Section A contains 6 questions of 1 mark each. Scroll down for explanatory answer and video solution for all six one mark questions. Questions appeared from Quadratic Equations, Real Numbers, Coordinate Geometry, Arithmetic Progressions, Trigonometry, and Triangles.
Find the value of a, for which point P(\\frac{a}{3}) , 2) is the mid-point of the line segment joining the points Q(-5,4) and R(-1,0).
Substitute the coordinate of the points Q and R in the midpoint formula and solve it to find the value for a.
Find the value of k, for which one root of the quadratic equation kx^{2} - 14x + 8 = 0 is 2.
One of the roots of the equation is 2. So, substituting x = 2 will satisfy the equation. By substituting the value of x as 2, we can find the value of k.
Find the value(s) of k for which the equation x^{2} + 5kx + 16 = 0 has real and equal roots.
Roots of the quadratic equation are real and equal.
If the roots are real and equal, the discriminant (D) of the equation is 0.
Express discriminant b^{2} - 4ac in terms of k and equate it to zero to compute the value of k.
Write the value of cot^{2}θ − \\frac{1}{sin^2θ})
Step 1:Express cot θ in terms of cos θ and sin θ
Step 2: The denominator of both the terms of the expression will become sin^{2} θ
Step 3>: Simplify the numerator and cancel like terms in the numerator and denominator to find the value of the expression.
If sin θ = cos θ, then find the value of 2 tanθ + cos^{2} θ
Step 1: Compute value of θ for which sin θ = cos θ
Step 2: Substitute value of θ computed in step 1 into the given expression to find the value of the expression.
If n^{th} term of an A.P. is (2n + 1), what is the sum of its first three terms?
Approach: Compute the 1st, 2nd, and 3rd term by susbtituting n = 1, n = 2, and n = 3 in the expression (2n + 1). Subsequently, find the sum of the 3 terms.
In figure if AD = 6cm, DB = 9cm, AE = 8cm and EC = 12cm and ∠ADE = 48°. Find ∠ABC.
Concept: Check whether Basic Proportionality theorem applies in this case.
Step 1: Compute ratio between AD and DB and between AE and EC
Step 2: If the ratios are same, by converse of Basic Proportionality Theorem, DE will be parallel to BC.
Step 3: If DE is parallel to BC, corresponding angles will be equal and the answer can be computed.
After how many decimal places will the decimal expansion of \\frac{23}{2^4 × 5^3}) terminate?
Step 1: Rewrite the fraction such that the powers of 2s and 5s in the denominator is the same. This ensures that we get the denominator as a power of 10
Step 2: The power of 10 in the denominator is the number of decimal places in the decimal expansion of the fraction.
Question 1: Find the value of a, for which point P(\\frac{a}{3}) , 2) is the mid-point of the line segment joining the points Q(-5,4) and R(-1,0).
Scroll down further for explanatory answer text
Midpoint of 2 points (x_{1}, y_{1}) and (x_{2}, y_{2}) = \\frac{x_1 + x_2}{2}), \\frac{y_1 + y_2}{2})
The points given here are Q(-5, 4) and R(-1, 0) and the midpoint is P(\\frac{a}{3}), 2)
∴\\frac{-5 + (-1)}{2}) = \\frac{a}{3}) and \\frac{4 + 0}{2}) = 2
\\frac{-6}{2}) = \\frac{a}{3})
a = -9
Question 2: Find the value of k, for which one root of the quadratic equation kx^{2} - 14x + 8 = 0 is 2.
Scroll down further for explanatory answer text
If 2 is one of the roots of the quadratic equation kx^{2} - 14x + 8 = 0, then it will satisfy the equation
Substitute x = 2 in the above equation.
∴ k(2)^{2} – 14(2) + 8 = 0
4k – 28 + 8 = 0
4k = 20
k = 5
Question 2 Internal Choice 2: Find the value(s) of k for which the equation x^{2 }+ 5kx + 16 = 0 has real and equal roots.
Scroll down further for explanatory answer text
If a quadratic equation has real and equal roots, then the discriminant (D) of the equation is 0.
Discriminant of a quadratic equation ax^{2} + bx + c = 0 is b^{2} – 4ac
In the given equation, a = 1, b = 5k, and c = 16
b^{2} – 4ac = 0
(5k)^{2} – 4(1)(16) = 0
25k^{2} – 64 = 0
k^{2} = \\frac{64}{25})
k = ± \\frac{8}{5})
Question 3 Internal Choice 1: Write the value of cot^{2}θ − \\frac{1}{sin^2θ})
Scroll down further for explanatory answer text
cot^{2} θ − \\frac{1 }{sin^2 θ})
= \\frac{cos^2 θ}{sin^2 θ}) − \\frac{1}{sin^2 θ}) [Because cot θ = \\frac{cos θ}{sin θ})]
= \\frac{cos^2 θ − 1}{sin^2 θ})
= \\frac{− sin^2 θ}{sin^2 θ}) [Because sin^{2} θ + cos^{2} θ = 1 => sin^{2} θ = 1 - cos^{2} θ]
= -1
Question 3 Internal Choice 2: If sin θ = cos θ, then find the value of 2tan θ + cos2 θ
Scroll down further for explanatory answer text
For what values of θ will sin θ = cos θ?
sin θ = cos θ only when θ = 45°
Substitute θ = 45° in given expression
2tan θ + cos^{2} θ = 2tan 45° + cos^{2} 45°
= 2tan (1) + cos^{2} (\\frac{1}{√2}))
= 2 + (\\frac{1}{2}))
= \\frac{5}{2})
Question 4: If nth term of an A.P. is (2n + 1), what is the sum of its first three terms?
Scroll down further for explanatory answer text
nth term of the given A.P. is (2n + 1)
Substitute n = 1 to find the first term of the A.P. a_{1} = 2(1) + 1 = 3
Substitute n = 2 to find the second term of the A.P. a_{2} = 2(2) + 1 = 5
Substitute n = 3 to find the third term of the A.P. a_{3} = 2(3) + 1 = 7
Sum of the first three terms = 3 + 5 + 7 = 15
Question 5: In figure if AD = 6 cm, DB = 9 cm, AE = 8 cm and EC = 12 cm and ∠ADE = 48°. Find ∠ABC.
Scroll down further for explanatory answer text
Basic Proportionality Theorem: If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.
AD = 6 cm, DB = 9 cm, AE = 8 cm and EC = 12 cm
\\frac{AD}{DB}) = \\frac{6}{9}) = \\frac{2}{3}); \\frac{AE}{EC}) = \\frac{8}{12}) = \\frac{2}{3})
\\frac{AD}{DB}) = \\frac{AE}{EC}) ∴ DE is parallel to BC
If DE is parallel to BC, corresponding angles will be equal.
∴ ∠ADE = ∠ABC = 48°
Question 6: After how many decimal places will the decimal expansion of \\frac{23}{2^4 × 5^3}) terminate?
Scroll down further for explanatory answer text
Any real number which has a decimal expansion that terminates can be expressed as a rational number whose denominator is a power of 10 i.e., power of 2 and 5.
Express the fraction in an equivalent form such that the powers of 2 and 5 of the denominator are equal. Ensures that the denominator is a power of 10.
\\frac{23}{2^4 × 5^3}) = \\frac{23}{2^4 × 5^3}) × \\frac{5}{5}) = \\frac{23 × 5}{2^4 × 5^4}) = \\frac{115}{10^4}) = 0.0115
The answer is 4 places.
Register in 2 easy steps and
Start learning in 5 minutes!
Copyrights © 2016 - 19 All Rights Reserved by Maxtute.com - An Ascent Education Initiative.
Privacy Policy | Terms & Conditions
Phone: (91) 44 4500 8484
Mobile: (91) 93822 48484
WhatsApp: WhatsApp Now
Email: learn@maxtute.com