Sample Paper: Math Class ten
Mathematics Class Tenth
sample paper math term-1 class ten mcqs-5
Case study (A) A group of Class X student goes to picnic during vacation. There were three different slides and three friends Anoop, Atul and Alkash are sliding in the three slides. The position of the three friends shown by P, Q and R in the three different slides are given below:
Consider O as origin, answer the question given below:
Question (41) The coordinates of the point Q which divides the line segment PR in the ratio 1 : 2 internally.
Answer : (d) `(4, 13/4)`
Explanation :
Given, PQ : QR = 1 : 2
From the given figure, P (2, 5) and R (8, 3)
Now, we know that the coordinates of the point p(x, y) which divides the line segment joining the points A (x1, y1) and B (x2, y2) in the ratio m1 : m2 are
Thus, option (a) `(4, 13/3)` is the correct answer.
Question (42) The distance PR
(a) √20
(b) √36
(c) √38
(d) 2 √10
Answer : (d) (d) 2 √10
Explanation :
From the distance formula, we know that, the distance between two points whose coordinates are given,
Thus, option (d) 2 √10 is the correct answer
Question (43) The coordinates of a point on the x-axis, which is at an equal distance from P and R
(a) (`11/3`, 0)
(b) 3, 0
(c) (`13/9`, 0)
(d) (1, 3)
Answer : (a) (`11/3`, 0)
Explanation :
Now, since the required point is lies on the x-axis, thus, its y coordinate will be equal to 0.
Thus, let the coordinate of required point z is (x, 0)
And given, P (2, 5) and R (8, 3)
Now, from the distance formula, we know that, the distance between two points whose coordinates are given,
Similarly,
Now, since Z is at the equal distance from P and R,
Thus PZ = RZ
⇒ x2 – 4x + 29 = x2 – 16 x + 73
⇒ – 4x + 29 = – 16 x + 73
⇒ – 4x + 16 x = 73 – 29
⇒ 12 x = 44
⇒ x = 44/12
⇒ x = 11/3
Thus, coordinate of point Z is 11/3, 0
Thus, option (a) (`11/3`, 0) is the correct answer.
Question (44) The coordinates of the midpoint of PR
(a) – 4, 5)
(b) (4, 5)
(c) (5, 4)
(d) (4, 4)
Answer : (c) (5, 4)
Explanation :
Let, Z is the mid-point of PR
We know that, the coordinate of mid-point of two points
⇒ Coordinates of mid-point of PR = (5, 4)
Thus, option (c) (5, 4) is the correct answer.
Question (45) If we shift origin O by 2 units towards the right and 1 unit towards North (as shown in the figure given), then the coordinates of point R.
(a) (2, 6)
(b) (6, 2)
(c) (–2, 6)
(d) (– 6, 2)
Answer : (6, 2)
Explanation :
From the given figure, it is clear that x-coordinate of point R is 6 and it's y-coordinate is 2
Thus, coordinate of point R = 6, 2
Thus, option (b) (6, 2) is the correct answer.
Case study (B)
Junk food is unhealthy food that is high in calories from sugar or fat, with little dietary fiber, protein, vitamins, minerals, or other important forms of nutritional value. A sample of a few students has been taken. If α be the number of students who take junk food, β be the number of students who take healthy food such that α > & beta; and &alpha and β are the zeroes of the quadratic polynomial P(x) = x2 – 9x + 18 then answer the following questions.
Question (46) The graphical representation of the above expression will
(a) Downward Parabola
(b) Straight Line
(c) Curve
(d) Upward Parabola
Answer : (d) Upward Parabola
Explanation :
Given, P(x) = x2 – 9x + 18
Thus, zeros of polynomial,
x2 – 9x + 18 = 0
⇒ x2 – 6x – 3x + 18 = 0
⇒ x (x – 6) – 3 (x – 3) = 0
⇒ (x – 3 ) (x – 6) = 0
Now, if x – 3 = 0
Thus, x = 3
And, if x – 6 = 0
Thus, x = 6
Thus, zeros of given polynomial = 3 and 6
Now, after putting the value of x 1 to 7, the values of y are
10, 4, 0, –2, – 2, 0, 4
Thus, by taking these values, a graph can be plotted as below:
Clearly, the graphical representation of the given expression is an upward parabola.
Thus, option (d) Upward Parabola is the correct answer.
Shortcut Method to know the UPWARD or DOWNWARD parabola of the graph for a given expression
If the coefficient of x2 is greater than zero (0), then graphical representation will be Upward parabola and if the coefficient of x2 is less than zero (0), then the graphical representation will be Downward parabola.
Here, since coefficient of x2 in the given equation is 1, which is greater than zero, thus graphical representation for the given expression is Upward Parabola.
Hence, option (d) Upward Parabola is the correct answer.
Question (47) The number of students who take Junk Food and Healthy Food are
(a) 2 and 9
(b) 9 and 2
(c) 6 and 3
(d) 3 and 6
Answer : (c) 6 and 3
Explanation :
Given, P(x) = x2 – 9x + 18
Thus, zeros of polynomial,
x2 – 9x + 18 = 0
⇒ x2 – 6x – 3x + 18 = 0
⇒ x (x – 6) – 3 (x – 3) = 0
⇒ (x – 3 ) (x – 6) = 0
Now, if x – 3 = 0
Thus, x = 3
And, if x – 6 = 0
Thus, x = 6
Thus, zeros of given polynomial = 3 and 6
Since, as given in the question, number of students who takes junk food is greater, thus, the number of students who take junk food = 6 and number of students who take healthy food = 3
Thus, option (c) 6 and 3 is the correct answer.
Question (48) For the above-given polynomial find the value of `1/alpha + 1/beta`
(a) 1/2
(b) 2/3
(c) 2
(d) –1/2
Answer : (a) 1/2
Explanation :
The α = 6 and β = 3 (as calculated in the above question)
Hence, option (a) 1/2 is the correct answer.
Question (49) The quadratic polynomial whose zeroes are √10 and –√ 10 is
(a) x2 + 10
(b) x2 – 2 radic; 10 x + 10
(c) x2 – 100
(d) x2 – 10
Answer : (d) x2 – 10
Explanation
Given, zeros of the polynomial = √10 & –√10
i.e. α = √10
And, β = – √10
Now, we know that, a quadratic polynomial
= x2 – (α + β ) x + α β
= x2 – [√10 + (– √10) x + √10 × (– √10)
= x2 – (√10 – √10) x + (– 10)
= x2 – (0) x – 10
= x2 – 10
Thus, option (d) x2 – 10 is the correct answer.
Question (50) See the graph given below:
The zeroes of the polynomial in this given graph is /are
(a) – 4, – 2, 2
(b) –2, 2
(c) – 4, 0
(d) – 4 only
Answer : (b) –2, 2
Explanation :
Since, graph intersect x-axis at two distinct point –2 and 2. Thus, the zeros of the given polynomial is –2, 2
Thus, option (b) –2, 2 is the correct answer
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