1 Mark questions for SA-2
1. The common difference of the A.P., 1/p, (1-p)/p, (1-2p)/p,,.... is :
(i) p (ii) -p
(iii) -1 (iv)1
Answer:(iii) -1
Common difference is the difference between any two consecutive terms of an A.P.
Therefore, Common difference here is :
(1-p)/p - 1/p = {(1-p)-1}/p = (1-p-1)/p = -p/p = -1
2.The angle of depression of a car, standing on the ground from the top of a 75 m high tower is 30° .The distance of the car from the base of the tower (in m ) is:
(i) 25 √3 (ii) 50√3
(iii) 75 √3 (iv)150
Answer:(i) 25 √3
Given that the angle of depression is 30° , then angle of elevation is (90 - 30 = 60°). Therefore,
We have,
tan60° = height of tower /distance of the car from the base of the tower
=> √3 = 75/distance of the car from the base of the tower
=> distance of the car from the base of the tower = 75/ √3 or 75√3/(√3√3) = 25√3
3.A dice is thrown once. The probability of getting a prime number is
(i) 2/3 (ii) 1/3
(iii) 1/2 (iv) 1/6
Answer:(iii) 1/2
A dice has six faces with each face having one of the numbers from 1,2,3,4,5,6.
Prime numbers are 2,3,5
Thus, probability of getting a prime numbers is = (3)/(6) = 1/2
4.In figure, DE and DF are tangents from the external point D to a circle with center A.If DE = 5 cm and if DE is perpendicular to DF, then the radius of the circle is:
(iii) 4 cm (iv) 6 cm
Answer: (ii) 5 cm
In the figure, Angle <AED and <AFD are of 90°. And given that <EDF = 90°. Thus AEDF is a rectangle. Again, AE=AF (radius of a circle)
Thus, AEDF is a square as adjacent sides are equal.
Thus AF = 5 cm.
5. The common difference of the A.P., 1/2q, (1-2q)/2q, (1-4q)/2q,,.... is :
(i) -1 (ii) 1
(iii) q (iv) 2q
Answer:(iii) -1
Common difference is the difference between any two consecutive terms of an A.P.
Therefore, Common difference here is :
(1-2q)/2q - 1/2q
= {1-2q-1}/2q
= -1
6. A box contains cards numbered 6 to 50. A card is drawn at random from the box. The probability that the drawn card has a number which is a perfect square, is
(i) 1/45 (ii) 2/45
(iii) 1/9 (iv) 4/45
Answer: (iii) 1/9
From the number 6 to 50 there are 5 perfect squares, they are 9,16,25,36,49
In total there are (50-6) +1 = 45 numbers.
Thus the probability of drawing a prime numbers is 5/45 = 1/9
7.The point on the x-axis which is equidistant from the points(-1,0) and (5,0) is
(a) (0,2) (b) (2,0)
(c) (3,0) (d) (0,3)
Answer:
The distance between two points is given by
for point (a): distances are √1+4 =√5 and √25+4 = √29. Not the correct one
for point (b): distances are √9+0 =√9 and √9+0 = √9. Hence, the correct one. No need to check further.
8. If π is taken as 22/7, the distance (in metres) covered by a wheel of diameter 35 cm, in one revolution, is
(a) 2.2 (b) 1.1
(c) 9.625 (d) 96.25
Answer:(a) 2.2
Here we just have to calculate the circumference of the circle = 2(π)r = 2(22/7)(35) = 220 cm or 2.2 m
9. A ladder 15 m long just reaches the top of a vertical wall. If the ladder make an angle of 60° with the wall, then the height of the wall is
(i) 15√3 m (ii) 15√3/2 m
(iii) 15/2 m (iv)15 m
Answer:(iii) 15/2 m
cos60°= height of the wall / length of the ladder
=> height of the wall = cos60°x(length of the ladder)
=>height of the wall = (1/2) x(15) = 15/2
10.A box contains 90 discs, numbered from 1 to 90. If one disc is drawn at random from the box, the probability that it bears a prime-number less than 23 is:
(i) 7/90 (ii) 10/90
(iii) 4/45 (iv) 9/89
Answer:(iii) 4/45
Prime numbers less than 23 are 2,3,5,7,11,13,17,19. They are 8 in numbers and in total there are 90 numbers in all.
Thus, the probability of drawing a prime number less than 23 from 1 to 90 are 8/90 = 4/45.
11. If the difference between the circumference and the radius of a circle is 37 cm, then using π = 22/7, the circumference (in cm ) of the circle is:
(a) 154 (b) 44
(c) 14 (d) 7
Answer: (b) 44
Given that the difference between the circumference and the radius of a circle is
2πr - r = 37
=> r{2(22/7) - 1} = 37
=> r {44/7 - 1 } = 37
=> r(37/7)=37
=> r = (37)(7/37)
r = 7cm
and therefore circumference = 2(π)r = 2(22/7)(7) = 44
12. If a die is rolled once, then the probability of obtaining a number greater than 3 is :
(a) 1/2 (b) 1/3
(c) 0 (d) 2/3
Answer:(a) 1/2
On a dice the number of numbers greater than 3 are 4,5,6.
Therefore, the probability of getting a number greater than 3 is 3/6 = 1/2
13. If α and β are the roots of the equation
(a+1)x² + (2a+3)x + (3a + 4) = 0
If αβ = 2 , then (α + β) is equal to
(i) 0 (ii) -1
(iii) 2 (iv) -2
Answer: (ii) -1
We know, for a quadratic equations Ax² + Bx + C = 0
Sum of roots = -B/A and product of roots = C/A
Thus, we have
(α + β) = -(2a+3)/(a+1)
and (αβ) = (3a+4)/(a+1) =2 (given)
=> 3a+4 = 2a + 2
=> a = -2
Again,
(α + β) = -(2a+3)/(a+1)
= -{2(-2) + 3}/(-2+1)
= -{-1}/(-1) = -1
14. If the nth term of an A,P, is (2n + 1), then the sum of n terms of an A,P, is
(a) n(n+1) (b) (n+1)(n+2)
(c) n(n+2) (d) n(n+3)
Answer: (c) n(n+2)
first term of an A.P. = a = (2.1 + 1) = 3
second term of an A,P. = a + (n-1)d = 3 + d
again , 3 + d = (2n+1) {given}
=> 3 +d = 2.2 + 1
=> d = 5 -3 =2
And the nth term is = 3 + (n-a)3
Thus, sum of n terms of an A.P. = (n/2){ sum of first and last tern}
= (n/2){ 3 + 3 + (n-1)2
= (n/2){3 + 3 -2 + 2n}
= (n/2) {4+2n} = n(n+2)
15. The radii of two circles are 9 cm and 12 cm. The radius of a circle whose area is equal to the sum of the areas of the two circles is
(a) 15 cm (b) 14 cm
(c) 13 cm (d) 12 cm
Answer:(a) 15 cm
Area of circle with radius 9 cm = πr²= π (9)² = 81π
Area of circle with radius 12 cm = πr² = π (12)² = 144π
Sum of areas of these 2 circles is = 144π + 81π = 225π
Thus, a radius of circle whose area is equal to the areas of both the given circles = πR² = 225π
=> πR² = 225π
= > R² = 225
=> R = 15 cm
16. The perimeter of a square circumscribing a circle with radius a ,is
(a) 8a (b) 4a
(c) 2a (d) 16a
Answer: (a) 8a
In such cases the side of the circumscribing square is equal to the diameter of the circle. Thus the side of the square is 2a.And perimeter is 4(2a) = 8a
17. A card is drawn from a well-shuffled deck of 52 playing card. The probability of not drawing a Black king is
(a) 1/13 (b) 12/13
(c) 1/26 (d) 25/26
Answer:
There are 2 black kings in a deck of 52 cards. Therefore, there will be 50 cards that are not king of spades. Therefore probability of not drawing a Black King is 50/52 = 25/26
18. The value of A30 - A20 for the A.P. 1,5,9,13,17,...
(a) 40 (b) 80
(c) 10 (d) 20
Answer:(a) 40
The A.P. is 1,5,9,13,17,.... The first term is 1 and the common difference is 5-1 = 4.
Therefore, A30 - A20 = (1 + 29.4) - (1+19.4) = 1 + 29.4 - 1 - 19.4
= 4(29-19)=4(10) = 40
19. A chord of a circle of radius 14 cm subtends a right angle at the centre of the circle. The area of the minor sector is:
(a) 147 cm² (b) 154 cm²
(c) 161 cm² (d) 140 cm²
Answer: (b) 154 cm²
Area of sector = (θ/360)π r²
Therefore, required area = (90/360)(22/7)(14)(14)
= (1/4) (22)(2)(14)
= (11)(14) = 154 cm²
20. The Value of k so that the sum of the roots of the quadratic equation 3x² + (2k+1)x -(k+5) =0
is equal to the product of the roots is
(a) 4 (b) 3
(c) 2 (d) 1
Answer: (a) 4
We know that for a quadratic equation of type Ax + Bx + C = 0 the
sum of roots = -B/A
and product of the roots = C/A
Using these formula for the given quadratic equation we have
-(2k+1)/3 = -(k+5)/3
=> (2k+1) = (k+5)
=> 2k-k = 5 - 1
=> k = 4
21.In figure , O is the centre of a circle, AB is a chord and AT is the tangent at A. If <AOB = 100°, then <BAT is equal to
 |
| Question 21 |
(c) 50° (d)90°
Answer:(c) 50°
Given that <AOB = 100°.
Since OA = OB (both are radius of the circle)
=> <OAB = <OBA --------(i)
Again, <AOB + <OBA + <OAB = 180° (sum of angles of triangle )
=> 100° + 2 <OBA = 180
=> < OBA = 40° = <OAB from (i)
Again,
Since, AT is tangent to the circle at A, we have
<OAT = 90°
=> <OAB + < BAT = 90°
=> 40° + <BAT = 90°
=> <BAT = 50°
22. Which of the following cannot be the probability of an event?
(a) 1.5 (b) 3/5
(c) 25% (d) 0.3
Answer: (a) 1.5
The probability of an event also lies between 0 (there is no chance for the event to occur) and 1 (the event will definitely occur).
23. If α and β are the roots of the equation x² + 3x + 4 =0, then the equation whose roots are α +1 and β + 1 is
(a) x² + x + 1 = 0 (b) x² + x + 2 = 0
(c) x² + 3x +1 = 0 (d) x² + 3x + 4 = 0
Answer: (b) x² + x + 2 = 0
For the given equation, the sum of roots = α + β = -3
and product of roots = α β = 4
The the sum of roots of the equation whose roots are (α+1),(β+1) = (α + β) + 2 = -3 + 2 = -1
and product of roots = (α+1)(β+1) = αβ + α + β +1 = 4 - 3 + 1 = 2
Therefore the required equation = x² - (Sum of roots)x + product of roots = 0
=> x² - (-1)x + 2 =0
=> x² + x + 2 = 0
24. The angle through which the minute hand of a clock moves from 8 A.M. to 8:35 AM is :
(a) 210° (b) 90°
(c) 60° (d) 45°
Answer:(a) 210°
A minute hand moves 6° in 1 minute.
Therefore , from 8 A.M. to 8:35 AM the minute hand had moved 35 x 6° = 210°
25. If P(a/2,4) is the mid-point of the line-segment joining the points A(-6,5) and B(-2,3), then the value of a is
(a) -8 (b) 3
(c) -4 (d) 4
Answer: (a) -8
Since P is the mid-point of the line-segment joining the points A and B , the the co-ordinate of P are given by
P=(x,y) = P{(-6-2)/2,(5+3)/2} = P(-4,4)
But given that P(a/2,4) = P(-4,4)
=> a/2 = -4
=> a = -8
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