193: Calculus in mathematics
studies change, motion, and accumulation through concepts like limits,
derivatives, and integrals. It is especially important for exams such as
JEE, SAT, and A-levels, where students solve problems involving rates of
change, graphs, optimization, and areas under curves. Calculus is widely
applied in physics, engineering, economics, computer science, and
advanced mathematical modeling.
Answer Explanation: .PNG)
The trigonometric ratio of cosine is the ratio of the length of the adjacent side divided by the length of the hypotenuse. The length of the adjacent side is the x−value in a point on the unit circle. The hypotenuse is the radius of the unit circle, so the hypotenuse is 1. Thus, the value of the cosine ratio of any angle in the unit circle is the x−value of the point on the unit circle that corresponds to that angle. The trigonometric ratio of tangent is the length of the opposite side divided by the length of the adjacent side. The length of the opposite side is the y−value in a point on the unit circle and the length of the adjacent side is the x−value in a point on the unit circle. The hypotenuse is the radius of the unit circle, so the hypotenuse is 1. Thus, the value of the tangent ratio of any angle in the unit circle is the ratio yx from the point on the unit circle that corresponds to that angle. In this question, tan(5π/4)=1. This ratio is taken from the point (−2/√2,−2/√2) that corresponds to the angle with a measure of 5π/4 radians. Thus, using the information above, the value of cos(5π4) is the same as the x−value in the point (−2/√2,−2/√2).Therefore, the value of cos(5π/4)=−2/√2.
Answer Explanation: The original circle F has its center at the point (−5,−6) with a radius of 4 units. The translated/dilated circle F’ has its center at the point (−5,4) with a radius of 1 units. This means the center was translated up 10 units. As a transformation, this translation is written as (x,y)→(x,y+10). Circle F was also dilated by a factor of 1/4 because the radius was reduced from 4 units to 1 units. As a transformation, this dilation is written as (x,y)→1/4(x,y). Putting the translation and dilation together, the rule is (x,y)→1/4(x,y+10).
Sample Question: A company ships spherical paperweights in cubic boxes. The circumference of the paperweight is 9π cm. If the box fits the sphere exactly with the sides of the sphere touching the box, what is the volume of the smallest box the company can use for shipping.
81 cm3
81 π cm3
729 cm3
1009 π cm3
Answer Explanation:
Grade 10 math Geometry – Modeling with Geometry
Notice that the diameter of the sphere will be the same as the side of the cubic box. Using the value of the circumference the diameter of the paperweight can be determined.
C = πd9π
cm = πd9
cm = d
Since the diameter is equal in measure to the sides{\dots}
V=s3
V=(9 cm)3
V=729 cm3
Standards: HSG.MG.A.3
Sample Question: A vector in standard form has components <3, 10>. What is the initial point?
- (0, 0)
- (3, 10)
- (6, 20)
- Not enough information given
Click here to practice: Grade 10 Math Number and Quantity – The Complex Number System Questions
Answer Explanation: i(7−i)=i*7−i*i=7i−i2=7i−(−1)=7i+1=1+7i
Sample Question: Which expression is equivalent to 9x2 – 16y2?
- -10
- 10
- 5
- -5
-
Sample Question: Which expression is equivalent to 9x2 – 16y2?
(3x – 4y) (3x – 4y)
(3x + 4y) (3x + 4y)
(3x + 4y) (3x – 4y)
(3x – 4y)2
Answer Explanation: Student must recognize the expression is the difference of two perfect squares
Standards: HSA.SSE.A.2
Click here to practice: Algebra – Seeing Structure in Expressions Questions for Grade 10 Math
Domain: Grade 10 >> Algebra – Arithmetic with Polynomials & Rational Expressions
Sample Question: Evaluate f(x)=−a3+6a−7 at a = – 1 and state the remainder.
-14
-12
14
12
Answer Explanation: student must substitute – 1 into the function as follows −(−1)3+6(−1)−7=−12 and find the value to get the remainder
Standards: HSA.APR.B.2
Click here to practice: Algebra – Arithmetic with Polynomials & Rational Expressions Questions for Grade 10 Math
Domain: Grade 10 >> Algebra – Creating Equations
Sample Question: The ratio of staff to guests at the gala was 3 to 5. There were a total of 576 people in the ballroom. How many guests were at the gala?
276
300
360
216
Answer Explanation: Setup a proportion of guests to the total number of people, 8/5 = x/576. Solve by cross multiplying. 8x = 2880. Divide both sides by 8. So x=360.
Standards: HSA.CED.A.3
Click here to practice: Algebra – Creating Equations Questions for Grade 10 Math
Domain: Grade 10 >> Algebra – Reasoning with Equations & Inequalities
Sample Question: Solve the quadratic x2+10x=−25.
-10
10
5
-5
Answer Explanation: This problem can be easily solved by rearranging the equation so that it is solved for zero and then factoring as shown:
x2+10x=−25
x2+10x+25=0
(x+5)(x+5)=0
Since both factors are exactly the same, you will only have one solution to this problem.
x+5=0
x=−5
Standards: HSA.REI.B.4
Domain: Grade 10 >> Functions – Interpreting Functions
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