infinitely Many Solutions:
A linear equation has infinitely many solutions when the equation represents a line that coincides with the coordinate plane. In this case, every point on the line satisfies the equation, so there are infinitely many points of intersection between the line and the coordinate plane.

Analysis:
The given linear equation is 3x - 5y = 2. This equation represents a line in the coordinate plane. To determine the number of solutions, we need to analyze the coefficients of x and y.

The coefficient of x is 3 and the coefficient of y is -5. Since both coefficients are non-zero and not equal, the line represented by the equation is not parallel to either the x-axis or the y-axis.

If the line is not parallel to the x-axis, it will intersect the x-axis at a unique point. Similarly, if the line is not parallel to the y-axis, it will intersect the y-axis at a unique point. Therefore, the line represented by the equation 3x - 5y = 2 will have a unique solution.

Conclusion:
The given assertion (A) that the linear equation 3x - 5y = 2 has a unique solution is false. However, the reason (R) that a linear equation in two variables has infinitely many solutions is true.

DIRECTION : In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion: The abscissa of a point (5, 2) is 5.
Reason: The perpendicular distance of a point from y-axis is called its abscissa.
  • a)
    Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
  • b)
    Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
  • c)
    Assertion (A) is true but reason (R) is false.
  • d)
    Assertion (A) is false but reason (R) is true.
Correct answer is option 'A'. Can you explain this answer?

Radha Iyer answered
We know that the perpendicular distance of a point from y-axis is called its x-coordinate or abscissa.
So, Reason is correct.
The x co-ordinate of the point (5, 2) is 5.
So, Assertion is also correct
Correct option is (a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).

DIRECTION : In the following questions, a statement of assertion (A) is followed by a
statement of reason (R). Mark the correct choice as:
Assertion : Sum of the pair of angles 1200 and 600 is supplementary.
Reason : Two angles, the sum of whose measures is 1800, are called supplementary angles.
  • a)
    Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
  • b)
    Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
  • c)
    Assertion (A) is true but reason (R) is false.
  • d)
    Assertion (A) is false but reason (R) is true.
Correct answer is option 'A'. Can you explain this answer?

Swati Verma answered
We know that two angles are said to be supplementary if their sum of measure of angles is 1800.
So, Reason is correct.
Now, 1200 + 600 = 1800 ⇒ Sum of the pair of angles 1200 and 600 is supplementary.
So, Assertion is also correct.
Also, reason (R) is the correct explanation of assertion (A).
Correct option is (a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).

Direction: In the following questions, a statement of Assertion is given followed by a corresponding statement of Reason just below it. Of the statements, mark the correct answer as
Assertion : The length of a chord which is at a distance of 5 cm from the centre of a circle of radius 10 cm is 17.32 cm.
Reason : The perpendicular from the centre of a circle to a chord bisects the chord.
  • a)
    Both assertion and reason are true and reason is the correct explanation of assertion.
  • b)
    Both assertion and reason are true but reason is not the correct explanation of assertion.
  • c)
    Assertion is true but reason is false.
  • d)
    Assertion is false but reason is true.
Correct answer is option 'A'. Can you explain this answer?

Aditi Sharma answered
Let, PQ be a chord of a circle with centre O and radius 10cm. Draw OR ⊥PQ = .
Now, OP = 10cm and OR = 5cm
In right triangle ORP, we get
OP2 = PR2 + OR2
PR2 = OP2 - OR2
PR2 = 102 - 52 = 75
Since, the perpendicular from the centre to a chord bisects the chord.
Therefore, PQ = 2 x PR
= 2 x 8.66 = 17.32 cm

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