1. -6,0
2. 4, 6
3. -30,-20
4. -6,6

View Answer
Correct Answer is (D).
The graphical meaning of the zero of a polynomial is the x-coordinate where the graph of the polynomial intersects or touches the x-axis. We can see from the graph touches the x-axis at two points i.e. (−6,0) and (6,0). So, the zeroes are (−6,6).

1. -10
2. -5
3. 5
4. 7

View Answer
Correct Answer is (B).
For an inconsistent solution, there is only one case: the graphs of the pair of equations are parallel i.e. the pair of equations has no solutions. We will now apply inconsistency condition which is:From the given equations :Now, we set up the ratio and solve for k.

1. A number of secants can be drawn at any point on the circle.
2. Only one tangent can be drawn at any point on a circle.
3. A chord is a line segment joining two points on the circle
4. From a point inside a circle only two tangents can be drawn.

View Answer
Correct Answer is (D).
Option d. is false as NO tangent can be drawn from a point inside the circle.
The other three options are absolutely correct.

1. 7
2. 7n
3. -4
4. 4

View Answer
Correct Answer is (A).
Given: t𝑛=7𝑛−4
On substituting the value of n as 1 we will get first term, on substituting n as 2 we get second term and so on.∴ A.P. is 3,10,17,…………….
To find out the common difference of A.P. we have to subtract any two consecutive term.

1. 5 cm
2. 20 cm
3. 10 cm
4. 4 cm

View Answer
Correct Answer is (B).
Given:
Radius of right circular cone = Radius of a sphere= 5cm Volume of cone = Volume of sphereOn solving the above equation we get h= 20 cm.

1. 𝟏𝟏/𝟗
2. 𝟑/𝟐
3. 𝟗/𝟏𝟏
4. 4

View Answer
Correct Answer is (A).

1. 110⁰
2. 70⁰
3. 140⁰
4. 55⁰

View Answer
Correct Answer is (C).
Statement∠𝐓𝐏𝐐=𝟏𝟏𝟎° ∠𝑶𝑷𝑻=𝟗𝟎° ∠𝑸𝑷𝑶=𝟏𝟏𝟎°−𝟗𝟎°= 𝟐𝟎° In OPQ 𝑶𝑷=𝑶𝑸 ∠ 𝑷=∠𝑸=𝟐𝟎° ∠𝑷𝑶𝑸=𝟏𝟖𝟎°−(𝟐𝟎°+𝟐𝟎°) = 𝟏𝟒𝟎°ReasonGiven
The tangent at any point on the on the circle is perpendicular to the radius at the point of contact.
From figure
Radii of same circle Angles opposite to equal sides are equal. Angle sum property.

1. 𝒙²−𝟓√𝟐 𝒙+𝟏
2. 𝟖𝒙²−𝟐𝟎
3. 𝟏𝟓𝒙²−𝟔
4. 𝒙²−𝟐√𝟓 𝒙−𝟏

View Answer
Correct Answer is (B).
Put the values in the general equation of the quadratic polynomial that is 𝑘 [𝑥² − (𝛼+𝛽) 𝑥 + (𝛼𝛽)] where k is any real number.Option a. and d. gets ruled out as there is no term of x in the above polynomial . Now let us see option c.We can see that the co-efficient of 𝑥² is 8. So, on substituting 8 in place of k we get 8𝑥²−20.

1. 20
2. 10
3. 30
4. 40

View Answer
Correct Answer is (C).

Class-Interval	Frequency	C.I.
0-10	5	5
10-20	9	14
20-30	15	29
30-40	10	39
40-50	6	45

𝑁=45.22.5^th term lies in the row having Cumulative frequency as 29. So, the median class is 20-30. ∴ Upper limit is 30.

1. equilateral and similar
2. isosceles and similar
3. isosceles but not similar
4. not similar

View Answer
Correct Answer is (B).
In ∆𝐴𝑂𝐷 the lengths of OA and OD are equal. ∴ ∆ AOD is isosceles. Also,
∠ 𝐴=∠𝐷 (Angles opposite to equal sides are equal.)
We will next calculate angles A and D by angle sum property .

∠𝐴+∠𝐷+∠𝑂=180°
⟹∠𝐴+∠𝐴+∠𝑂=180° (∵∠𝐴=∠𝐷)
⟹2∠𝐴+80°=180°
⟹2∠𝐴=180°−∠80°
⟹2∠𝐴=100°
⟹∠𝐴=(1𝑜𝑜°)/2
⟹∠𝐴=50°
∴∠𝐴=∠𝐷=50°
Now from one of the theorems of circle that angles in the same segment are equal.
∠𝐴𝐷𝑂=∠𝑂𝐵𝐶
∴∠𝑂𝐵𝐶=50°
∠𝐴𝑂𝐷 and ∠𝐵𝑂𝐶 are equal as they are vertically opposite angles.
∴∠𝐵𝑂𝐶=80°
Now, In ∆𝑂𝐵𝐶
∠𝑂+∠𝐵+∠𝐶=180°
⟹80°+50°+∠𝐶=180°
⟹130°+∠𝐶=180°
⟹∠𝐶=180°−130°
⟹∠𝐶=50°
So, in ∆𝐵𝑂𝐶 we see that ∠𝐵=∠𝐶=50°
∴ ∆𝐵𝑂𝐶 is isosceles.
Both the triangles are isosceles.
Now, let us see for similarity.
In ∆𝑂𝐷𝐴 and ∆𝑂𝐵𝐶
∠𝐷𝑂𝐴=∠𝐵𝑂𝐶=80°
∠𝑂𝐷𝐴=∠𝑂𝐵𝐶=50°
∴∆𝑂𝐷𝐴~∆𝑂𝐵𝐶 (A.A. Criteria)
So, the triangles are isosceles and similar.

STATEMENT	REASON
∠BOC = 80°	Given
∠BOC = 80°	Given
In ΔAOD OA = OD ∠A = ∠D   = 500	Radii of same circle Angles opposite to equal sides are equal.
∠A = ∠D =	Angle sum property.
∠ADO = ∠OBC ∴ ΔODA = ΔOCB	Angles in the same segment A.A. criteria

The triangles are isosceles and similar

1. Irrational and distinct
2. not real
3. rational and distinct
4. real and equal

View Answer
Correct Answer is (A).
Given equation: 𝑥²+𝑥+1=0 where: a=1;b=1 and c=1 To find the nature of roots we have to calculate the discriminant. 𝐷=𝑏^2−4𝑎𝑐
𝐷=1²−4×1×−1
𝐷=1+4
𝐷=5
i.e. D>0
The roots are real and distinct. Now real roots means both rational and irrational.
To find whether the roots are rational or irrational, we will find out the roots.
∴ The roots are irrational and distinct.

1. 1
2. 
3. 
4. Not defined

View Answer
Correct Answer is (C).

1. 396/7
2. 594/7
3. 549/7
4. 604/7

View Answer
Correct Answer is (B).
Given: Volume of a hemisphere =396/7 𝑐𝑚³
Volume of a hemisphere is 2/3 𝜋𝑟³

On solving the above equation we get the value of r as 3.

1. 1/6
2. 3/8
3. 11/24
4. 5/8

View Answer
Correct Answer is (B).
No. of balls= 24
No. of blue balls = 4
No. of green balls = 11
∴ No. of white balls = 24 – (4+11)
= 24 – 15
= 9

1. (0, 0)
2. (-4, 0)
3. (-5, 0)
4. (√𝟒𝟏, 0)

View Answer
Correct Answer is (B).
One way to solve this is by taking out the distance of (−4,−5) and each of the given points in option.

1. Median
2. Mean
3. Range
4. Mode

View Answer
Correct Answer is (A).
Among all the measures of central tendency (mean, median and mode) MEDIAN gives the middle most observation of the data.
Let us also discuss about other options –
Mean – The mean is defined as the sum of all the values in a data set divided by the number of values in that data set. It represents the central or average value around which the data is distributed.
Range – It is the difference between the highest value and the lowest value in a given set of data.
Mode – The mode is the value that appears most frequently in a data set.

1. (4, 0)
2. (7/3, 3/2)
3. (3,0)
4. (0,3)

View Answer
Correct Answer is (C).
Given - AP:PB = 1:2
The point P lies on X - axis i.e. ordinate will be zero. Let the co-ordinates of point P be(x,0). We will take out the co- ordinates of point P usinh the section - formula:

1. 3/13
2. 1/2
3. 3/52
4. 3/26

View Answer
Correct Answer is (D).
No. of face cards =4(Jack) + 4(Queen) + 4(King)

∴ No. of red face cards = 6
Total number of cards = 52
Probability of red face cards =

1. Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
2. Both assertion (A) and reason (R) are true and reason (R) is not the correct explanation of assertion (A).
3. Assertion (A) is true but reason (R) is false.
4. Assertion (A) is false but reason (R) is true.

View Answer
Correct Answer is (B).
Let us take some even natural numbers
2,4,6,8,10,12,14,16,18,20,22,... ... ...
Now take any two number from consecutive series
Say we have 10,12.
The H.C.F of 10,12 is 2.
Next take any other two consecutive numbers.
This time we take 18,20.
The H.C.F is again 2.
We go on continuing this process and we will see that the H.C.F. is always 1.
∴ Assertion(A) is correct.
The reason is also correct as we know even numbers are divisible by 2, but reason is not correct explanation of reason. OPTION B is correct.

1. Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
2. Both assertion (A) and reason (R) are true and reason (R) is not the correct explanation of assertion (A).
3. Assertion (A) is true but reason (R) is false.
4. Assertion (A) is false but reason (R) is true.

View Answer
Correct Answer is (D).

View Solution for Option A
Solution: (A).
Step 1. Prime Factorization:
            480 = 2⁵ × 3 × 5
            720 = 2⁴ × 3² × 5
Step 2. Finding the H.C.F (Highest Common Factor):
            The H.C.F is obtained by taking the lowest powers of common prime factors.
            Common prime factors: 2, 3, 5
            Lowest powers: 2⁴, 3¹, 5¹
            H.C.F = 2⁴ × 3 × 5 = 240
Step 3. Finding the L.C.M (Lowest Common Multiple):
            The L.C.M is obtained by taking the highest powers of all prime factors.
            Highest powers: 2⁵, 3², 5¹
            Lowest powers: 2⁴, 3¹, 5¹
            L.C.M = 2⁵ × 3² × 5 = 1440
Final Answer:
H.C.F = 240
L.C.M = 1440

View Solution for Option B
Solution: (B).
Step 1. Prime Factorization:
            Prime factorization of 85: 85 = 5 × 17
            Prime factorization of 238: 238 = 2 × 7 × 17
Step 2. Finding the H.C.F (Highest Common Factor):
            The common factor between 85 and 238 is 17. Hence, the H.C.F is 17.
Step 3. Expressing the H.C.F in the Given Form
            The equation is:
            85m - 238 = 17
            Rearranging for m:
            85m = 17 + 238
            85m = 255
            m = 255 ÷ 85 = 3
Final Answer: The value of m is 3.

View Solution for Option A
Solution: (A).
Step 1. Understand the conditions for an odd product:
            A product is odd if and only if both numbers rolled are odd.
Step 2. Identify odd numbers on the dice:
            Odd numbers are: 7, 9, 11 (3 out of 6 numbers on each die).
Step 3. Total outcomes:
             Total outcomes = 6 × 6 = 36
Step 4. Total outcomes:
             Total outcomes = 6 × 6 = 36
Step 5. Probability
             P(Odd product) =
Final Answer: Probability =

View Solution for Option B
Solution: (B).
There are a total of 900 three-digit numbers, ranging from 100 to 999
We havt to find the count and probability of selecting a three-digit number where:
1. The hundred&aposs digit is 8.
2. The unit&aposs digit is 5.
Such numbers will have the form 8X5, where X is the ten&aposs digit, which can range from 0 to 9.
This gives us 10 possible numbers: 805, 815, 825, ..., 895.
The total count of such numbers is 10.
Therefore, the probability of randomly selecting one of these numbers out of all three-digit numbers is

View Answer
Substitute the values : 

View Answer
Step 1. Use the distance formula:
                
Step 2. Simplify the equation by squaring both sides:
            41         = (x - 8)² + 25
            (x - 8)² = 16
            x - 8     = ±4
Step 3. Solve for x
            x = 8 + 4 = 12 or x = 8 - 4 = 4
Final Answer: Points are (12, 0) and (4, 0)

View Answer
Step 1. Calculate the lengths of the sides using the distance formula:
            
Step 2. Compare the side lengths
            AB = BC, so triangle ABC is isosceles.
            (x - 8)² = 16
Final Answer: Triangle ABC is isosceles.

View Answer for Option (A)
From Mid - point theorem we know line joining mid-points of two sides of a traiangle is parallel to the 3rd side and half of it.
In ∆ ∆ABC,
F and E are mid-points of AB and AC respectively,
∴ FE || BC and FE =
So, FE || BD also and FE = BD (Since, BD = )
We know if any one pair of opposite sides of quadrilateral is parallel and equal it is a Parallelogram.
∴ DBEF is a parallelogram.
Similarity, DCEF is a parallelogram.

1. In △DEF and △FBD
∠DFE = ∠FDB[Alternate angles]
∠DFE = ∠FBD[Opposite sides of a parallelogram are equal]
∴ ∆DEF ∼ ∆FBD[AA Similarity Criterion]

2. In △DEF and △ABC
    ∠ FED = ∠ABC[Opposite sides of a parallelogram are equal]
    ∠ EFD = ∠ACB[Opposite sides of a parallelogram are equal]
    ∴ ∆DEF ∼ ∆ABC[AA Similarity Criterion]

View Answer for Option (B)
Solution: (B).
Given:
1. ΔABC, where D is the midpoint of BC (i.e., AD is the median), so BD=CD.
2. P and Q are points on AB and AC, respectively.
3. PQ is parallel to BC.
To Prove:
AD bisects PQ, i.e., PR = RQ.
Proof:
In ∆ APR and ∆ABD
∠APR=∠ABD             Corresponding angles
∠ARP=∠ADB             Corresponding angles
∴ ∆APR~∆ABD          A.A. criteria
                     C.P.S.T….(i)
In ∆ ARQ and ∆ADC
∠ARQ=∠ADC             Corresponding angles
∠ARQ=∠ACD             Corresponding angles
∴ ∆ARQ~∆ADC          A.A. criteria
                     C.P.S.T….(ii)
From (i) and (ii)
As BD=CD
∴ PR=QR
Hence, AD bisects PQ

View Answer
Let the two numbers be x and y.
Given:
x + y = 18 ...(1)
....(2)
Simplify equation (2):

Substituting x + y = 18 from equation (1):

...(3)

From Equation 1
y = 18 - x
Substituting the value of y in Equation 3
xy = 80
x(18 - x) = 80
18x - x² = 80
x² -18x + 80 = 0
x² -10x - 8x + 80 = 0
x(x-10) - 8(x-10) = 0
(x-8)(x-10) = 0
x=8 or 10.
Ans : The numbers are 8 and 10 or 10 and 8.

View Answer
Given polynomial: 6x² - 5x + 1

Sum of squares of roots:

Product of squares of roots: 

Quadratic polynomial:
t² - (sum of roots)t + product of roots = 0;

Multiply through by 36:
36t² - 13t + 1 = 0

View Answer
Start with the given equation:
cos²θ + sin²θ = 1 ...(1)
Let A = cosθ - sinθ.
Square both sides:
A² = (cosθ - sinθ)²
     = cos²θ - 2cosθsinθ + sin²θ
Using equation (1):
From trigonometric identity:
sin(2θ) = 2cosθsinθ
A² = 1 - sin(2θ)
For cosθ - sinθ = ±1:
A² = 1 ⇒ 1 - sin(2θ) = 1 sin(2θ) = 0
Hence, cosθ - sinθ = ±1.

View Solution for Option A
Solution: (A).
Length of the minute hand (radius): r = 18 cm
The minute hand moves round the clock once in 60 minutes .
Angle swept by the minute hand in 60 min = 360°
Angle swept by the minute hand in 1 min =
Angle swept by the minute hand in 35 min = 6°×35=210°
Area of sector:

View Solution for Option B
Solution: (B).
Radius of the circle: r = 14 cm
Central angle: θ = 60°
Area of sector:

Area of triangle:

Area of minor segment:
A_segment = A_sector - A_triangle
                    = 102.67 - 84.77
                    = 17.9 cm²

View Answer
Assume √3 is rational. where p and q are coprime integers, q ≠ 0.
Squaring both sides:

Let p = 3k.
Substituting into p² = 3q²
(3k)² = 3q²
9k²   = 3q²
q²     = 3k²
Hence, 3 divides q.
But this contradicts the assumption that p and q are coprime.
Therefore, √3 is irrational.